diff --git "a/Extracted/Mathlib/TacticPrediction/Mathlib_Algebra_Algebra_NonUnitalSubalgebra.jsonl" "b/Extracted/Mathlib/TacticPrediction/Mathlib_Algebra_Algebra_NonUnitalSubalgebra.jsonl"
deleted file mode 100644--- "a/Extracted/Mathlib/TacticPrediction/Mathlib_Algebra_Algebra_NonUnitalSubalgebra.jsonl"
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-{"state":"F : Type v'\nR' : Type u'\nR : Type u\nA : Type v\nB : Type w\nC : Type w'\ninst✝⁶ : CommSemiring R\ninst✝⁵ : NonUnitalNonAssocSemiring A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : NonUnitalNonAssocSemiring C\ninst✝² : Module R A\ninst✝¹ : Module R B\ninst✝ : Module R C\np q : NonUnitalSubalgebra R A\nh : (fun s => s.carrier) p = (fun s => s.carrier) q\n⊢ p = q","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by ","nextTactic":"cases p","declUpToTactic":"instance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.65_0.jQvdAcOcyVirh23","decl":"instance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s "}
-{"state":"case mk\nF : Type v'\nR' : Type u'\nR : Type u\nA : Type v\nB : Type w\nC : Type w'\ninst✝⁶ : CommSemiring R\ninst✝⁵ : NonUnitalNonAssocSemiring A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : NonUnitalNonAssocSemiring C\ninst✝² : Module R A\ninst✝¹ : Module R B\ninst✝ : Module R C\nq : NonUnitalSubalgebra R A\ntoNonUnitalSubsemiring✝ : NonUnitalSubsemiring A\nsmul_mem'✝ : ∀ (c : R) {x : A}, x ∈ toNonUnitalSubsemiring✝.carrier → c • x ∈ toNonUnitalSubsemiring✝.carrier\nh :\n  (fun s => s.carrier) { toNonUnitalSubsemiring := toNonUnitalSubsemiring✝, smul_mem' := smul_mem'✝ } =\n    (fun s => s.carrier) q\n⊢ { toNonUnitalSubsemiring := toNonUnitalSubsemiring✝, smul_mem' := smul_mem'✝ } = q","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; ","nextTactic":"cases q","declUpToTactic":"instance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.65_0.jQvdAcOcyVirh23","decl":"instance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s "}
-{"state":"case mk.mk\nF : Type v'\nR' : Type u'\nR : Type u\nA : Type v\nB : Type w\nC : Type w'\ninst✝⁶ : CommSemiring R\ninst✝⁵ : NonUnitalNonAssocSemiring A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : NonUnitalNonAssocSemiring C\ninst✝² : Module R A\ninst✝¹ : Module R B\ninst✝ : Module R C\ntoNonUnitalSubsemiring✝¹ : NonUnitalSubsemiring A\nsmul_mem'✝¹ : ∀ (c : R) {x : A}, x ∈ toNonUnitalSubsemiring✝¹.carrier → c • x ∈ toNonUnitalSubsemiring✝¹.carrier\ntoNonUnitalSubsemiring✝ : NonUnitalSubsemiring A\nsmul_mem'✝ : ∀ (c : R) {x : A}, x ∈ toNonUnitalSubsemiring✝.carrier → c • x ∈ toNonUnitalSubsemiring✝.carrier\nh :\n  (fun s => s.carrier) { toNonUnitalSubsemiring := toNonUnitalSubsemiring✝¹, smul_mem' := smul_mem'✝¹ } =\n    (fun s => s.carrier) { toNonUnitalSubsemiring := toNonUnitalSubsemiring✝, smul_mem' := smul_mem'✝ }\n⊢ { toNonUnitalSubsemiring := toNonUnitalSubsemiring✝¹, smul_mem' := smul_mem'✝¹ } =\n    { toNonUnitalSubsemiring := toNonUnitalSubsemiring✝, smul_mem' := smul_mem'✝ }","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; ","nextTactic":"congr","declUpToTactic":"instance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.65_0.jQvdAcOcyVirh23","decl":"instance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s "}
-{"state":"case mk.mk.e_toNonUnitalSubsemiring\nF : Type v'\nR' : Type u'\nR : Type u\nA : Type v\nB : Type w\nC : Type w'\ninst✝⁶ : CommSemiring R\ninst✝⁵ : NonUnitalNonAssocSemiring A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : NonUnitalNonAssocSemiring C\ninst✝² : Module R A\ninst✝¹ : Module R B\ninst✝ : Module R C\ntoNonUnitalSubsemiring✝¹ : NonUnitalSubsemiring A\nsmul_mem'✝¹ : ∀ (c : R) {x : A}, x ∈ toNonUnitalSubsemiring✝¹.carrier → c • x ∈ toNonUnitalSubsemiring✝¹.carrier\ntoNonUnitalSubsemiring✝ : NonUnitalSubsemiring A\nsmul_mem'✝ : ∀ (c : R) {x : A}, x ∈ toNonUnitalSubsemiring✝.carrier → c • x ∈ toNonUnitalSubsemiring✝.carrier\nh :\n  (fun s => s.carrier) { toNonUnitalSubsemiring := toNonUnitalSubsemiring✝¹, smul_mem' := smul_mem'✝¹ } =\n    (fun s => s.carrier) { toNonUnitalSubsemiring := toNonUnitalSubsemiring✝, smul_mem' := smul_mem'✝ }\n⊢ toNonUnitalSubsemiring✝¹ = toNonUnitalSubsemiring✝","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; ","nextTactic":"exact SetLike.coe_injective h","declUpToTactic":"instance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.65_0.jQvdAcOcyVirh23","decl":"instance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s "}
-{"state":"F : Type v'\nR' : Type u'\nR : Type u\nA : Type v\nB : Type w\nC : Type w'\ninst✝⁶ : CommSemiring R\ninst✝⁵ : NonUnitalNonAssocSemiring A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : NonUnitalNonAssocSemiring C\ninst✝² : Module R A\ninst✝¹ : Module R B\ninst✝ : Module R C\nS T : NonUnitalSubalgebra R A\nh : S.toNonUnitalSubsemiring = T.toNonUnitalSubsemiring\nx : A\n⊢ x ∈ S ↔ x ∈ T","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by ","nextTactic":"rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]","declUpToTactic":"theorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.96_0.jQvdAcOcyVirh23","decl":"theorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) "}
-{"state":"F : Type v'\nR' : Type u'\nR : Type u\nA : Type v\nB : Type w\nC : Type w'\ninst✝⁶ : CommSemiring R\ninst✝⁵ : NonUnitalNonAssocSemiring A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : NonUnitalNonAssocSemiring C\ninst✝² : Module R A\ninst✝¹ : Module R B\ninst✝ : Module R C\nS T : NonUnitalSubalgebra R A\nh : toSubmodule S = toSubmodule T\nx : A\n⊢ x ∈ S ↔ x ∈ T","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by ","nextTactic":"rw [← mem_toSubmodule, ← mem_toSubmodule, h]","declUpToTactic":"theorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.113_0.jQvdAcOcyVirh23","decl":"theorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) "}
-{"state":"F : Type v'\nR' : Type u'\nR : Type u\nA : Type v\nB : Type w\nC : Type w'\ninst✝⁶ : CommSemiring R\ninst✝⁵ : NonUnitalNonAssocSemiring A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : NonUnitalNonAssocSemiring C\ninst✝² : Module R A\ninst✝¹ : Module R B\ninst✝ : Module R C\nS : NonUnitalSubalgebra R A\ns : Set A\nhs : s = ↑S\nsrc✝ : NonUnitalSubsemiring A := NonUnitalSubsemiring.copy S.toNonUnitalSubsemiring s hs\nr : R\na : A\nha : a ∈ s\n⊢ r • a ∈\n    { toAddSubmonoid := src✝.toAddSubmonoid,\n            mul_mem' :=\n              (_ :\n                ∀ {a b : A},\n                  a ∈ src✝.carrier → b ∈ src✝.carrier → a * b ∈ src✝.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      ","nextTactic":"show r • a ∈ s","declUpToTactic":"/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.120_0.jQvdAcOcyVirh23","decl":"/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A "}
-{"state":"F : Type v'\nR' : Type u'\nR : Type u\nA : Type v\nB : Type w\nC : Type w'\ninst✝⁶ : CommSemiring R\ninst✝⁵ : NonUnitalNonAssocSemiring A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : NonUnitalNonAssocSemiring C\ninst✝² : Module R A\ninst✝¹ : Module R B\ninst✝ : Module R C\nS : NonUnitalSubalgebra R A\ns : Set A\nhs : s = ↑S\nsrc✝ : NonUnitalSubsemiring A := NonUnitalSubsemiring.copy S.toNonUnitalSubsemiring s hs\nr : R\na : A\nha : a ∈ s\n⊢ r • a ∈ s","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      ","nextTactic":"rw [hs] at ha ⊢","declUpToTactic":"/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.120_0.jQvdAcOcyVirh23","decl":"/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A "}
-{"state":"F : Type v'\nR' : Type u'\nR : Type u\nA : Type v\nB : Type w\nC : Type w'\ninst✝⁶ : CommSemiring R\ninst✝⁵ : NonUnitalNonAssocSemiring A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : NonUnitalNonAssocSemiring C\ninst✝² : Module R A\ninst✝¹ : Module R B\ninst✝ : Module R C\nS : NonUnitalSubalgebra R A\ns : Set A\nhs : s = ↑S\nsrc✝ : NonUnitalSubsemiring A := NonUnitalSubsemiring.copy S.toNonUnitalSubsemiring s hs\nr : R\na : A\nha : a ∈ ↑S\n⊢ r • a ∈ ↑S","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      ","nextTactic":"exact S.smul_mem' r ha","declUpToTactic":"/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.120_0.jQvdAcOcyVirh23","decl":"/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A "}
-{"state":"F : Type v'\nR' : Type u'\nR : Type u\nA : Type v\nB : Type w\nC : Type w'\ninst✝⁶ : CommRing R\ninst✝⁵ : NonUnitalNonAssocRing A\ninst✝⁴ : NonUnitalNonAssocRing B\ninst✝³ : NonUnitalNonAssocRing C\ninst✝² : Module R A\ninst✝¹ : Module R B\ninst✝ : Module R C\nS T : NonUnitalSubalgebra R A\nh : toNonUnitalSubring S = toNonUnitalSubring T\nx : A\n⊢ x ∈ S ↔ x ∈ T","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by ","nextTactic":"rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]","declUpToTactic":"theorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.167_0.jQvdAcOcyVirh23","decl":"theorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) "}
-{"state":"F : Type v'\nR' : Type u'\nR : Type u\nA : Type v\nB : Type w\nC : Type w'\ninst✝² : CommSemiring R\ninst✝¹ : NonUnitalNonAssocSemiring A\ninst✝ : Module R A\nS T : NonUnitalSubalgebra R A\nh : (fun S => toSubmodule S) S = (fun S => toSubmodule S) T\n⊢ ∀ (x : A), x ∈ S ↔ x ∈ T","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by ","nextTactic":"apply SetLike.ext_iff.1 h","declUpToTactic":"/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.209_0.jQvdAcOcyVirh23","decl":"/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding "}
-{"state":"F : Type v'\nR' : Type u'\nR : Type u\nA : Type v\nB : Type w\nC : Type w'\ninst✝² : CommSemiring R\ninst✝¹ : NonUnitalNonAssocSemiring A\ninst✝ : Module R A\nS T : NonUnitalSubalgebra R A\nh : (fun S => S.toNonUnitalSubsemiring) S = (fun S => S.toNonUnitalSubsemiring) T\n⊢ ∀ (x : A), x ∈ S ↔ x ∈ T","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by ","nextTactic":"apply SetLike.ext_iff.1 h","declUpToTactic":"/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.217_0.jQvdAcOcyVirh23","decl":"/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding "}
-{"state":"F : Type v'\nR' : Type u'\nR : Type u\nA : Type v\nB : Type w\nC : Type w'\ninst✝² : CommRing R\ninst✝¹ : NonUnitalNonAssocRing A\ninst✝ : Module R A\nS T : NonUnitalSubalgebra R A\nh : (fun S => toNonUnitalSubring S) S = (fun S => toNonUnitalSubring S) T\n⊢ ∀ (x : A), x ∈ S ↔ x ∈ T","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by ","nextTactic":"apply SetLike.ext_iff.1 h","declUpToTactic":"/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.226_0.jQvdAcOcyVirh23","decl":"/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding "}
-{"state":"F : Type v'\nR' : Type u'\nR : Type u\nA : Type v\nB : Type w\nC : Type w'\ninst✝⁷ : CommSemiring R\ninst✝⁶ : NonUnitalNonAssocSemiring A\ninst✝⁵ : NonUnitalNonAssocSemiring B\ninst✝⁴ : NonUnitalNonAssocSemiring C\ninst✝³ : Module R A\ninst✝² : Module R B\ninst✝¹ : Module R C\nS✝ : NonUnitalSubalgebra R A\ninst✝ : NonUnitalAlgHomClass F R A B\nf : F\nS : NonUnitalSubalgebra R A\nsrc✝ : NonUnitalSubsemiring B := NonUnitalSubsemiring.map (↑f) S.toNonUnitalSubsemiring\nr : R\nb : B\nhb :\n  b ∈\n    { toAddSubmonoid := src✝.toAddSubmonoid,\n            mul_mem' :=\n              (_ :\n                ∀ {a b : B},\n                  a ∈ src✝.carrier → b ∈ src✝.carrier → a * b ∈ src✝.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier\n⊢ r • b ∈\n    { toAddSubmonoid := src✝.toAddSubmonoid,\n            mul_mem' :=\n              (_ :\n                ∀ {a b : B},\n                  a ∈ src✝.carrier → b ∈ src✝.carrier → a * b ∈ src✝.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      ","nextTactic":"rcases hb with ⟨a, ha, rfl⟩","declUpToTactic":"/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.314_0.jQvdAcOcyVirh23","decl":"/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B "}
-{"state":"case intro.intro\nF : Type v'\nR' : Type u'\nR : Type u\nA : Type v\nB : Type w\nC : Type w'\ninst✝⁷ : CommSemiring R\ninst✝⁶ : NonUnitalNonAssocSemiring A\ninst✝⁵ : NonUnitalNonAssocSemiring B\ninst✝⁴ : NonUnitalNonAssocSemiring C\ninst✝³ : Module R A\ninst✝² : Module R B\ninst✝¹ : Module R C\nS✝ : NonUnitalSubalgebra R A\ninst✝ : NonUnitalAlgHomClass F R A B\nf : F\nS : NonUnitalSubalgebra R A\nsrc✝ : NonUnitalSubsemiring B := NonUnitalSubsemiring.map (↑f) S.toNonUnitalSubsemiring\nr : R\na : A\nha : a ∈ ↑S.toNonUnitalSubsemiring\n⊢ r • ↑f a ∈\n    { toAddSubmonoid := src✝.toAddSubmonoid,\n            mul_mem' :=\n              (_ :\n                ∀ {a b : B},\n                  a ∈ src✝.carrier → b ∈ src✝.carrier → a * b ∈ src✝.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      ","nextTactic":"exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha)","declUpToTactic":"/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.314_0.jQvdAcOcyVirh23","decl":"/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B "}
-{"state":"F : Type v'\nR' : Type u'\nR : Type u\nA : Type v\nB : Type w\nC : Type w'\ninst✝⁷ : CommSemiring R\ninst✝⁶ : NonUnitalNonAssocSemiring A\ninst✝⁵ : Module R A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : NonUnitalNonAssocSemiring C\ninst✝¹ : Module R C\ninst✝ : NonUnitalAlgHomClass F R A B\nφ : F\nr : R\na : B\n⊢ a ∈ (NonUnitalRingHom.srange ↑φ).toAddSubmonoid.toAddSubsemigroup.carrier →\n    r • a ∈ (NonUnitalRingHom.srange ↑φ).toAddSubmonoid.toAddSubsemigroup.carrier","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by ","nextTactic":"rintro ⟨a, rfl⟩","declUpToTactic":"/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.427_0.jQvdAcOcyVirh23","decl":"/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring "}
-{"state":"case intro\nF : Type v'\nR' : Type u'\nR : Type u\nA : Type v\nB : Type w\nC : Type w'\ninst✝⁷ : CommSemiring R\ninst✝⁶ : NonUnitalNonAssocSemiring A\ninst✝⁵ : Module R A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : NonUnitalNonAssocSemiring C\ninst✝¹ : Module R C\ninst✝ : NonUnitalAlgHomClass F R A B\nφ : F\nr : R\na : A\n⊢ r • ↑φ a ∈ (NonUnitalRingHom.srange ↑φ).toAddSubmonoid.toAddSubsemigroup.carrier","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; ","nextTactic":"exact ⟨r • a, map_smul φ r a⟩","declUpToTactic":"/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.427_0.jQvdAcOcyVirh23","decl":"/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring "}
-{"state":"F : Type v'\nR' : Type u'\nR : Type u\nA : Type v\nB : Type w\nC : Type w'\ninst✝⁷ : CommSemiring R\ninst✝⁶ : NonUnitalNonAssocSemiring A\ninst✝⁵ : Module R A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : NonUnitalNonAssocSemiring C\ninst✝¹ : Module R C\ninst✝ : NonUnitalAlgHomClass F R A B\nφ : F\n⊢ ↑(NonUnitalAlgHom.range φ) = Set.range ⇑φ","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ","nextTactic":"ext","declUpToTactic":"@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.441_0.jQvdAcOcyVirh23","decl":"@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) "}
-{"state":"case h\nF : Type v'\nR' : Type u'\nR : Type u\nA : Type v\nB : Type w\nC : Type w'\ninst✝⁷ : CommSemiring R\ninst✝⁶ : NonUnitalNonAssocSemiring A\ninst✝⁵ : Module R A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : NonUnitalNonAssocSemiring C\ninst✝¹ : Module R C\ninst✝ : NonUnitalAlgHomClass F R A B\nφ : F\nx✝ : B\n⊢ x✝ ∈ ↑(NonUnitalAlgHom.range φ) ↔ x✝ ∈ Set.range ⇑φ","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  ","nextTactic":"rw [SetLike.mem_coe]","declUpToTactic":"@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.441_0.jQvdAcOcyVirh23","decl":"@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) "}
-{"state":"case h\nF : Type v'\nR' : Type u'\nR : Type u\nA : Type v\nB : Type w\nC : Type w'\ninst✝⁷ : CommSemiring R\ninst✝⁶ : NonUnitalNonAssocSemiring A\ninst✝⁵ : Module R A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : NonUnitalNonAssocSemiring C\ninst✝¹ : Module R C\ninst✝ : NonUnitalAlgHomClass F R A B\nφ : F\nx✝ : B\n⊢ x✝ ∈ NonUnitalAlgHom.range φ ↔ x✝ ∈ Set.range ⇑φ","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ��� x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  ","nextTactic":"rw [mem_range]","declUpToTactic":"@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.441_0.jQvdAcOcyVirh23","decl":"@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) "}
-{"state":"case h\nF : Type v'\nR' : Type u'\nR : Type u\nA : Type v\nB : Type w\nC : Type w'\ninst✝⁷ : CommSemiring R\ninst✝⁶ : NonUnitalNonAssocSemiring A\ninst✝⁵ : Module R A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : NonUnitalNonAssocSemiring C\ninst✝¹ : Module R C\ninst✝ : NonUnitalAlgHomClass F R A B\nφ : F\nx✝ : B\n⊢ (∃ x, φ x = x✝) ↔ x✝ ∈ Set.range ⇑φ","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  ","nextTactic":"rfl","declUpToTactic":"@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.441_0.jQvdAcOcyVirh23","decl":"@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) "}
-{"state":"F : Type v'\nR' : Type u'\nR : Type u\nA : Type v\nB : Type w\nC : Type w'\ninst✝⁷ : CommSemiring R\ninst✝⁶ : NonUnitalNonAssocSemiring A\ninst✝⁵ : Module R A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : NonUnitalNonAssocSemiring C\ninst✝¹ : Module R C\ninst✝ : NonUnitalAlgHomClass F R A B\nϕ ψ : F\nx y : A\nhx : ϕ x = ψ x\nhy : ϕ y = ψ y\n⊢ x + y ∈ {a | ϕ a = ψ a}","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    ","nextTactic":"rw [Set.mem_setOf_eq]","declUpToTactic":"/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.483_0.jQvdAcOcyVirh23","decl":"/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier "}
-{"state":"F : Type v'\nR' : Type u'\nR : Type u\nA : Type v\nB : Type w\nC : Type w'\ninst✝⁷ : CommSemiring R\ninst✝⁶ : NonUnitalNonAssocSemiring A\ninst✝⁵ : Module R A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : NonUnitalNonAssocSemiring C\ninst✝¹ : Module R C\ninst✝ : NonUnitalAlgHomClass F R A B\nϕ ψ : F\nx y : A\nhx : ϕ x = ψ x\nhy : ϕ y = ψ y\n⊢ ϕ (x + y) = ψ (x + y)","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    ","nextTactic":"rw [map_add]","declUpToTactic":"/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.483_0.jQvdAcOcyVirh23","decl":"/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier "}
-{"state":"F : Type v'\nR' : Type u'\nR : Type u\nA : Type v\nB : Type w\nC : Type w'\ninst✝⁷ : CommSemiring R\ninst✝⁶ : NonUnitalNonAssocSemiring A\ninst✝⁵ : Module R A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : NonUnitalNonAssocSemiring C\ninst✝¹ : Module R C\ninst✝ : NonUnitalAlgHomClass F R A B\nϕ ψ : F\nx y : A\nhx : ϕ x = ψ x\nhy : ϕ y = ψ y\n⊢ ϕ x + ϕ y = ψ (x + y)","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    ","nextTactic":"rw [map_add]","declUpToTactic":"/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.483_0.jQvdAcOcyVirh23","decl":"/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier "}
-{"state":"F : Type v'\nR' : Type u'\nR : Type u\nA : Type v\nB : Type w\nC : Type w'\ninst✝⁷ : CommSemiring R\ninst✝⁶ : NonUnitalNonAssocSemiring A\ninst✝⁵ : Module R A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : NonUnitalNonAssocSemiring C\ninst✝¹ : Module R C\ninst✝ : NonUnitalAlgHomClass F R A B\nϕ ψ : F\nx y : A\nhx : ϕ x = ψ x\nhy : ϕ y = ψ y\n⊢ ϕ x + ϕ y = ψ x + ψ y","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    ","nextTactic":"rw [hx]","declUpToTactic":"/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.483_0.jQvdAcOcyVirh23","decl":"/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier "}
-{"state":"F : Type v'\nR' : Type u'\nR : Type u\nA : Type v\nB : Type w\nC : Type w'\ninst✝⁷ : CommSemiring R\ninst✝⁶ : NonUnitalNonAssocSemiring A\ninst✝⁵ : Module R A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : NonUnitalNonAssocSemiring C\ninst✝¹ : Module R C\ninst✝ : NonUnitalAlgHomClass F R A B\nϕ ψ : F\nx y : A\nhx : ϕ x = ψ x\nhy : ϕ y = ψ y\n⊢ ψ x + ϕ y = ψ x + ψ y","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    ","nextTactic":"rw [hy]","declUpToTactic":"/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.483_0.jQvdAcOcyVirh23","decl":"/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier "}
-{"state":"F : Type v'\nR' : Type u'\nR : Type u\nA : Type v\nB : Type w\nC : Type w'\ninst✝⁷ : CommSemiring R\ninst✝⁶ : NonUnitalNonAssocSemiring A\ninst✝⁵ : Module R A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : NonUnitalNonAssocSemiring C\ninst✝¹ : Module R C\ninst✝ : NonUnitalAlgHomClass F R A B\nϕ ψ : F\n⊢ 0 ∈\n    { carrier := {a | ϕ a = ψ a},\n        add_mem' := (_ : ∀ {x y : A}, ϕ x = ψ x → ϕ y = ψ y → x + y ∈ {a | ϕ a = ψ a}) }.carrier","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by ","nextTactic":"rw [Set.mem_setOf_eq, map_zero, map_zero]","declUpToTactic":"/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.483_0.jQvdAcOcyVirh23","decl":"/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier "}
-{"state":"F : Type v'\nR' : Type u'\nR : Type u\nA : Type v\nB : Type w\nC : Type w'\ninst✝⁷ : CommSemiring R\ninst✝⁶ : NonUnitalNonAssocSemiring A\ninst✝⁵ : Module R A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : NonUnitalNonAssocSemiring C\ninst✝¹ : Module R C\ninst✝ : NonUnitalAlgHomClass F R A B\nϕ ψ : F\nx y : A\nhx : ϕ x = ψ x\nhy : ϕ y = ψ y\n⊢ x * y ∈\n    {\n          toAddSubsemigroup :=\n            { carrier := {a | ϕ a = ψ a},\n              add_mem' := (_ : ∀ {x y : A}, ϕ x = ψ x → ϕ y = ψ y → x + y ∈ {a | ϕ a = ψ a}) },\n          zero_mem' :=\n            (_ :\n              0 ∈\n                { carrier := {a | ϕ a = ψ a},\n                    add_mem' :=\n                      (_ :\n                        ∀ {x y : A},\n                          ϕ x = ψ x → ϕ y = ψ y → x + y ∈ {a | ϕ a = ψ a}) }.carrier) }.toAddSubsemigroup.carrier","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    ","nextTactic":"rw [Set.mem_setOf_eq]","declUpToTactic":"/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.483_0.jQvdAcOcyVirh23","decl":"/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier "}
-{"state":"F : Type v'\nR' : Type u'\nR : Type u\nA : Type v\nB : Type w\nC : Type w'\ninst✝⁷ : CommSemiring R\ninst✝⁶ : NonUnitalNonAssocSemiring A\ninst✝⁵ : Module R A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : NonUnitalNonAssocSemiring C\ninst✝¹ : Module R C\ninst✝ : NonUnitalAlgHomClass F R A B\nϕ ψ : F\nx y : A\nhx : ϕ x = ψ x\nhy : ϕ y = ψ y\n⊢ ϕ (x * y) = ψ (x * y)","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    ","nextTactic":"rw [map_mul]","declUpToTactic":"/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.483_0.jQvdAcOcyVirh23","decl":"/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier "}
-{"state":"F : Type v'\nR' : Type u'\nR : Type u\nA : Type v\nB : Type w\nC : Type w'\ninst✝⁷ : CommSemiring R\ninst✝⁶ : NonUnitalNonAssocSemiring A\ninst✝⁵ : Module R A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : NonUnitalNonAssocSemiring C\ninst✝¹ : Module R C\ninst✝ : NonUnitalAlgHomClass F R A B\nϕ ψ : F\nx y : A\nhx : ϕ x = ψ x\nhy : ϕ y = ψ y\n⊢ ϕ x * ϕ y = ψ (x * y)","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    ","nextTactic":"rw [map_mul]","declUpToTactic":"/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.483_0.jQvdAcOcyVirh23","decl":"/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier "}
-{"state":"F : Type v'\nR' : Type u'\nR : Type u\nA : Type v\nB : Type w\nC : Type w'\ninst✝⁷ : CommSemiring R\ninst✝⁶ : NonUnitalNonAssocSemiring A\ninst✝⁵ : Module R A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : NonUnitalNonAssocSemiring C\ninst✝¹ : Module R C\ninst✝ : NonUnitalAlgHomClass F R A B\nϕ ψ : F\nx y : A\nhx : ϕ x = ψ x\nhy : ϕ y = ψ y\n⊢ ϕ x * ϕ y = ψ x * ψ y","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    ","nextTactic":"rw [hx]","declUpToTactic":"/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.483_0.jQvdAcOcyVirh23","decl":"/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (�� ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier "}
-{"state":"F : Type v'\nR' : Type u'\nR : Type u\nA : Type v\nB : Type w\nC : Type w'\ninst✝⁷ : CommSemiring R\ninst✝⁶ : NonUnitalNonAssocSemiring A\ninst✝⁵ : Module R A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : NonUnitalNonAssocSemiring C\ninst✝¹ : Module R C\ninst✝ : NonUnitalAlgHomClass F R A B\nϕ ψ : F\nx y : A\nhx : ϕ x = ψ x\nhy : ϕ y = ψ y\n⊢ ψ x * ϕ y = ψ x * ψ y","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    ","nextTactic":"rw [hy]","declUpToTactic":"/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.483_0.jQvdAcOcyVirh23","decl":"/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier "}
-{"state":"F : Type v'\nR' : Type u'\nR : Type u\nA : Type v\nB : Type w\nC : Type w'\ninst✝⁷ : CommSemiring R\ninst✝⁶ : NonUnitalNonAssocSemiring A\ninst✝⁵ : Module R A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : NonUnitalNonAssocSemiring C\ninst✝¹ : Module R C\ninst✝ : NonUnitalAlgHomClass F R A B\nϕ ψ : F\nr : R\nx : A\nhx : ϕ x = ψ x\n⊢ r • x ∈\n    {\n            toAddSubmonoid :=\n              {\n                toAddSubsemigroup :=\n                  { carrier := {a | ϕ a = ψ a},\n                    add_mem' := (_ : ∀ {x y : A}, ϕ x = ψ x → ϕ y = ψ y → x + y ∈ {a | ϕ a = ψ a}) },\n                zero_mem' :=\n                  (_ :\n                    0 ∈\n                      { carrier := {a | ϕ a = ψ a},\n                          add_mem' := (_ : ∀ {x y : A}, ϕ x = ψ x → ϕ y = ψ y → x + y ∈ {a | ϕ a = ψ a}) }.carrier) },\n            mul_mem' :=\n              (_ :\n                ∀ {x y : A},\n                  ϕ x = ψ x →\n                    ϕ y = ψ y →\n                      x * y ∈\n                        {\n                              toAddSubsemigroup :=\n                                { carrier := {a | ϕ a = ψ a},\n                                  add_mem' := (_ : ∀ {x y : A}, ϕ x = ψ x → ϕ y = ψ y → x + y ∈ {a | ϕ a = ψ a}) },\n                              zero_mem' :=\n                                (_ :\n                                  0 ∈\n                                    { carrier := {a | ϕ a = ψ a},\n                                        add_mem' :=\n                                          (_ :\n                                            ∀ {x y : A},\n                                              ϕ x = ψ x →\n                                                ϕ y = ψ y →\n                                                  x + y ∈\n                                                    {a |\n                                                      ϕ a =\n                                                        ψ\n                                                          a}) }.carrier) }.toAddSubsemigroup.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by ","nextTactic":"rw [Set.mem_setOf_eq, map_smul, map_smul, hx]","declUpToTactic":"/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.483_0.jQvdAcOcyVirh23","decl":"/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier "}
-{"state":"F : Type u_1\nR : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\ninst✝ : NonUnitalAlgHomClass F R A B\ns : Set A\nsrc✝ : Submodule R A := Submodule.span R ↑(NonUnitalSubsemiring.closure s)\na b : A\nha : a ∈ Submodule.span R ↑(NonUnitalSubsemiring.closure s)\nhb : b ∈ Submodule.span R ↑(NonUnitalSubsemiring.closure s)\n⊢ a * b ∈ Submodule.span R ↑(NonUnitalSubsemiring.closure s)","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        ","nextTactic":"refine' Submodule.span_induction ha _ _ _ _","declUpToTactic":"/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.524_0.jQvdAcOcyVirh23","decl":"/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A "}
-{"state":"case refine'_1\nF : Type u_1\nR : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\ninst✝ : NonUnitalAlgHomClass F R A B\ns : Set A\nsrc✝ : Submodule R A := Submodule.span R ↑(NonUnitalSubsemiring.closure s)\na b : A\nha : a ∈ Submodule.span R ↑(NonUnitalSubsemiring.closure s)\nhb : b ∈ Submodule.span R ↑(NonUnitalSubsemiring.closure s)\n⊢ ∀ x ∈ ↑(NonUnitalSubsemiring.closure s), x * b ∈ Submodule.span R ↑(NonUnitalSubsemiring.closure s)","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · ","nextTactic":"refine' Submodule.span_induction hb _ _ _ _","declUpToTactic":"/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.524_0.jQvdAcOcyVirh23","decl":"/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A "}
-{"state":"case refine'_1.refine'_1\nF : Type u_1\nR : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\ninst✝ : NonUnitalAlgHomClass F R A B\ns : Set A\nsrc✝ : Submodule R A := Submodule.span R ↑(NonUnitalSubsemiring.closure s)\na b : A\nha : a ∈ Submodule.span R ↑(NonUnitalSubsemiring.closure s)\nhb : b ∈ Submodule.span R ↑(NonUnitalSubsemiring.closure s)\n⊢ ∀ x ∈ ↑(NonUnitalSubsemiring.closure s),\n    ∀ x_1 ∈ ↑(NonUnitalSubsemiring.closure s), x_1 * x ∈ Submodule.span R ↑(NonUnitalSubsemiring.closure s)","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · ","nextTactic":"exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)","declUpToTactic":"/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.524_0.jQvdAcOcyVirh23","decl":"/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A "}
-{"state":"case refine'_1.refine'_2\nF : Type u_1\nR : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\ninst✝ : NonUnitalAlgHomClass F R A B\ns : Set A\nsrc✝ : Submodule R A := Submodule.span R ↑(NonUnitalSubsemiring.closure s)\na b : A\nha : a ∈ Submodule.span R ↑(NonUnitalSubsemiring.closure s)\nhb : b ∈ Submodule.span R ↑(NonUnitalSubsemiring.closure s)\n⊢ ∀ x ∈ ↑(NonUnitalSubsemiring.closure s), x * 0 ∈ Submodule.span R ↑(NonUnitalSubsemiring.closure s)","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · ","nextTactic":"exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _","declUpToTactic":"/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.524_0.jQvdAcOcyVirh23","decl":"/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A "}
-{"state":"case refine'_1.refine'_3\nF : Type u_1\nR : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\ninst✝ : NonUnitalAlgHomClass F R A B\ns : Set A\nsrc✝ : Submodule R A := Submodule.span R ↑(NonUnitalSubsemiring.closure s)\na b : A\nha : a ∈ Submodule.span R ↑(NonUnitalSubsemiring.closure s)\nhb : b ∈ Submodule.span R ↑(NonUnitalSubsemiring.closure s)\n⊢ ∀ (x y : A),\n    (∀ x_1 ∈ ↑(NonUnitalSubsemiring.closure s), x_1 * x ∈ Submodule.span R ↑(NonUnitalSubsemiring.closure s)) →\n      (∀ x ∈ ↑(NonUnitalSubsemiring.closure s), x * y ∈ Submodule.span R ↑(NonUnitalSubsemiring.closure s)) →\n        ∀ x_1 ∈ ↑(NonUnitalSubsemiring.closure s), x_1 * (x + y) ∈ Submodule.span R ↑(NonUnitalSubsemiring.closure s)","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · ","nextTactic":"exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)","declUpToTactic":"/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.524_0.jQvdAcOcyVirh23","decl":"/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A "}
-{"state":"case refine'_1.refine'_4\nF : Type u_1\nR : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\ninst✝ : NonUnitalAlgHomClass F R A B\ns : Set A\nsrc✝ : Submodule R A := Submodule.span R ↑(NonUnitalSubsemiring.closure s)\na b : A\nha : a ∈ Submodule.span R ↑(NonUnitalSubsemiring.closure s)\nhb : b ∈ Submodule.span R ↑(NonUnitalSubsemiring.closure s)\n⊢ ∀ (a : R) (x : A),\n    (∀ x_1 ∈ ↑(NonUnitalSubsemiring.closure s), x_1 * x ∈ Submodule.span R ↑(NonUnitalSubsemiring.closure s)) →\n      ∀ x_1 ∈ ↑(NonUnitalSubsemiring.closure s), x_1 * a • x ∈ Submodule.span R ↑(NonUnitalSubsemiring.closure s)","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · ","nextTactic":"exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)","declUpToTactic":"/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.524_0.jQvdAcOcyVirh23","decl":"/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A "}
-{"state":"case refine'_2\nF : Type u_1\nR : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\ninst✝ : NonUnitalAlgHomClass F R A B\ns : Set A\nsrc✝ : Submodule R A := Submodule.span R ↑(NonUnitalSubsemiring.closure s)\na b : A\nha : a ∈ Submodule.span R ↑(NonUnitalSubsemiring.closure s)\nhb : b ∈ Submodule.span R ↑(NonUnitalSubsemiring.closure s)\n⊢ 0 * b ∈ Submodule.span R ↑(NonUnitalSubsemiring.closure s)","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · ","nextTactic":"exact (zero_mul b).symm ▸ Submodule.zero_mem _","declUpToTactic":"/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.524_0.jQvdAcOcyVirh23","decl":"/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A "}
-{"state":"case refine'_3\nF : Type u_1\nR : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\ninst✝ : NonUnitalAlgHomClass F R A B\ns : Set A\nsrc✝ : Submodule R A := Submodule.span R ↑(NonUnitalSubsemiring.closure s)\na b : A\nha : a ��� Submodule.span R ↑(NonUnitalSubsemiring.closure s)\nhb : b ∈ Submodule.span R ↑(NonUnitalSubsemiring.closure s)\n⊢ ∀ (x y : A),\n    x * b ∈ Submodule.span R ↑(NonUnitalSubsemiring.closure s) →\n      y * b ∈ Submodule.span R ↑(NonUnitalSubsemiring.closure s) →\n        (x + y) * b ∈ Submodule.span R ↑(NonUnitalSubsemiring.closure s)","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · ","nextTactic":"exact fun x y => (add_mul x y b).symm ▸ add_mem","declUpToTactic":"/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.524_0.jQvdAcOcyVirh23","decl":"/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A "}
-{"state":"case refine'_4\nF : Type u_1\nR : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\ninst✝ : NonUnitalAlgHomClass F R A B\ns : Set A\nsrc✝ : Submodule R A := Submodule.span R ↑(NonUnitalSubsemiring.closure s)\na b : A\nha : a ∈ Submodule.span R ↑(NonUnitalSubsemiring.closure s)\nhb : b ∈ Submodule.span R ↑(NonUnitalSubsemiring.closure s)\n⊢ ∀ (a : R) (x : A),\n    x * b ∈ Submodule.span R ↑(NonUnitalSubsemiring.closure s) →\n      a • x * b ∈ Submodule.span R ↑(NonUnitalSubsemiring.closure s)","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · ","nextTactic":"exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx","declUpToTactic":"/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.524_0.jQvdAcOcyVirh23","decl":"/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A "}
-{"state":"F : Type u_1\nR : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\ninst✝ : NonUnitalAlgHomClass F R A B\ns : Set A\np : ↥(adjoin R s) → Prop\na : ↥(adjoin R s)\nHs : ∀ (x : A) (h : x ∈ s), p { val := x, property := (_ : x ∈ ↑(adjoin R s)) }\nHadd : ∀ (x y : ↥(adjoin R s)), p x → p y → p (x + y)\nH0 : p 0\nHmul : ∀ (x y : ↥(adjoin R s)), p x → p y → p (x * y)\nHsmul : ∀ (r : R) (x : ↥(adjoin R s)), p x → p (r • x)\nb : A\nhb : b ∈ adjoin R s\n⊢ p { val := b, property := hb }","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    ","nextTactic":"refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)","declUpToTactic":"/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.577_0.jQvdAcOcyVirh23","decl":"/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a "}
-{"state":"F : Type u_1\nR : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\ninst✝ : NonUnitalAlgHomClass F R A B\ns : Set A\np : ↥(adjoin R s) → Prop\na : ↥(adjoin R s)\nHs : ∀ (x : A) (h : x ∈ s), p { val := x, property := (_ : x ∈ ↑(adjoin R s)) }\nHadd : ∀ (x y : ↥(adjoin R s)), p x → p y → p (x + y)\nH0 : p 0\nHmul : ∀ (x y : ↥(adjoin R s)), p x → p y → p (x * y)\nHsmul : ∀ (r : R) (x : ↥(adjoin R s)), p x → p (r • x)\nb : A\nhb : b ∈ adjoin R s\n⊢ ∃ (x : b ∈ adjoin R s), p { val := b, property := x }","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →���[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    ","nextTactic":"apply adjoin_induction hb","declUpToTactic":"/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.577_0.jQvdAcOcyVirh23","decl":"/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a "}
-{"state":"case Hs\nF : Type u_1\nR : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\ninst✝ : NonUnitalAlgHomClass F R A B\ns : Set A\np : ↥(adjoin R s) → Prop\na : ↥(adjoin R s)\nHs : ∀ (x : A) (h : x ∈ s), p { val := x, property := (_ : x ∈ ↑(adjoin R s)) }\nHadd : ∀ (x y : ↥(adjoin R s)), p x → p y → p (x + y)\nH0 : p 0\nHmul : ∀ (x y : ↥(adjoin R s)), p x → p y → p (x * y)\nHsmul : ∀ (r : R) (x : ↥(adjoin R s)), p x → p (r • x)\nb : A\nhb : b ∈ adjoin R s\n⊢ ∀ x ∈ s, ∃ (x_1 : x ∈ adjoin R s), p { val := x, property := x_1 }","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙ���[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · ","nextTactic":"exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩","declUpToTactic":"/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.577_0.jQvdAcOcyVirh23","decl":"/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a "}
-{"state":"case Hadd\nF : Type u_1\nR : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\ninst✝ : NonUnitalAlgHomClass F R A B\ns : Set A\np : ↥(adjoin R s) → Prop\na : ↥(adjoin R s)\nHs : ∀ (x : A) (h : x ∈ s), p { val := x, property := (_ : x ∈ ↑(adjoin R s)) }\nHadd : ∀ (x y : ↥(adjoin R s)), p x → p y → p (x + y)\nH0 : p 0\nHmul : ∀ (x y : ↥(adjoin R s)), p x → p y → p (x * y)\nHsmul : ∀ (r : R) (x : ↥(adjoin R s)), p x → p (r • x)\nb : A\nhb : b ∈ adjoin R s\n⊢ ∀ (x y : A),\n    (∃ (x_1 : x ∈ adjoin R s), p { val := x, property := x_1 }) →\n      (∃ (x : y ∈ adjoin R s), p { val := y, property := x }) →\n        ∃ (x_1 : x + y ∈ adjoin R s), p { val := x + y, property := x_1 }","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · ","nextTactic":"exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩","declUpToTactic":"/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.577_0.jQvdAcOcyVirh23","decl":"/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a "}
-{"state":"case H0\nF : Type u_1\nR : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\ninst✝ : NonUnitalAlgHomClass F R A B\ns : Set A\np : ↥(adjoin R s) → Prop\na : ↥(adjoin R s)\nHs : ∀ (x : A) (h : x ∈ s), p { val := x, property := (_ : x ∈ ↑(adjoin R s)) }\nHadd : ∀ (x y : ↥(adjoin R s)), p x → p y → p (x + y)\nH0 : p 0\nHmul : ∀ (x y : ↥(adjoin R s)), p x → p y → p (x * y)\nHsmul : ∀ (r : R) (x : ↥(adjoin R s)), p x → p (r • x)\nb : A\nhb : b ∈ adjoin R s\n⊢ ∃ (x : 0 ∈ adjoin R s), p { val := 0, property := x }","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · ","nextTactic":"exact ⟨_, H0⟩","declUpToTactic":"/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.577_0.jQvdAcOcyVirh23","decl":"/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a "}
-{"state":"case Hmul\nF : Type u_1\nR : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\ninst✝ : NonUnitalAlgHomClass F R A B\ns : Set A\np : ↥(adjoin R s) → Prop\na : ↥(adjoin R s)\nHs : ∀ (x : A) (h : x ∈ s), p { val := x, property := (_ : x ∈ ↑(adjoin R s)) }\nHadd : ∀ (x y : ↥(adjoin R s)), p x → p y → p (x + y)\nH0 : p 0\nHmul : ∀ (x y : ↥(adjoin R s)), p x → p y → p (x * y)\nHsmul : ∀ (r : R) (x : ↥(adjoin R s)), p x → p (r • x)\nb : A\nhb : b ∈ adjoin R s\n⊢ ∀ (x y : A),\n    (∃ (x_1 : x ∈ adjoin R s), p { val := x, property := x_1 }) →\n      (∃ (x : y ∈ adjoin R s), p { val := y, property := x }) →\n        ∃ (x_1 : x * y ∈ adjoin R s), p { val := x * y, property := x_1 }","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)��\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · ","nextTactic":"exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩","declUpToTactic":"/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.577_0.jQvdAcOcyVirh23","decl":"/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a "}
-{"state":"case Hsmul\nF : Type u_1\nR : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\ninst✝ : NonUnitalAlgHomClass F R A B\ns : Set A\np : ↥(adjoin R s) → Prop\na : ↥(adjoin R s)\nHs : ∀ (x : A) (h : x ∈ s), p { val := x, property := (_ : x ∈ ↑(adjoin R s)) }\nHadd : ∀ (x y : ↥(adjoin R s)), p x → p y → p (x + y)\nH0 : p 0\nHmul : ∀ (x y : ↥(adjoin R s)), p x → p y → p (x * y)\nHsmul : ∀ (r : R) (x : ↥(adjoin R s)), p x → p (r • x)\nb : A\nhb : b ∈ adjoin R s\n⊢ ∀ (r : R) (x : A),\n    (∃ (x_1 : x ∈ adjoin R s), p { val := x, property := x_1 }) →\n      ∃ (x_1 : r • x ∈ adjoin R s), p { val := r • x, property := x_1 }","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · ","nextTactic":"exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩","declUpToTactic":"/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.577_0.jQvdAcOcyVirh23","decl":"/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a "}
-{"state":"F : Type u_1\nR : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\ninst✝ : NonUnitalAlgHomClass F R A B\n⊢ adjoin R ⊥ = ⊥","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by ","nextTactic":"apply GaloisConnection.l_bot","declUpToTactic":"@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.623_0.jQvdAcOcyVirh23","decl":"@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ "}
-{"state":"case gc\nF : Type u_1\nR : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\ninst✝ : NonUnitalAlgHomClass F R A B\n⊢ GaloisConnection (adjoin R) ?u\ncase u\nF : Type u_1\nR : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\ninst✝ : NonUnitalAlgHomClass F R A B\n⊢ NonUnitalSubalgebra R A → Set A","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; ","nextTactic":"exact NonUnitalAlgebra.gc","declUpToTactic":"@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.623_0.jQvdAcOcyVirh23","decl":"@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ "}
-{"state":"F : Type u_1\nR : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\ninst✝ : NonUnitalAlgHomClass F R A B\nS T : NonUnitalSubalgebra R A\n⊢ ∀ {x : A}, x ∈ S → x ∈ S ⊔ T","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact NonUnitalAlgebra.gc\n\n@[simp]\ntheorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=\n  eq_top_iff.2 fun _x hx => subset_adjoin R hx\n\nvariable {R A}\n\n@[simp]\ntheorem coe_top : (↑(⊤ : NonUnitalSubalgebra R A) : Set A) = Set.univ :=\n  rfl\n\n@[simp]\ntheorem mem_top {x : A} : x ∈ (⊤ : NonUnitalSubalgebra R A) :=\n  Set.mem_univ x\n\n@[simp]\ntheorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toNonUnitalSubsemiring : (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubsemiring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toSubring {R A : Type*} [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    [IsScalarTower R A A] [SMulCommClass R A A] :\n    (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem toSubmodule_eq_top {S : NonUnitalSubalgebra R A} : S.toSubmodule = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toSubmodule'.injective.eq_iff' top_toSubmodule\n\n@[simp]\ntheorem toNonUnitalSubsemiring_eq_top {S : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubsemiring_injective.eq_iff' top_toNonUnitalSubsemiring\n\n@[simp]\ntheorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubring_injective.eq_iff' top_toSubring\n\ntheorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  ","nextTactic":"rw [← SetLike.le_def]","declUpToTactic":"theorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.669_0.jQvdAcOcyVirh23","decl":"theorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T "}
-{"state":"F : Type u_1\nR : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\ninst✝ : NonUnitalAlgHomClass F R A B\nS T : NonUnitalSubalgebra R A\n⊢ S ≤ S ⊔ T","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact NonUnitalAlgebra.gc\n\n@[simp]\ntheorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=\n  eq_top_iff.2 fun _x hx => subset_adjoin R hx\n\nvariable {R A}\n\n@[simp]\ntheorem coe_top : (↑(⊤ : NonUnitalSubalgebra R A) : Set A) = Set.univ :=\n  rfl\n\n@[simp]\ntheorem mem_top {x : A} : x ∈ (⊤ : NonUnitalSubalgebra R A) :=\n  Set.mem_univ x\n\n@[simp]\ntheorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toNonUnitalSubsemiring : (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubsemiring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toSubring {R A : Type*} [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    [IsScalarTower R A A] [SMulCommClass R A A] :\n    (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem toSubmodule_eq_top {S : NonUnitalSubalgebra R A} : S.toSubmodule = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toSubmodule'.injective.eq_iff' top_toSubmodule\n\n@[simp]\ntheorem toNonUnitalSubsemiring_eq_top {S : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubsemiring_injective.eq_iff' top_toNonUnitalSubsemiring\n\n@[simp]\ntheorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubring_injective.eq_iff' top_toSubring\n\ntheorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  ","nextTactic":"exact le_sup_left","declUpToTactic":"theorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.669_0.jQvdAcOcyVirh23","decl":"theorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T "}
-{"state":"F : Type u_1\nR : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\ninst✝ : NonUnitalAlgHomClass F R A B\nS T : NonUnitalSubalgebra R A\n⊢ ∀ {x : A}, x ∈ T → x ∈ S ⊔ T","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy���\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact NonUnitalAlgebra.gc\n\n@[simp]\ntheorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=\n  eq_top_iff.2 fun _x hx => subset_adjoin R hx\n\nvariable {R A}\n\n@[simp]\ntheorem coe_top : (↑(⊤ : NonUnitalSubalgebra R A) : Set A) = Set.univ :=\n  rfl\n\n@[simp]\ntheorem mem_top {x : A} : x ∈ (⊤ : NonUnitalSubalgebra R A) :=\n  Set.mem_univ x\n\n@[simp]\ntheorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toNonUnitalSubsemiring : (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubsemiring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toSubring {R A : Type*} [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    [IsScalarTower R A A] [SMulCommClass R A A] :\n    (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem toSubmodule_eq_top {S : NonUnitalSubalgebra R A} : S.toSubmodule = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toSubmodule'.injective.eq_iff' top_toSubmodule\n\n@[simp]\ntheorem toNonUnitalSubsemiring_eq_top {S : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubsemiring_injective.eq_iff' top_toNonUnitalSubsemiring\n\n@[simp]\ntheorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubring_injective.eq_iff' top_toSubring\n\ntheorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_left\n\ntheorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by\n  ","nextTactic":"rw [← SetLike.le_def]","declUpToTactic":"theorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.673_0.jQvdAcOcyVirh23","decl":"theorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T "}
-{"state":"F : Type u_1\nR : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\ninst✝ : NonUnitalAlgHomClass F R A B\nS T : NonUnitalSubalgebra R A\n⊢ T ≤ S ⊔ T","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact NonUnitalAlgebra.gc\n\n@[simp]\ntheorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=\n  eq_top_iff.2 fun _x hx => subset_adjoin R hx\n\nvariable {R A}\n\n@[simp]\ntheorem coe_top : (↑(⊤ : NonUnitalSubalgebra R A) : Set A) = Set.univ :=\n  rfl\n\n@[simp]\ntheorem mem_top {x : A} : x ∈ (⊤ : NonUnitalSubalgebra R A) :=\n  Set.mem_univ x\n\n@[simp]\ntheorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toNonUnitalSubsemiring : (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubsemiring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toSubring {R A : Type*} [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    [IsScalarTower R A A] [SMulCommClass R A A] :\n    (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem toSubmodule_eq_top {S : NonUnitalSubalgebra R A} : S.toSubmodule = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toSubmodule'.injective.eq_iff' top_toSubmodule\n\n@[simp]\ntheorem toNonUnitalSubsemiring_eq_top {S : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubsemiring_injective.eq_iff' top_toNonUnitalSubsemiring\n\n@[simp]\ntheorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubring_injective.eq_iff' top_toSubring\n\ntheorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_left\n\ntheorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  ","nextTactic":"exact le_sup_right","declUpToTactic":"theorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.673_0.jQvdAcOcyVirh23","decl":"theorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T "}
-{"state":"F : Type u_1\nR : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\ninst✝ : NonUnitalAlgHomClass F R A B\nS : Set (NonUnitalSubalgebra R A)\nx : A\n⊢ x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact NonUnitalAlgebra.gc\n\n@[simp]\ntheorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=\n  eq_top_iff.2 fun _x hx => subset_adjoin R hx\n\nvariable {R A}\n\n@[simp]\ntheorem coe_top : (↑(⊤ : NonUnitalSubalgebra R A) : Set A) = Set.univ :=\n  rfl\n\n@[simp]\ntheorem mem_top {x : A} : x ∈ (⊤ : NonUnitalSubalgebra R A) :=\n  Set.mem_univ x\n\n@[simp]\ntheorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toNonUnitalSubsemiring : (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubsemiring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toSubring {R A : Type*} [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    [IsScalarTower R A A] [SMulCommClass R A A] :\n    (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem toSubmodule_eq_top {S : NonUnitalSubalgebra R A} : S.toSubmodule = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toSubmodule'.injective.eq_iff' top_toSubmodule\n\n@[simp]\ntheorem toNonUnitalSubsemiring_eq_top {S : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubsemiring_injective.eq_iff' top_toNonUnitalSubsemiring\n\n@[simp]\ntheorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubring_injective.eq_iff' top_toSubring\n\ntheorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_left\n\ntheorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_right\n\ntheorem mul_mem_sup {S T : NonUnitalSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) :\n    x * y ∈ S ⊔ T :=\n  mul_mem (mem_sup_left hx) (mem_sup_right hy)\n\ntheorem map_sup (f : F) (S T : NonUnitalSubalgebra R A) :\n    ((S ⊔ T).map f : NonUnitalSubalgebra R B) = S.map f ⊔ T.map f :=\n  (@NonUnitalSubalgebra.gc_map_comap F R A B _ _ _ _ _ _ f).l_sup\n\n@[simp, norm_cast]\ntheorem coe_inf (S T : NonUnitalSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T :=\n  rfl\n\n@[simp]\ntheorem mem_inf {S T : NonUnitalSubalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T :=\n  Iff.rfl\n\n@[simp]\ntheorem inf_toSubmodule (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toSubmodule = S.toSubmodule ⊓ T.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem inf_toNonUnitalSubsemiring (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring ⊓ T.toNonUnitalSubsemiring :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sInf (S : Set (NonUnitalSubalgebra R A)) : (↑(sInf S) : Set A) = ⋂ s ∈ S, ↑s :=\n  sInf_image\n\ntheorem mem_sInf {S : Set (NonUnitalSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p := by\n  ","nextTactic":"simp only [← SetLike.mem_coe, coe_sInf, Set.mem_iInter₂]","declUpToTactic":"theorem mem_sInf {S : Set (NonUnitalSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.707_0.jQvdAcOcyVirh23","decl":"theorem mem_sInf {S : Set (NonUnitalSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p "}
-{"state":"F : Type u_1\nR : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\ninst✝ : NonUnitalAlgHomClass F R A B\nS : Set (NonUnitalSubalgebra R A)\n⊢ ↑(NonUnitalSubalgebra.toSubmodule (sInf S)) = ↑(sInf (NonUnitalSubalgebra.toSubmodule '' S))","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙ���[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact NonUnitalAlgebra.gc\n\n@[simp]\ntheorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=\n  eq_top_iff.2 fun _x hx => subset_adjoin R hx\n\nvariable {R A}\n\n@[simp]\ntheorem coe_top : (↑(⊤ : NonUnitalSubalgebra R A) : Set A) = Set.univ :=\n  rfl\n\n@[simp]\ntheorem mem_top {x : A} : x ∈ (⊤ : NonUnitalSubalgebra R A) :=\n  Set.mem_univ x\n\n@[simp]\ntheorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toNonUnitalSubsemiring : (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubsemiring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toSubring {R A : Type*} [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    [IsScalarTower R A A] [SMulCommClass R A A] :\n    (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem toSubmodule_eq_top {S : NonUnitalSubalgebra R A} : S.toSubmodule = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toSubmodule'.injective.eq_iff' top_toSubmodule\n\n@[simp]\ntheorem toNonUnitalSubsemiring_eq_top {S : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubsemiring_injective.eq_iff' top_toNonUnitalSubsemiring\n\n@[simp]\ntheorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubring_injective.eq_iff' top_toSubring\n\ntheorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_left\n\ntheorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_right\n\ntheorem mul_mem_sup {S T : NonUnitalSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) :\n    x * y ∈ S ⊔ T :=\n  mul_mem (mem_sup_left hx) (mem_sup_right hy)\n\ntheorem map_sup (f : F) (S T : NonUnitalSubalgebra R A) :\n    ((S ⊔ T).map f : NonUnitalSubalgebra R B) = S.map f ⊔ T.map f :=\n  (@NonUnitalSubalgebra.gc_map_comap F R A B _ _ _ _ _ _ f).l_sup\n\n@[simp, norm_cast]\ntheorem coe_inf (S T : NonUnitalSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T :=\n  rfl\n\n@[simp]\ntheorem mem_inf {S T : NonUnitalSubalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T :=\n  Iff.rfl\n\n@[simp]\ntheorem inf_toSubmodule (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toSubmodule = S.toSubmodule ⊓ T.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem inf_toNonUnitalSubsemiring (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring ⊓ T.toNonUnitalSubsemiring :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sInf (S : Set (NonUnitalSubalgebra R A)) : (↑(sInf S) : Set A) = ⋂ s ∈ S, ↑s :=\n  sInf_image\n\ntheorem mem_sInf {S : Set (NonUnitalSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p := by\n  simp only [← SetLike.mem_coe, coe_sInf, Set.mem_iInter₂]\n\n@[simp]\ntheorem sInf_toSubmodule (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toSubmodule = sInf (NonUnitalSubalgebra.toSubmodule '' S) :=\n  SetLike.coe_injective <| by ","nextTactic":"simp","declUpToTactic":"@[simp]\ntheorem sInf_toSubmodule (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toSubmodule = sInf (NonUnitalSubalgebra.toSubmodule '' S) :=\n  SetLike.coe_injective <| by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.710_0.jQvdAcOcyVirh23","decl":"@[simp]\ntheorem sInf_toSubmodule (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toSubmodule = sInf (NonUnitalSubalgebra.toSubmodule '' S) "}
-{"state":"F : Type u_1\nR : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\ninst✝ : NonUnitalAlgHomClass F R A B\nS : Set (NonUnitalSubalgebra R A)\n⊢ ↑(sInf S).toNonUnitalSubsemiring = ↑(sInf (NonUnitalSubalgebra.toNonUnitalSubsemiring '' S))","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact NonUnitalAlgebra.gc\n\n@[simp]\ntheorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=\n  eq_top_iff.2 fun _x hx => subset_adjoin R hx\n\nvariable {R A}\n\n@[simp]\ntheorem coe_top : (↑(⊤ : NonUnitalSubalgebra R A) : Set A) = Set.univ :=\n  rfl\n\n@[simp]\ntheorem mem_top {x : A} : x ∈ (⊤ : NonUnitalSubalgebra R A) :=\n  Set.mem_univ x\n\n@[simp]\ntheorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toNonUnitalSubsemiring : (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubsemiring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toSubring {R A : Type*} [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    [IsScalarTower R A A] [SMulCommClass R A A] :\n    (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem toSubmodule_eq_top {S : NonUnitalSubalgebra R A} : S.toSubmodule = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toSubmodule'.injective.eq_iff' top_toSubmodule\n\n@[simp]\ntheorem toNonUnitalSubsemiring_eq_top {S : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubsemiring_injective.eq_iff' top_toNonUnitalSubsemiring\n\n@[simp]\ntheorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubring_injective.eq_iff' top_toSubring\n\ntheorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_left\n\ntheorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_right\n\ntheorem mul_mem_sup {S T : NonUnitalSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) :\n    x * y ∈ S ⊔ T :=\n  mul_mem (mem_sup_left hx) (mem_sup_right hy)\n\ntheorem map_sup (f : F) (S T : NonUnitalSubalgebra R A) :\n    ((S ⊔ T).map f : NonUnitalSubalgebra R B) = S.map f ⊔ T.map f :=\n  (@NonUnitalSubalgebra.gc_map_comap F R A B _ _ _ _ _ _ f).l_sup\n\n@[simp, norm_cast]\ntheorem coe_inf (S T : NonUnitalSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T :=\n  rfl\n\n@[simp]\ntheorem mem_inf {S T : NonUnitalSubalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T :=\n  Iff.rfl\n\n@[simp]\ntheorem inf_toSubmodule (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toSubmodule = S.toSubmodule ⊓ T.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem inf_toNonUnitalSubsemiring (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring ⊓ T.toNonUnitalSubsemiring :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sInf (S : Set (NonUnitalSubalgebra R A)) : (↑(sInf S) : Set A) = ⋂ s ∈ S, ↑s :=\n  sInf_image\n\ntheorem mem_sInf {S : Set (NonUnitalSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p := by\n  simp only [← SetLike.mem_coe, coe_sInf, Set.mem_iInter₂]\n\n@[simp]\ntheorem sInf_toSubmodule (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toSubmodule = sInf (NonUnitalSubalgebra.toSubmodule '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp]\ntheorem sInf_toNonUnitalSubsemiring (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toNonUnitalSubsemiring = sInf (NonUnitalSubalgebra.toNonUnitalSubsemiring '' S) :=\n  SetLike.coe_injective <| by ","nextTactic":"simp","declUpToTactic":"@[simp]\ntheorem sInf_toNonUnitalSubsemiring (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toNonUnitalSubsemiring = sInf (NonUnitalSubalgebra.toNonUnitalSubsemiring '' S) :=\n  SetLike.coe_injective <| by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.715_0.jQvdAcOcyVirh23","decl":"@[simp]\ntheorem sInf_toNonUnitalSubsemiring (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toNonUnitalSubsemiring = sInf (NonUnitalSubalgebra.toNonUnitalSubsemiring '' S) "}
-{"state":"F : Type u_1\nR : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\ninst✝ : NonUnitalAlgHomClass F R A B\nι : Sort u_2\nS : ι → NonUnitalSubalgebra R A\n⊢ ↑(⨅ i, S i) = ⋂ i, ↑(S i)","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact NonUnitalAlgebra.gc\n\n@[simp]\ntheorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=\n  eq_top_iff.2 fun _x hx => subset_adjoin R hx\n\nvariable {R A}\n\n@[simp]\ntheorem coe_top : (↑(⊤ : NonUnitalSubalgebra R A) : Set A) = Set.univ :=\n  rfl\n\n@[simp]\ntheorem mem_top {x : A} : x ∈ (⊤ : NonUnitalSubalgebra R A) :=\n  Set.mem_univ x\n\n@[simp]\ntheorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toNonUnitalSubsemiring : (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubsemiring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toSubring {R A : Type*} [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    [IsScalarTower R A A] [SMulCommClass R A A] :\n    (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem toSubmodule_eq_top {S : NonUnitalSubalgebra R A} : S.toSubmodule = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toSubmodule'.injective.eq_iff' top_toSubmodule\n\n@[simp]\ntheorem toNonUnitalSubsemiring_eq_top {S : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubsemiring_injective.eq_iff' top_toNonUnitalSubsemiring\n\n@[simp]\ntheorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubring_injective.eq_iff' top_toSubring\n\ntheorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_left\n\ntheorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_right\n\ntheorem mul_mem_sup {S T : NonUnitalSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) :\n    x * y ∈ S ⊔ T :=\n  mul_mem (mem_sup_left hx) (mem_sup_right hy)\n\ntheorem map_sup (f : F) (S T : NonUnitalSubalgebra R A) :\n    ((S ⊔ T).map f : NonUnitalSubalgebra R B) = S.map f ⊔ T.map f :=\n  (@NonUnitalSubalgebra.gc_map_comap F R A B _ _ _ _ _ _ f).l_sup\n\n@[simp, norm_cast]\ntheorem coe_inf (S T : NonUnitalSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T :=\n  rfl\n\n@[simp]\ntheorem mem_inf {S T : NonUnitalSubalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T :=\n  Iff.rfl\n\n@[simp]\ntheorem inf_toSubmodule (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toSubmodule = S.toSubmodule ⊓ T.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem inf_toNonUnitalSubsemiring (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring ⊓ T.toNonUnitalSubsemiring :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sInf (S : Set (NonUnitalSubalgebra R A)) : (↑(sInf S) : Set A) = ⋂ s ∈ S, ↑s :=\n  sInf_image\n\ntheorem mem_sInf {S : Set (NonUnitalSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p := by\n  simp only [← SetLike.mem_coe, coe_sInf, Set.mem_iInter₂]\n\n@[simp]\ntheorem sInf_toSubmodule (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toSubmodule = sInf (NonUnitalSubalgebra.toSubmodule '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp]\ntheorem sInf_toNonUnitalSubsemiring (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toNonUnitalSubsemiring = sInf (NonUnitalSubalgebra.toNonUnitalSubsemiring '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp, norm_cast]\ntheorem coe_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} :\n    (↑(⨅ i, S i) : Set A) = ⋂ i, S i := by ","nextTactic":"simp [iInf]","declUpToTactic":"@[simp, norm_cast]\ntheorem coe_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} :\n    (↑(⨅ i, S i) : Set A) = ⋂ i, S i := by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.720_0.jQvdAcOcyVirh23","decl":"@[simp, norm_cast]\ntheorem coe_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} :\n    (↑(⨅ i, S i) : Set A) = ⋂ i, S i "}
-{"state":"F : Type u_1\nR : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\ninst✝ : NonUnitalAlgHomClass F R A B\nι : Sort u_2\nS : ι → NonUnitalSubalgebra R A\nx : A\n⊢ x ∈ ⨅ i, S i ↔ ∀ (i : ι), x ∈ S i","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact NonUnitalAlgebra.gc\n\n@[simp]\ntheorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=\n  eq_top_iff.2 fun _x hx => subset_adjoin R hx\n\nvariable {R A}\n\n@[simp]\ntheorem coe_top : (↑(⊤ : NonUnitalSubalgebra R A) : Set A) = Set.univ :=\n  rfl\n\n@[simp]\ntheorem mem_top {x : A} : x ∈ (⊤ : NonUnitalSubalgebra R A) :=\n  Set.mem_univ x\n\n@[simp]\ntheorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toNonUnitalSubsemiring : (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubsemiring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toSubring {R A : Type*} [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    [IsScalarTower R A A] [SMulCommClass R A A] :\n    (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem toSubmodule_eq_top {S : NonUnitalSubalgebra R A} : S.toSubmodule = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toSubmodule'.injective.eq_iff' top_toSubmodule\n\n@[simp]\ntheorem toNonUnitalSubsemiring_eq_top {S : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubsemiring_injective.eq_iff' top_toNonUnitalSubsemiring\n\n@[simp]\ntheorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubring_injective.eq_iff' top_toSubring\n\ntheorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_left\n\ntheorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_right\n\ntheorem mul_mem_sup {S T : NonUnitalSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) :\n    x * y ∈ S ⊔ T :=\n  mul_mem (mem_sup_left hx) (mem_sup_right hy)\n\ntheorem map_sup (f : F) (S T : NonUnitalSubalgebra R A) :\n    ((S ⊔ T).map f : NonUnitalSubalgebra R B) = S.map f ⊔ T.map f :=\n  (@NonUnitalSubalgebra.gc_map_comap F R A B _ _ _ _ _ _ f).l_sup\n\n@[simp, norm_cast]\ntheorem coe_inf (S T : NonUnitalSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T :=\n  rfl\n\n@[simp]\ntheorem mem_inf {S T : NonUnitalSubalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T :=\n  Iff.rfl\n\n@[simp]\ntheorem inf_toSubmodule (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toSubmodule = S.toSubmodule ⊓ T.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem inf_toNonUnitalSubsemiring (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring ⊓ T.toNonUnitalSubsemiring :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sInf (S : Set (NonUnitalSubalgebra R A)) : (↑(sInf S) : Set A) = ⋂ s ∈ S, ↑s :=\n  sInf_image\n\ntheorem mem_sInf {S : Set (NonUnitalSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p := by\n  simp only [← SetLike.mem_coe, coe_sInf, Set.mem_iInter₂]\n\n@[simp]\ntheorem sInf_toSubmodule (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toSubmodule = sInf (NonUnitalSubalgebra.toSubmodule '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp]\ntheorem sInf_toNonUnitalSubsemiring (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toNonUnitalSubsemiring = sInf (NonUnitalSubalgebra.toNonUnitalSubsemiring '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp, norm_cast]\ntheorem coe_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} :\n    (↑(⨅ i, S i) : Set A) = ⋂ i, S i := by simp [iInf]\n\ntheorem mem_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} {x : A} :\n    (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by ","nextTactic":"simp only [iInf, mem_sInf, Set.forall_range_iff]","declUpToTactic":"theorem mem_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} {x : A} :\n    (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.724_0.jQvdAcOcyVirh23","decl":"theorem mem_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} {x : A} :\n    (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i "}
-{"state":"F : Type u_1\nR : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\ninst✝ : NonUnitalAlgHomClass F R A B\nι : Sort u_2\nS : ι → NonUnitalSubalgebra R A\n⊢ ↑(NonUnitalSubalgebra.toSubmodule (⨅ i, S i)) = ↑(⨅ i, NonUnitalSubalgebra.toSubmodule (S i))","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact NonUnitalAlgebra.gc\n\n@[simp]\ntheorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=\n  eq_top_iff.2 fun _x hx => subset_adjoin R hx\n\nvariable {R A}\n\n@[simp]\ntheorem coe_top : (↑(⊤ : NonUnitalSubalgebra R A) : Set A) = Set.univ :=\n  rfl\n\n@[simp]\ntheorem mem_top {x : A} : x ∈ (⊤ : NonUnitalSubalgebra R A) :=\n  Set.mem_univ x\n\n@[simp]\ntheorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toNonUnitalSubsemiring : (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubsemiring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toSubring {R A : Type*} [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    [IsScalarTower R A A] [SMulCommClass R A A] :\n    (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem toSubmodule_eq_top {S : NonUnitalSubalgebra R A} : S.toSubmodule = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toSubmodule'.injective.eq_iff' top_toSubmodule\n\n@[simp]\ntheorem toNonUnitalSubsemiring_eq_top {S : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubsemiring_injective.eq_iff' top_toNonUnitalSubsemiring\n\n@[simp]\ntheorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubring_injective.eq_iff' top_toSubring\n\ntheorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_left\n\ntheorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_right\n\ntheorem mul_mem_sup {S T : NonUnitalSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) :\n    x * y ∈ S ⊔ T :=\n  mul_mem (mem_sup_left hx) (mem_sup_right hy)\n\ntheorem map_sup (f : F) (S T : NonUnitalSubalgebra R A) :\n    ((S ⊔ T).map f : NonUnitalSubalgebra R B) = S.map f ⊔ T.map f :=\n  (@NonUnitalSubalgebra.gc_map_comap F R A B _ _ _ _ _ _ f).l_sup\n\n@[simp, norm_cast]\ntheorem coe_inf (S T : NonUnitalSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T :=\n  rfl\n\n@[simp]\ntheorem mem_inf {S T : NonUnitalSubalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T :=\n  Iff.rfl\n\n@[simp]\ntheorem inf_toSubmodule (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toSubmodule = S.toSubmodule ⊓ T.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem inf_toNonUnitalSubsemiring (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring ⊓ T.toNonUnitalSubsemiring :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sInf (S : Set (NonUnitalSubalgebra R A)) : (↑(sInf S) : Set A) = ⋂ s ∈ S, ↑s :=\n  sInf_image\n\ntheorem mem_sInf {S : Set (NonUnitalSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p := by\n  simp only [← SetLike.mem_coe, coe_sInf, Set.mem_iInter₂]\n\n@[simp]\ntheorem sInf_toSubmodule (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toSubmodule = sInf (NonUnitalSubalgebra.toSubmodule '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp]\ntheorem sInf_toNonUnitalSubsemiring (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toNonUnitalSubsemiring = sInf (NonUnitalSubalgebra.toNonUnitalSubsemiring '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp, norm_cast]\ntheorem coe_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} :\n    (↑(⨅ i, S i) : Set A) = ⋂ i, S i := by simp [iInf]\n\ntheorem mem_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} {x : A} :\n    (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by simp only [iInf, mem_sInf, Set.forall_range_iff]\n\n@[simp]\ntheorem iInf_toSubmodule {ι : Sort*} (S : ι → NonUnitalSubalgebra R A) :\n    (⨅ i, S i).toSubmodule = ⨅ i, (S i).toSubmodule :=\n  SetLike.coe_injective <| by ","nextTactic":"simp","declUpToTactic":"@[simp]\ntheorem iInf_toSubmodule {ι : Sort*} (S : ι → NonUnitalSubalgebra R A) :\n    (⨅ i, S i).toSubmodule = ⨅ i, (S i).toSubmodule :=\n  SetLike.coe_injective <| by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.727_0.jQvdAcOcyVirh23","decl":"@[simp]\ntheorem iInf_toSubmodule {ι : Sort*} (S : ι → NonUnitalSubalgebra R A) :\n    (⨅ i, S i).toSubmodule = ⨅ i, (S i).toSubmodule "}
-{"state":"F : Type u_1\nR : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\ninst✝ : NonUnitalAlgHomClass F R A B\nx : A\n⊢ x ∈ Submodule.span R ↑(NonUnitalSubsemiring.closure ∅) ↔ x = 0","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : �� x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact NonUnitalAlgebra.gc\n\n@[simp]\ntheorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=\n  eq_top_iff.2 fun _x hx => subset_adjoin R hx\n\nvariable {R A}\n\n@[simp]\ntheorem coe_top : (↑(⊤ : NonUnitalSubalgebra R A) : Set A) = Set.univ :=\n  rfl\n\n@[simp]\ntheorem mem_top {x : A} : x ∈ (⊤ : NonUnitalSubalgebra R A) :=\n  Set.mem_univ x\n\n@[simp]\ntheorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toNonUnitalSubsemiring : (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubsemiring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toSubring {R A : Type*} [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    [IsScalarTower R A A] [SMulCommClass R A A] :\n    (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem toSubmodule_eq_top {S : NonUnitalSubalgebra R A} : S.toSubmodule = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toSubmodule'.injective.eq_iff' top_toSubmodule\n\n@[simp]\ntheorem toNonUnitalSubsemiring_eq_top {S : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubsemiring_injective.eq_iff' top_toNonUnitalSubsemiring\n\n@[simp]\ntheorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubring_injective.eq_iff' top_toSubring\n\ntheorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_left\n\ntheorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_right\n\ntheorem mul_mem_sup {S T : NonUnitalSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) :\n    x * y ∈ S ⊔ T :=\n  mul_mem (mem_sup_left hx) (mem_sup_right hy)\n\ntheorem map_sup (f : F) (S T : NonUnitalSubalgebra R A) :\n    ((S ⊔ T).map f : NonUnitalSubalgebra R B) = S.map f ⊔ T.map f :=\n  (@NonUnitalSubalgebra.gc_map_comap F R A B _ _ _ _ _ _ f).l_sup\n\n@[simp, norm_cast]\ntheorem coe_inf (S T : NonUnitalSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T :=\n  rfl\n\n@[simp]\ntheorem mem_inf {S T : NonUnitalSubalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T :=\n  Iff.rfl\n\n@[simp]\ntheorem inf_toSubmodule (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toSubmodule = S.toSubmodule ⊓ T.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem inf_toNonUnitalSubsemiring (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring ⊓ T.toNonUnitalSubsemiring :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sInf (S : Set (NonUnitalSubalgebra R A)) : (↑(sInf S) : Set A) = ⋂ s ∈ S, ↑s :=\n  sInf_image\n\ntheorem mem_sInf {S : Set (NonUnitalSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p := by\n  simp only [← SetLike.mem_coe, coe_sInf, Set.mem_iInter₂]\n\n@[simp]\ntheorem sInf_toSubmodule (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toSubmodule = sInf (NonUnitalSubalgebra.toSubmodule '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp]\ntheorem sInf_toNonUnitalSubsemiring (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toNonUnitalSubsemiring = sInf (NonUnitalSubalgebra.toNonUnitalSubsemiring '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp, norm_cast]\ntheorem coe_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} :\n    (↑(⨅ i, S i) : Set A) = ⋂ i, S i := by simp [iInf]\n\ntheorem mem_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} {x : A} :\n    (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by simp only [iInf, mem_sInf, Set.forall_range_iff]\n\n@[simp]\ntheorem iInf_toSubmodule {ι : Sort*} (S : ι → NonUnitalSubalgebra R A) :\n    (⨅ i, S i).toSubmodule = ⨅ i, (S i).toSubmodule :=\n  SetLike.coe_injective <| by simp\n\ninstance : Inhabited (NonUnitalSubalgebra R A) :=\n  ⟨⊥⟩\n\ntheorem mem_bot {x : A} : x ∈ (⊥ : NonUnitalSubalgebra R A) ↔ x = 0 :=\n  show x ∈ Submodule.span R (NonUnitalSubsemiring.closure (∅ : Set A) : Set A) ↔ x = 0 by\n    ","nextTactic":"rw [NonUnitalSubsemiring.closure_empty, NonUnitalSubsemiring.coe_bot,\n      Submodule.span_zero_singleton, Submodule.mem_bot]","declUpToTactic":"theorem mem_bot {x : A} : x ∈ (⊥ : NonUnitalSubalgebra R A) ↔ x = 0 :=\n  show x ∈ Submodule.span R (NonUnitalSubsemiring.closure (∅ : Set A) : Set A) ↔ x = 0 by\n    ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.735_0.jQvdAcOcyVirh23","decl":"theorem mem_bot {x : A} : x ∈ (⊥ : NonUnitalSubalgebra R A) ↔ x = 0 "}
-{"state":"F : Type u_1\nR : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\ninst✝ : NonUnitalAlgHomClass F R A B\n⊢ NonUnitalSubalgebra.toSubmodule ⊥ = ⊥","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y �� p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact NonUnitalAlgebra.gc\n\n@[simp]\ntheorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=\n  eq_top_iff.2 fun _x hx => subset_adjoin R hx\n\nvariable {R A}\n\n@[simp]\ntheorem coe_top : (↑(⊤ : NonUnitalSubalgebra R A) : Set A) = Set.univ :=\n  rfl\n\n@[simp]\ntheorem mem_top {x : A} : x ∈ (⊤ : NonUnitalSubalgebra R A) :=\n  Set.mem_univ x\n\n@[simp]\ntheorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toNonUnitalSubsemiring : (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubsemiring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toSubring {R A : Type*} [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    [IsScalarTower R A A] [SMulCommClass R A A] :\n    (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem toSubmodule_eq_top {S : NonUnitalSubalgebra R A} : S.toSubmodule = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toSubmodule'.injective.eq_iff' top_toSubmodule\n\n@[simp]\ntheorem toNonUnitalSubsemiring_eq_top {S : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubsemiring_injective.eq_iff' top_toNonUnitalSubsemiring\n\n@[simp]\ntheorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubring_injective.eq_iff' top_toSubring\n\ntheorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_left\n\ntheorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_right\n\ntheorem mul_mem_sup {S T : NonUnitalSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) :\n    x * y ∈ S ⊔ T :=\n  mul_mem (mem_sup_left hx) (mem_sup_right hy)\n\ntheorem map_sup (f : F) (S T : NonUnitalSubalgebra R A) :\n    ((S ⊔ T).map f : NonUnitalSubalgebra R B) = S.map f ⊔ T.map f :=\n  (@NonUnitalSubalgebra.gc_map_comap F R A B _ _ _ _ _ _ f).l_sup\n\n@[simp, norm_cast]\ntheorem coe_inf (S T : NonUnitalSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T :=\n  rfl\n\n@[simp]\ntheorem mem_inf {S T : NonUnitalSubalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T :=\n  Iff.rfl\n\n@[simp]\ntheorem inf_toSubmodule (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toSubmodule = S.toSubmodule ⊓ T.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem inf_toNonUnitalSubsemiring (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring ⊓ T.toNonUnitalSubsemiring :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sInf (S : Set (NonUnitalSubalgebra R A)) : (↑(sInf S) : Set A) = ⋂ s ∈ S, ↑s :=\n  sInf_image\n\ntheorem mem_sInf {S : Set (NonUnitalSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p := by\n  simp only [← SetLike.mem_coe, coe_sInf, Set.mem_iInter₂]\n\n@[simp]\ntheorem sInf_toSubmodule (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toSubmodule = sInf (NonUnitalSubalgebra.toSubmodule '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp]\ntheorem sInf_toNonUnitalSubsemiring (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toNonUnitalSubsemiring = sInf (NonUnitalSubalgebra.toNonUnitalSubsemiring '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp, norm_cast]\ntheorem coe_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} :\n    (↑(⨅ i, S i) : Set A) = ⋂ i, S i := by simp [iInf]\n\ntheorem mem_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} {x : A} :\n    (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by simp only [iInf, mem_sInf, Set.forall_range_iff]\n\n@[simp]\ntheorem iInf_toSubmodule {ι : Sort*} (S : ι → NonUnitalSubalgebra R A) :\n    (⨅ i, S i).toSubmodule = ⨅ i, (S i).toSubmodule :=\n  SetLike.coe_injective <| by simp\n\ninstance : Inhabited (NonUnitalSubalgebra R A) :=\n  ⟨⊥⟩\n\ntheorem mem_bot {x : A} : x ∈ (⊥ : NonUnitalSubalgebra R A) ↔ x = 0 :=\n  show x ∈ Submodule.span R (NonUnitalSubsemiring.closure (∅ : Set A) : Set A) ↔ x = 0 by\n    rw [NonUnitalSubsemiring.closure_empty, NonUnitalSubsemiring.coe_bot,\n      Submodule.span_zero_singleton, Submodule.mem_bot]\n\ntheorem toSubmodule_bot : (⊥ : NonUnitalSubalgebra R A).toSubmodule = ⊥ := by\n  ","nextTactic":"ext","declUpToTactic":"theorem toSubmodule_bot : (⊥ : NonUnitalSubalgebra R A).toSubmodule = ⊥ := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.740_0.jQvdAcOcyVirh23","decl":"theorem toSubmodule_bot : (⊥ : NonUnitalSubalgebra R A).toSubmodule = ⊥ "}
-{"state":"case h\nF : Type u_1\nR : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\ninst✝ : NonUnitalAlgHomClass F R A B\nx✝ : A\n⊢ x✝ ∈ NonUnitalSubalgebra.toSubmodule ⊥ ↔ x✝ ∈ ⊥","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact NonUnitalAlgebra.gc\n\n@[simp]\ntheorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=\n  eq_top_iff.2 fun _x hx => subset_adjoin R hx\n\nvariable {R A}\n\n@[simp]\ntheorem coe_top : (↑(⊤ : NonUnitalSubalgebra R A) : Set A) = Set.univ :=\n  rfl\n\n@[simp]\ntheorem mem_top {x : A} : x ∈ (⊤ : NonUnitalSubalgebra R A) :=\n  Set.mem_univ x\n\n@[simp]\ntheorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toNonUnitalSubsemiring : (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubsemiring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toSubring {R A : Type*} [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    [IsScalarTower R A A] [SMulCommClass R A A] :\n    (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem toSubmodule_eq_top {S : NonUnitalSubalgebra R A} : S.toSubmodule = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toSubmodule'.injective.eq_iff' top_toSubmodule\n\n@[simp]\ntheorem toNonUnitalSubsemiring_eq_top {S : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubsemiring_injective.eq_iff' top_toNonUnitalSubsemiring\n\n@[simp]\ntheorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubring_injective.eq_iff' top_toSubring\n\ntheorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_left\n\ntheorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_right\n\ntheorem mul_mem_sup {S T : NonUnitalSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) :\n    x * y ∈ S ⊔ T :=\n  mul_mem (mem_sup_left hx) (mem_sup_right hy)\n\ntheorem map_sup (f : F) (S T : NonUnitalSubalgebra R A) :\n    ((S ⊔ T).map f : NonUnitalSubalgebra R B) = S.map f ⊔ T.map f :=\n  (@NonUnitalSubalgebra.gc_map_comap F R A B _ _ _ _ _ _ f).l_sup\n\n@[simp, norm_cast]\ntheorem coe_inf (S T : NonUnitalSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T :=\n  rfl\n\n@[simp]\ntheorem mem_inf {S T : NonUnitalSubalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T :=\n  Iff.rfl\n\n@[simp]\ntheorem inf_toSubmodule (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toSubmodule = S.toSubmodule ⊓ T.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem inf_toNonUnitalSubsemiring (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring ⊓ T.toNonUnitalSubsemiring :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sInf (S : Set (NonUnitalSubalgebra R A)) : (↑(sInf S) : Set A) = ⋂ s ∈ S, ↑s :=\n  sInf_image\n\ntheorem mem_sInf {S : Set (NonUnitalSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p := by\n  simp only [← SetLike.mem_coe, coe_sInf, Set.mem_iInter₂]\n\n@[simp]\ntheorem sInf_toSubmodule (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toSubmodule = sInf (NonUnitalSubalgebra.toSubmodule '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp]\ntheorem sInf_toNonUnitalSubsemiring (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toNonUnitalSubsemiring = sInf (NonUnitalSubalgebra.toNonUnitalSubsemiring '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp, norm_cast]\ntheorem coe_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} :\n    (↑(⨅ i, S i) : Set A) = ⋂ i, S i := by simp [iInf]\n\ntheorem mem_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} {x : A} :\n    (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by simp only [iInf, mem_sInf, Set.forall_range_iff]\n\n@[simp]\ntheorem iInf_toSubmodule {ι : Sort*} (S : ι → NonUnitalSubalgebra R A) :\n    (⨅ i, S i).toSubmodule = ⨅ i, (S i).toSubmodule :=\n  SetLike.coe_injective <| by simp\n\ninstance : Inhabited (NonUnitalSubalgebra R A) :=\n  ⟨⊥⟩\n\ntheorem mem_bot {x : A} : x ∈ (⊥ : NonUnitalSubalgebra R A) ↔ x = 0 :=\n  show x ∈ Submodule.span R (NonUnitalSubsemiring.closure (∅ : Set A) : Set A) ↔ x = 0 by\n    rw [NonUnitalSubsemiring.closure_empty, NonUnitalSubsemiring.coe_bot,\n      Submodule.span_zero_singleton, Submodule.mem_bot]\n\ntheorem toSubmodule_bot : (⊥ : NonUnitalSubalgebra R A).toSubmodule = ⊥ := by\n  ext\n  ","nextTactic":"simp only [mem_bot, NonUnitalSubalgebra.mem_toSubmodule, Submodule.mem_bot]","declUpToTactic":"theorem toSubmodule_bot : (⊥ : NonUnitalSubalgebra R A).toSubmodule = ⊥ := by\n  ext\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.740_0.jQvdAcOcyVirh23","decl":"theorem toSubmodule_bot : (⊥ : NonUnitalSubalgebra R A).toSubmodule = ⊥ "}
-{"state":"F : Type u_1\nR : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\ninst✝ : NonUnitalAlgHomClass F R A B\n⊢ ↑⊥ = {0}","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact NonUnitalAlgebra.gc\n\n@[simp]\ntheorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=\n  eq_top_iff.2 fun _x hx => subset_adjoin R hx\n\nvariable {R A}\n\n@[simp]\ntheorem coe_top : (↑(⊤ : NonUnitalSubalgebra R A) : Set A) = Set.univ :=\n  rfl\n\n@[simp]\ntheorem mem_top {x : A} : x ∈ (⊤ : NonUnitalSubalgebra R A) :=\n  Set.mem_univ x\n\n@[simp]\ntheorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toNonUnitalSubsemiring : (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubsemiring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toSubring {R A : Type*} [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    [IsScalarTower R A A] [SMulCommClass R A A] :\n    (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem toSubmodule_eq_top {S : NonUnitalSubalgebra R A} : S.toSubmodule = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toSubmodule'.injective.eq_iff' top_toSubmodule\n\n@[simp]\ntheorem toNonUnitalSubsemiring_eq_top {S : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubsemiring_injective.eq_iff' top_toNonUnitalSubsemiring\n\n@[simp]\ntheorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubring_injective.eq_iff' top_toSubring\n\ntheorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_left\n\ntheorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_right\n\ntheorem mul_mem_sup {S T : NonUnitalSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) :\n    x * y ∈ S ⊔ T :=\n  mul_mem (mem_sup_left hx) (mem_sup_right hy)\n\ntheorem map_sup (f : F) (S T : NonUnitalSubalgebra R A) :\n    ((S ⊔ T).map f : NonUnitalSubalgebra R B) = S.map f ⊔ T.map f :=\n  (@NonUnitalSubalgebra.gc_map_comap F R A B _ _ _ _ _ _ f).l_sup\n\n@[simp, norm_cast]\ntheorem coe_inf (S T : NonUnitalSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T :=\n  rfl\n\n@[simp]\ntheorem mem_inf {S T : NonUnitalSubalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T :=\n  Iff.rfl\n\n@[simp]\ntheorem inf_toSubmodule (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toSubmodule = S.toSubmodule ⊓ T.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem inf_toNonUnitalSubsemiring (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring ⊓ T.toNonUnitalSubsemiring :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sInf (S : Set (NonUnitalSubalgebra R A)) : (↑(sInf S) : Set A) = ⋂ s ∈ S, ↑s :=\n  sInf_image\n\ntheorem mem_sInf {S : Set (NonUnitalSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p := by\n  simp only [← SetLike.mem_coe, coe_sInf, Set.mem_iInter₂]\n\n@[simp]\ntheorem sInf_toSubmodule (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toSubmodule = sInf (NonUnitalSubalgebra.toSubmodule '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp]\ntheorem sInf_toNonUnitalSubsemiring (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toNonUnitalSubsemiring = sInf (NonUnitalSubalgebra.toNonUnitalSubsemiring '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp, norm_cast]\ntheorem coe_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} :\n    (↑(⨅ i, S i) : Set A) = ⋂ i, S i := by simp [iInf]\n\ntheorem mem_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} {x : A} :\n    (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by simp only [iInf, mem_sInf, Set.forall_range_iff]\n\n@[simp]\ntheorem iInf_toSubmodule {ι : Sort*} (S : ι → NonUnitalSubalgebra R A) :\n    (⨅ i, S i).toSubmodule = ⨅ i, (S i).toSubmodule :=\n  SetLike.coe_injective <| by simp\n\ninstance : Inhabited (NonUnitalSubalgebra R A) :=\n  ⟨⊥⟩\n\ntheorem mem_bot {x : A} : x ∈ (⊥ : NonUnitalSubalgebra R A) ↔ x = 0 :=\n  show x ∈ Submodule.span R (NonUnitalSubsemiring.closure (∅ : Set A) : Set A) ↔ x = 0 by\n    rw [NonUnitalSubsemiring.closure_empty, NonUnitalSubsemiring.coe_bot,\n      Submodule.span_zero_singleton, Submodule.mem_bot]\n\ntheorem toSubmodule_bot : (⊥ : NonUnitalSubalgebra R A).toSubmodule = ⊥ := by\n  ext\n  simp only [mem_bot, NonUnitalSubalgebra.mem_toSubmodule, Submodule.mem_bot]\n\n@[simp]\ntheorem coe_bot : ((⊥ : NonUnitalSubalgebra R A) : Set A) = {0} := by\n  ","nextTactic":"simp [Set.ext_iff, NonUnitalAlgebra.mem_bot]","declUpToTactic":"@[simp]\ntheorem coe_bot : ((⊥ : NonUnitalSubalgebra R A) : Set A) = {0} := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.744_0.jQvdAcOcyVirh23","decl":"@[simp]\ntheorem coe_bot : ((⊥ : NonUnitalSubalgebra R A) : Set A) = {0} "}
-{"state":"F : Type u_1\nR : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\ninst✝ : NonUnitalAlgHomClass F R A B\nS : NonUnitalSubalgebra R A\nh : S = ⊤\nx : A\n⊢ x ∈ S","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact NonUnitalAlgebra.gc\n\n@[simp]\ntheorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=\n  eq_top_iff.2 fun _x hx => subset_adjoin R hx\n\nvariable {R A}\n\n@[simp]\ntheorem coe_top : (↑(⊤ : NonUnitalSubalgebra R A) : Set A) = Set.univ :=\n  rfl\n\n@[simp]\ntheorem mem_top {x : A} : x ∈ (⊤ : NonUnitalSubalgebra R A) :=\n  Set.mem_univ x\n\n@[simp]\ntheorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toNonUnitalSubsemiring : (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubsemiring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toSubring {R A : Type*} [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    [IsScalarTower R A A] [SMulCommClass R A A] :\n    (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem toSubmodule_eq_top {S : NonUnitalSubalgebra R A} : S.toSubmodule = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toSubmodule'.injective.eq_iff' top_toSubmodule\n\n@[simp]\ntheorem toNonUnitalSubsemiring_eq_top {S : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubsemiring_injective.eq_iff' top_toNonUnitalSubsemiring\n\n@[simp]\ntheorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubring_injective.eq_iff' top_toSubring\n\ntheorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_left\n\ntheorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_right\n\ntheorem mul_mem_sup {S T : NonUnitalSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) :\n    x * y ∈ S ⊔ T :=\n  mul_mem (mem_sup_left hx) (mem_sup_right hy)\n\ntheorem map_sup (f : F) (S T : NonUnitalSubalgebra R A) :\n    ((S ⊔ T).map f : NonUnitalSubalgebra R B) = S.map f ⊔ T.map f :=\n  (@NonUnitalSubalgebra.gc_map_comap F R A B _ _ _ _ _ _ f).l_sup\n\n@[simp, norm_cast]\ntheorem coe_inf (S T : NonUnitalSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T :=\n  rfl\n\n@[simp]\ntheorem mem_inf {S T : NonUnitalSubalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T :=\n  Iff.rfl\n\n@[simp]\ntheorem inf_toSubmodule (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toSubmodule = S.toSubmodule ⊓ T.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem inf_toNonUnitalSubsemiring (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring ⊓ T.toNonUnitalSubsemiring :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sInf (S : Set (NonUnitalSubalgebra R A)) : (↑(sInf S) : Set A) = ⋂ s ∈ S, ↑s :=\n  sInf_image\n\ntheorem mem_sInf {S : Set (NonUnitalSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p := by\n  simp only [← SetLike.mem_coe, coe_sInf, Set.mem_iInter₂]\n\n@[simp]\ntheorem sInf_toSubmodule (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toSubmodule = sInf (NonUnitalSubalgebra.toSubmodule '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp]\ntheorem sInf_toNonUnitalSubsemiring (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toNonUnitalSubsemiring = sInf (NonUnitalSubalgebra.toNonUnitalSubsemiring '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp, norm_cast]\ntheorem coe_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} :\n    (↑(⨅ i, S i) : Set A) = ⋂ i, S i := by simp [iInf]\n\ntheorem mem_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} {x : A} :\n    (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by simp only [iInf, mem_sInf, Set.forall_range_iff]\n\n@[simp]\ntheorem iInf_toSubmodule {ι : Sort*} (S : ι → NonUnitalSubalgebra R A) :\n    (⨅ i, S i).toSubmodule = ⨅ i, (S i).toSubmodule :=\n  SetLike.coe_injective <| by simp\n\ninstance : Inhabited (NonUnitalSubalgebra R A) :=\n  ⟨⊥⟩\n\ntheorem mem_bot {x : A} : x ∈ (⊥ : NonUnitalSubalgebra R A) ↔ x = 0 :=\n  show x ∈ Submodule.span R (NonUnitalSubsemiring.closure (∅ : Set A) : Set A) ↔ x = 0 by\n    rw [NonUnitalSubsemiring.closure_empty, NonUnitalSubsemiring.coe_bot,\n      Submodule.span_zero_singleton, Submodule.mem_bot]\n\ntheorem toSubmodule_bot : (⊥ : NonUnitalSubalgebra R A).toSubmodule = ⊥ := by\n  ext\n  simp only [mem_bot, NonUnitalSubalgebra.mem_toSubmodule, Submodule.mem_bot]\n\n@[simp]\ntheorem coe_bot : ((⊥ : NonUnitalSubalgebra R A) : Set A) = {0} := by\n  simp [Set.ext_iff, NonUnitalAlgebra.mem_bot]\n\ntheorem eq_top_iff {S : NonUnitalSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S :=\n  ⟨fun h x => by ","nextTactic":"rw [h]","declUpToTactic":"theorem eq_top_iff {S : NonUnitalSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S :=\n  ⟨fun h x => by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.748_0.jQvdAcOcyVirh23","decl":"theorem eq_top_iff {S : NonUnitalSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S "}
-{"state":"F : Type u_1\nR : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\ninst✝ : NonUnitalAlgHomClass F R A B\nS : NonUnitalSubalgebra R A\nh : S = ⊤\nx : A\n⊢ x ∈ ⊤","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact NonUnitalAlgebra.gc\n\n@[simp]\ntheorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=\n  eq_top_iff.2 fun _x hx => subset_adjoin R hx\n\nvariable {R A}\n\n@[simp]\ntheorem coe_top : (↑(⊤ : NonUnitalSubalgebra R A) : Set A) = Set.univ :=\n  rfl\n\n@[simp]\ntheorem mem_top {x : A} : x ∈ (⊤ : NonUnitalSubalgebra R A) :=\n  Set.mem_univ x\n\n@[simp]\ntheorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toNonUnitalSubsemiring : (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubsemiring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toSubring {R A : Type*} [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    [IsScalarTower R A A] [SMulCommClass R A A] :\n    (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem toSubmodule_eq_top {S : NonUnitalSubalgebra R A} : S.toSubmodule = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toSubmodule'.injective.eq_iff' top_toSubmodule\n\n@[simp]\ntheorem toNonUnitalSubsemiring_eq_top {S : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubsemiring_injective.eq_iff' top_toNonUnitalSubsemiring\n\n@[simp]\ntheorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubring_injective.eq_iff' top_toSubring\n\ntheorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_left\n\ntheorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_right\n\ntheorem mul_mem_sup {S T : NonUnitalSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) :\n    x * y ∈ S ⊔ T :=\n  mul_mem (mem_sup_left hx) (mem_sup_right hy)\n\ntheorem map_sup (f : F) (S T : NonUnitalSubalgebra R A) :\n    ((S ⊔ T).map f : NonUnitalSubalgebra R B) = S.map f ⊔ T.map f :=\n  (@NonUnitalSubalgebra.gc_map_comap F R A B _ _ _ _ _ _ f).l_sup\n\n@[simp, norm_cast]\ntheorem coe_inf (S T : NonUnitalSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T :=\n  rfl\n\n@[simp]\ntheorem mem_inf {S T : NonUnitalSubalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T :=\n  Iff.rfl\n\n@[simp]\ntheorem inf_toSubmodule (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toSubmodule = S.toSubmodule ⊓ T.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem inf_toNonUnitalSubsemiring (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring ⊓ T.toNonUnitalSubsemiring :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sInf (S : Set (NonUnitalSubalgebra R A)) : (↑(sInf S) : Set A) = ⋂ s ∈ S, ↑s :=\n  sInf_image\n\ntheorem mem_sInf {S : Set (NonUnitalSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p := by\n  simp only [← SetLike.mem_coe, coe_sInf, Set.mem_iInter₂]\n\n@[simp]\ntheorem sInf_toSubmodule (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toSubmodule = sInf (NonUnitalSubalgebra.toSubmodule '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp]\ntheorem sInf_toNonUnitalSubsemiring (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toNonUnitalSubsemiring = sInf (NonUnitalSubalgebra.toNonUnitalSubsemiring '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp, norm_cast]\ntheorem coe_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} :\n    (↑(⨅ i, S i) : Set A) = ⋂ i, S i := by simp [iInf]\n\ntheorem mem_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} {x : A} :\n    (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by simp only [iInf, mem_sInf, Set.forall_range_iff]\n\n@[simp]\ntheorem iInf_toSubmodule {ι : Sort*} (S : ι → NonUnitalSubalgebra R A) :\n    (⨅ i, S i).toSubmodule = ⨅ i, (S i).toSubmodule :=\n  SetLike.coe_injective <| by simp\n\ninstance : Inhabited (NonUnitalSubalgebra R A) :=\n  ⟨⊥⟩\n\ntheorem mem_bot {x : A} : x ∈ (⊥ : NonUnitalSubalgebra R A) ↔ x = 0 :=\n  show x ∈ Submodule.span R (NonUnitalSubsemiring.closure (∅ : Set A) : Set A) ↔ x = 0 by\n    rw [NonUnitalSubsemiring.closure_empty, NonUnitalSubsemiring.coe_bot,\n      Submodule.span_zero_singleton, Submodule.mem_bot]\n\ntheorem toSubmodule_bot : (⊥ : NonUnitalSubalgebra R A).toSubmodule = ⊥ := by\n  ext\n  simp only [mem_bot, NonUnitalSubalgebra.mem_toSubmodule, Submodule.mem_bot]\n\n@[simp]\ntheorem coe_bot : ((⊥ : NonUnitalSubalgebra R A) : Set A) = {0} := by\n  simp [Set.ext_iff, NonUnitalAlgebra.mem_bot]\n\ntheorem eq_top_iff {S : NonUnitalSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S :=\n  ⟨fun h x => by rw [h]; ","nextTactic":"exact mem_top","declUpToTactic":"theorem eq_top_iff {S : NonUnitalSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S :=\n  ⟨fun h x => by rw [h]; ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.748_0.jQvdAcOcyVirh23","decl":"theorem eq_top_iff {S : NonUnitalSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S "}
-{"state":"F : Type u_1\nR : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\ninst✝ : NonUnitalAlgHomClass F R A B\nS : NonUnitalSubalgebra R A\nh : ∀ (x : A), x ∈ S\n⊢ S = ⊤","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact NonUnitalAlgebra.gc\n\n@[simp]\ntheorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=\n  eq_top_iff.2 fun _x hx => subset_adjoin R hx\n\nvariable {R A}\n\n@[simp]\ntheorem coe_top : (↑(⊤ : NonUnitalSubalgebra R A) : Set A) = Set.univ :=\n  rfl\n\n@[simp]\ntheorem mem_top {x : A} : x ∈ (⊤ : NonUnitalSubalgebra R A) :=\n  Set.mem_univ x\n\n@[simp]\ntheorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toNonUnitalSubsemiring : (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubsemiring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toSubring {R A : Type*} [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    [IsScalarTower R A A] [SMulCommClass R A A] :\n    (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem toSubmodule_eq_top {S : NonUnitalSubalgebra R A} : S.toSubmodule = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toSubmodule'.injective.eq_iff' top_toSubmodule\n\n@[simp]\ntheorem toNonUnitalSubsemiring_eq_top {S : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubsemiring_injective.eq_iff' top_toNonUnitalSubsemiring\n\n@[simp]\ntheorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubring_injective.eq_iff' top_toSubring\n\ntheorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_left\n\ntheorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_right\n\ntheorem mul_mem_sup {S T : NonUnitalSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) :\n    x * y ∈ S ⊔ T :=\n  mul_mem (mem_sup_left hx) (mem_sup_right hy)\n\ntheorem map_sup (f : F) (S T : NonUnitalSubalgebra R A) :\n    ((S ⊔ T).map f : NonUnitalSubalgebra R B) = S.map f ⊔ T.map f :=\n  (@NonUnitalSubalgebra.gc_map_comap F R A B _ _ _ _ _ _ f).l_sup\n\n@[simp, norm_cast]\ntheorem coe_inf (S T : NonUnitalSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T :=\n  rfl\n\n@[simp]\ntheorem mem_inf {S T : NonUnitalSubalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T :=\n  Iff.rfl\n\n@[simp]\ntheorem inf_toSubmodule (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toSubmodule = S.toSubmodule ⊓ T.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem inf_toNonUnitalSubsemiring (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring ⊓ T.toNonUnitalSubsemiring :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sInf (S : Set (NonUnitalSubalgebra R A)) : (↑(sInf S) : Set A) = ⋂ s ∈ S, ↑s :=\n  sInf_image\n\ntheorem mem_sInf {S : Set (NonUnitalSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p := by\n  simp only [← SetLike.mem_coe, coe_sInf, Set.mem_iInter₂]\n\n@[simp]\ntheorem sInf_toSubmodule (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toSubmodule = sInf (NonUnitalSubalgebra.toSubmodule '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp]\ntheorem sInf_toNonUnitalSubsemiring (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toNonUnitalSubsemiring = sInf (NonUnitalSubalgebra.toNonUnitalSubsemiring '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp, norm_cast]\ntheorem coe_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} :\n    (↑(⨅ i, S i) : Set A) = ⋂ i, S i := by simp [iInf]\n\ntheorem mem_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} {x : A} :\n    (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by simp only [iInf, mem_sInf, Set.forall_range_iff]\n\n@[simp]\ntheorem iInf_toSubmodule {ι : Sort*} (S : ι → NonUnitalSubalgebra R A) :\n    (⨅ i, S i).toSubmodule = ⨅ i, (S i).toSubmodule :=\n  SetLike.coe_injective <| by simp\n\ninstance : Inhabited (NonUnitalSubalgebra R A) :=\n  ⟨⊥⟩\n\ntheorem mem_bot {x : A} : x ∈ (⊥ : NonUnitalSubalgebra R A) ↔ x = 0 :=\n  show x ∈ Submodule.span R (NonUnitalSubsemiring.closure (∅ : Set A) : Set A) ↔ x = 0 by\n    rw [NonUnitalSubsemiring.closure_empty, NonUnitalSubsemiring.coe_bot,\n      Submodule.span_zero_singleton, Submodule.mem_bot]\n\ntheorem toSubmodule_bot : (⊥ : NonUnitalSubalgebra R A).toSubmodule = ⊥ := by\n  ext\n  simp only [mem_bot, NonUnitalSubalgebra.mem_toSubmodule, Submodule.mem_bot]\n\n@[simp]\ntheorem coe_bot : ((⊥ : NonUnitalSubalgebra R A) : Set A) = {0} := by\n  simp [Set.ext_iff, NonUnitalAlgebra.mem_bot]\n\ntheorem eq_top_iff {S : NonUnitalSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S :=\n  ⟨fun h x => by rw [h]; exact mem_top, fun h => by ","nextTactic":"ext x","declUpToTactic":"theorem eq_top_iff {S : NonUnitalSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S :=\n  ⟨fun h x => by rw [h]; exact mem_top, fun h => by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.748_0.jQvdAcOcyVirh23","decl":"theorem eq_top_iff {S : NonUnitalSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S "}
-{"state":"case h\nF : Type u_1\nR : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\ninst✝ : NonUnitalAlgHomClass F R A B\nS : NonUnitalSubalgebra R A\nh : ∀ (x : A), x ∈ S\nx : A\n⊢ x ∈ S ↔ x ∈ ⊤","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact NonUnitalAlgebra.gc\n\n@[simp]\ntheorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=\n  eq_top_iff.2 fun _x hx => subset_adjoin R hx\n\nvariable {R A}\n\n@[simp]\ntheorem coe_top : (↑(⊤ : NonUnitalSubalgebra R A) : Set A) = Set.univ :=\n  rfl\n\n@[simp]\ntheorem mem_top {x : A} : x ∈ (⊤ : NonUnitalSubalgebra R A) :=\n  Set.mem_univ x\n\n@[simp]\ntheorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toNonUnitalSubsemiring : (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubsemiring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toSubring {R A : Type*} [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    [IsScalarTower R A A] [SMulCommClass R A A] :\n    (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem toSubmodule_eq_top {S : NonUnitalSubalgebra R A} : S.toSubmodule = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toSubmodule'.injective.eq_iff' top_toSubmodule\n\n@[simp]\ntheorem toNonUnitalSubsemiring_eq_top {S : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubsemiring_injective.eq_iff' top_toNonUnitalSubsemiring\n\n@[simp]\ntheorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubring_injective.eq_iff' top_toSubring\n\ntheorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_left\n\ntheorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_right\n\ntheorem mul_mem_sup {S T : NonUnitalSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) :\n    x * y ∈ S ⊔ T :=\n  mul_mem (mem_sup_left hx) (mem_sup_right hy)\n\ntheorem map_sup (f : F) (S T : NonUnitalSubalgebra R A) :\n    ((S ⊔ T).map f : NonUnitalSubalgebra R B) = S.map f ⊔ T.map f :=\n  (@NonUnitalSubalgebra.gc_map_comap F R A B _ _ _ _ _ _ f).l_sup\n\n@[simp, norm_cast]\ntheorem coe_inf (S T : NonUnitalSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T :=\n  rfl\n\n@[simp]\ntheorem mem_inf {S T : NonUnitalSubalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T :=\n  Iff.rfl\n\n@[simp]\ntheorem inf_toSubmodule (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toSubmodule = S.toSubmodule ⊓ T.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem inf_toNonUnitalSubsemiring (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring ⊓ T.toNonUnitalSubsemiring :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sInf (S : Set (NonUnitalSubalgebra R A)) : (↑(sInf S) : Set A) = ⋂ s ∈ S, ↑s :=\n  sInf_image\n\ntheorem mem_sInf {S : Set (NonUnitalSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p := by\n  simp only [← SetLike.mem_coe, coe_sInf, Set.mem_iInter₂]\n\n@[simp]\ntheorem sInf_toSubmodule (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toSubmodule = sInf (NonUnitalSubalgebra.toSubmodule '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp]\ntheorem sInf_toNonUnitalSubsemiring (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toNonUnitalSubsemiring = sInf (NonUnitalSubalgebra.toNonUnitalSubsemiring '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp, norm_cast]\ntheorem coe_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} :\n    (↑(⨅ i, S i) : Set A) = ⋂ i, S i := by simp [iInf]\n\ntheorem mem_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} {x : A} :\n    (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by simp only [iInf, mem_sInf, Set.forall_range_iff]\n\n@[simp]\ntheorem iInf_toSubmodule {ι : Sort*} (S : ι → NonUnitalSubalgebra R A) :\n    (⨅ i, S i).toSubmodule = ⨅ i, (S i).toSubmodule :=\n  SetLike.coe_injective <| by simp\n\ninstance : Inhabited (NonUnitalSubalgebra R A) :=\n  ⟨⊥⟩\n\ntheorem mem_bot {x : A} : x ∈ (⊥ : NonUnitalSubalgebra R A) ↔ x = 0 :=\n  show x ∈ Submodule.span R (NonUnitalSubsemiring.closure (∅ : Set A) : Set A) ↔ x = 0 by\n    rw [NonUnitalSubsemiring.closure_empty, NonUnitalSubsemiring.coe_bot,\n      Submodule.span_zero_singleton, Submodule.mem_bot]\n\ntheorem toSubmodule_bot : (⊥ : NonUnitalSubalgebra R A).toSubmodule = ⊥ := by\n  ext\n  simp only [mem_bot, NonUnitalSubalgebra.mem_toSubmodule, Submodule.mem_bot]\n\n@[simp]\ntheorem coe_bot : ((⊥ : NonUnitalSubalgebra R A) : Set A) = {0} := by\n  simp [Set.ext_iff, NonUnitalAlgebra.mem_bot]\n\ntheorem eq_top_iff {S : NonUnitalSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S :=\n  ⟨fun h x => by rw [h]; exact mem_top, fun h => by ext x; ","nextTactic":"exact ⟨fun _ => mem_top, fun _ => h x⟩","declUpToTactic":"theorem eq_top_iff {S : NonUnitalSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S :=\n  ⟨fun h x => by rw [h]; exact mem_top, fun h => by ext x; ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.748_0.jQvdAcOcyVirh23","decl":"theorem eq_top_iff {S : NonUnitalSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S "}
-{"state":"F : Type u_1\nR : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\ninst✝ : NonUnitalAlgHomClass F R A B\nf : A →ₙₐ[R] B\n⊢ ↑(NonUnitalSubalgebra.map f ⊥) = ↑⊥","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact NonUnitalAlgebra.gc\n\n@[simp]\ntheorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=\n  eq_top_iff.2 fun _x hx => subset_adjoin R hx\n\nvariable {R A}\n\n@[simp]\ntheorem coe_top : (↑(⊤ : NonUnitalSubalgebra R A) : Set A) = Set.univ :=\n  rfl\n\n@[simp]\ntheorem mem_top {x : A} : x ∈ (⊤ : NonUnitalSubalgebra R A) :=\n  Set.mem_univ x\n\n@[simp]\ntheorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toNonUnitalSubsemiring : (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubsemiring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toSubring {R A : Type*} [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    [IsScalarTower R A A] [SMulCommClass R A A] :\n    (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem toSubmodule_eq_top {S : NonUnitalSubalgebra R A} : S.toSubmodule = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toSubmodule'.injective.eq_iff' top_toSubmodule\n\n@[simp]\ntheorem toNonUnitalSubsemiring_eq_top {S : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubsemiring_injective.eq_iff' top_toNonUnitalSubsemiring\n\n@[simp]\ntheorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubring_injective.eq_iff' top_toSubring\n\ntheorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_left\n\ntheorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_right\n\ntheorem mul_mem_sup {S T : NonUnitalSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) :\n    x * y ∈ S ⊔ T :=\n  mul_mem (mem_sup_left hx) (mem_sup_right hy)\n\ntheorem map_sup (f : F) (S T : NonUnitalSubalgebra R A) :\n    ((S ⊔ T).map f : NonUnitalSubalgebra R B) = S.map f ⊔ T.map f :=\n  (@NonUnitalSubalgebra.gc_map_comap F R A B _ _ _ _ _ _ f).l_sup\n\n@[simp, norm_cast]\ntheorem coe_inf (S T : NonUnitalSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T :=\n  rfl\n\n@[simp]\ntheorem mem_inf {S T : NonUnitalSubalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T :=\n  Iff.rfl\n\n@[simp]\ntheorem inf_toSubmodule (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toSubmodule = S.toSubmodule ⊓ T.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem inf_toNonUnitalSubsemiring (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring ⊓ T.toNonUnitalSubsemiring :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sInf (S : Set (NonUnitalSubalgebra R A)) : (↑(sInf S) : Set A) = ⋂ s ∈ S, ↑s :=\n  sInf_image\n\ntheorem mem_sInf {S : Set (NonUnitalSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p := by\n  simp only [← SetLike.mem_coe, coe_sInf, Set.mem_iInter₂]\n\n@[simp]\ntheorem sInf_toSubmodule (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toSubmodule = sInf (NonUnitalSubalgebra.toSubmodule '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp]\ntheorem sInf_toNonUnitalSubsemiring (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toNonUnitalSubsemiring = sInf (NonUnitalSubalgebra.toNonUnitalSubsemiring '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp, norm_cast]\ntheorem coe_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} :\n    (↑(⨅ i, S i) : Set A) = ⋂ i, S i := by simp [iInf]\n\ntheorem mem_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} {x : A} :\n    (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by simp only [iInf, mem_sInf, Set.forall_range_iff]\n\n@[simp]\ntheorem iInf_toSubmodule {ι : Sort*} (S : ι → NonUnitalSubalgebra R A) :\n    (⨅ i, S i).toSubmodule = ⨅ i, (S i).toSubmodule :=\n  SetLike.coe_injective <| by simp\n\ninstance : Inhabited (NonUnitalSubalgebra R A) :=\n  ⟨⊥⟩\n\ntheorem mem_bot {x : A} : x ∈ (⊥ : NonUnitalSubalgebra R A) ↔ x = 0 :=\n  show x ∈ Submodule.span R (NonUnitalSubsemiring.closure (∅ : Set A) : Set A) ↔ x = 0 by\n    rw [NonUnitalSubsemiring.closure_empty, NonUnitalSubsemiring.coe_bot,\n      Submodule.span_zero_singleton, Submodule.mem_bot]\n\ntheorem toSubmodule_bot : (⊥ : NonUnitalSubalgebra R A).toSubmodule = ⊥ := by\n  ext\n  simp only [mem_bot, NonUnitalSubalgebra.mem_toSubmodule, Submodule.mem_bot]\n\n@[simp]\ntheorem coe_bot : ((⊥ : NonUnitalSubalgebra R A) : Set A) = {0} := by\n  simp [Set.ext_iff, NonUnitalAlgebra.mem_bot]\n\ntheorem eq_top_iff {S : NonUnitalSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S :=\n  ⟨fun h x => by rw [h]; exact mem_top, fun h => by ext x; exact ⟨fun _ => mem_top, fun _ => h x⟩⟩\n\ntheorem range_top_iff_surjective (f : A →ₙₐ[R] B) :\n    NonUnitalAlgHom.range f = (⊤ : NonUnitalSubalgebra R B) ↔ Function.Surjective f :=\n  NonUnitalAlgebra.eq_top_iff\n\n@[simp]\ntheorem range_id : NonUnitalAlgHom.range (NonUnitalAlgHom.id R A) = ⊤ :=\n  SetLike.coe_injective Set.range_id\n\n@[simp]\ntheorem map_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R A).map f = NonUnitalAlgHom.range f :=\n  SetLike.coe_injective Set.image_univ\n\n@[simp]\ntheorem map_bot (f : A →ₙₐ[R] B) : (⊥ : NonUnitalSubalgebra R A).map f = ⊥ :=\n  SetLike.coe_injective <| by ","nextTactic":"simp [NonUnitalAlgebra.coe_bot, NonUnitalSubalgebra.coe_map]","declUpToTactic":"@[simp]\ntheorem map_bot (f : A →ₙₐ[R] B) : (⊥ : NonUnitalSubalgebra R A).map f = ⊥ :=\n  SetLike.coe_injective <| by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.763_0.jQvdAcOcyVirh23","decl":"@[simp]\ntheorem map_bot (f : A →ₙₐ[R] B) : (⊥ : NonUnitalSubalgebra R A).map f = ⊥ "}
-{"state":"R : Type u\nA : Type v\nB✝ : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B✝\ninst✝³ : Module R B✝\ninst✝² : IsScalarTower R B✝ B✝\ninst✝¹ : SMulCommClass R B✝ B✝\nS : NonUnitalSubalgebra R A\ninst✝ : Subsingleton A\nB C : NonUnitalSubalgebra R A\nx : A\n⊢ x ∈ B ↔ x ∈ C","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact NonUnitalAlgebra.gc\n\n@[simp]\ntheorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=\n  eq_top_iff.2 fun _x hx => subset_adjoin R hx\n\nvariable {R A}\n\n@[simp]\ntheorem coe_top : (↑(⊤ : NonUnitalSubalgebra R A) : Set A) = Set.univ :=\n  rfl\n\n@[simp]\ntheorem mem_top {x : A} : x ∈ (⊤ : NonUnitalSubalgebra R A) :=\n  Set.mem_univ x\n\n@[simp]\ntheorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toNonUnitalSubsemiring : (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubsemiring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toSubring {R A : Type*} [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    [IsScalarTower R A A] [SMulCommClass R A A] :\n    (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem toSubmodule_eq_top {S : NonUnitalSubalgebra R A} : S.toSubmodule = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toSubmodule'.injective.eq_iff' top_toSubmodule\n\n@[simp]\ntheorem toNonUnitalSubsemiring_eq_top {S : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubsemiring_injective.eq_iff' top_toNonUnitalSubsemiring\n\n@[simp]\ntheorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubring_injective.eq_iff' top_toSubring\n\ntheorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_left\n\ntheorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_right\n\ntheorem mul_mem_sup {S T : NonUnitalSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) :\n    x * y ∈ S ⊔ T :=\n  mul_mem (mem_sup_left hx) (mem_sup_right hy)\n\ntheorem map_sup (f : F) (S T : NonUnitalSubalgebra R A) :\n    ((S ⊔ T).map f : NonUnitalSubalgebra R B) = S.map f ⊔ T.map f :=\n  (@NonUnitalSubalgebra.gc_map_comap F R A B _ _ _ _ _ _ f).l_sup\n\n@[simp, norm_cast]\ntheorem coe_inf (S T : NonUnitalSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T :=\n  rfl\n\n@[simp]\ntheorem mem_inf {S T : NonUnitalSubalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T :=\n  Iff.rfl\n\n@[simp]\ntheorem inf_toSubmodule (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toSubmodule = S.toSubmodule ⊓ T.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem inf_toNonUnitalSubsemiring (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring ⊓ T.toNonUnitalSubsemiring :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sInf (S : Set (NonUnitalSubalgebra R A)) : (↑(sInf S) : Set A) = ⋂ s ∈ S, ↑s :=\n  sInf_image\n\ntheorem mem_sInf {S : Set (NonUnitalSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p := by\n  simp only [← SetLike.mem_coe, coe_sInf, Set.mem_iInter₂]\n\n@[simp]\ntheorem sInf_toSubmodule (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toSubmodule = sInf (NonUnitalSubalgebra.toSubmodule '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp]\ntheorem sInf_toNonUnitalSubsemiring (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toNonUnitalSubsemiring = sInf (NonUnitalSubalgebra.toNonUnitalSubsemiring '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp, norm_cast]\ntheorem coe_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} :\n    (↑(⨅ i, S i) : Set A) = ⋂ i, S i := by simp [iInf]\n\ntheorem mem_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} {x : A} :\n    (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by simp only [iInf, mem_sInf, Set.forall_range_iff]\n\n@[simp]\ntheorem iInf_toSubmodule {ι : Sort*} (S : ι → NonUnitalSubalgebra R A) :\n    (⨅ i, S i).toSubmodule = ⨅ i, (S i).toSubmodule :=\n  SetLike.coe_injective <| by simp\n\ninstance : Inhabited (NonUnitalSubalgebra R A) :=\n  ⟨⊥⟩\n\ntheorem mem_bot {x : A} : x ∈ (⊥ : NonUnitalSubalgebra R A) ↔ x = 0 :=\n  show x ∈ Submodule.span R (NonUnitalSubsemiring.closure (∅ : Set A) : Set A) ↔ x = 0 by\n    rw [NonUnitalSubsemiring.closure_empty, NonUnitalSubsemiring.coe_bot,\n      Submodule.span_zero_singleton, Submodule.mem_bot]\n\ntheorem toSubmodule_bot : (⊥ : NonUnitalSubalgebra R A).toSubmodule = ⊥ := by\n  ext\n  simp only [mem_bot, NonUnitalSubalgebra.mem_toSubmodule, Submodule.mem_bot]\n\n@[simp]\ntheorem coe_bot : ((⊥ : NonUnitalSubalgebra R A) : Set A) = {0} := by\n  simp [Set.ext_iff, NonUnitalAlgebra.mem_bot]\n\ntheorem eq_top_iff {S : NonUnitalSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S :=\n  ⟨fun h x => by rw [h]; exact mem_top, fun h => by ext x; exact ⟨fun _ => mem_top, fun _ => h x⟩⟩\n\ntheorem range_top_iff_surjective (f : A →ₙₐ[R] B) :\n    NonUnitalAlgHom.range f = (⊤ : NonUnitalSubalgebra R B) ↔ Function.Surjective f :=\n  NonUnitalAlgebra.eq_top_iff\n\n@[simp]\ntheorem range_id : NonUnitalAlgHom.range (NonUnitalAlgHom.id R A) = ⊤ :=\n  SetLike.coe_injective Set.range_id\n\n@[simp]\ntheorem map_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R A).map f = NonUnitalAlgHom.range f :=\n  SetLike.coe_injective Set.image_univ\n\n@[simp]\ntheorem map_bot (f : A →ₙₐ[R] B) : (⊥ : NonUnitalSubalgebra R A).map f = ⊥ :=\n  SetLike.coe_injective <| by simp [NonUnitalAlgebra.coe_bot, NonUnitalSubalgebra.coe_map]\n\n@[simp]\ntheorem comap_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R B).comap f = ⊤ :=\n  eq_top_iff.2 fun _ => mem_top\n\n/-- `NonUnitalAlgHom` to `⊤ : NonUnitalSubalgebra R A`. -/\ndef toTop : A →ₙₐ[R] (⊤ : NonUnitalSubalgebra R A) :=\n  NonUnitalAlgHom.codRestrict (NonUnitalAlgHom.id R A) ⊤ fun _ => mem_top\n\nend NonUnitalAlgebra\n\nnamespace NonUnitalSubalgebra\n\nopen NonUnitalAlgebra\n\nsection NonAssoc\n\nvariable {R : Type u} {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable (S : NonUnitalSubalgebra R A)\n\ninstance subsingleton_of_subsingleton [Subsingleton A] : Subsingleton (NonUnitalSubalgebra R A) :=\n  ⟨fun B C => ext fun x => by ","nextTactic":"simp only [Subsingleton.elim x 0, zero_mem B, zero_mem C]","declUpToTactic":"instance subsingleton_of_subsingleton [Subsingleton A] : Subsingleton (NonUnitalSubalgebra R A) :=\n  ⟨fun B C => ext fun x => by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.789_0.jQvdAcOcyVirh23","decl":"instance subsingleton_of_subsingleton [Subsingleton A] : Subsingleton (NonUnitalSubalgebra R A) "}
-{"state":"R : Type u\nA : Type v\nB : Type w\ninst✝⁸ : CommSemiring R\ninst✝⁷ : NonUnitalNonAssocSemiring A\ninst✝⁶ : Module R A\ninst✝⁵ : IsScalarTower R A A\ninst✝⁴ : SMulCommClass R A A\ninst✝³ : NonUnitalNonAssocSemiring B\ninst✝² : Module R B\ninst✝¹ : IsScalarTower R B B\ninst✝ : SMulCommClass R B B\nS : NonUnitalSubalgebra R A\nS₁ : NonUnitalSubalgebra R B\n⊢ prod ⊤ ⊤ = ⊤","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact NonUnitalAlgebra.gc\n\n@[simp]\ntheorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=\n  eq_top_iff.2 fun _x hx => subset_adjoin R hx\n\nvariable {R A}\n\n@[simp]\ntheorem coe_top : (↑(⊤ : NonUnitalSubalgebra R A) : Set A) = Set.univ :=\n  rfl\n\n@[simp]\ntheorem mem_top {x : A} : x ∈ (⊤ : NonUnitalSubalgebra R A) :=\n  Set.mem_univ x\n\n@[simp]\ntheorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toNonUnitalSubsemiring : (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubsemiring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toSubring {R A : Type*} [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    [IsScalarTower R A A] [SMulCommClass R A A] :\n    (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem toSubmodule_eq_top {S : NonUnitalSubalgebra R A} : S.toSubmodule = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toSubmodule'.injective.eq_iff' top_toSubmodule\n\n@[simp]\ntheorem toNonUnitalSubsemiring_eq_top {S : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubsemiring_injective.eq_iff' top_toNonUnitalSubsemiring\n\n@[simp]\ntheorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubring_injective.eq_iff' top_toSubring\n\ntheorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_left\n\ntheorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_right\n\ntheorem mul_mem_sup {S T : NonUnitalSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) :\n    x * y ∈ S ⊔ T :=\n  mul_mem (mem_sup_left hx) (mem_sup_right hy)\n\ntheorem map_sup (f : F) (S T : NonUnitalSubalgebra R A) :\n    ((S ⊔ T).map f : NonUnitalSubalgebra R B) = S.map f ⊔ T.map f :=\n  (@NonUnitalSubalgebra.gc_map_comap F R A B _ _ _ _ _ _ f).l_sup\n\n@[simp, norm_cast]\ntheorem coe_inf (S T : NonUnitalSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T :=\n  rfl\n\n@[simp]\ntheorem mem_inf {S T : NonUnitalSubalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T :=\n  Iff.rfl\n\n@[simp]\ntheorem inf_toSubmodule (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toSubmodule = S.toSubmodule ⊓ T.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem inf_toNonUnitalSubsemiring (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring ⊓ T.toNonUnitalSubsemiring :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sInf (S : Set (NonUnitalSubalgebra R A)) : (↑(sInf S) : Set A) = ⋂ s ∈ S, ↑s :=\n  sInf_image\n\ntheorem mem_sInf {S : Set (NonUnitalSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p := by\n  simp only [← SetLike.mem_coe, coe_sInf, Set.mem_iInter₂]\n\n@[simp]\ntheorem sInf_toSubmodule (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toSubmodule = sInf (NonUnitalSubalgebra.toSubmodule '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp]\ntheorem sInf_toNonUnitalSubsemiring (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toNonUnitalSubsemiring = sInf (NonUnitalSubalgebra.toNonUnitalSubsemiring '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp, norm_cast]\ntheorem coe_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} :\n    (↑(⨅ i, S i) : Set A) = ⋂ i, S i := by simp [iInf]\n\ntheorem mem_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} {x : A} :\n    (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by simp only [iInf, mem_sInf, Set.forall_range_iff]\n\n@[simp]\ntheorem iInf_toSubmodule {ι : Sort*} (S : ι → NonUnitalSubalgebra R A) :\n    (⨅ i, S i).toSubmodule = ⨅ i, (S i).toSubmodule :=\n  SetLike.coe_injective <| by simp\n\ninstance : Inhabited (NonUnitalSubalgebra R A) :=\n  ⟨⊥⟩\n\ntheorem mem_bot {x : A} : x ∈ (⊥ : NonUnitalSubalgebra R A) ↔ x = 0 :=\n  show x ∈ Submodule.span R (NonUnitalSubsemiring.closure (∅ : Set A) : Set A) ↔ x = 0 by\n    rw [NonUnitalSubsemiring.closure_empty, NonUnitalSubsemiring.coe_bot,\n      Submodule.span_zero_singleton, Submodule.mem_bot]\n\ntheorem toSubmodule_bot : (⊥ : NonUnitalSubalgebra R A).toSubmodule = ⊥ := by\n  ext\n  simp only [mem_bot, NonUnitalSubalgebra.mem_toSubmodule, Submodule.mem_bot]\n\n@[simp]\ntheorem coe_bot : ((⊥ : NonUnitalSubalgebra R A) : Set A) = {0} := by\n  simp [Set.ext_iff, NonUnitalAlgebra.mem_bot]\n\ntheorem eq_top_iff {S : NonUnitalSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S :=\n  ⟨fun h x => by rw [h]; exact mem_top, fun h => by ext x; exact ⟨fun _ => mem_top, fun _ => h x⟩⟩\n\ntheorem range_top_iff_surjective (f : A →ₙₐ[R] B) :\n    NonUnitalAlgHom.range f = (⊤ : NonUnitalSubalgebra R B) ↔ Function.Surjective f :=\n  NonUnitalAlgebra.eq_top_iff\n\n@[simp]\ntheorem range_id : NonUnitalAlgHom.range (NonUnitalAlgHom.id R A) = ⊤ :=\n  SetLike.coe_injective Set.range_id\n\n@[simp]\ntheorem map_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R A).map f = NonUnitalAlgHom.range f :=\n  SetLike.coe_injective Set.image_univ\n\n@[simp]\ntheorem map_bot (f : A →ₙₐ[R] B) : (⊥ : NonUnitalSubalgebra R A).map f = ⊥ :=\n  SetLike.coe_injective <| by simp [NonUnitalAlgebra.coe_bot, NonUnitalSubalgebra.coe_map]\n\n@[simp]\ntheorem comap_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R B).comap f = ⊤ :=\n  eq_top_iff.2 fun _ => mem_top\n\n/-- `NonUnitalAlgHom` to `⊤ : NonUnitalSubalgebra R A`. -/\ndef toTop : A →ₙₐ[R] (⊤ : NonUnitalSubalgebra R A) :=\n  NonUnitalAlgHom.codRestrict (NonUnitalAlgHom.id R A) ⊤ fun _ => mem_top\n\nend NonUnitalAlgebra\n\nnamespace NonUnitalSubalgebra\n\nopen NonUnitalAlgebra\n\nsection NonAssoc\n\nvariable {R : Type u} {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable (S : NonUnitalSubalgebra R A)\n\ninstance subsingleton_of_subsingleton [Subsingleton A] : Subsingleton (NonUnitalSubalgebra R A) :=\n  ⟨fun B C => ext fun x => by simp only [Subsingleton.elim x 0, zero_mem B, zero_mem C]⟩\n\ninstance _root_.NonUnitalAlgHom.subsingleton [Subsingleton (NonUnitalSubalgebra R A)] :\n    Subsingleton (A →ₙₐ[R] B) :=\n  ⟨fun f g =>\n    NonUnitalAlgHom.ext fun a =>\n      have : a ∈ (⊥ : NonUnitalSubalgebra R A) :=\n        Subsingleton.elim (⊤ : NonUnitalSubalgebra R A) ⊥ ▸ mem_top\n      (mem_bot.mp this).symm ▸ (map_zero f).trans (map_zero g).symm⟩\n\ntheorem range_val : NonUnitalAlgHom.range (NonUnitalSubalgebraClass.subtype S) = S :=\n  ext <| Set.ext_iff.1 <| (NonUnitalSubalgebraClass.subtype S).coe_range.trans Subtype.range_val\n\n/-- The map `S → T` when `S` is a non-unital subalgebra contained in the non-unital subalgebra `T`.\n\nThis is the non-unital subalgebra version of `Submodule.inclusion`, or `Subring.inclusion`  -/\ndef inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) : S →ₙₐ[R] T\n    where\n  toFun := Set.inclusion h\n  map_add' _ _ := rfl\n  map_mul' _ _ := rfl\n  map_zero' := rfl\n  map_smul' _ _ := rfl\n\ntheorem inclusion_injective {S T : NonUnitalSubalgebra R A} (h : S ≤ T) :\n    Function.Injective (inclusion h) := fun _ _ => Subtype.ext ∘ Subtype.mk.inj\n\n@[simp]\ntheorem inclusion_self {S : NonUnitalSubalgebra R A} :\n    inclusion (le_refl S) = NonUnitalAlgHom.id R S :=\n  NonUnitalAlgHom.ext fun _ => Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_mk {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : A) (hx : x ∈ S) :\n    inclusion h ⟨x, hx⟩ = ⟨x, h hx⟩ :=\n  rfl\n\ntheorem inclusion_right {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : T) (m : (x : A) ∈ S) :\n    inclusion h ⟨x, m⟩ = x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_inclusion {S T U : NonUnitalSubalgebra R A} (hst : S ≤ T) (htu : T ≤ U) (x : S) :\n    inclusion htu (inclusion hst x) = inclusion (le_trans hst htu) x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem coe_inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (s : S) :\n    (inclusion h s : A) = s :=\n  rfl\n\nsection Prod\n\nvariable (S₁ : NonUnitalSubalgebra R B)\n\n/-- The product of two non-unital subalgebras is a non-unital subalgebra. -/\ndef prod : NonUnitalSubalgebra R (A × B) :=\n  { S.toNonUnitalSubsemiring.prod S₁.toNonUnitalSubsemiring with\n    carrier := S ×ˢ S₁\n    smul_mem' := fun r _x hx => ⟨SMulMemClass.smul_mem r hx.1, SMulMemClass.smul_mem r hx.2⟩ }\n\n@[simp]\ntheorem coe_prod : (prod S S₁ : Set (A × B)) = (S : Set A) ×ˢ S₁ :=\n  rfl\n\ntheorem prod_toSubmodule : (S.prod S₁).toSubmodule = S.toSubmodule.prod S₁.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem mem_prod {S : NonUnitalSubalgebra R A} {S₁ : NonUnitalSubalgebra R B} {x : A × B} :\n    x ∈ prod S S₁ ↔ x.1 ∈ S ∧ x.2 ∈ S₁ :=\n  Set.mem_prod\n\n@[simp]\ntheorem prod_top : (prod ⊤ ⊤ : NonUnitalSubalgebra R (A × B)) = ⊤ := by ","nextTactic":"ext","declUpToTactic":"@[simp]\ntheorem prod_top : (prod ⊤ ⊤ : NonUnitalSubalgebra R (A × B)) = ⊤ := by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.863_0.jQvdAcOcyVirh23","decl":"@[simp]\ntheorem prod_top : (prod ⊤ ⊤ : NonUnitalSubalgebra R (A × B)) = ⊤ "}
-{"state":"case h\nR : Type u\nA : Type v\nB : Type w\ninst✝⁸ : CommSemiring R\ninst✝⁷ : NonUnitalNonAssocSemiring A\ninst✝⁶ : Module R A\ninst✝⁵ : IsScalarTower R A A\ninst✝⁴ : SMulCommClass R A A\ninst✝³ : NonUnitalNonAssocSemiring B\ninst✝² : Module R B\ninst✝¹ : IsScalarTower R B B\ninst✝ : SMulCommClass R B B\nS : NonUnitalSubalgebra R A\nS₁ : NonUnitalSubalgebra R B\nx✝ : A × B\n⊢ x✝ ∈ prod ⊤ ⊤ ↔ x✝ ∈ ⊤","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B ��ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact NonUnitalAlgebra.gc\n\n@[simp]\ntheorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=\n  eq_top_iff.2 fun _x hx => subset_adjoin R hx\n\nvariable {R A}\n\n@[simp]\ntheorem coe_top : (↑(⊤ : NonUnitalSubalgebra R A) : Set A) = Set.univ :=\n  rfl\n\n@[simp]\ntheorem mem_top {x : A} : x ∈ (⊤ : NonUnitalSubalgebra R A) :=\n  Set.mem_univ x\n\n@[simp]\ntheorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toNonUnitalSubsemiring : (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubsemiring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toSubring {R A : Type*} [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    [IsScalarTower R A A] [SMulCommClass R A A] :\n    (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem toSubmodule_eq_top {S : NonUnitalSubalgebra R A} : S.toSubmodule = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toSubmodule'.injective.eq_iff' top_toSubmodule\n\n@[simp]\ntheorem toNonUnitalSubsemiring_eq_top {S : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubsemiring_injective.eq_iff' top_toNonUnitalSubsemiring\n\n@[simp]\ntheorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubring_injective.eq_iff' top_toSubring\n\ntheorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_left\n\ntheorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_right\n\ntheorem mul_mem_sup {S T : NonUnitalSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) :\n    x * y ∈ S ⊔ T :=\n  mul_mem (mem_sup_left hx) (mem_sup_right hy)\n\ntheorem map_sup (f : F) (S T : NonUnitalSubalgebra R A) :\n    ((S ⊔ T).map f : NonUnitalSubalgebra R B) = S.map f ⊔ T.map f :=\n  (@NonUnitalSubalgebra.gc_map_comap F R A B _ _ _ _ _ _ f).l_sup\n\n@[simp, norm_cast]\ntheorem coe_inf (S T : NonUnitalSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T :=\n  rfl\n\n@[simp]\ntheorem mem_inf {S T : NonUnitalSubalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T :=\n  Iff.rfl\n\n@[simp]\ntheorem inf_toSubmodule (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toSubmodule = S.toSubmodule ⊓ T.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem inf_toNonUnitalSubsemiring (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring ⊓ T.toNonUnitalSubsemiring :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sInf (S : Set (NonUnitalSubalgebra R A)) : (↑(sInf S) : Set A) = ⋂ s ∈ S, ↑s :=\n  sInf_image\n\ntheorem mem_sInf {S : Set (NonUnitalSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p := by\n  simp only [← SetLike.mem_coe, coe_sInf, Set.mem_iInter₂]\n\n@[simp]\ntheorem sInf_toSubmodule (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toSubmodule = sInf (NonUnitalSubalgebra.toSubmodule '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp]\ntheorem sInf_toNonUnitalSubsemiring (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toNonUnitalSubsemiring = sInf (NonUnitalSubalgebra.toNonUnitalSubsemiring '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp, norm_cast]\ntheorem coe_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} :\n    (↑(⨅ i, S i) : Set A) = ⋂ i, S i := by simp [iInf]\n\ntheorem mem_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} {x : A} :\n    (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by simp only [iInf, mem_sInf, Set.forall_range_iff]\n\n@[simp]\ntheorem iInf_toSubmodule {ι : Sort*} (S : ι → NonUnitalSubalgebra R A) :\n    (⨅ i, S i).toSubmodule = ⨅ i, (S i).toSubmodule :=\n  SetLike.coe_injective <| by simp\n\ninstance : Inhabited (NonUnitalSubalgebra R A) :=\n  ⟨⊥⟩\n\ntheorem mem_bot {x : A} : x ∈ (⊥ : NonUnitalSubalgebra R A) ↔ x = 0 :=\n  show x ∈ Submodule.span R (NonUnitalSubsemiring.closure (∅ : Set A) : Set A) ↔ x = 0 by\n    rw [NonUnitalSubsemiring.closure_empty, NonUnitalSubsemiring.coe_bot,\n      Submodule.span_zero_singleton, Submodule.mem_bot]\n\ntheorem toSubmodule_bot : (⊥ : NonUnitalSubalgebra R A).toSubmodule = ⊥ := by\n  ext\n  simp only [mem_bot, NonUnitalSubalgebra.mem_toSubmodule, Submodule.mem_bot]\n\n@[simp]\ntheorem coe_bot : ((⊥ : NonUnitalSubalgebra R A) : Set A) = {0} := by\n  simp [Set.ext_iff, NonUnitalAlgebra.mem_bot]\n\ntheorem eq_top_iff {S : NonUnitalSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S :=\n  ⟨fun h x => by rw [h]; exact mem_top, fun h => by ext x; exact ⟨fun _ => mem_top, fun _ => h x⟩⟩\n\ntheorem range_top_iff_surjective (f : A →ₙₐ[R] B) :\n    NonUnitalAlgHom.range f = (⊤ : NonUnitalSubalgebra R B) ↔ Function.Surjective f :=\n  NonUnitalAlgebra.eq_top_iff\n\n@[simp]\ntheorem range_id : NonUnitalAlgHom.range (NonUnitalAlgHom.id R A) = ⊤ :=\n  SetLike.coe_injective Set.range_id\n\n@[simp]\ntheorem map_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R A).map f = NonUnitalAlgHom.range f :=\n  SetLike.coe_injective Set.image_univ\n\n@[simp]\ntheorem map_bot (f : A →ₙₐ[R] B) : (⊥ : NonUnitalSubalgebra R A).map f = ⊥ :=\n  SetLike.coe_injective <| by simp [NonUnitalAlgebra.coe_bot, NonUnitalSubalgebra.coe_map]\n\n@[simp]\ntheorem comap_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R B).comap f = ⊤ :=\n  eq_top_iff.2 fun _ => mem_top\n\n/-- `NonUnitalAlgHom` to `⊤ : NonUnitalSubalgebra R A`. -/\ndef toTop : A →ₙₐ[R] (⊤ : NonUnitalSubalgebra R A) :=\n  NonUnitalAlgHom.codRestrict (NonUnitalAlgHom.id R A) ⊤ fun _ => mem_top\n\nend NonUnitalAlgebra\n\nnamespace NonUnitalSubalgebra\n\nopen NonUnitalAlgebra\n\nsection NonAssoc\n\nvariable {R : Type u} {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable (S : NonUnitalSubalgebra R A)\n\ninstance subsingleton_of_subsingleton [Subsingleton A] : Subsingleton (NonUnitalSubalgebra R A) :=\n  ⟨fun B C => ext fun x => by simp only [Subsingleton.elim x 0, zero_mem B, zero_mem C]⟩\n\ninstance _root_.NonUnitalAlgHom.subsingleton [Subsingleton (NonUnitalSubalgebra R A)] :\n    Subsingleton (A →ₙₐ[R] B) :=\n  ⟨fun f g =>\n    NonUnitalAlgHom.ext fun a =>\n      have : a ∈ (⊥ : NonUnitalSubalgebra R A) :=\n        Subsingleton.elim (⊤ : NonUnitalSubalgebra R A) ⊥ ▸ mem_top\n      (mem_bot.mp this).symm ▸ (map_zero f).trans (map_zero g).symm⟩\n\ntheorem range_val : NonUnitalAlgHom.range (NonUnitalSubalgebraClass.subtype S) = S :=\n  ext <| Set.ext_iff.1 <| (NonUnitalSubalgebraClass.subtype S).coe_range.trans Subtype.range_val\n\n/-- The map `S → T` when `S` is a non-unital subalgebra contained in the non-unital subalgebra `T`.\n\nThis is the non-unital subalgebra version of `Submodule.inclusion`, or `Subring.inclusion`  -/\ndef inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) : S →ₙₐ[R] T\n    where\n  toFun := Set.inclusion h\n  map_add' _ _ := rfl\n  map_mul' _ _ := rfl\n  map_zero' := rfl\n  map_smul' _ _ := rfl\n\ntheorem inclusion_injective {S T : NonUnitalSubalgebra R A} (h : S ≤ T) :\n    Function.Injective (inclusion h) := fun _ _ => Subtype.ext ∘ Subtype.mk.inj\n\n@[simp]\ntheorem inclusion_self {S : NonUnitalSubalgebra R A} :\n    inclusion (le_refl S) = NonUnitalAlgHom.id R S :=\n  NonUnitalAlgHom.ext fun _ => Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_mk {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : A) (hx : x ∈ S) :\n    inclusion h ⟨x, hx⟩ = ⟨x, h hx⟩ :=\n  rfl\n\ntheorem inclusion_right {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : T) (m : (x : A) ∈ S) :\n    inclusion h ⟨x, m⟩ = x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_inclusion {S T U : NonUnitalSubalgebra R A} (hst : S ≤ T) (htu : T ≤ U) (x : S) :\n    inclusion htu (inclusion hst x) = inclusion (le_trans hst htu) x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem coe_inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (s : S) :\n    (inclusion h s : A) = s :=\n  rfl\n\nsection Prod\n\nvariable (S₁ : NonUnitalSubalgebra R B)\n\n/-- The product of two non-unital subalgebras is a non-unital subalgebra. -/\ndef prod : NonUnitalSubalgebra R (A × B) :=\n  { S.toNonUnitalSubsemiring.prod S₁.toNonUnitalSubsemiring with\n    carrier := S ×ˢ S₁\n    smul_mem' := fun r _x hx => ⟨SMulMemClass.smul_mem r hx.1, SMulMemClass.smul_mem r hx.2⟩ }\n\n@[simp]\ntheorem coe_prod : (prod S S₁ : Set (A × B)) = (S : Set A) ×ˢ S₁ :=\n  rfl\n\ntheorem prod_toSubmodule : (S.prod S₁).toSubmodule = S.toSubmodule.prod S₁.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem mem_prod {S : NonUnitalSubalgebra R A} {S₁ : NonUnitalSubalgebra R B} {x : A × B} :\n    x ∈ prod S S₁ ↔ x.1 ∈ S ∧ x.2 ∈ S₁ :=\n  Set.mem_prod\n\n@[simp]\ntheorem prod_top : (prod ⊤ ⊤ : NonUnitalSubalgebra R (A × B)) = ⊤ := by ext; ","nextTactic":"simp","declUpToTactic":"@[simp]\ntheorem prod_top : (prod ⊤ ⊤ : NonUnitalSubalgebra R (A × B)) = ⊤ := by ext; ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.863_0.jQvdAcOcyVirh23","decl":"@[simp]\ntheorem prod_top : (prod ⊤ ⊤ : NonUnitalSubalgebra R (A × B)) = ⊤ "}
-{"state":"R : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\nS : NonUnitalSubalgebra R A\nι : Type u_1\ninst✝ : Nonempty ι\nK : ι → NonUnitalSubalgebra R A\ndir : Directed (fun x x_1 => x ≤ x_1) K\nf : (i : ι) → ↥(K i) →ₙₐ[R] B\nhf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalAlgHom.comp (f j) (inclusion h)\nT : NonUnitalSubalgebra R A\nhT : T = iSup K\n⊢ ↥T →ₙₐ[R] B","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact NonUnitalAlgebra.gc\n\n@[simp]\ntheorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=\n  eq_top_iff.2 fun _x hx => subset_adjoin R hx\n\nvariable {R A}\n\n@[simp]\ntheorem coe_top : (↑(⊤ : NonUnitalSubalgebra R A) : Set A) = Set.univ :=\n  rfl\n\n@[simp]\ntheorem mem_top {x : A} : x ∈ (⊤ : NonUnitalSubalgebra R A) :=\n  Set.mem_univ x\n\n@[simp]\ntheorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toNonUnitalSubsemiring : (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubsemiring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toSubring {R A : Type*} [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    [IsScalarTower R A A] [SMulCommClass R A A] :\n    (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem toSubmodule_eq_top {S : NonUnitalSubalgebra R A} : S.toSubmodule = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toSubmodule'.injective.eq_iff' top_toSubmodule\n\n@[simp]\ntheorem toNonUnitalSubsemiring_eq_top {S : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubsemiring_injective.eq_iff' top_toNonUnitalSubsemiring\n\n@[simp]\ntheorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubring_injective.eq_iff' top_toSubring\n\ntheorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_left\n\ntheorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_right\n\ntheorem mul_mem_sup {S T : NonUnitalSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) :\n    x * y ∈ S ⊔ T :=\n  mul_mem (mem_sup_left hx) (mem_sup_right hy)\n\ntheorem map_sup (f : F) (S T : NonUnitalSubalgebra R A) :\n    ((S ⊔ T).map f : NonUnitalSubalgebra R B) = S.map f ⊔ T.map f :=\n  (@NonUnitalSubalgebra.gc_map_comap F R A B _ _ _ _ _ _ f).l_sup\n\n@[simp, norm_cast]\ntheorem coe_inf (S T : NonUnitalSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T :=\n  rfl\n\n@[simp]\ntheorem mem_inf {S T : NonUnitalSubalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T :=\n  Iff.rfl\n\n@[simp]\ntheorem inf_toSubmodule (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toSubmodule = S.toSubmodule ⊓ T.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem inf_toNonUnitalSubsemiring (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring ⊓ T.toNonUnitalSubsemiring :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sInf (S : Set (NonUnitalSubalgebra R A)) : (↑(sInf S) : Set A) = ⋂ s ∈ S, ↑s :=\n  sInf_image\n\ntheorem mem_sInf {S : Set (NonUnitalSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p := by\n  simp only [← SetLike.mem_coe, coe_sInf, Set.mem_iInter₂]\n\n@[simp]\ntheorem sInf_toSubmodule (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toSubmodule = sInf (NonUnitalSubalgebra.toSubmodule '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp]\ntheorem sInf_toNonUnitalSubsemiring (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toNonUnitalSubsemiring = sInf (NonUnitalSubalgebra.toNonUnitalSubsemiring '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp, norm_cast]\ntheorem coe_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} :\n    (↑(⨅ i, S i) : Set A) = ⋂ i, S i := by simp [iInf]\n\ntheorem mem_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} {x : A} :\n    (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by simp only [iInf, mem_sInf, Set.forall_range_iff]\n\n@[simp]\ntheorem iInf_toSubmodule {ι : Sort*} (S : ι → NonUnitalSubalgebra R A) :\n    (⨅ i, S i).toSubmodule = ⨅ i, (S i).toSubmodule :=\n  SetLike.coe_injective <| by simp\n\ninstance : Inhabited (NonUnitalSubalgebra R A) :=\n  ⟨⊥⟩\n\ntheorem mem_bot {x : A} : x ∈ (⊥ : NonUnitalSubalgebra R A) ↔ x = 0 :=\n  show x ∈ Submodule.span R (NonUnitalSubsemiring.closure (∅ : Set A) : Set A) ↔ x = 0 by\n    rw [NonUnitalSubsemiring.closure_empty, NonUnitalSubsemiring.coe_bot,\n      Submodule.span_zero_singleton, Submodule.mem_bot]\n\ntheorem toSubmodule_bot : (⊥ : NonUnitalSubalgebra R A).toSubmodule = ⊥ := by\n  ext\n  simp only [mem_bot, NonUnitalSubalgebra.mem_toSubmodule, Submodule.mem_bot]\n\n@[simp]\ntheorem coe_bot : ((⊥ : NonUnitalSubalgebra R A) : Set A) = {0} := by\n  simp [Set.ext_iff, NonUnitalAlgebra.mem_bot]\n\ntheorem eq_top_iff {S : NonUnitalSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S :=\n  ⟨fun h x => by rw [h]; exact mem_top, fun h => by ext x; exact ⟨fun _ => mem_top, fun _ => h x⟩⟩\n\ntheorem range_top_iff_surjective (f : A →ₙₐ[R] B) :\n    NonUnitalAlgHom.range f = (⊤ : NonUnitalSubalgebra R B) ↔ Function.Surjective f :=\n  NonUnitalAlgebra.eq_top_iff\n\n@[simp]\ntheorem range_id : NonUnitalAlgHom.range (NonUnitalAlgHom.id R A) = ⊤ :=\n  SetLike.coe_injective Set.range_id\n\n@[simp]\ntheorem map_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R A).map f = NonUnitalAlgHom.range f :=\n  SetLike.coe_injective Set.image_univ\n\n@[simp]\ntheorem map_bot (f : A →ₙₐ[R] B) : (⊥ : NonUnitalSubalgebra R A).map f = ⊥ :=\n  SetLike.coe_injective <| by simp [NonUnitalAlgebra.coe_bot, NonUnitalSubalgebra.coe_map]\n\n@[simp]\ntheorem comap_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R B).comap f = ⊤ :=\n  eq_top_iff.2 fun _ => mem_top\n\n/-- `NonUnitalAlgHom` to `⊤ : NonUnitalSubalgebra R A`. -/\ndef toTop : A →ₙₐ[R] (⊤ : NonUnitalSubalgebra R A) :=\n  NonUnitalAlgHom.codRestrict (NonUnitalAlgHom.id R A) ⊤ fun _ => mem_top\n\nend NonUnitalAlgebra\n\nnamespace NonUnitalSubalgebra\n\nopen NonUnitalAlgebra\n\nsection NonAssoc\n\nvariable {R : Type u} {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable (S : NonUnitalSubalgebra R A)\n\ninstance subsingleton_of_subsingleton [Subsingleton A] : Subsingleton (NonUnitalSubalgebra R A) :=\n  ⟨fun B C => ext fun x => by simp only [Subsingleton.elim x 0, zero_mem B, zero_mem C]⟩\n\ninstance _root_.NonUnitalAlgHom.subsingleton [Subsingleton (NonUnitalSubalgebra R A)] :\n    Subsingleton (A →ₙₐ[R] B) :=\n  ⟨fun f g =>\n    NonUnitalAlgHom.ext fun a =>\n      have : a ∈ (⊥ : NonUnitalSubalgebra R A) :=\n        Subsingleton.elim (⊤ : NonUnitalSubalgebra R A) ⊥ ▸ mem_top\n      (mem_bot.mp this).symm ▸ (map_zero f).trans (map_zero g).symm⟩\n\ntheorem range_val : NonUnitalAlgHom.range (NonUnitalSubalgebraClass.subtype S) = S :=\n  ext <| Set.ext_iff.1 <| (NonUnitalSubalgebraClass.subtype S).coe_range.trans Subtype.range_val\n\n/-- The map `S → T` when `S` is a non-unital subalgebra contained in the non-unital subalgebra `T`.\n\nThis is the non-unital subalgebra version of `Submodule.inclusion`, or `Subring.inclusion`  -/\ndef inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) : S →ₙₐ[R] T\n    where\n  toFun := Set.inclusion h\n  map_add' _ _ := rfl\n  map_mul' _ _ := rfl\n  map_zero' := rfl\n  map_smul' _ _ := rfl\n\ntheorem inclusion_injective {S T : NonUnitalSubalgebra R A} (h : S ≤ T) :\n    Function.Injective (inclusion h) := fun _ _ => Subtype.ext ∘ Subtype.mk.inj\n\n@[simp]\ntheorem inclusion_self {S : NonUnitalSubalgebra R A} :\n    inclusion (le_refl S) = NonUnitalAlgHom.id R S :=\n  NonUnitalAlgHom.ext fun _ => Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_mk {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : A) (hx : x ∈ S) :\n    inclusion h ⟨x, hx⟩ = ⟨x, h hx⟩ :=\n  rfl\n\ntheorem inclusion_right {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : T) (m : (x : A) ∈ S) :\n    inclusion h ⟨x, m⟩ = x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_inclusion {S T U : NonUnitalSubalgebra R A} (hst : S ≤ T) (htu : T ≤ U) (x : S) :\n    inclusion htu (inclusion hst x) = inclusion (le_trans hst htu) x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem coe_inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (s : S) :\n    (inclusion h s : A) = s :=\n  rfl\n\nsection Prod\n\nvariable (S₁ : NonUnitalSubalgebra R B)\n\n/-- The product of two non-unital subalgebras is a non-unital subalgebra. -/\ndef prod : NonUnitalSubalgebra R (A × B) :=\n  { S.toNonUnitalSubsemiring.prod S₁.toNonUnitalSubsemiring with\n    carrier := S ×ˢ S₁\n    smul_mem' := fun r _x hx => ⟨SMulMemClass.smul_mem r hx.1, SMulMemClass.smul_mem r hx.2⟩ }\n\n@[simp]\ntheorem coe_prod : (prod S S₁ : Set (A × B)) = (S : Set A) ×ˢ S₁ :=\n  rfl\n\ntheorem prod_toSubmodule : (S.prod S₁).toSubmodule = S.toSubmodule.prod S₁.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem mem_prod {S : NonUnitalSubalgebra R A} {S₁ : NonUnitalSubalgebra R B} {x : A × B} :\n    x ∈ prod S S₁ ↔ x.1 ∈ S ∧ x.2 ∈ S₁ :=\n  Set.mem_prod\n\n@[simp]\ntheorem prod_top : (prod ⊤ ⊤ : NonUnitalSubalgebra R (A × B)) = ⊤ := by ext; simp\n\ntheorem prod_mono {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S ≤ T → S₁ ≤ T₁ → prod S S₁ ≤ prod T T₁ :=\n  Set.prod_mono\n\n@[simp]\ntheorem prod_inf_prod {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S.prod S₁ ⊓ T.prod T₁ = (S ⊓ T).prod (S₁ ⊓ T₁) :=\n  SetLike.coe_injective Set.prod_inter_prod\n\nend Prod\n\nsection SuprLift\n\nvariable {ι : Type*}\n\ntheorem coe_iSup_of_directed [Nonempty ι] {S : ι → NonUnitalSubalgebra R A}\n    (dir : Directed (· ≤ ·) S) : ↑(iSup S) = ⋃ i, (S i : Set A) :=\n  let K : NonUnitalSubalgebra R A :=\n    { __ := NonUnitalSubsemiring.copy _ _ (NonUnitalSubsemiring.coe_iSup_of_directed dir).symm\n      smul_mem' := fun r _x hx ↦\n        let ⟨i, hi⟩ := Set.mem_iUnion.1 hx\n        Set.mem_iUnion.2 ⟨i, (S i).smul_mem' r hi⟩ }\n  have : iSup S = K := le_antisymm\n    (iSup_le fun i ↦ le_iSup (fun i ↦ (S i : Set A)) i) (Set.iUnion_subset fun _ ↦ le_iSup S _)\n  this.symm ▸ rfl\n\n/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  ","nextTactic":"subst hT","declUpToTactic":"/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.892_0.jQvdAcOcyVirh23","decl":"/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B "}
-{"state":"R : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\nS : NonUnitalSubalgebra R A\nι : Type u_1\ninst✝ : Nonempty ι\nK : ι → NonUnitalSubalgebra R A\ndir : Directed (fun x x_1 => x ≤ x_1) K\nf : (i : ι) → ↥(K i) →ₙₐ[R] B\nhf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalAlgHom.comp (f j) (inclusion h)\n⊢ ↥(iSup K) →ₙₐ[R] B","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b �� adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact NonUnitalAlgebra.gc\n\n@[simp]\ntheorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=\n  eq_top_iff.2 fun _x hx => subset_adjoin R hx\n\nvariable {R A}\n\n@[simp]\ntheorem coe_top : (↑(⊤ : NonUnitalSubalgebra R A) : Set A) = Set.univ :=\n  rfl\n\n@[simp]\ntheorem mem_top {x : A} : x ∈ (⊤ : NonUnitalSubalgebra R A) :=\n  Set.mem_univ x\n\n@[simp]\ntheorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toNonUnitalSubsemiring : (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubsemiring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toSubring {R A : Type*} [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    [IsScalarTower R A A] [SMulCommClass R A A] :\n    (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem toSubmodule_eq_top {S : NonUnitalSubalgebra R A} : S.toSubmodule = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toSubmodule'.injective.eq_iff' top_toSubmodule\n\n@[simp]\ntheorem toNonUnitalSubsemiring_eq_top {S : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubsemiring_injective.eq_iff' top_toNonUnitalSubsemiring\n\n@[simp]\ntheorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubring_injective.eq_iff' top_toSubring\n\ntheorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_left\n\ntheorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_right\n\ntheorem mul_mem_sup {S T : NonUnitalSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) :\n    x * y ∈ S ⊔ T :=\n  mul_mem (mem_sup_left hx) (mem_sup_right hy)\n\ntheorem map_sup (f : F) (S T : NonUnitalSubalgebra R A) :\n    ((S ⊔ T).map f : NonUnitalSubalgebra R B) = S.map f ⊔ T.map f :=\n  (@NonUnitalSubalgebra.gc_map_comap F R A B _ _ _ _ _ _ f).l_sup\n\n@[simp, norm_cast]\ntheorem coe_inf (S T : NonUnitalSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T :=\n  rfl\n\n@[simp]\ntheorem mem_inf {S T : NonUnitalSubalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T :=\n  Iff.rfl\n\n@[simp]\ntheorem inf_toSubmodule (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toSubmodule = S.toSubmodule ⊓ T.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem inf_toNonUnitalSubsemiring (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring ⊓ T.toNonUnitalSubsemiring :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sInf (S : Set (NonUnitalSubalgebra R A)) : (↑(sInf S) : Set A) = ⋂ s ∈ S, ↑s :=\n  sInf_image\n\ntheorem mem_sInf {S : Set (NonUnitalSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p := by\n  simp only [← SetLike.mem_coe, coe_sInf, Set.mem_iInter₂]\n\n@[simp]\ntheorem sInf_toSubmodule (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toSubmodule = sInf (NonUnitalSubalgebra.toSubmodule '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp]\ntheorem sInf_toNonUnitalSubsemiring (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toNonUnitalSubsemiring = sInf (NonUnitalSubalgebra.toNonUnitalSubsemiring '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp, norm_cast]\ntheorem coe_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} :\n    (↑(⨅ i, S i) : Set A) = ⋂ i, S i := by simp [iInf]\n\ntheorem mem_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} {x : A} :\n    (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by simp only [iInf, mem_sInf, Set.forall_range_iff]\n\n@[simp]\ntheorem iInf_toSubmodule {ι : Sort*} (S : ι → NonUnitalSubalgebra R A) :\n    (⨅ i, S i).toSubmodule = ⨅ i, (S i).toSubmodule :=\n  SetLike.coe_injective <| by simp\n\ninstance : Inhabited (NonUnitalSubalgebra R A) :=\n  ⟨⊥⟩\n\ntheorem mem_bot {x : A} : x ∈ (⊥ : NonUnitalSubalgebra R A) ↔ x = 0 :=\n  show x ∈ Submodule.span R (NonUnitalSubsemiring.closure (∅ : Set A) : Set A) ↔ x = 0 by\n    rw [NonUnitalSubsemiring.closure_empty, NonUnitalSubsemiring.coe_bot,\n      Submodule.span_zero_singleton, Submodule.mem_bot]\n\ntheorem toSubmodule_bot : (⊥ : NonUnitalSubalgebra R A).toSubmodule = ⊥ := by\n  ext\n  simp only [mem_bot, NonUnitalSubalgebra.mem_toSubmodule, Submodule.mem_bot]\n\n@[simp]\ntheorem coe_bot : ((⊥ : NonUnitalSubalgebra R A) : Set A) = {0} := by\n  simp [Set.ext_iff, NonUnitalAlgebra.mem_bot]\n\ntheorem eq_top_iff {S : NonUnitalSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S :=\n  ⟨fun h x => by rw [h]; exact mem_top, fun h => by ext x; exact ⟨fun _ => mem_top, fun _ => h x⟩⟩\n\ntheorem range_top_iff_surjective (f : A →ₙₐ[R] B) :\n    NonUnitalAlgHom.range f = (⊤ : NonUnitalSubalgebra R B) ↔ Function.Surjective f :=\n  NonUnitalAlgebra.eq_top_iff\n\n@[simp]\ntheorem range_id : NonUnitalAlgHom.range (NonUnitalAlgHom.id R A) = ⊤ :=\n  SetLike.coe_injective Set.range_id\n\n@[simp]\ntheorem map_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R A).map f = NonUnitalAlgHom.range f :=\n  SetLike.coe_injective Set.image_univ\n\n@[simp]\ntheorem map_bot (f : A →ₙₐ[R] B) : (⊥ : NonUnitalSubalgebra R A).map f = ⊥ :=\n  SetLike.coe_injective <| by simp [NonUnitalAlgebra.coe_bot, NonUnitalSubalgebra.coe_map]\n\n@[simp]\ntheorem comap_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R B).comap f = ⊤ :=\n  eq_top_iff.2 fun _ => mem_top\n\n/-- `NonUnitalAlgHom` to `⊤ : NonUnitalSubalgebra R A`. -/\ndef toTop : A →ₙₐ[R] (⊤ : NonUnitalSubalgebra R A) :=\n  NonUnitalAlgHom.codRestrict (NonUnitalAlgHom.id R A) ⊤ fun _ => mem_top\n\nend NonUnitalAlgebra\n\nnamespace NonUnitalSubalgebra\n\nopen NonUnitalAlgebra\n\nsection NonAssoc\n\nvariable {R : Type u} {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable (S : NonUnitalSubalgebra R A)\n\ninstance subsingleton_of_subsingleton [Subsingleton A] : Subsingleton (NonUnitalSubalgebra R A) :=\n  ⟨fun B C => ext fun x => by simp only [Subsingleton.elim x 0, zero_mem B, zero_mem C]⟩\n\ninstance _root_.NonUnitalAlgHom.subsingleton [Subsingleton (NonUnitalSubalgebra R A)] :\n    Subsingleton (A →ₙₐ[R] B) :=\n  ⟨fun f g =>\n    NonUnitalAlgHom.ext fun a =>\n      have : a ∈ (⊥ : NonUnitalSubalgebra R A) :=\n        Subsingleton.elim (⊤ : NonUnitalSubalgebra R A) ⊥ ▸ mem_top\n      (mem_bot.mp this).symm ▸ (map_zero f).trans (map_zero g).symm⟩\n\ntheorem range_val : NonUnitalAlgHom.range (NonUnitalSubalgebraClass.subtype S) = S :=\n  ext <| Set.ext_iff.1 <| (NonUnitalSubalgebraClass.subtype S).coe_range.trans Subtype.range_val\n\n/-- The map `S → T` when `S` is a non-unital subalgebra contained in the non-unital subalgebra `T`.\n\nThis is the non-unital subalgebra version of `Submodule.inclusion`, or `Subring.inclusion`  -/\ndef inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) : S →ₙₐ[R] T\n    where\n  toFun := Set.inclusion h\n  map_add' _ _ := rfl\n  map_mul' _ _ := rfl\n  map_zero' := rfl\n  map_smul' _ _ := rfl\n\ntheorem inclusion_injective {S T : NonUnitalSubalgebra R A} (h : S ≤ T) :\n    Function.Injective (inclusion h) := fun _ _ => Subtype.ext ∘ Subtype.mk.inj\n\n@[simp]\ntheorem inclusion_self {S : NonUnitalSubalgebra R A} :\n    inclusion (le_refl S) = NonUnitalAlgHom.id R S :=\n  NonUnitalAlgHom.ext fun _ => Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_mk {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : A) (hx : x ∈ S) :\n    inclusion h ⟨x, hx⟩ = ⟨x, h hx⟩ :=\n  rfl\n\ntheorem inclusion_right {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : T) (m : (x : A) ∈ S) :\n    inclusion h ⟨x, m⟩ = x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_inclusion {S T U : NonUnitalSubalgebra R A} (hst : S ≤ T) (htu : T ≤ U) (x : S) :\n    inclusion htu (inclusion hst x) = inclusion (le_trans hst htu) x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem coe_inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (s : S) :\n    (inclusion h s : A) = s :=\n  rfl\n\nsection Prod\n\nvariable (S₁ : NonUnitalSubalgebra R B)\n\n/-- The product of two non-unital subalgebras is a non-unital subalgebra. -/\ndef prod : NonUnitalSubalgebra R (A × B) :=\n  { S.toNonUnitalSubsemiring.prod S₁.toNonUnitalSubsemiring with\n    carrier := S ×ˢ S₁\n    smul_mem' := fun r _x hx => ⟨SMulMemClass.smul_mem r hx.1, SMulMemClass.smul_mem r hx.2⟩ }\n\n@[simp]\ntheorem coe_prod : (prod S S₁ : Set (A × B)) = (S : Set A) ×ˢ S₁ :=\n  rfl\n\ntheorem prod_toSubmodule : (S.prod S₁).toSubmodule = S.toSubmodule.prod S₁.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem mem_prod {S : NonUnitalSubalgebra R A} {S₁ : NonUnitalSubalgebra R B} {x : A × B} :\n    x ∈ prod S S₁ ↔ x.1 ∈ S ∧ x.2 ∈ S₁ :=\n  Set.mem_prod\n\n@[simp]\ntheorem prod_top : (prod ⊤ ⊤ : NonUnitalSubalgebra R (A × B)) = ⊤ := by ext; simp\n\ntheorem prod_mono {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S ≤ T → S₁ ≤ T₁ → prod S S₁ ≤ prod T T₁ :=\n  Set.prod_mono\n\n@[simp]\ntheorem prod_inf_prod {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S.prod S₁ ⊓ T.prod T₁ = (S ⊓ T).prod (S₁ ⊓ T₁) :=\n  SetLike.coe_injective Set.prod_inter_prod\n\nend Prod\n\nsection SuprLift\n\nvariable {ι : Type*}\n\ntheorem coe_iSup_of_directed [Nonempty ι] {S : ι → NonUnitalSubalgebra R A}\n    (dir : Directed (· ≤ ·) S) : ↑(iSup S) = ⋃ i, (S i : Set A) :=\n  let K : NonUnitalSubalgebra R A :=\n    { __ := NonUnitalSubsemiring.copy _ _ (NonUnitalSubsemiring.coe_iSup_of_directed dir).symm\n      smul_mem' := fun r _x hx ↦\n        let ⟨i, hi⟩ := Set.mem_iUnion.1 hx\n        Set.mem_iUnion.2 ⟨i, (S i).smul_mem' r hi⟩ }\n  have : iSup S = K := le_antisymm\n    (iSup_le fun i ↦ le_iSup (fun i ↦ (S i : Set A)) i) (Set.iUnion_subset fun _ ↦ le_iSup S _)\n  this.symm ▸ rfl\n\n/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  ","nextTactic":"exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              rw [hf j k hjk]\n              rfl)\n            (↑(iSup K)) (by rw [coe_iSup_of_directed dir])\n        map_zero' := by\n          dsimp\n          exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl) _ (by simp)\n        map_mul' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· * ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_add' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· + ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_smul' := fun r => by\n          dsimp\n          apply Set.iUnionLift_unary (coe_iSup_of_directed dir) _ (fun _ x => r • x)\n            (fun _ _ => rfl)\n          on_goal 2 => rw [coe_iSup_of_directed dir]\n          all_goals simp }","declUpToTactic":"/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.892_0.jQvdAcOcyVirh23","decl":"/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B "}
-{"state":"R : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\nS : NonUnitalSubalgebra R A\nι : Type u_1\ninst✝ : Nonempty ι\nK : ι → NonUnitalSubalgebra R A\ndir : Directed (fun x x_1 => x ≤ x_1) K\nf : (i : ι) → ↥(K i) →ₙₐ[R] B\nhf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalAlgHom.comp (f j) (inclusion h)\ni j : ι\nx : A\nhxi : x ∈ (fun i => ↑(K i)) i\nhxj : x ∈ (fun i => ↑(K i)) j\n⊢ (fun i x => (f i) x) i { val := x, property := hxi } = (fun i x => (f i) x) j { val := x, property := hxj }","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact NonUnitalAlgebra.gc\n\n@[simp]\ntheorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=\n  eq_top_iff.2 fun _x hx => subset_adjoin R hx\n\nvariable {R A}\n\n@[simp]\ntheorem coe_top : (↑(⊤ : NonUnitalSubalgebra R A) : Set A) = Set.univ :=\n  rfl\n\n@[simp]\ntheorem mem_top {x : A} : x ∈ (⊤ : NonUnitalSubalgebra R A) :=\n  Set.mem_univ x\n\n@[simp]\ntheorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toNonUnitalSubsemiring : (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubsemiring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toSubring {R A : Type*} [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    [IsScalarTower R A A] [SMulCommClass R A A] :\n    (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem toSubmodule_eq_top {S : NonUnitalSubalgebra R A} : S.toSubmodule = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toSubmodule'.injective.eq_iff' top_toSubmodule\n\n@[simp]\ntheorem toNonUnitalSubsemiring_eq_top {S : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubsemiring_injective.eq_iff' top_toNonUnitalSubsemiring\n\n@[simp]\ntheorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubring_injective.eq_iff' top_toSubring\n\ntheorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_left\n\ntheorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_right\n\ntheorem mul_mem_sup {S T : NonUnitalSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) :\n    x * y ∈ S ⊔ T :=\n  mul_mem (mem_sup_left hx) (mem_sup_right hy)\n\ntheorem map_sup (f : F) (S T : NonUnitalSubalgebra R A) :\n    ((S ⊔ T).map f : NonUnitalSubalgebra R B) = S.map f ⊔ T.map f :=\n  (@NonUnitalSubalgebra.gc_map_comap F R A B _ _ _ _ _ _ f).l_sup\n\n@[simp, norm_cast]\ntheorem coe_inf (S T : NonUnitalSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T :=\n  rfl\n\n@[simp]\ntheorem mem_inf {S T : NonUnitalSubalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T :=\n  Iff.rfl\n\n@[simp]\ntheorem inf_toSubmodule (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toSubmodule = S.toSubmodule ⊓ T.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem inf_toNonUnitalSubsemiring (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring ⊓ T.toNonUnitalSubsemiring :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sInf (S : Set (NonUnitalSubalgebra R A)) : (↑(sInf S) : Set A) = ⋂ s ∈ S, ↑s :=\n  sInf_image\n\ntheorem mem_sInf {S : Set (NonUnitalSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p := by\n  simp only [← SetLike.mem_coe, coe_sInf, Set.mem_iInter₂]\n\n@[simp]\ntheorem sInf_toSubmodule (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toSubmodule = sInf (NonUnitalSubalgebra.toSubmodule '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp]\ntheorem sInf_toNonUnitalSubsemiring (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toNonUnitalSubsemiring = sInf (NonUnitalSubalgebra.toNonUnitalSubsemiring '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp, norm_cast]\ntheorem coe_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} :\n    (↑(⨅ i, S i) : Set A) = ⋂ i, S i := by simp [iInf]\n\ntheorem mem_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} {x : A} :\n    (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by simp only [iInf, mem_sInf, Set.forall_range_iff]\n\n@[simp]\ntheorem iInf_toSubmodule {ι : Sort*} (S : ι → NonUnitalSubalgebra R A) :\n    (⨅ i, S i).toSubmodule = ⨅ i, (S i).toSubmodule :=\n  SetLike.coe_injective <| by simp\n\ninstance : Inhabited (NonUnitalSubalgebra R A) :=\n  ⟨⊥⟩\n\ntheorem mem_bot {x : A} : x ∈ (⊥ : NonUnitalSubalgebra R A) ↔ x = 0 :=\n  show x ∈ Submodule.span R (NonUnitalSubsemiring.closure (∅ : Set A) : Set A) ↔ x = 0 by\n    rw [NonUnitalSubsemiring.closure_empty, NonUnitalSubsemiring.coe_bot,\n      Submodule.span_zero_singleton, Submodule.mem_bot]\n\ntheorem toSubmodule_bot : (⊥ : NonUnitalSubalgebra R A).toSubmodule = ⊥ := by\n  ext\n  simp only [mem_bot, NonUnitalSubalgebra.mem_toSubmodule, Submodule.mem_bot]\n\n@[simp]\ntheorem coe_bot : ((⊥ : NonUnitalSubalgebra R A) : Set A) = {0} := by\n  simp [Set.ext_iff, NonUnitalAlgebra.mem_bot]\n\ntheorem eq_top_iff {S : NonUnitalSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S :=\n  ⟨fun h x => by rw [h]; exact mem_top, fun h => by ext x; exact ⟨fun _ => mem_top, fun _ => h x⟩⟩\n\ntheorem range_top_iff_surjective (f : A →ₙₐ[R] B) :\n    NonUnitalAlgHom.range f = (⊤ : NonUnitalSubalgebra R B) ↔ Function.Surjective f :=\n  NonUnitalAlgebra.eq_top_iff\n\n@[simp]\ntheorem range_id : NonUnitalAlgHom.range (NonUnitalAlgHom.id R A) = ⊤ :=\n  SetLike.coe_injective Set.range_id\n\n@[simp]\ntheorem map_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R A).map f = NonUnitalAlgHom.range f :=\n  SetLike.coe_injective Set.image_univ\n\n@[simp]\ntheorem map_bot (f : A →ₙₐ[R] B) : (⊥ : NonUnitalSubalgebra R A).map f = ⊥ :=\n  SetLike.coe_injective <| by simp [NonUnitalAlgebra.coe_bot, NonUnitalSubalgebra.coe_map]\n\n@[simp]\ntheorem comap_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R B).comap f = ⊤ :=\n  eq_top_iff.2 fun _ => mem_top\n\n/-- `NonUnitalAlgHom` to `⊤ : NonUnitalSubalgebra R A`. -/\ndef toTop : A →ₙₐ[R] (⊤ : NonUnitalSubalgebra R A) :=\n  NonUnitalAlgHom.codRestrict (NonUnitalAlgHom.id R A) ⊤ fun _ => mem_top\n\nend NonUnitalAlgebra\n\nnamespace NonUnitalSubalgebra\n\nopen NonUnitalAlgebra\n\nsection NonAssoc\n\nvariable {R : Type u} {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable (S : NonUnitalSubalgebra R A)\n\ninstance subsingleton_of_subsingleton [Subsingleton A] : Subsingleton (NonUnitalSubalgebra R A) :=\n  ⟨fun B C => ext fun x => by simp only [Subsingleton.elim x 0, zero_mem B, zero_mem C]⟩\n\ninstance _root_.NonUnitalAlgHom.subsingleton [Subsingleton (NonUnitalSubalgebra R A)] :\n    Subsingleton (A →ₙₐ[R] B) :=\n  ⟨fun f g =>\n    NonUnitalAlgHom.ext fun a =>\n      have : a ∈ (⊥ : NonUnitalSubalgebra R A) :=\n        Subsingleton.elim (⊤ : NonUnitalSubalgebra R A) ⊥ ▸ mem_top\n      (mem_bot.mp this).symm ▸ (map_zero f).trans (map_zero g).symm⟩\n\ntheorem range_val : NonUnitalAlgHom.range (NonUnitalSubalgebraClass.subtype S) = S :=\n  ext <| Set.ext_iff.1 <| (NonUnitalSubalgebraClass.subtype S).coe_range.trans Subtype.range_val\n\n/-- The map `S → T` when `S` is a non-unital subalgebra contained in the non-unital subalgebra `T`.\n\nThis is the non-unital subalgebra version of `Submodule.inclusion`, or `Subring.inclusion`  -/\ndef inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) : S →ₙₐ[R] T\n    where\n  toFun := Set.inclusion h\n  map_add' _ _ := rfl\n  map_mul' _ _ := rfl\n  map_zero' := rfl\n  map_smul' _ _ := rfl\n\ntheorem inclusion_injective {S T : NonUnitalSubalgebra R A} (h : S ≤ T) :\n    Function.Injective (inclusion h) := fun _ _ => Subtype.ext ∘ Subtype.mk.inj\n\n@[simp]\ntheorem inclusion_self {S : NonUnitalSubalgebra R A} :\n    inclusion (le_refl S) = NonUnitalAlgHom.id R S :=\n  NonUnitalAlgHom.ext fun _ => Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_mk {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : A) (hx : x ∈ S) :\n    inclusion h ⟨x, hx⟩ = ⟨x, h hx⟩ :=\n  rfl\n\ntheorem inclusion_right {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : T) (m : (x : A) ∈ S) :\n    inclusion h ⟨x, m⟩ = x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_inclusion {S T U : NonUnitalSubalgebra R A} (hst : S ≤ T) (htu : T ≤ U) (x : S) :\n    inclusion htu (inclusion hst x) = inclusion (le_trans hst htu) x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem coe_inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (s : S) :\n    (inclusion h s : A) = s :=\n  rfl\n\nsection Prod\n\nvariable (S₁ : NonUnitalSubalgebra R B)\n\n/-- The product of two non-unital subalgebras is a non-unital subalgebra. -/\ndef prod : NonUnitalSubalgebra R (A × B) :=\n  { S.toNonUnitalSubsemiring.prod S₁.toNonUnitalSubsemiring with\n    carrier := S ×ˢ S₁\n    smul_mem' := fun r _x hx => ⟨SMulMemClass.smul_mem r hx.1, SMulMemClass.smul_mem r hx.2⟩ }\n\n@[simp]\ntheorem coe_prod : (prod S S₁ : Set (A × B)) = (S : Set A) ×ˢ S₁ :=\n  rfl\n\ntheorem prod_toSubmodule : (S.prod S₁).toSubmodule = S.toSubmodule.prod S₁.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem mem_prod {S : NonUnitalSubalgebra R A} {S₁ : NonUnitalSubalgebra R B} {x : A × B} :\n    x ∈ prod S S₁ ↔ x.1 ∈ S ∧ x.2 ∈ S₁ :=\n  Set.mem_prod\n\n@[simp]\ntheorem prod_top : (prod ⊤ ⊤ : NonUnitalSubalgebra R (A × B)) = ⊤ := by ext; simp\n\ntheorem prod_mono {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S ≤ T → S₁ ≤ T₁ → prod S S₁ ≤ prod T T₁ :=\n  Set.prod_mono\n\n@[simp]\ntheorem prod_inf_prod {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S.prod S₁ ⊓ T.prod T₁ = (S ⊓ T).prod (S₁ ⊓ T₁) :=\n  SetLike.coe_injective Set.prod_inter_prod\n\nend Prod\n\nsection SuprLift\n\nvariable {ι : Type*}\n\ntheorem coe_iSup_of_directed [Nonempty ι] {S : ι → NonUnitalSubalgebra R A}\n    (dir : Directed (· ≤ ·) S) : ↑(iSup S) = ⋃ i, (S i : Set A) :=\n  let K : NonUnitalSubalgebra R A :=\n    { __ := NonUnitalSubsemiring.copy _ _ (NonUnitalSubsemiring.coe_iSup_of_directed dir).symm\n      smul_mem' := fun r _x hx ↦\n        let ⟨i, hi⟩ := Set.mem_iUnion.1 hx\n        Set.mem_iUnion.2 ⟨i, (S i).smul_mem' r hi⟩ }\n  have : iSup S = K := le_antisymm\n    (iSup_le fun i ↦ le_iSup (fun i ↦ (S i : Set A)) i) (Set.iUnion_subset fun _ ↦ le_iSup S _)\n  this.symm ▸ rfl\n\n/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              ","nextTactic":"let ⟨k, hik, hjk⟩ := dir i j","declUpToTactic":"/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.892_0.jQvdAcOcyVirh23","decl":"/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B "}
-{"state":"R : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\nS : NonUnitalSubalgebra R A\nι : Type u_1\ninst✝ : Nonempty ι\nK : ι → NonUnitalSubalgebra R A\ndir : Directed (fun x x_1 => x ≤ x_1) K\nf : (i : ι) → ↥(K i) →ₙₐ[R] B\nhf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalAlgHom.comp (f j) (inclusion h)\ni j : ι\nx : A\nhxi : x ∈ (fun i => ↑(K i)) i\nhxj : x ∈ (fun i => ↑(K i)) j\nk : ι\nhik : (fun x x_1 => x ≤ x_1) (K i) (K k)\nhjk : (fun x x_1 => x ≤ x_1) (K j) (K k)\n⊢ (fun i x => (f i) x) i { val := x, property := hxi } = (fun i x => (f i) x) j { val := x, property := hxj }","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact NonUnitalAlgebra.gc\n\n@[simp]\ntheorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=\n  eq_top_iff.2 fun _x hx => subset_adjoin R hx\n\nvariable {R A}\n\n@[simp]\ntheorem coe_top : (↑(⊤ : NonUnitalSubalgebra R A) : Set A) = Set.univ :=\n  rfl\n\n@[simp]\ntheorem mem_top {x : A} : x ∈ (⊤ : NonUnitalSubalgebra R A) :=\n  Set.mem_univ x\n\n@[simp]\ntheorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toNonUnitalSubsemiring : (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubsemiring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toSubring {R A : Type*} [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    [IsScalarTower R A A] [SMulCommClass R A A] :\n    (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem toSubmodule_eq_top {S : NonUnitalSubalgebra R A} : S.toSubmodule = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toSubmodule'.injective.eq_iff' top_toSubmodule\n\n@[simp]\ntheorem toNonUnitalSubsemiring_eq_top {S : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubsemiring_injective.eq_iff' top_toNonUnitalSubsemiring\n\n@[simp]\ntheorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubring_injective.eq_iff' top_toSubring\n\ntheorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_left\n\ntheorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_right\n\ntheorem mul_mem_sup {S T : NonUnitalSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) :\n    x * y ∈ S ⊔ T :=\n  mul_mem (mem_sup_left hx) (mem_sup_right hy)\n\ntheorem map_sup (f : F) (S T : NonUnitalSubalgebra R A) :\n    ((S ⊔ T).map f : NonUnitalSubalgebra R B) = S.map f ⊔ T.map f :=\n  (@NonUnitalSubalgebra.gc_map_comap F R A B _ _ _ _ _ _ f).l_sup\n\n@[simp, norm_cast]\ntheorem coe_inf (S T : NonUnitalSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T :=\n  rfl\n\n@[simp]\ntheorem mem_inf {S T : NonUnitalSubalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T :=\n  Iff.rfl\n\n@[simp]\ntheorem inf_toSubmodule (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toSubmodule = S.toSubmodule ⊓ T.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem inf_toNonUnitalSubsemiring (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring ⊓ T.toNonUnitalSubsemiring :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sInf (S : Set (NonUnitalSubalgebra R A)) : (↑(sInf S) : Set A) = ⋂ s ∈ S, ↑s :=\n  sInf_image\n\ntheorem mem_sInf {S : Set (NonUnitalSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p := by\n  simp only [← SetLike.mem_coe, coe_sInf, Set.mem_iInter₂]\n\n@[simp]\ntheorem sInf_toSubmodule (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toSubmodule = sInf (NonUnitalSubalgebra.toSubmodule '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp]\ntheorem sInf_toNonUnitalSubsemiring (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toNonUnitalSubsemiring = sInf (NonUnitalSubalgebra.toNonUnitalSubsemiring '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp, norm_cast]\ntheorem coe_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} :\n    (↑(⨅ i, S i) : Set A) = ⋂ i, S i := by simp [iInf]\n\ntheorem mem_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} {x : A} :\n    (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by simp only [iInf, mem_sInf, Set.forall_range_iff]\n\n@[simp]\ntheorem iInf_toSubmodule {ι : Sort*} (S : ι → NonUnitalSubalgebra R A) :\n    (⨅ i, S i).toSubmodule = ⨅ i, (S i).toSubmodule :=\n  SetLike.coe_injective <| by simp\n\ninstance : Inhabited (NonUnitalSubalgebra R A) :=\n  ⟨⊥⟩\n\ntheorem mem_bot {x : A} : x ∈ (⊥ : NonUnitalSubalgebra R A) ↔ x = 0 :=\n  show x ∈ Submodule.span R (NonUnitalSubsemiring.closure (∅ : Set A) : Set A) ↔ x = 0 by\n    rw [NonUnitalSubsemiring.closure_empty, NonUnitalSubsemiring.coe_bot,\n      Submodule.span_zero_singleton, Submodule.mem_bot]\n\ntheorem toSubmodule_bot : (⊥ : NonUnitalSubalgebra R A).toSubmodule = ⊥ := by\n  ext\n  simp only [mem_bot, NonUnitalSubalgebra.mem_toSubmodule, Submodule.mem_bot]\n\n@[simp]\ntheorem coe_bot : ((⊥ : NonUnitalSubalgebra R A) : Set A) = {0} := by\n  simp [Set.ext_iff, NonUnitalAlgebra.mem_bot]\n\ntheorem eq_top_iff {S : NonUnitalSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S :=\n  ⟨fun h x => by rw [h]; exact mem_top, fun h => by ext x; exact ⟨fun _ => mem_top, fun _ => h x⟩⟩\n\ntheorem range_top_iff_surjective (f : A →ₙₐ[R] B) :\n    NonUnitalAlgHom.range f = (⊤ : NonUnitalSubalgebra R B) ↔ Function.Surjective f :=\n  NonUnitalAlgebra.eq_top_iff\n\n@[simp]\ntheorem range_id : NonUnitalAlgHom.range (NonUnitalAlgHom.id R A) = ⊤ :=\n  SetLike.coe_injective Set.range_id\n\n@[simp]\ntheorem map_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R A).map f = NonUnitalAlgHom.range f :=\n  SetLike.coe_injective Set.image_univ\n\n@[simp]\ntheorem map_bot (f : A →ₙₐ[R] B) : (⊥ : NonUnitalSubalgebra R A).map f = ⊥ :=\n  SetLike.coe_injective <| by simp [NonUnitalAlgebra.coe_bot, NonUnitalSubalgebra.coe_map]\n\n@[simp]\ntheorem comap_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R B).comap f = ⊤ :=\n  eq_top_iff.2 fun _ => mem_top\n\n/-- `NonUnitalAlgHom` to `⊤ : NonUnitalSubalgebra R A`. -/\ndef toTop : A →ₙₐ[R] (⊤ : NonUnitalSubalgebra R A) :=\n  NonUnitalAlgHom.codRestrict (NonUnitalAlgHom.id R A) ⊤ fun _ => mem_top\n\nend NonUnitalAlgebra\n\nnamespace NonUnitalSubalgebra\n\nopen NonUnitalAlgebra\n\nsection NonAssoc\n\nvariable {R : Type u} {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable (S : NonUnitalSubalgebra R A)\n\ninstance subsingleton_of_subsingleton [Subsingleton A] : Subsingleton (NonUnitalSubalgebra R A) :=\n  ⟨fun B C => ext fun x => by simp only [Subsingleton.elim x 0, zero_mem B, zero_mem C]⟩\n\ninstance _root_.NonUnitalAlgHom.subsingleton [Subsingleton (NonUnitalSubalgebra R A)] :\n    Subsingleton (A →ₙₐ[R] B) :=\n  ⟨fun f g =>\n    NonUnitalAlgHom.ext fun a =>\n      have : a ∈ (⊥ : NonUnitalSubalgebra R A) :=\n        Subsingleton.elim (⊤ : NonUnitalSubalgebra R A) ⊥ ▸ mem_top\n      (mem_bot.mp this).symm ▸ (map_zero f).trans (map_zero g).symm⟩\n\ntheorem range_val : NonUnitalAlgHom.range (NonUnitalSubalgebraClass.subtype S) = S :=\n  ext <| Set.ext_iff.1 <| (NonUnitalSubalgebraClass.subtype S).coe_range.trans Subtype.range_val\n\n/-- The map `S → T` when `S` is a non-unital subalgebra contained in the non-unital subalgebra `T`.\n\nThis is the non-unital subalgebra version of `Submodule.inclusion`, or `Subring.inclusion`  -/\ndef inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) : S →ₙₐ[R] T\n    where\n  toFun := Set.inclusion h\n  map_add' _ _ := rfl\n  map_mul' _ _ := rfl\n  map_zero' := rfl\n  map_smul' _ _ := rfl\n\ntheorem inclusion_injective {S T : NonUnitalSubalgebra R A} (h : S ≤ T) :\n    Function.Injective (inclusion h) := fun _ _ => Subtype.ext ∘ Subtype.mk.inj\n\n@[simp]\ntheorem inclusion_self {S : NonUnitalSubalgebra R A} :\n    inclusion (le_refl S) = NonUnitalAlgHom.id R S :=\n  NonUnitalAlgHom.ext fun _ => Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_mk {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : A) (hx : x ∈ S) :\n    inclusion h ⟨x, hx⟩ = ⟨x, h hx⟩ :=\n  rfl\n\ntheorem inclusion_right {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : T) (m : (x : A) ∈ S) :\n    inclusion h ⟨x, m⟩ = x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_inclusion {S T U : NonUnitalSubalgebra R A} (hst : S ≤ T) (htu : T ≤ U) (x : S) :\n    inclusion htu (inclusion hst x) = inclusion (le_trans hst htu) x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem coe_inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (s : S) :\n    (inclusion h s : A) = s :=\n  rfl\n\nsection Prod\n\nvariable (S₁ : NonUnitalSubalgebra R B)\n\n/-- The product of two non-unital subalgebras is a non-unital subalgebra. -/\ndef prod : NonUnitalSubalgebra R (A × B) :=\n  { S.toNonUnitalSubsemiring.prod S₁.toNonUnitalSubsemiring with\n    carrier := S ×ˢ S₁\n    smul_mem' := fun r _x hx => ⟨SMulMemClass.smul_mem r hx.1, SMulMemClass.smul_mem r hx.2⟩ }\n\n@[simp]\ntheorem coe_prod : (prod S S₁ : Set (A × B)) = (S : Set A) ×ˢ S₁ :=\n  rfl\n\ntheorem prod_toSubmodule : (S.prod S₁).toSubmodule = S.toSubmodule.prod S₁.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem mem_prod {S : NonUnitalSubalgebra R A} {S₁ : NonUnitalSubalgebra R B} {x : A × B} :\n    x ∈ prod S S₁ ↔ x.1 ∈ S ∧ x.2 ∈ S₁ :=\n  Set.mem_prod\n\n@[simp]\ntheorem prod_top : (prod ⊤ ⊤ : NonUnitalSubalgebra R (A × B)) = ⊤ := by ext; simp\n\ntheorem prod_mono {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S ≤ T → S₁ ≤ T₁ → prod S S₁ ≤ prod T T₁ :=\n  Set.prod_mono\n\n@[simp]\ntheorem prod_inf_prod {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S.prod S₁ ⊓ T.prod T₁ = (S ⊓ T).prod (S₁ ⊓ T₁) :=\n  SetLike.coe_injective Set.prod_inter_prod\n\nend Prod\n\nsection SuprLift\n\nvariable {ι : Type*}\n\ntheorem coe_iSup_of_directed [Nonempty ι] {S : ι → NonUnitalSubalgebra R A}\n    (dir : Directed (· ≤ ·) S) : ↑(iSup S) = ⋃ i, (S i : Set A) :=\n  let K : NonUnitalSubalgebra R A :=\n    { __ := NonUnitalSubsemiring.copy _ _ (NonUnitalSubsemiring.coe_iSup_of_directed dir).symm\n      smul_mem' := fun r _x hx ↦\n        let ⟨i, hi⟩ := Set.mem_iUnion.1 hx\n        Set.mem_iUnion.2 ⟨i, (S i).smul_mem' r hi⟩ }\n  have : iSup S = K := le_antisymm\n    (iSup_le fun i ↦ le_iSup (fun i ↦ (S i : Set A)) i) (Set.iUnion_subset fun _ ↦ le_iSup S _)\n  this.symm ▸ rfl\n\n/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              ","nextTactic":"simp only","declUpToTactic":"/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.892_0.jQvdAcOcyVirh23","decl":"/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B "}
-{"state":"R : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\nS : NonUnitalSubalgebra R A\nι : Type u_1\ninst✝ : Nonempty ι\nK : ι → NonUnitalSubalgebra R A\ndir : Directed (fun x x_1 => x ≤ x_1) K\nf : (i : ι) → ↥(K i) →ₙₐ[R] B\nhf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalAlgHom.comp (f j) (inclusion h)\ni j : ι\nx : A\nhxi : x ∈ (fun i => ↑(K i)) i\nhxj : x ∈ (fun i => ↑(K i)) j\nk : ι\nhik : (fun x x_1 => x ≤ x_1) (K i) (K k)\nhjk : (fun x x_1 => x ≤ x_1) (K j) (K k)\n⊢ (f i) { val := x, property := hxi } = (f j) { val := x, property := hxj }","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact NonUnitalAlgebra.gc\n\n@[simp]\ntheorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=\n  eq_top_iff.2 fun _x hx => subset_adjoin R hx\n\nvariable {R A}\n\n@[simp]\ntheorem coe_top : (↑(⊤ : NonUnitalSubalgebra R A) : Set A) = Set.univ :=\n  rfl\n\n@[simp]\ntheorem mem_top {x : A} : x ∈ (⊤ : NonUnitalSubalgebra R A) :=\n  Set.mem_univ x\n\n@[simp]\ntheorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toNonUnitalSubsemiring : (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubsemiring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toSubring {R A : Type*} [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    [IsScalarTower R A A] [SMulCommClass R A A] :\n    (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem toSubmodule_eq_top {S : NonUnitalSubalgebra R A} : S.toSubmodule = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toSubmodule'.injective.eq_iff' top_toSubmodule\n\n@[simp]\ntheorem toNonUnitalSubsemiring_eq_top {S : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubsemiring_injective.eq_iff' top_toNonUnitalSubsemiring\n\n@[simp]\ntheorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubring_injective.eq_iff' top_toSubring\n\ntheorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_left\n\ntheorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_right\n\ntheorem mul_mem_sup {S T : NonUnitalSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) :\n    x * y ∈ S ⊔ T :=\n  mul_mem (mem_sup_left hx) (mem_sup_right hy)\n\ntheorem map_sup (f : F) (S T : NonUnitalSubalgebra R A) :\n    ((S ⊔ T).map f : NonUnitalSubalgebra R B) = S.map f ⊔ T.map f :=\n  (@NonUnitalSubalgebra.gc_map_comap F R A B _ _ _ _ _ _ f).l_sup\n\n@[simp, norm_cast]\ntheorem coe_inf (S T : NonUnitalSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T :=\n  rfl\n\n@[simp]\ntheorem mem_inf {S T : NonUnitalSubalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T :=\n  Iff.rfl\n\n@[simp]\ntheorem inf_toSubmodule (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toSubmodule = S.toSubmodule ⊓ T.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem inf_toNonUnitalSubsemiring (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring ⊓ T.toNonUnitalSubsemiring :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sInf (S : Set (NonUnitalSubalgebra R A)) : (↑(sInf S) : Set A) = ⋂ s ∈ S, ↑s :=\n  sInf_image\n\ntheorem mem_sInf {S : Set (NonUnitalSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p := by\n  simp only [← SetLike.mem_coe, coe_sInf, Set.mem_iInter₂]\n\n@[simp]\ntheorem sInf_toSubmodule (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toSubmodule = sInf (NonUnitalSubalgebra.toSubmodule '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp]\ntheorem sInf_toNonUnitalSubsemiring (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toNonUnitalSubsemiring = sInf (NonUnitalSubalgebra.toNonUnitalSubsemiring '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp, norm_cast]\ntheorem coe_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} :\n    (↑(⨅ i, S i) : Set A) = ⋂ i, S i := by simp [iInf]\n\ntheorem mem_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} {x : A} :\n    (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by simp only [iInf, mem_sInf, Set.forall_range_iff]\n\n@[simp]\ntheorem iInf_toSubmodule {ι : Sort*} (S : ι → NonUnitalSubalgebra R A) :\n    (⨅ i, S i).toSubmodule = ⨅ i, (S i).toSubmodule :=\n  SetLike.coe_injective <| by simp\n\ninstance : Inhabited (NonUnitalSubalgebra R A) :=\n  ⟨⊥⟩\n\ntheorem mem_bot {x : A} : x ∈ (⊥ : NonUnitalSubalgebra R A) ↔ x = 0 :=\n  show x ∈ Submodule.span R (NonUnitalSubsemiring.closure (∅ : Set A) : Set A) ↔ x = 0 by\n    rw [NonUnitalSubsemiring.closure_empty, NonUnitalSubsemiring.coe_bot,\n      Submodule.span_zero_singleton, Submodule.mem_bot]\n\ntheorem toSubmodule_bot : (⊥ : NonUnitalSubalgebra R A).toSubmodule = ⊥ := by\n  ext\n  simp only [mem_bot, NonUnitalSubalgebra.mem_toSubmodule, Submodule.mem_bot]\n\n@[simp]\ntheorem coe_bot : ((⊥ : NonUnitalSubalgebra R A) : Set A) = {0} := by\n  simp [Set.ext_iff, NonUnitalAlgebra.mem_bot]\n\ntheorem eq_top_iff {S : NonUnitalSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S :=\n  ⟨fun h x => by rw [h]; exact mem_top, fun h => by ext x; exact ⟨fun _ => mem_top, fun _ => h x⟩⟩\n\ntheorem range_top_iff_surjective (f : A →ₙₐ[R] B) :\n    NonUnitalAlgHom.range f = (⊤ : NonUnitalSubalgebra R B) ↔ Function.Surjective f :=\n  NonUnitalAlgebra.eq_top_iff\n\n@[simp]\ntheorem range_id : NonUnitalAlgHom.range (NonUnitalAlgHom.id R A) = ⊤ :=\n  SetLike.coe_injective Set.range_id\n\n@[simp]\ntheorem map_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R A).map f = NonUnitalAlgHom.range f :=\n  SetLike.coe_injective Set.image_univ\n\n@[simp]\ntheorem map_bot (f : A →ₙₐ[R] B) : (⊥ : NonUnitalSubalgebra R A).map f = ⊥ :=\n  SetLike.coe_injective <| by simp [NonUnitalAlgebra.coe_bot, NonUnitalSubalgebra.coe_map]\n\n@[simp]\ntheorem comap_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R B).comap f = ⊤ :=\n  eq_top_iff.2 fun _ => mem_top\n\n/-- `NonUnitalAlgHom` to `⊤ : NonUnitalSubalgebra R A`. -/\ndef toTop : A →ₙₐ[R] (⊤ : NonUnitalSubalgebra R A) :=\n  NonUnitalAlgHom.codRestrict (NonUnitalAlgHom.id R A) ⊤ fun _ => mem_top\n\nend NonUnitalAlgebra\n\nnamespace NonUnitalSubalgebra\n\nopen NonUnitalAlgebra\n\nsection NonAssoc\n\nvariable {R : Type u} {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable (S : NonUnitalSubalgebra R A)\n\ninstance subsingleton_of_subsingleton [Subsingleton A] : Subsingleton (NonUnitalSubalgebra R A) :=\n  ⟨fun B C => ext fun x => by simp only [Subsingleton.elim x 0, zero_mem B, zero_mem C]⟩\n\ninstance _root_.NonUnitalAlgHom.subsingleton [Subsingleton (NonUnitalSubalgebra R A)] :\n    Subsingleton (A →ₙₐ[R] B) :=\n  ⟨fun f g =>\n    NonUnitalAlgHom.ext fun a =>\n      have : a ∈ (⊥ : NonUnitalSubalgebra R A) :=\n        Subsingleton.elim (⊤ : NonUnitalSubalgebra R A) ⊥ ▸ mem_top\n      (mem_bot.mp this).symm ▸ (map_zero f).trans (map_zero g).symm⟩\n\ntheorem range_val : NonUnitalAlgHom.range (NonUnitalSubalgebraClass.subtype S) = S :=\n  ext <| Set.ext_iff.1 <| (NonUnitalSubalgebraClass.subtype S).coe_range.trans Subtype.range_val\n\n/-- The map `S → T` when `S` is a non-unital subalgebra contained in the non-unital subalgebra `T`.\n\nThis is the non-unital subalgebra version of `Submodule.inclusion`, or `Subring.inclusion`  -/\ndef inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) : S →ₙₐ[R] T\n    where\n  toFun := Set.inclusion h\n  map_add' _ _ := rfl\n  map_mul' _ _ := rfl\n  map_zero' := rfl\n  map_smul' _ _ := rfl\n\ntheorem inclusion_injective {S T : NonUnitalSubalgebra R A} (h : S ≤ T) :\n    Function.Injective (inclusion h) := fun _ _ => Subtype.ext ∘ Subtype.mk.inj\n\n@[simp]\ntheorem inclusion_self {S : NonUnitalSubalgebra R A} :\n    inclusion (le_refl S) = NonUnitalAlgHom.id R S :=\n  NonUnitalAlgHom.ext fun _ => Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_mk {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : A) (hx : x ∈ S) :\n    inclusion h ⟨x, hx⟩ = ⟨x, h hx⟩ :=\n  rfl\n\ntheorem inclusion_right {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : T) (m : (x : A) ∈ S) :\n    inclusion h ⟨x, m⟩ = x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_inclusion {S T U : NonUnitalSubalgebra R A} (hst : S ≤ T) (htu : T ≤ U) (x : S) :\n    inclusion htu (inclusion hst x) = inclusion (le_trans hst htu) x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem coe_inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (s : S) :\n    (inclusion h s : A) = s :=\n  rfl\n\nsection Prod\n\nvariable (S₁ : NonUnitalSubalgebra R B)\n\n/-- The product of two non-unital subalgebras is a non-unital subalgebra. -/\ndef prod : NonUnitalSubalgebra R (A × B) :=\n  { S.toNonUnitalSubsemiring.prod S₁.toNonUnitalSubsemiring with\n    carrier := S ×ˢ S₁\n    smul_mem' := fun r _x hx => ⟨SMulMemClass.smul_mem r hx.1, SMulMemClass.smul_mem r hx.2⟩ }\n\n@[simp]\ntheorem coe_prod : (prod S S₁ : Set (A × B)) = (S : Set A) ×ˢ S₁ :=\n  rfl\n\ntheorem prod_toSubmodule : (S.prod S₁).toSubmodule = S.toSubmodule.prod S₁.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem mem_prod {S : NonUnitalSubalgebra R A} {S₁ : NonUnitalSubalgebra R B} {x : A × B} :\n    x ∈ prod S S₁ ↔ x.1 ∈ S ∧ x.2 ∈ S₁ :=\n  Set.mem_prod\n\n@[simp]\ntheorem prod_top : (prod ⊤ ⊤ : NonUnitalSubalgebra R (A × B)) = ⊤ := by ext; simp\n\ntheorem prod_mono {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S ≤ T → S₁ ≤ T₁ → prod S S₁ ≤ prod T T₁ :=\n  Set.prod_mono\n\n@[simp]\ntheorem prod_inf_prod {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S.prod S₁ ⊓ T.prod T₁ = (S ⊓ T).prod (S₁ ⊓ T₁) :=\n  SetLike.coe_injective Set.prod_inter_prod\n\nend Prod\n\nsection SuprLift\n\nvariable {ι : Type*}\n\ntheorem coe_iSup_of_directed [Nonempty ι] {S : ι → NonUnitalSubalgebra R A}\n    (dir : Directed (· ≤ ·) S) : ↑(iSup S) = ⋃ i, (S i : Set A) :=\n  let K : NonUnitalSubalgebra R A :=\n    { __ := NonUnitalSubsemiring.copy _ _ (NonUnitalSubsemiring.coe_iSup_of_directed dir).symm\n      smul_mem' := fun r _x hx ↦\n        let ⟨i, hi⟩ := Set.mem_iUnion.1 hx\n        Set.mem_iUnion.2 ⟨i, (S i).smul_mem' r hi⟩ }\n  have : iSup S = K := le_antisymm\n    (iSup_le fun i ↦ le_iSup (fun i ↦ (S i : Set A)) i) (Set.iUnion_subset fun _ ↦ le_iSup S _)\n  this.symm ▸ rfl\n\n/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              ","nextTactic":"rw [hf i k hik]","declUpToTactic":"/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.892_0.jQvdAcOcyVirh23","decl":"/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B "}
-{"state":"R : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\nS : NonUnitalSubalgebra R A\nι : Type u_1\ninst✝ : Nonempty ι\nK : ι → NonUnitalSubalgebra R A\ndir : Directed (fun x x_1 => x ≤ x_1) K\nf : (i : ι) → ↥(K i) →ₙₐ[R] B\nhf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalAlgHom.comp (f j) (inclusion h)\ni j : ι\nx : A\nhxi : x ∈ (fun i => ↑(K i)) i\nhxj : x ∈ (fun i => ↑(K i)) j\nk : ι\nhik : (fun x x_1 => x ≤ x_1) (K i) (K k)\nhjk : (fun x x_1 => x ≤ x_1) (K j) (K k)\n⊢ (NonUnitalAlgHom.comp (f k) (inclusion hik)) { val := x, property := hxi } = (f j) { val := x, property := hxj }","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact NonUnitalAlgebra.gc\n\n@[simp]\ntheorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=\n  eq_top_iff.2 fun _x hx => subset_adjoin R hx\n\nvariable {R A}\n\n@[simp]\ntheorem coe_top : (↑(⊤ : NonUnitalSubalgebra R A) : Set A) = Set.univ :=\n  rfl\n\n@[simp]\ntheorem mem_top {x : A} : x ∈ (⊤ : NonUnitalSubalgebra R A) :=\n  Set.mem_univ x\n\n@[simp]\ntheorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toNonUnitalSubsemiring : (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubsemiring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toSubring {R A : Type*} [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    [IsScalarTower R A A] [SMulCommClass R A A] :\n    (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem toSubmodule_eq_top {S : NonUnitalSubalgebra R A} : S.toSubmodule = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toSubmodule'.injective.eq_iff' top_toSubmodule\n\n@[simp]\ntheorem toNonUnitalSubsemiring_eq_top {S : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubsemiring_injective.eq_iff' top_toNonUnitalSubsemiring\n\n@[simp]\ntheorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubring_injective.eq_iff' top_toSubring\n\ntheorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_left\n\ntheorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_right\n\ntheorem mul_mem_sup {S T : NonUnitalSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) :\n    x * y ∈ S ⊔ T :=\n  mul_mem (mem_sup_left hx) (mem_sup_right hy)\n\ntheorem map_sup (f : F) (S T : NonUnitalSubalgebra R A) :\n    ((S ⊔ T).map f : NonUnitalSubalgebra R B) = S.map f ⊔ T.map f :=\n  (@NonUnitalSubalgebra.gc_map_comap F R A B _ _ _ _ _ _ f).l_sup\n\n@[simp, norm_cast]\ntheorem coe_inf (S T : NonUnitalSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T :=\n  rfl\n\n@[simp]\ntheorem mem_inf {S T : NonUnitalSubalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T :=\n  Iff.rfl\n\n@[simp]\ntheorem inf_toSubmodule (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toSubmodule = S.toSubmodule ⊓ T.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem inf_toNonUnitalSubsemiring (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring ⊓ T.toNonUnitalSubsemiring :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sInf (S : Set (NonUnitalSubalgebra R A)) : (↑(sInf S) : Set A) = ⋂ s ∈ S, ↑s :=\n  sInf_image\n\ntheorem mem_sInf {S : Set (NonUnitalSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p := by\n  simp only [← SetLike.mem_coe, coe_sInf, Set.mem_iInter₂]\n\n@[simp]\ntheorem sInf_toSubmodule (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toSubmodule = sInf (NonUnitalSubalgebra.toSubmodule '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp]\ntheorem sInf_toNonUnitalSubsemiring (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toNonUnitalSubsemiring = sInf (NonUnitalSubalgebra.toNonUnitalSubsemiring '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp, norm_cast]\ntheorem coe_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} :\n    (↑(⨅ i, S i) : Set A) = ⋂ i, S i := by simp [iInf]\n\ntheorem mem_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} {x : A} :\n    (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by simp only [iInf, mem_sInf, Set.forall_range_iff]\n\n@[simp]\ntheorem iInf_toSubmodule {ι : Sort*} (S : ι → NonUnitalSubalgebra R A) :\n    (⨅ i, S i).toSubmodule = ⨅ i, (S i).toSubmodule :=\n  SetLike.coe_injective <| by simp\n\ninstance : Inhabited (NonUnitalSubalgebra R A) :=\n  ⟨⊥⟩\n\ntheorem mem_bot {x : A} : x ∈ (⊥ : NonUnitalSubalgebra R A) ↔ x = 0 :=\n  show x ∈ Submodule.span R (NonUnitalSubsemiring.closure (∅ : Set A) : Set A) ↔ x = 0 by\n    rw [NonUnitalSubsemiring.closure_empty, NonUnitalSubsemiring.coe_bot,\n      Submodule.span_zero_singleton, Submodule.mem_bot]\n\ntheorem toSubmodule_bot : (⊥ : NonUnitalSubalgebra R A).toSubmodule = ⊥ := by\n  ext\n  simp only [mem_bot, NonUnitalSubalgebra.mem_toSubmodule, Submodule.mem_bot]\n\n@[simp]\ntheorem coe_bot : ((⊥ : NonUnitalSubalgebra R A) : Set A) = {0} := by\n  simp [Set.ext_iff, NonUnitalAlgebra.mem_bot]\n\ntheorem eq_top_iff {S : NonUnitalSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S :=\n  ⟨fun h x => by rw [h]; exact mem_top, fun h => by ext x; exact ⟨fun _ => mem_top, fun _ => h x⟩⟩\n\ntheorem range_top_iff_surjective (f : A →ₙₐ[R] B) :\n    NonUnitalAlgHom.range f = (⊤ : NonUnitalSubalgebra R B) ↔ Function.Surjective f :=\n  NonUnitalAlgebra.eq_top_iff\n\n@[simp]\ntheorem range_id : NonUnitalAlgHom.range (NonUnitalAlgHom.id R A) = ⊤ :=\n  SetLike.coe_injective Set.range_id\n\n@[simp]\ntheorem map_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R A).map f = NonUnitalAlgHom.range f :=\n  SetLike.coe_injective Set.image_univ\n\n@[simp]\ntheorem map_bot (f : A →ₙₐ[R] B) : (⊥ : NonUnitalSubalgebra R A).map f = ⊥ :=\n  SetLike.coe_injective <| by simp [NonUnitalAlgebra.coe_bot, NonUnitalSubalgebra.coe_map]\n\n@[simp]\ntheorem comap_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R B).comap f = ⊤ :=\n  eq_top_iff.2 fun _ => mem_top\n\n/-- `NonUnitalAlgHom` to `⊤ : NonUnitalSubalgebra R A`. -/\ndef toTop : A →ₙₐ[R] (⊤ : NonUnitalSubalgebra R A) :=\n  NonUnitalAlgHom.codRestrict (NonUnitalAlgHom.id R A) ⊤ fun _ => mem_top\n\nend NonUnitalAlgebra\n\nnamespace NonUnitalSubalgebra\n\nopen NonUnitalAlgebra\n\nsection NonAssoc\n\nvariable {R : Type u} {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable (S : NonUnitalSubalgebra R A)\n\ninstance subsingleton_of_subsingleton [Subsingleton A] : Subsingleton (NonUnitalSubalgebra R A) :=\n  ⟨fun B C => ext fun x => by simp only [Subsingleton.elim x 0, zero_mem B, zero_mem C]⟩\n\ninstance _root_.NonUnitalAlgHom.subsingleton [Subsingleton (NonUnitalSubalgebra R A)] :\n    Subsingleton (A →ₙₐ[R] B) :=\n  ⟨fun f g =>\n    NonUnitalAlgHom.ext fun a =>\n      have : a ∈ (⊥ : NonUnitalSubalgebra R A) :=\n        Subsingleton.elim (⊤ : NonUnitalSubalgebra R A) ⊥ ▸ mem_top\n      (mem_bot.mp this).symm ▸ (map_zero f).trans (map_zero g).symm⟩\n\ntheorem range_val : NonUnitalAlgHom.range (NonUnitalSubalgebraClass.subtype S) = S :=\n  ext <| Set.ext_iff.1 <| (NonUnitalSubalgebraClass.subtype S).coe_range.trans Subtype.range_val\n\n/-- The map `S → T` when `S` is a non-unital subalgebra contained in the non-unital subalgebra `T`.\n\nThis is the non-unital subalgebra version of `Submodule.inclusion`, or `Subring.inclusion`  -/\ndef inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) : S →ₙₐ[R] T\n    where\n  toFun := Set.inclusion h\n  map_add' _ _ := rfl\n  map_mul' _ _ := rfl\n  map_zero' := rfl\n  map_smul' _ _ := rfl\n\ntheorem inclusion_injective {S T : NonUnitalSubalgebra R A} (h : S ≤ T) :\n    Function.Injective (inclusion h) := fun _ _ => Subtype.ext ∘ Subtype.mk.inj\n\n@[simp]\ntheorem inclusion_self {S : NonUnitalSubalgebra R A} :\n    inclusion (le_refl S) = NonUnitalAlgHom.id R S :=\n  NonUnitalAlgHom.ext fun _ => Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_mk {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : A) (hx : x ∈ S) :\n    inclusion h ⟨x, hx⟩ = ⟨x, h hx⟩ :=\n  rfl\n\ntheorem inclusion_right {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : T) (m : (x : A) ∈ S) :\n    inclusion h ⟨x, m⟩ = x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_inclusion {S T U : NonUnitalSubalgebra R A} (hst : S ≤ T) (htu : T ≤ U) (x : S) :\n    inclusion htu (inclusion hst x) = inclusion (le_trans hst htu) x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem coe_inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (s : S) :\n    (inclusion h s : A) = s :=\n  rfl\n\nsection Prod\n\nvariable (S₁ : NonUnitalSubalgebra R B)\n\n/-- The product of two non-unital subalgebras is a non-unital subalgebra. -/\ndef prod : NonUnitalSubalgebra R (A × B) :=\n  { S.toNonUnitalSubsemiring.prod S₁.toNonUnitalSubsemiring with\n    carrier := S ×ˢ S₁\n    smul_mem' := fun r _x hx => ⟨SMulMemClass.smul_mem r hx.1, SMulMemClass.smul_mem r hx.2⟩ }\n\n@[simp]\ntheorem coe_prod : (prod S S₁ : Set (A × B)) = (S : Set A) ×ˢ S₁ :=\n  rfl\n\ntheorem prod_toSubmodule : (S.prod S₁).toSubmodule = S.toSubmodule.prod S₁.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem mem_prod {S : NonUnitalSubalgebra R A} {S₁ : NonUnitalSubalgebra R B} {x : A × B} :\n    x ∈ prod S S₁ ↔ x.1 ∈ S ∧ x.2 ∈ S₁ :=\n  Set.mem_prod\n\n@[simp]\ntheorem prod_top : (prod ⊤ ⊤ : NonUnitalSubalgebra R (A × B)) = ⊤ := by ext; simp\n\ntheorem prod_mono {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S ≤ T → S₁ ≤ T₁ → prod S S₁ ≤ prod T T₁ :=\n  Set.prod_mono\n\n@[simp]\ntheorem prod_inf_prod {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S.prod S₁ ⊓ T.prod T₁ = (S ⊓ T).prod (S₁ ⊓ T₁) :=\n  SetLike.coe_injective Set.prod_inter_prod\n\nend Prod\n\nsection SuprLift\n\nvariable {ι : Type*}\n\ntheorem coe_iSup_of_directed [Nonempty ι] {S : ι → NonUnitalSubalgebra R A}\n    (dir : Directed (· ≤ ·) S) : ↑(iSup S) = ⋃ i, (S i : Set A) :=\n  let K : NonUnitalSubalgebra R A :=\n    { __ := NonUnitalSubsemiring.copy _ _ (NonUnitalSubsemiring.coe_iSup_of_directed dir).symm\n      smul_mem' := fun r _x hx ↦\n        let ⟨i, hi⟩ := Set.mem_iUnion.1 hx\n        Set.mem_iUnion.2 ⟨i, (S i).smul_mem' r hi⟩ }\n  have : iSup S = K := le_antisymm\n    (iSup_le fun i ↦ le_iSup (fun i ↦ (S i : Set A)) i) (Set.iUnion_subset fun _ ↦ le_iSup S _)\n  this.symm ▸ rfl\n\n/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              ","nextTactic":"rw [hf j k hjk]","declUpToTactic":"/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.892_0.jQvdAcOcyVirh23","decl":"/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B "}
-{"state":"R : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\nS : NonUnitalSubalgebra R A\nι : Type u_1\ninst✝ : Nonempty ι\nK : ι → NonUnitalSubalgebra R A\ndir : Directed (fun x x_1 => x ≤ x_1) K\nf : (i : ι) → ↥(K i) →ₙₐ[R] B\nhf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalAlgHom.comp (f j) (inclusion h)\ni j : ι\nx : A\nhxi : x ∈ (fun i => ↑(K i)) i\nhxj : x ∈ (fun i => ↑(K i)) j\nk : ι\nhik : (fun x x_1 => x ≤ x_1) (K i) (K k)\nhjk : (fun x x_1 => x ≤ x_1) (K j) (K k)\n⊢ (NonUnitalAlgHom.comp (f k) (inclusion hik)) { val := x, property := hxi } =\n    (NonUnitalAlgHom.comp (f k) (inclusion hjk)) { val := x, property := hxj }","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact NonUnitalAlgebra.gc\n\n@[simp]\ntheorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=\n  eq_top_iff.2 fun _x hx => subset_adjoin R hx\n\nvariable {R A}\n\n@[simp]\ntheorem coe_top : (↑(⊤ : NonUnitalSubalgebra R A) : Set A) = Set.univ :=\n  rfl\n\n@[simp]\ntheorem mem_top {x : A} : x ∈ (⊤ : NonUnitalSubalgebra R A) :=\n  Set.mem_univ x\n\n@[simp]\ntheorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toNonUnitalSubsemiring : (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubsemiring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toSubring {R A : Type*} [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    [IsScalarTower R A A] [SMulCommClass R A A] :\n    (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem toSubmodule_eq_top {S : NonUnitalSubalgebra R A} : S.toSubmodule = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toSubmodule'.injective.eq_iff' top_toSubmodule\n\n@[simp]\ntheorem toNonUnitalSubsemiring_eq_top {S : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubsemiring_injective.eq_iff' top_toNonUnitalSubsemiring\n\n@[simp]\ntheorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubring_injective.eq_iff' top_toSubring\n\ntheorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_left\n\ntheorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_right\n\ntheorem mul_mem_sup {S T : NonUnitalSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) :\n    x * y ∈ S ⊔ T :=\n  mul_mem (mem_sup_left hx) (mem_sup_right hy)\n\ntheorem map_sup (f : F) (S T : NonUnitalSubalgebra R A) :\n    ((S ⊔ T).map f : NonUnitalSubalgebra R B) = S.map f ⊔ T.map f :=\n  (@NonUnitalSubalgebra.gc_map_comap F R A B _ _ _ _ _ _ f).l_sup\n\n@[simp, norm_cast]\ntheorem coe_inf (S T : NonUnitalSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T :=\n  rfl\n\n@[simp]\ntheorem mem_inf {S T : NonUnitalSubalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T :=\n  Iff.rfl\n\n@[simp]\ntheorem inf_toSubmodule (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toSubmodule = S.toSubmodule ⊓ T.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem inf_toNonUnitalSubsemiring (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring ⊓ T.toNonUnitalSubsemiring :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sInf (S : Set (NonUnitalSubalgebra R A)) : (↑(sInf S) : Set A) = ⋂ s ∈ S, ↑s :=\n  sInf_image\n\ntheorem mem_sInf {S : Set (NonUnitalSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p := by\n  simp only [← SetLike.mem_coe, coe_sInf, Set.mem_iInter₂]\n\n@[simp]\ntheorem sInf_toSubmodule (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toSubmodule = sInf (NonUnitalSubalgebra.toSubmodule '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp]\ntheorem sInf_toNonUnitalSubsemiring (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toNonUnitalSubsemiring = sInf (NonUnitalSubalgebra.toNonUnitalSubsemiring '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp, norm_cast]\ntheorem coe_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} :\n    (↑(⨅ i, S i) : Set A) = ⋂ i, S i := by simp [iInf]\n\ntheorem mem_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} {x : A} :\n    (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by simp only [iInf, mem_sInf, Set.forall_range_iff]\n\n@[simp]\ntheorem iInf_toSubmodule {ι : Sort*} (S : ι → NonUnitalSubalgebra R A) :\n    (⨅ i, S i).toSubmodule = ⨅ i, (S i).toSubmodule :=\n  SetLike.coe_injective <| by simp\n\ninstance : Inhabited (NonUnitalSubalgebra R A) :=\n  ⟨⊥⟩\n\ntheorem mem_bot {x : A} : x ∈ (⊥ : NonUnitalSubalgebra R A) ↔ x = 0 :=\n  show x ∈ Submodule.span R (NonUnitalSubsemiring.closure (∅ : Set A) : Set A) ↔ x = 0 by\n    rw [NonUnitalSubsemiring.closure_empty, NonUnitalSubsemiring.coe_bot,\n      Submodule.span_zero_singleton, Submodule.mem_bot]\n\ntheorem toSubmodule_bot : (⊥ : NonUnitalSubalgebra R A).toSubmodule = ⊥ := by\n  ext\n  simp only [mem_bot, NonUnitalSubalgebra.mem_toSubmodule, Submodule.mem_bot]\n\n@[simp]\ntheorem coe_bot : ((⊥ : NonUnitalSubalgebra R A) : Set A) = {0} := by\n  simp [Set.ext_iff, NonUnitalAlgebra.mem_bot]\n\ntheorem eq_top_iff {S : NonUnitalSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S :=\n  ⟨fun h x => by rw [h]; exact mem_top, fun h => by ext x; exact ⟨fun _ => mem_top, fun _ => h x⟩⟩\n\ntheorem range_top_iff_surjective (f : A →ₙₐ[R] B) :\n    NonUnitalAlgHom.range f = (⊤ : NonUnitalSubalgebra R B) ↔ Function.Surjective f :=\n  NonUnitalAlgebra.eq_top_iff\n\n@[simp]\ntheorem range_id : NonUnitalAlgHom.range (NonUnitalAlgHom.id R A) = ⊤ :=\n  SetLike.coe_injective Set.range_id\n\n@[simp]\ntheorem map_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R A).map f = NonUnitalAlgHom.range f :=\n  SetLike.coe_injective Set.image_univ\n\n@[simp]\ntheorem map_bot (f : A →ₙₐ[R] B) : (⊥ : NonUnitalSubalgebra R A).map f = ⊥ :=\n  SetLike.coe_injective <| by simp [NonUnitalAlgebra.coe_bot, NonUnitalSubalgebra.coe_map]\n\n@[simp]\ntheorem comap_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R B).comap f = ⊤ :=\n  eq_top_iff.2 fun _ => mem_top\n\n/-- `NonUnitalAlgHom` to `⊤ : NonUnitalSubalgebra R A`. -/\ndef toTop : A →ₙₐ[R] (⊤ : NonUnitalSubalgebra R A) :=\n  NonUnitalAlgHom.codRestrict (NonUnitalAlgHom.id R A) ⊤ fun _ => mem_top\n\nend NonUnitalAlgebra\n\nnamespace NonUnitalSubalgebra\n\nopen NonUnitalAlgebra\n\nsection NonAssoc\n\nvariable {R : Type u} {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable (S : NonUnitalSubalgebra R A)\n\ninstance subsingleton_of_subsingleton [Subsingleton A] : Subsingleton (NonUnitalSubalgebra R A) :=\n  ⟨fun B C => ext fun x => by simp only [Subsingleton.elim x 0, zero_mem B, zero_mem C]⟩\n\ninstance _root_.NonUnitalAlgHom.subsingleton [Subsingleton (NonUnitalSubalgebra R A)] :\n    Subsingleton (A →ₙₐ[R] B) :=\n  ⟨fun f g =>\n    NonUnitalAlgHom.ext fun a =>\n      have : a ∈ (⊥ : NonUnitalSubalgebra R A) :=\n        Subsingleton.elim (⊤ : NonUnitalSubalgebra R A) ⊥ ▸ mem_top\n      (mem_bot.mp this).symm ▸ (map_zero f).trans (map_zero g).symm⟩\n\ntheorem range_val : NonUnitalAlgHom.range (NonUnitalSubalgebraClass.subtype S) = S :=\n  ext <| Set.ext_iff.1 <| (NonUnitalSubalgebraClass.subtype S).coe_range.trans Subtype.range_val\n\n/-- The map `S → T` when `S` is a non-unital subalgebra contained in the non-unital subalgebra `T`.\n\nThis is the non-unital subalgebra version of `Submodule.inclusion`, or `Subring.inclusion`  -/\ndef inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) : S →ₙₐ[R] T\n    where\n  toFun := Set.inclusion h\n  map_add' _ _ := rfl\n  map_mul' _ _ := rfl\n  map_zero' := rfl\n  map_smul' _ _ := rfl\n\ntheorem inclusion_injective {S T : NonUnitalSubalgebra R A} (h : S ≤ T) :\n    Function.Injective (inclusion h) := fun _ _ => Subtype.ext ∘ Subtype.mk.inj\n\n@[simp]\ntheorem inclusion_self {S : NonUnitalSubalgebra R A} :\n    inclusion (le_refl S) = NonUnitalAlgHom.id R S :=\n  NonUnitalAlgHom.ext fun _ => Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_mk {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : A) (hx : x ∈ S) :\n    inclusion h ⟨x, hx⟩ = ⟨x, h hx⟩ :=\n  rfl\n\ntheorem inclusion_right {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : T) (m : (x : A) ∈ S) :\n    inclusion h ⟨x, m⟩ = x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_inclusion {S T U : NonUnitalSubalgebra R A} (hst : S ≤ T) (htu : T ≤ U) (x : S) :\n    inclusion htu (inclusion hst x) = inclusion (le_trans hst htu) x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem coe_inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (s : S) :\n    (inclusion h s : A) = s :=\n  rfl\n\nsection Prod\n\nvariable (S₁ : NonUnitalSubalgebra R B)\n\n/-- The product of two non-unital subalgebras is a non-unital subalgebra. -/\ndef prod : NonUnitalSubalgebra R (A × B) :=\n  { S.toNonUnitalSubsemiring.prod S₁.toNonUnitalSubsemiring with\n    carrier := S ×ˢ S₁\n    smul_mem' := fun r _x hx => ⟨SMulMemClass.smul_mem r hx.1, SMulMemClass.smul_mem r hx.2⟩ }\n\n@[simp]\ntheorem coe_prod : (prod S S₁ : Set (A × B)) = (S : Set A) ×ˢ S₁ :=\n  rfl\n\ntheorem prod_toSubmodule : (S.prod S₁).toSubmodule = S.toSubmodule.prod S₁.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem mem_prod {S : NonUnitalSubalgebra R A} {S₁ : NonUnitalSubalgebra R B} {x : A × B} :\n    x ∈ prod S S₁ ↔ x.1 ∈ S ∧ x.2 ∈ S₁ :=\n  Set.mem_prod\n\n@[simp]\ntheorem prod_top : (prod ⊤ ⊤ : NonUnitalSubalgebra R (A × B)) = ⊤ := by ext; simp\n\ntheorem prod_mono {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S ≤ T → S₁ ≤ T₁ → prod S S₁ ≤ prod T T₁ :=\n  Set.prod_mono\n\n@[simp]\ntheorem prod_inf_prod {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S.prod S₁ ⊓ T.prod T₁ = (S ⊓ T).prod (S₁ ⊓ T₁) :=\n  SetLike.coe_injective Set.prod_inter_prod\n\nend Prod\n\nsection SuprLift\n\nvariable {ι : Type*}\n\ntheorem coe_iSup_of_directed [Nonempty ι] {S : ι → NonUnitalSubalgebra R A}\n    (dir : Directed (· ≤ ·) S) : ↑(iSup S) = ⋃ i, (S i : Set A) :=\n  let K : NonUnitalSubalgebra R A :=\n    { __ := NonUnitalSubsemiring.copy _ _ (NonUnitalSubsemiring.coe_iSup_of_directed dir).symm\n      smul_mem' := fun r _x hx ↦\n        let ⟨i, hi⟩ := Set.mem_iUnion.1 hx\n        Set.mem_iUnion.2 ⟨i, (S i).smul_mem' r hi⟩ }\n  have : iSup S = K := le_antisymm\n    (iSup_le fun i ↦ le_iSup (fun i ↦ (S i : Set A)) i) (Set.iUnion_subset fun _ ↦ le_iSup S _)\n  this.symm ▸ rfl\n\n/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              rw [hf j k hjk]\n              ","nextTactic":"rfl","declUpToTactic":"/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              rw [hf j k hjk]\n              ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.892_0.jQvdAcOcyVirh23","decl":"/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B "}
-{"state":"R : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\nS : NonUnitalSubalgebra R A\nι : Type u_1\ninst✝ : Nonempty ι\nK : ι → NonUnitalSubalgebra R A\ndir : Directed (fun x x_1 => x ≤ x_1) K\nf : (i : ι) → ↥(K i) →ₙₐ[R] B\nhf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalAlgHom.comp (f j) (inclusion h)\n⊢ ↑(iSup K) ⊆ ⋃ i, ↑(K i)","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact NonUnitalAlgebra.gc\n\n@[simp]\ntheorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=\n  eq_top_iff.2 fun _x hx => subset_adjoin R hx\n\nvariable {R A}\n\n@[simp]\ntheorem coe_top : (↑(⊤ : NonUnitalSubalgebra R A) : Set A) = Set.univ :=\n  rfl\n\n@[simp]\ntheorem mem_top {x : A} : x ∈ (⊤ : NonUnitalSubalgebra R A) :=\n  Set.mem_univ x\n\n@[simp]\ntheorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toNonUnitalSubsemiring : (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubsemiring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toSubring {R A : Type*} [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    [IsScalarTower R A A] [SMulCommClass R A A] :\n    (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem toSubmodule_eq_top {S : NonUnitalSubalgebra R A} : S.toSubmodule = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toSubmodule'.injective.eq_iff' top_toSubmodule\n\n@[simp]\ntheorem toNonUnitalSubsemiring_eq_top {S : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubsemiring_injective.eq_iff' top_toNonUnitalSubsemiring\n\n@[simp]\ntheorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubring_injective.eq_iff' top_toSubring\n\ntheorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_left\n\ntheorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_right\n\ntheorem mul_mem_sup {S T : NonUnitalSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) :\n    x * y ∈ S ⊔ T :=\n  mul_mem (mem_sup_left hx) (mem_sup_right hy)\n\ntheorem map_sup (f : F) (S T : NonUnitalSubalgebra R A) :\n    ((S ⊔ T).map f : NonUnitalSubalgebra R B) = S.map f ⊔ T.map f :=\n  (@NonUnitalSubalgebra.gc_map_comap F R A B _ _ _ _ _ _ f).l_sup\n\n@[simp, norm_cast]\ntheorem coe_inf (S T : NonUnitalSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T :=\n  rfl\n\n@[simp]\ntheorem mem_inf {S T : NonUnitalSubalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T :=\n  Iff.rfl\n\n@[simp]\ntheorem inf_toSubmodule (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toSubmodule = S.toSubmodule ⊓ T.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem inf_toNonUnitalSubsemiring (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring ⊓ T.toNonUnitalSubsemiring :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sInf (S : Set (NonUnitalSubalgebra R A)) : (↑(sInf S) : Set A) = ⋂ s ∈ S, ↑s :=\n  sInf_image\n\ntheorem mem_sInf {S : Set (NonUnitalSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p := by\n  simp only [← SetLike.mem_coe, coe_sInf, Set.mem_iInter₂]\n\n@[simp]\ntheorem sInf_toSubmodule (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toSubmodule = sInf (NonUnitalSubalgebra.toSubmodule '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp]\ntheorem sInf_toNonUnitalSubsemiring (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toNonUnitalSubsemiring = sInf (NonUnitalSubalgebra.toNonUnitalSubsemiring '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp, norm_cast]\ntheorem coe_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} :\n    (↑(⨅ i, S i) : Set A) = ⋂ i, S i := by simp [iInf]\n\ntheorem mem_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} {x : A} :\n    (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by simp only [iInf, mem_sInf, Set.forall_range_iff]\n\n@[simp]\ntheorem iInf_toSubmodule {ι : Sort*} (S : ι → NonUnitalSubalgebra R A) :\n    (⨅ i, S i).toSubmodule = ⨅ i, (S i).toSubmodule :=\n  SetLike.coe_injective <| by simp\n\ninstance : Inhabited (NonUnitalSubalgebra R A) :=\n  ⟨⊥⟩\n\ntheorem mem_bot {x : A} : x ∈ (⊥ : NonUnitalSubalgebra R A) ↔ x = 0 :=\n  show x ∈ Submodule.span R (NonUnitalSubsemiring.closure (∅ : Set A) : Set A) ↔ x = 0 by\n    rw [NonUnitalSubsemiring.closure_empty, NonUnitalSubsemiring.coe_bot,\n      Submodule.span_zero_singleton, Submodule.mem_bot]\n\ntheorem toSubmodule_bot : (⊥ : NonUnitalSubalgebra R A).toSubmodule = ⊥ := by\n  ext\n  simp only [mem_bot, NonUnitalSubalgebra.mem_toSubmodule, Submodule.mem_bot]\n\n@[simp]\ntheorem coe_bot : ((⊥ : NonUnitalSubalgebra R A) : Set A) = {0} := by\n  simp [Set.ext_iff, NonUnitalAlgebra.mem_bot]\n\ntheorem eq_top_iff {S : NonUnitalSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S :=\n  ⟨fun h x => by rw [h]; exact mem_top, fun h => by ext x; exact ⟨fun _ => mem_top, fun _ => h x⟩⟩\n\ntheorem range_top_iff_surjective (f : A →ₙₐ[R] B) :\n    NonUnitalAlgHom.range f = (⊤ : NonUnitalSubalgebra R B) ↔ Function.Surjective f :=\n  NonUnitalAlgebra.eq_top_iff\n\n@[simp]\ntheorem range_id : NonUnitalAlgHom.range (NonUnitalAlgHom.id R A) = ⊤ :=\n  SetLike.coe_injective Set.range_id\n\n@[simp]\ntheorem map_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R A).map f = NonUnitalAlgHom.range f :=\n  SetLike.coe_injective Set.image_univ\n\n@[simp]\ntheorem map_bot (f : A →ₙₐ[R] B) : (⊥ : NonUnitalSubalgebra R A).map f = ⊥ :=\n  SetLike.coe_injective <| by simp [NonUnitalAlgebra.coe_bot, NonUnitalSubalgebra.coe_map]\n\n@[simp]\ntheorem comap_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R B).comap f = ⊤ :=\n  eq_top_iff.2 fun _ => mem_top\n\n/-- `NonUnitalAlgHom` to `⊤ : NonUnitalSubalgebra R A`. -/\ndef toTop : A →ₙₐ[R] (⊤ : NonUnitalSubalgebra R A) :=\n  NonUnitalAlgHom.codRestrict (NonUnitalAlgHom.id R A) ⊤ fun _ => mem_top\n\nend NonUnitalAlgebra\n\nnamespace NonUnitalSubalgebra\n\nopen NonUnitalAlgebra\n\nsection NonAssoc\n\nvariable {R : Type u} {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable (S : NonUnitalSubalgebra R A)\n\ninstance subsingleton_of_subsingleton [Subsingleton A] : Subsingleton (NonUnitalSubalgebra R A) :=\n  ⟨fun B C => ext fun x => by simp only [Subsingleton.elim x 0, zero_mem B, zero_mem C]⟩\n\ninstance _root_.NonUnitalAlgHom.subsingleton [Subsingleton (NonUnitalSubalgebra R A)] :\n    Subsingleton (A →ₙₐ[R] B) :=\n  ⟨fun f g =>\n    NonUnitalAlgHom.ext fun a =>\n      have : a ∈ (⊥ : NonUnitalSubalgebra R A) :=\n        Subsingleton.elim (⊤ : NonUnitalSubalgebra R A) ⊥ ▸ mem_top\n      (mem_bot.mp this).symm ▸ (map_zero f).trans (map_zero g).symm⟩\n\ntheorem range_val : NonUnitalAlgHom.range (NonUnitalSubalgebraClass.subtype S) = S :=\n  ext <| Set.ext_iff.1 <| (NonUnitalSubalgebraClass.subtype S).coe_range.trans Subtype.range_val\n\n/-- The map `S → T` when `S` is a non-unital subalgebra contained in the non-unital subalgebra `T`.\n\nThis is the non-unital subalgebra version of `Submodule.inclusion`, or `Subring.inclusion`  -/\ndef inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) : S →ₙₐ[R] T\n    where\n  toFun := Set.inclusion h\n  map_add' _ _ := rfl\n  map_mul' _ _ := rfl\n  map_zero' := rfl\n  map_smul' _ _ := rfl\n\ntheorem inclusion_injective {S T : NonUnitalSubalgebra R A} (h : S ≤ T) :\n    Function.Injective (inclusion h) := fun _ _ => Subtype.ext ∘ Subtype.mk.inj\n\n@[simp]\ntheorem inclusion_self {S : NonUnitalSubalgebra R A} :\n    inclusion (le_refl S) = NonUnitalAlgHom.id R S :=\n  NonUnitalAlgHom.ext fun _ => Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_mk {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : A) (hx : x ∈ S) :\n    inclusion h ⟨x, hx⟩ = ⟨x, h hx⟩ :=\n  rfl\n\ntheorem inclusion_right {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : T) (m : (x : A) ∈ S) :\n    inclusion h ⟨x, m⟩ = x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_inclusion {S T U : NonUnitalSubalgebra R A} (hst : S ≤ T) (htu : T ≤ U) (x : S) :\n    inclusion htu (inclusion hst x) = inclusion (le_trans hst htu) x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem coe_inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (s : S) :\n    (inclusion h s : A) = s :=\n  rfl\n\nsection Prod\n\nvariable (S₁ : NonUnitalSubalgebra R B)\n\n/-- The product of two non-unital subalgebras is a non-unital subalgebra. -/\ndef prod : NonUnitalSubalgebra R (A × B) :=\n  { S.toNonUnitalSubsemiring.prod S₁.toNonUnitalSubsemiring with\n    carrier := S ×ˢ S₁\n    smul_mem' := fun r _x hx => ⟨SMulMemClass.smul_mem r hx.1, SMulMemClass.smul_mem r hx.2⟩ }\n\n@[simp]\ntheorem coe_prod : (prod S S₁ : Set (A × B)) = (S : Set A) ×ˢ S₁ :=\n  rfl\n\ntheorem prod_toSubmodule : (S.prod S₁).toSubmodule = S.toSubmodule.prod S₁.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem mem_prod {S : NonUnitalSubalgebra R A} {S₁ : NonUnitalSubalgebra R B} {x : A × B} :\n    x ∈ prod S S₁ ↔ x.1 ∈ S ∧ x.2 ∈ S₁ :=\n  Set.mem_prod\n\n@[simp]\ntheorem prod_top : (prod ⊤ ⊤ : NonUnitalSubalgebra R (A × B)) = ⊤ := by ext; simp\n\ntheorem prod_mono {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S ≤ T → S₁ ≤ T₁ → prod S S₁ ≤ prod T T₁ :=\n  Set.prod_mono\n\n@[simp]\ntheorem prod_inf_prod {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S.prod S₁ ⊓ T.prod T₁ = (S ⊓ T).prod (S₁ ⊓ T₁) :=\n  SetLike.coe_injective Set.prod_inter_prod\n\nend Prod\n\nsection SuprLift\n\nvariable {ι : Type*}\n\ntheorem coe_iSup_of_directed [Nonempty ι] {S : ι → NonUnitalSubalgebra R A}\n    (dir : Directed (· ≤ ·) S) : ↑(iSup S) = ⋃ i, (S i : Set A) :=\n  let K : NonUnitalSubalgebra R A :=\n    { __ := NonUnitalSubsemiring.copy _ _ (NonUnitalSubsemiring.coe_iSup_of_directed dir).symm\n      smul_mem' := fun r _x hx ↦\n        let ⟨i, hi⟩ := Set.mem_iUnion.1 hx\n        Set.mem_iUnion.2 ⟨i, (S i).smul_mem' r hi⟩ }\n  have : iSup S = K := le_antisymm\n    (iSup_le fun i ↦ le_iSup (fun i ↦ (S i : Set A)) i) (Set.iUnion_subset fun _ ↦ le_iSup S _)\n  this.symm ▸ rfl\n\n/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              rw [hf j k hjk]\n              rfl)\n            (↑(iSup K)) (by ","nextTactic":"rw [coe_iSup_of_directed dir]","declUpToTactic":"/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              rw [hf j k hjk]\n              rfl)\n            (↑(iSup K)) (by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.892_0.jQvdAcOcyVirh23","decl":"/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B "}
-{"state":"R : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\nS : NonUnitalSubalgebra R A\nι : Type u_1\ninst✝ : Nonempty ι\nK : ι → NonUnitalSubalgebra R A\ndir : Directed (fun x x_1 => x ≤ x_1) K\nf : (i : ι) → ↥(K i) →ₙₐ[R] B\nhf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalAlgHom.comp (f j) (inclusion h)\nr : R\n⊢ ∀ (x : ↥(iSup K)),\n    Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x)\n        (_ :\n          ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),\n            (fun i x => (f i) x) i { val := x, property := hxi } = (fun i x => (f i) x) j { val := x, property := hxj })\n        ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ i, ↑(K i)) (r • x) =\n      r •\n        Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x)\n          (_ :\n            ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),\n              (fun i x => (f i) x) i { val := x, property := hxi } =\n                (fun i x => (f i) x) j { val := x, property := hxj })\n          ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ i, ↑(K i)) x","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact NonUnitalAlgebra.gc\n\n@[simp]\ntheorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=\n  eq_top_iff.2 fun _x hx => subset_adjoin R hx\n\nvariable {R A}\n\n@[simp]\ntheorem coe_top : (↑(⊤ : NonUnitalSubalgebra R A) : Set A) = Set.univ :=\n  rfl\n\n@[simp]\ntheorem mem_top {x : A} : x ∈ (⊤ : NonUnitalSubalgebra R A) :=\n  Set.mem_univ x\n\n@[simp]\ntheorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toNonUnitalSubsemiring : (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubsemiring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toSubring {R A : Type*} [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    [IsScalarTower R A A] [SMulCommClass R A A] :\n    (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem toSubmodule_eq_top {S : NonUnitalSubalgebra R A} : S.toSubmodule = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toSubmodule'.injective.eq_iff' top_toSubmodule\n\n@[simp]\ntheorem toNonUnitalSubsemiring_eq_top {S : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubsemiring_injective.eq_iff' top_toNonUnitalSubsemiring\n\n@[simp]\ntheorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubring_injective.eq_iff' top_toSubring\n\ntheorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_left\n\ntheorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_right\n\ntheorem mul_mem_sup {S T : NonUnitalSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) :\n    x * y ∈ S ⊔ T :=\n  mul_mem (mem_sup_left hx) (mem_sup_right hy)\n\ntheorem map_sup (f : F) (S T : NonUnitalSubalgebra R A) :\n    ((S ⊔ T).map f : NonUnitalSubalgebra R B) = S.map f ⊔ T.map f :=\n  (@NonUnitalSubalgebra.gc_map_comap F R A B _ _ _ _ _ _ f).l_sup\n\n@[simp, norm_cast]\ntheorem coe_inf (S T : NonUnitalSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T :=\n  rfl\n\n@[simp]\ntheorem mem_inf {S T : NonUnitalSubalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T :=\n  Iff.rfl\n\n@[simp]\ntheorem inf_toSubmodule (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toSubmodule = S.toSubmodule ⊓ T.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem inf_toNonUnitalSubsemiring (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring ⊓ T.toNonUnitalSubsemiring :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sInf (S : Set (NonUnitalSubalgebra R A)) : (↑(sInf S) : Set A) = ⋂ s ∈ S, ↑s :=\n  sInf_image\n\ntheorem mem_sInf {S : Set (NonUnitalSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p := by\n  simp only [← SetLike.mem_coe, coe_sInf, Set.mem_iInter₂]\n\n@[simp]\ntheorem sInf_toSubmodule (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toSubmodule = sInf (NonUnitalSubalgebra.toSubmodule '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp]\ntheorem sInf_toNonUnitalSubsemiring (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toNonUnitalSubsemiring = sInf (NonUnitalSubalgebra.toNonUnitalSubsemiring '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp, norm_cast]\ntheorem coe_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} :\n    (↑(⨅ i, S i) : Set A) = ⋂ i, S i := by simp [iInf]\n\ntheorem mem_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} {x : A} :\n    (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by simp only [iInf, mem_sInf, Set.forall_range_iff]\n\n@[simp]\ntheorem iInf_toSubmodule {ι : Sort*} (S : ι → NonUnitalSubalgebra R A) :\n    (⨅ i, S i).toSubmodule = ⨅ i, (S i).toSubmodule :=\n  SetLike.coe_injective <| by simp\n\ninstance : Inhabited (NonUnitalSubalgebra R A) :=\n  ⟨⊥⟩\n\ntheorem mem_bot {x : A} : x ∈ (⊥ : NonUnitalSubalgebra R A) ↔ x = 0 :=\n  show x ∈ Submodule.span R (NonUnitalSubsemiring.closure (∅ : Set A) : Set A) ↔ x = 0 by\n    rw [NonUnitalSubsemiring.closure_empty, NonUnitalSubsemiring.coe_bot,\n      Submodule.span_zero_singleton, Submodule.mem_bot]\n\ntheorem toSubmodule_bot : (⊥ : NonUnitalSubalgebra R A).toSubmodule = ⊥ := by\n  ext\n  simp only [mem_bot, NonUnitalSubalgebra.mem_toSubmodule, Submodule.mem_bot]\n\n@[simp]\ntheorem coe_bot : ((⊥ : NonUnitalSubalgebra R A) : Set A) = {0} := by\n  simp [Set.ext_iff, NonUnitalAlgebra.mem_bot]\n\ntheorem eq_top_iff {S : NonUnitalSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S :=\n  ⟨fun h x => by rw [h]; exact mem_top, fun h => by ext x; exact ⟨fun _ => mem_top, fun _ => h x⟩⟩\n\ntheorem range_top_iff_surjective (f : A →ₙₐ[R] B) :\n    NonUnitalAlgHom.range f = (⊤ : NonUnitalSubalgebra R B) ↔ Function.Surjective f :=\n  NonUnitalAlgebra.eq_top_iff\n\n@[simp]\ntheorem range_id : NonUnitalAlgHom.range (NonUnitalAlgHom.id R A) = ⊤ :=\n  SetLike.coe_injective Set.range_id\n\n@[simp]\ntheorem map_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R A).map f = NonUnitalAlgHom.range f :=\n  SetLike.coe_injective Set.image_univ\n\n@[simp]\ntheorem map_bot (f : A →ₙₐ[R] B) : (⊥ : NonUnitalSubalgebra R A).map f = ⊥ :=\n  SetLike.coe_injective <| by simp [NonUnitalAlgebra.coe_bot, NonUnitalSubalgebra.coe_map]\n\n@[simp]\ntheorem comap_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R B).comap f = ⊤ :=\n  eq_top_iff.2 fun _ => mem_top\n\n/-- `NonUnitalAlgHom` to `⊤ : NonUnitalSubalgebra R A`. -/\ndef toTop : A →ₙₐ[R] (⊤ : NonUnitalSubalgebra R A) :=\n  NonUnitalAlgHom.codRestrict (NonUnitalAlgHom.id R A) ⊤ fun _ => mem_top\n\nend NonUnitalAlgebra\n\nnamespace NonUnitalSubalgebra\n\nopen NonUnitalAlgebra\n\nsection NonAssoc\n\nvariable {R : Type u} {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable (S : NonUnitalSubalgebra R A)\n\ninstance subsingleton_of_subsingleton [Subsingleton A] : Subsingleton (NonUnitalSubalgebra R A) :=\n  ⟨fun B C => ext fun x => by simp only [Subsingleton.elim x 0, zero_mem B, zero_mem C]⟩\n\ninstance _root_.NonUnitalAlgHom.subsingleton [Subsingleton (NonUnitalSubalgebra R A)] :\n    Subsingleton (A →ₙₐ[R] B) :=\n  ⟨fun f g =>\n    NonUnitalAlgHom.ext fun a =>\n      have : a ∈ (⊥ : NonUnitalSubalgebra R A) :=\n        Subsingleton.elim (⊤ : NonUnitalSubalgebra R A) ⊥ ▸ mem_top\n      (mem_bot.mp this).symm ▸ (map_zero f).trans (map_zero g).symm⟩\n\ntheorem range_val : NonUnitalAlgHom.range (NonUnitalSubalgebraClass.subtype S) = S :=\n  ext <| Set.ext_iff.1 <| (NonUnitalSubalgebraClass.subtype S).coe_range.trans Subtype.range_val\n\n/-- The map `S → T` when `S` is a non-unital subalgebra contained in the non-unital subalgebra `T`.\n\nThis is the non-unital subalgebra version of `Submodule.inclusion`, or `Subring.inclusion`  -/\ndef inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) : S →ₙₐ[R] T\n    where\n  toFun := Set.inclusion h\n  map_add' _ _ := rfl\n  map_mul' _ _ := rfl\n  map_zero' := rfl\n  map_smul' _ _ := rfl\n\ntheorem inclusion_injective {S T : NonUnitalSubalgebra R A} (h : S ≤ T) :\n    Function.Injective (inclusion h) := fun _ _ => Subtype.ext ∘ Subtype.mk.inj\n\n@[simp]\ntheorem inclusion_self {S : NonUnitalSubalgebra R A} :\n    inclusion (le_refl S) = NonUnitalAlgHom.id R S :=\n  NonUnitalAlgHom.ext fun _ => Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_mk {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : A) (hx : x ∈ S) :\n    inclusion h ⟨x, hx⟩ = ⟨x, h hx⟩ :=\n  rfl\n\ntheorem inclusion_right {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : T) (m : (x : A) ∈ S) :\n    inclusion h ⟨x, m⟩ = x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_inclusion {S T U : NonUnitalSubalgebra R A} (hst : S ≤ T) (htu : T ≤ U) (x : S) :\n    inclusion htu (inclusion hst x) = inclusion (le_trans hst htu) x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem coe_inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (s : S) :\n    (inclusion h s : A) = s :=\n  rfl\n\nsection Prod\n\nvariable (S₁ : NonUnitalSubalgebra R B)\n\n/-- The product of two non-unital subalgebras is a non-unital subalgebra. -/\ndef prod : NonUnitalSubalgebra R (A × B) :=\n  { S.toNonUnitalSubsemiring.prod S₁.toNonUnitalSubsemiring with\n    carrier := S ×ˢ S₁\n    smul_mem' := fun r _x hx => ⟨SMulMemClass.smul_mem r hx.1, SMulMemClass.smul_mem r hx.2⟩ }\n\n@[simp]\ntheorem coe_prod : (prod S S₁ : Set (A × B)) = (S : Set A) ×ˢ S₁ :=\n  rfl\n\ntheorem prod_toSubmodule : (S.prod S₁).toSubmodule = S.toSubmodule.prod S₁.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem mem_prod {S : NonUnitalSubalgebra R A} {S₁ : NonUnitalSubalgebra R B} {x : A × B} :\n    x ∈ prod S S₁ ↔ x.1 ∈ S ∧ x.2 ∈ S₁ :=\n  Set.mem_prod\n\n@[simp]\ntheorem prod_top : (prod ⊤ ⊤ : NonUnitalSubalgebra R (A × B)) = ⊤ := by ext; simp\n\ntheorem prod_mono {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S ≤ T → S₁ ≤ T₁ → prod S S₁ ≤ prod T T₁ :=\n  Set.prod_mono\n\n@[simp]\ntheorem prod_inf_prod {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S.prod S₁ ⊓ T.prod T₁ = (S ⊓ T).prod (S₁ ⊓ T₁) :=\n  SetLike.coe_injective Set.prod_inter_prod\n\nend Prod\n\nsection SuprLift\n\nvariable {ι : Type*}\n\ntheorem coe_iSup_of_directed [Nonempty ι] {S : ι → NonUnitalSubalgebra R A}\n    (dir : Directed (· ≤ ·) S) : ↑(iSup S) = ⋃ i, (S i : Set A) :=\n  let K : NonUnitalSubalgebra R A :=\n    { __ := NonUnitalSubsemiring.copy _ _ (NonUnitalSubsemiring.coe_iSup_of_directed dir).symm\n      smul_mem' := fun r _x hx ↦\n        let ⟨i, hi⟩ := Set.mem_iUnion.1 hx\n        Set.mem_iUnion.2 ⟨i, (S i).smul_mem' r hi⟩ }\n  have : iSup S = K := le_antisymm\n    (iSup_le fun i ↦ le_iSup (fun i ↦ (S i : Set A)) i) (Set.iUnion_subset fun _ ↦ le_iSup S _)\n  this.symm ▸ rfl\n\n/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              rw [hf j k hjk]\n              rfl)\n            (↑(iSup K)) (by rw [coe_iSup_of_directed dir])\n        map_zero' := by\n          dsimp\n          exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl) _ (by simp)\n        map_mul' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· * ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_add' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· + ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_smul' := fun r => by\n          ","nextTactic":"dsimp","declUpToTactic":"/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              rw [hf j k hjk]\n              rfl)\n            (↑(iSup K)) (by rw [coe_iSup_of_directed dir])\n        map_zero' := by\n          dsimp\n          exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl) _ (by simp)\n        map_mul' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· * ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_add' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· + ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_smul' := fun r => by\n          ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.892_0.jQvdAcOcyVirh23","decl":"/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B "}
-{"state":"R : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\nS : NonUnitalSubalgebra R A\nι : Type u_1\ninst✝ : Nonempty ι\nK : ι → NonUnitalSubalgebra R A\ndir : Directed (fun x x_1 => x ≤ x_1) K\nf : (i : ι) → ↥(K i) →ₙₐ[R] B\nhf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalAlgHom.comp (f j) (inclusion h)\nr : R\n⊢ ∀ (x : ↥(iSup K)),\n    Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x)\n        (_ :\n          ∀ (i j : ι) (x : A) (hxi : x ∈ ↑(K i)) (hxj : x ∈ ↑(K j)),\n            (f i) { val := x, property := hxi } = (f j) { val := x, property := hxj })\n        ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ i, ↑(K i)) (r • x) =\n      r •\n        Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x)\n          (_ :\n            ∀ (i j : ι) (x : A) (hxi : x ∈ ↑(K i)) (hxj : x ∈ ↑(K j)),\n              (f i) { val := x, property := hxi } = (f j) { val := x, property := hxj })\n          ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ i, ↑(K i)) x","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact NonUnitalAlgebra.gc\n\n@[simp]\ntheorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=\n  eq_top_iff.2 fun _x hx => subset_adjoin R hx\n\nvariable {R A}\n\n@[simp]\ntheorem coe_top : (↑(⊤ : NonUnitalSubalgebra R A) : Set A) = Set.univ :=\n  rfl\n\n@[simp]\ntheorem mem_top {x : A} : x ∈ (⊤ : NonUnitalSubalgebra R A) :=\n  Set.mem_univ x\n\n@[simp]\ntheorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toNonUnitalSubsemiring : (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubsemiring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toSubring {R A : Type*} [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    [IsScalarTower R A A] [SMulCommClass R A A] :\n    (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem toSubmodule_eq_top {S : NonUnitalSubalgebra R A} : S.toSubmodule = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toSubmodule'.injective.eq_iff' top_toSubmodule\n\n@[simp]\ntheorem toNonUnitalSubsemiring_eq_top {S : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubsemiring_injective.eq_iff' top_toNonUnitalSubsemiring\n\n@[simp]\ntheorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubring_injective.eq_iff' top_toSubring\n\ntheorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_left\n\ntheorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_right\n\ntheorem mul_mem_sup {S T : NonUnitalSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) :\n    x * y ∈ S ⊔ T :=\n  mul_mem (mem_sup_left hx) (mem_sup_right hy)\n\ntheorem map_sup (f : F) (S T : NonUnitalSubalgebra R A) :\n    ((S ⊔ T).map f : NonUnitalSubalgebra R B) = S.map f ⊔ T.map f :=\n  (@NonUnitalSubalgebra.gc_map_comap F R A B _ _ _ _ _ _ f).l_sup\n\n@[simp, norm_cast]\ntheorem coe_inf (S T : NonUnitalSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T :=\n  rfl\n\n@[simp]\ntheorem mem_inf {S T : NonUnitalSubalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T :=\n  Iff.rfl\n\n@[simp]\ntheorem inf_toSubmodule (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toSubmodule = S.toSubmodule ⊓ T.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem inf_toNonUnitalSubsemiring (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring ⊓ T.toNonUnitalSubsemiring :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sInf (S : Set (NonUnitalSubalgebra R A)) : (↑(sInf S) : Set A) = ⋂ s ∈ S, ↑s :=\n  sInf_image\n\ntheorem mem_sInf {S : Set (NonUnitalSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p := by\n  simp only [← SetLike.mem_coe, coe_sInf, Set.mem_iInter₂]\n\n@[simp]\ntheorem sInf_toSubmodule (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toSubmodule = sInf (NonUnitalSubalgebra.toSubmodule '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp]\ntheorem sInf_toNonUnitalSubsemiring (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toNonUnitalSubsemiring = sInf (NonUnitalSubalgebra.toNonUnitalSubsemiring '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp, norm_cast]\ntheorem coe_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} :\n    (↑(⨅ i, S i) : Set A) = ⋂ i, S i := by simp [iInf]\n\ntheorem mem_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} {x : A} :\n    (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by simp only [iInf, mem_sInf, Set.forall_range_iff]\n\n@[simp]\ntheorem iInf_toSubmodule {ι : Sort*} (S : ι → NonUnitalSubalgebra R A) :\n    (⨅ i, S i).toSubmodule = ⨅ i, (S i).toSubmodule :=\n  SetLike.coe_injective <| by simp\n\ninstance : Inhabited (NonUnitalSubalgebra R A) :=\n  ⟨⊥⟩\n\ntheorem mem_bot {x : A} : x ∈ (⊥ : NonUnitalSubalgebra R A) ↔ x = 0 :=\n  show x ∈ Submodule.span R (NonUnitalSubsemiring.closure (∅ : Set A) : Set A) ↔ x = 0 by\n    rw [NonUnitalSubsemiring.closure_empty, NonUnitalSubsemiring.coe_bot,\n      Submodule.span_zero_singleton, Submodule.mem_bot]\n\ntheorem toSubmodule_bot : (⊥ : NonUnitalSubalgebra R A).toSubmodule = ⊥ := by\n  ext\n  simp only [mem_bot, NonUnitalSubalgebra.mem_toSubmodule, Submodule.mem_bot]\n\n@[simp]\ntheorem coe_bot : ((⊥ : NonUnitalSubalgebra R A) : Set A) = {0} := by\n  simp [Set.ext_iff, NonUnitalAlgebra.mem_bot]\n\ntheorem eq_top_iff {S : NonUnitalSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S :=\n  ⟨fun h x => by rw [h]; exact mem_top, fun h => by ext x; exact ⟨fun _ => mem_top, fun _ => h x⟩⟩\n\ntheorem range_top_iff_surjective (f : A →ₙₐ[R] B) :\n    NonUnitalAlgHom.range f = (⊤ : NonUnitalSubalgebra R B) ↔ Function.Surjective f :=\n  NonUnitalAlgebra.eq_top_iff\n\n@[simp]\ntheorem range_id : NonUnitalAlgHom.range (NonUnitalAlgHom.id R A) = ⊤ :=\n  SetLike.coe_injective Set.range_id\n\n@[simp]\ntheorem map_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R A).map f = NonUnitalAlgHom.range f :=\n  SetLike.coe_injective Set.image_univ\n\n@[simp]\ntheorem map_bot (f : A →ₙₐ[R] B) : (⊥ : NonUnitalSubalgebra R A).map f = ⊥ :=\n  SetLike.coe_injective <| by simp [NonUnitalAlgebra.coe_bot, NonUnitalSubalgebra.coe_map]\n\n@[simp]\ntheorem comap_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R B).comap f = ⊤ :=\n  eq_top_iff.2 fun _ => mem_top\n\n/-- `NonUnitalAlgHom` to `⊤ : NonUnitalSubalgebra R A`. -/\ndef toTop : A →ₙₐ[R] (⊤ : NonUnitalSubalgebra R A) :=\n  NonUnitalAlgHom.codRestrict (NonUnitalAlgHom.id R A) ⊤ fun _ => mem_top\n\nend NonUnitalAlgebra\n\nnamespace NonUnitalSubalgebra\n\nopen NonUnitalAlgebra\n\nsection NonAssoc\n\nvariable {R : Type u} {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable (S : NonUnitalSubalgebra R A)\n\ninstance subsingleton_of_subsingleton [Subsingleton A] : Subsingleton (NonUnitalSubalgebra R A) :=\n  ⟨fun B C => ext fun x => by simp only [Subsingleton.elim x 0, zero_mem B, zero_mem C]⟩\n\ninstance _root_.NonUnitalAlgHom.subsingleton [Subsingleton (NonUnitalSubalgebra R A)] :\n    Subsingleton (A →ₙₐ[R] B) :=\n  ⟨fun f g =>\n    NonUnitalAlgHom.ext fun a =>\n      have : a ∈ (⊥ : NonUnitalSubalgebra R A) :=\n        Subsingleton.elim (⊤ : NonUnitalSubalgebra R A) ⊥ ▸ mem_top\n      (mem_bot.mp this).symm ▸ (map_zero f).trans (map_zero g).symm⟩\n\ntheorem range_val : NonUnitalAlgHom.range (NonUnitalSubalgebraClass.subtype S) = S :=\n  ext <| Set.ext_iff.1 <| (NonUnitalSubalgebraClass.subtype S).coe_range.trans Subtype.range_val\n\n/-- The map `S → T` when `S` is a non-unital subalgebra contained in the non-unital subalgebra `T`.\n\nThis is the non-unital subalgebra version of `Submodule.inclusion`, or `Subring.inclusion`  -/\ndef inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) : S →ₙₐ[R] T\n    where\n  toFun := Set.inclusion h\n  map_add' _ _ := rfl\n  map_mul' _ _ := rfl\n  map_zero' := rfl\n  map_smul' _ _ := rfl\n\ntheorem inclusion_injective {S T : NonUnitalSubalgebra R A} (h : S ≤ T) :\n    Function.Injective (inclusion h) := fun _ _ => Subtype.ext ∘ Subtype.mk.inj\n\n@[simp]\ntheorem inclusion_self {S : NonUnitalSubalgebra R A} :\n    inclusion (le_refl S) = NonUnitalAlgHom.id R S :=\n  NonUnitalAlgHom.ext fun _ => Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_mk {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : A) (hx : x ∈ S) :\n    inclusion h ⟨x, hx⟩ = ⟨x, h hx⟩ :=\n  rfl\n\ntheorem inclusion_right {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : T) (m : (x : A) ∈ S) :\n    inclusion h ⟨x, m⟩ = x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_inclusion {S T U : NonUnitalSubalgebra R A} (hst : S ≤ T) (htu : T ≤ U) (x : S) :\n    inclusion htu (inclusion hst x) = inclusion (le_trans hst htu) x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem coe_inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (s : S) :\n    (inclusion h s : A) = s :=\n  rfl\n\nsection Prod\n\nvariable (S₁ : NonUnitalSubalgebra R B)\n\n/-- The product of two non-unital subalgebras is a non-unital subalgebra. -/\ndef prod : NonUnitalSubalgebra R (A × B) :=\n  { S.toNonUnitalSubsemiring.prod S₁.toNonUnitalSubsemiring with\n    carrier := S ×ˢ S₁\n    smul_mem' := fun r _x hx => ⟨SMulMemClass.smul_mem r hx.1, SMulMemClass.smul_mem r hx.2⟩ }\n\n@[simp]\ntheorem coe_prod : (prod S S₁ : Set (A × B)) = (S : Set A) ×ˢ S₁ :=\n  rfl\n\ntheorem prod_toSubmodule : (S.prod S₁).toSubmodule = S.toSubmodule.prod S₁.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem mem_prod {S : NonUnitalSubalgebra R A} {S₁ : NonUnitalSubalgebra R B} {x : A × B} :\n    x ∈ prod S S₁ ↔ x.1 ∈ S ∧ x.2 ∈ S₁ :=\n  Set.mem_prod\n\n@[simp]\ntheorem prod_top : (prod ⊤ ⊤ : NonUnitalSubalgebra R (A × B)) = ⊤ := by ext; simp\n\ntheorem prod_mono {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S ≤ T → S₁ ≤ T₁ → prod S S₁ ≤ prod T T₁ :=\n  Set.prod_mono\n\n@[simp]\ntheorem prod_inf_prod {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S.prod S₁ ⊓ T.prod T₁ = (S ⊓ T).prod (S₁ ⊓ T₁) :=\n  SetLike.coe_injective Set.prod_inter_prod\n\nend Prod\n\nsection SuprLift\n\nvariable {ι : Type*}\n\ntheorem coe_iSup_of_directed [Nonempty ι] {S : ι → NonUnitalSubalgebra R A}\n    (dir : Directed (· ≤ ·) S) : ↑(iSup S) = ⋃ i, (S i : Set A) :=\n  let K : NonUnitalSubalgebra R A :=\n    { __ := NonUnitalSubsemiring.copy _ _ (NonUnitalSubsemiring.coe_iSup_of_directed dir).symm\n      smul_mem' := fun r _x hx ↦\n        let ⟨i, hi⟩ := Set.mem_iUnion.1 hx\n        Set.mem_iUnion.2 ⟨i, (S i).smul_mem' r hi⟩ }\n  have : iSup S = K := le_antisymm\n    (iSup_le fun i ↦ le_iSup (fun i ↦ (S i : Set A)) i) (Set.iUnion_subset fun _ ↦ le_iSup S _)\n  this.symm ▸ rfl\n\n/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              rw [hf j k hjk]\n              rfl)\n            (↑(iSup K)) (by rw [coe_iSup_of_directed dir])\n        map_zero' := by\n          dsimp\n          exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl) _ (by simp)\n        map_mul' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· * ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_add' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· + ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_smul' := fun r => by\n          dsimp\n          ","nextTactic":"apply Set.iUnionLift_unary (coe_iSup_of_directed dir) _ (fun _ x => r • x)\n            (fun _ _ => rfl)","declUpToTactic":"/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              rw [hf j k hjk]\n              rfl)\n            (↑(iSup K)) (by rw [coe_iSup_of_directed dir])\n        map_zero' := by\n          dsimp\n          exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl) _ (by simp)\n        map_mul' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· * ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_add' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· + ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_smul' := fun r => by\n          dsimp\n          ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.892_0.jQvdAcOcyVirh23","decl":"/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B "}
-{"state":"case h\nR : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\nS : NonUnitalSubalgebra R A\nι : Type u_1\ninst✝ : Nonempty ι\nK : ι → NonUnitalSubalgebra R A\ndir : Directed (fun x x_1 => x ≤ x_1) K\nf : (i : ι) → ↥(K i) →ₙₐ[R] B\nhf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalAlgHom.comp (f j) (inclusion h)\nr : R\n⊢ ∀ (i : ι) (x : ↑↑(K i)), (f i) (r • x) = r • (f i) x\nR : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\nS : NonUnitalSubalgebra R A\nι : Type u_1\ninst✝ : Nonempty ι\nK : ι → NonUnitalSubalgebra R A\ndir : Directed (fun x x_1 => x ≤ x_1) K\nf : (i : ι) → ↥(K i) →ₙₐ[R] B\nhf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalAlgHom.comp (f j) (inclusion h)\nr : R\n⊢ ↑(iSup K) ⊆ ⋃ i, ↑(K i)","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact NonUnitalAlgebra.gc\n\n@[simp]\ntheorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=\n  eq_top_iff.2 fun _x hx => subset_adjoin R hx\n\nvariable {R A}\n\n@[simp]\ntheorem coe_top : (↑(⊤ : NonUnitalSubalgebra R A) : Set A) = Set.univ :=\n  rfl\n\n@[simp]\ntheorem mem_top {x : A} : x ∈ (⊤ : NonUnitalSubalgebra R A) :=\n  Set.mem_univ x\n\n@[simp]\ntheorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toNonUnitalSubsemiring : (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubsemiring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toSubring {R A : Type*} [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    [IsScalarTower R A A] [SMulCommClass R A A] :\n    (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem toSubmodule_eq_top {S : NonUnitalSubalgebra R A} : S.toSubmodule = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toSubmodule'.injective.eq_iff' top_toSubmodule\n\n@[simp]\ntheorem toNonUnitalSubsemiring_eq_top {S : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubsemiring_injective.eq_iff' top_toNonUnitalSubsemiring\n\n@[simp]\ntheorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubring_injective.eq_iff' top_toSubring\n\ntheorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_left\n\ntheorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_right\n\ntheorem mul_mem_sup {S T : NonUnitalSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) :\n    x * y ∈ S ⊔ T :=\n  mul_mem (mem_sup_left hx) (mem_sup_right hy)\n\ntheorem map_sup (f : F) (S T : NonUnitalSubalgebra R A) :\n    ((S ⊔ T).map f : NonUnitalSubalgebra R B) = S.map f ⊔ T.map f :=\n  (@NonUnitalSubalgebra.gc_map_comap F R A B _ _ _ _ _ _ f).l_sup\n\n@[simp, norm_cast]\ntheorem coe_inf (S T : NonUnitalSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T :=\n  rfl\n\n@[simp]\ntheorem mem_inf {S T : NonUnitalSubalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T :=\n  Iff.rfl\n\n@[simp]\ntheorem inf_toSubmodule (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toSubmodule = S.toSubmodule ⊓ T.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem inf_toNonUnitalSubsemiring (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring ⊓ T.toNonUnitalSubsemiring :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sInf (S : Set (NonUnitalSubalgebra R A)) : (↑(sInf S) : Set A) = ⋂ s ∈ S, ↑s :=\n  sInf_image\n\ntheorem mem_sInf {S : Set (NonUnitalSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p := by\n  simp only [← SetLike.mem_coe, coe_sInf, Set.mem_iInter₂]\n\n@[simp]\ntheorem sInf_toSubmodule (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toSubmodule = sInf (NonUnitalSubalgebra.toSubmodule '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp]\ntheorem sInf_toNonUnitalSubsemiring (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toNonUnitalSubsemiring = sInf (NonUnitalSubalgebra.toNonUnitalSubsemiring '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp, norm_cast]\ntheorem coe_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} :\n    (↑(⨅ i, S i) : Set A) = ⋂ i, S i := by simp [iInf]\n\ntheorem mem_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} {x : A} :\n    (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by simp only [iInf, mem_sInf, Set.forall_range_iff]\n\n@[simp]\ntheorem iInf_toSubmodule {ι : Sort*} (S : ι → NonUnitalSubalgebra R A) :\n    (⨅ i, S i).toSubmodule = ⨅ i, (S i).toSubmodule :=\n  SetLike.coe_injective <| by simp\n\ninstance : Inhabited (NonUnitalSubalgebra R A) :=\n  ⟨⊥⟩\n\ntheorem mem_bot {x : A} : x ∈ (⊥ : NonUnitalSubalgebra R A) ↔ x = 0 :=\n  show x ∈ Submodule.span R (NonUnitalSubsemiring.closure (∅ : Set A) : Set A) ↔ x = 0 by\n    rw [NonUnitalSubsemiring.closure_empty, NonUnitalSubsemiring.coe_bot,\n      Submodule.span_zero_singleton, Submodule.mem_bot]\n\ntheorem toSubmodule_bot : (⊥ : NonUnitalSubalgebra R A).toSubmodule = ⊥ := by\n  ext\n  simp only [mem_bot, NonUnitalSubalgebra.mem_toSubmodule, Submodule.mem_bot]\n\n@[simp]\ntheorem coe_bot : ((⊥ : NonUnitalSubalgebra R A) : Set A) = {0} := by\n  simp [Set.ext_iff, NonUnitalAlgebra.mem_bot]\n\ntheorem eq_top_iff {S : NonUnitalSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S :=\n  ⟨fun h x => by rw [h]; exact mem_top, fun h => by ext x; exact ⟨fun _ => mem_top, fun _ => h x⟩⟩\n\ntheorem range_top_iff_surjective (f : A →ₙₐ[R] B) :\n    NonUnitalAlgHom.range f = (⊤ : NonUnitalSubalgebra R B) ↔ Function.Surjective f :=\n  NonUnitalAlgebra.eq_top_iff\n\n@[simp]\ntheorem range_id : NonUnitalAlgHom.range (NonUnitalAlgHom.id R A) = ⊤ :=\n  SetLike.coe_injective Set.range_id\n\n@[simp]\ntheorem map_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R A).map f = NonUnitalAlgHom.range f :=\n  SetLike.coe_injective Set.image_univ\n\n@[simp]\ntheorem map_bot (f : A →ₙₐ[R] B) : (⊥ : NonUnitalSubalgebra R A).map f = ⊥ :=\n  SetLike.coe_injective <| by simp [NonUnitalAlgebra.coe_bot, NonUnitalSubalgebra.coe_map]\n\n@[simp]\ntheorem comap_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R B).comap f = ⊤ :=\n  eq_top_iff.2 fun _ => mem_top\n\n/-- `NonUnitalAlgHom` to `⊤ : NonUnitalSubalgebra R A`. -/\ndef toTop : A →ₙₐ[R] (⊤ : NonUnitalSubalgebra R A) :=\n  NonUnitalAlgHom.codRestrict (NonUnitalAlgHom.id R A) ⊤ fun _ => mem_top\n\nend NonUnitalAlgebra\n\nnamespace NonUnitalSubalgebra\n\nopen NonUnitalAlgebra\n\nsection NonAssoc\n\nvariable {R : Type u} {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable (S : NonUnitalSubalgebra R A)\n\ninstance subsingleton_of_subsingleton [Subsingleton A] : Subsingleton (NonUnitalSubalgebra R A) :=\n  ⟨fun B C => ext fun x => by simp only [Subsingleton.elim x 0, zero_mem B, zero_mem C]⟩\n\ninstance _root_.NonUnitalAlgHom.subsingleton [Subsingleton (NonUnitalSubalgebra R A)] :\n    Subsingleton (A →ₙₐ[R] B) :=\n  ⟨fun f g =>\n    NonUnitalAlgHom.ext fun a =>\n      have : a ∈ (⊥ : NonUnitalSubalgebra R A) :=\n        Subsingleton.elim (⊤ : NonUnitalSubalgebra R A) ⊥ ▸ mem_top\n      (mem_bot.mp this).symm ▸ (map_zero f).trans (map_zero g).symm⟩\n\ntheorem range_val : NonUnitalAlgHom.range (NonUnitalSubalgebraClass.subtype S) = S :=\n  ext <| Set.ext_iff.1 <| (NonUnitalSubalgebraClass.subtype S).coe_range.trans Subtype.range_val\n\n/-- The map `S → T` when `S` is a non-unital subalgebra contained in the non-unital subalgebra `T`.\n\nThis is the non-unital subalgebra version of `Submodule.inclusion`, or `Subring.inclusion`  -/\ndef inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) : S →ₙₐ[R] T\n    where\n  toFun := Set.inclusion h\n  map_add' _ _ := rfl\n  map_mul' _ _ := rfl\n  map_zero' := rfl\n  map_smul' _ _ := rfl\n\ntheorem inclusion_injective {S T : NonUnitalSubalgebra R A} (h : S ≤ T) :\n    Function.Injective (inclusion h) := fun _ _ => Subtype.ext ∘ Subtype.mk.inj\n\n@[simp]\ntheorem inclusion_self {S : NonUnitalSubalgebra R A} :\n    inclusion (le_refl S) = NonUnitalAlgHom.id R S :=\n  NonUnitalAlgHom.ext fun _ => Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_mk {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : A) (hx : x ∈ S) :\n    inclusion h ⟨x, hx⟩ = ⟨x, h hx⟩ :=\n  rfl\n\ntheorem inclusion_right {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : T) (m : (x : A) ∈ S) :\n    inclusion h ⟨x, m⟩ = x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_inclusion {S T U : NonUnitalSubalgebra R A} (hst : S ≤ T) (htu : T ≤ U) (x : S) :\n    inclusion htu (inclusion hst x) = inclusion (le_trans hst htu) x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem coe_inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (s : S) :\n    (inclusion h s : A) = s :=\n  rfl\n\nsection Prod\n\nvariable (S₁ : NonUnitalSubalgebra R B)\n\n/-- The product of two non-unital subalgebras is a non-unital subalgebra. -/\ndef prod : NonUnitalSubalgebra R (A × B) :=\n  { S.toNonUnitalSubsemiring.prod S₁.toNonUnitalSubsemiring with\n    carrier := S ×ˢ S₁\n    smul_mem' := fun r _x hx => ⟨SMulMemClass.smul_mem r hx.1, SMulMemClass.smul_mem r hx.2⟩ }\n\n@[simp]\ntheorem coe_prod : (prod S S₁ : Set (A × B)) = (S : Set A) ×ˢ S₁ :=\n  rfl\n\ntheorem prod_toSubmodule : (S.prod S₁).toSubmodule = S.toSubmodule.prod S₁.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem mem_prod {S : NonUnitalSubalgebra R A} {S₁ : NonUnitalSubalgebra R B} {x : A × B} :\n    x ∈ prod S S₁ ↔ x.1 ∈ S ∧ x.2 ∈ S₁ :=\n  Set.mem_prod\n\n@[simp]\ntheorem prod_top : (prod ⊤ ⊤ : NonUnitalSubalgebra R (A × B)) = ⊤ := by ext; simp\n\ntheorem prod_mono {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S ≤ T → S₁ ≤ T₁ → prod S S₁ ≤ prod T T₁ :=\n  Set.prod_mono\n\n@[simp]\ntheorem prod_inf_prod {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S.prod S₁ ⊓ T.prod T₁ = (S ⊓ T).prod (S₁ ⊓ T₁) :=\n  SetLike.coe_injective Set.prod_inter_prod\n\nend Prod\n\nsection SuprLift\n\nvariable {ι : Type*}\n\ntheorem coe_iSup_of_directed [Nonempty ι] {S : ι → NonUnitalSubalgebra R A}\n    (dir : Directed (· ≤ ·) S) : ↑(iSup S) = ⋃ i, (S i : Set A) :=\n  let K : NonUnitalSubalgebra R A :=\n    { __ := NonUnitalSubsemiring.copy _ _ (NonUnitalSubsemiring.coe_iSup_of_directed dir).symm\n      smul_mem' := fun r _x hx ↦\n        let ⟨i, hi⟩ := Set.mem_iUnion.1 hx\n        Set.mem_iUnion.2 ⟨i, (S i).smul_mem' r hi⟩ }\n  have : iSup S = K := le_antisymm\n    (iSup_le fun i ↦ le_iSup (fun i ↦ (S i : Set A)) i) (Set.iUnion_subset fun _ ↦ le_iSup S _)\n  this.symm ▸ rfl\n\n/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              rw [hf j k hjk]\n              rfl)\n            (↑(iSup K)) (by rw [coe_iSup_of_directed dir])\n        map_zero' := by\n          dsimp\n          exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl) _ (by simp)\n        map_mul' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· * ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_add' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· + ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_smul' := fun r => by\n          dsimp\n          apply Set.iUnionLift_unary (coe_iSup_of_directed dir) _ (fun _ x => r • x)\n            (fun _ _ => rfl)\n          ","nextTactic":"on_goal 2 => rw [coe_iSup_of_directed dir]","declUpToTactic":"/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              rw [hf j k hjk]\n              rfl)\n            (↑(iSup K)) (by rw [coe_iSup_of_directed dir])\n        map_zero' := by\n          dsimp\n          exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl) _ (by simp)\n        map_mul' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· * ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_add' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· + ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_smul' := fun r => by\n          dsimp\n          apply Set.iUnionLift_unary (coe_iSup_of_directed dir) _ (fun _ x => r • x)\n            (fun _ _ => rfl)\n          ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.892_0.jQvdAcOcyVirh23","decl":"/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B "}
-{"state":"case h\nR : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\nS : NonUnitalSubalgebra R A\nι : Type u_1\ninst✝ : Nonempty ι\nK : ι → NonUnitalSubalgebra R A\ndir : Directed (fun x x_1 => x ≤ x_1) K\nf : (i : ι) → ↥(K i) →ₙₐ[R] B\nhf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalAlgHom.comp (f j) (inclusion h)\nr : R\n⊢ ∀ (i : ι) (x : ↑↑(K i)), (f i) (r • x) = r • (f i) x\nR : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\nS : NonUnitalSubalgebra R A\nι : Type u_1\ninst✝ : Nonempty ι\nK : ι → NonUnitalSubalgebra R A\ndir : Directed (fun x x_1 => x ≤ x_1) K\nf : (i : ι) → ↥(K i) →ₙₐ[R] B\nhf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalAlgHom.comp (f j) (inclusion h)\nr : R\n⊢ ↑(iSup K) ⊆ ⋃ i, ↑(K i)","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact NonUnitalAlgebra.gc\n\n@[simp]\ntheorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=\n  eq_top_iff.2 fun _x hx => subset_adjoin R hx\n\nvariable {R A}\n\n@[simp]\ntheorem coe_top : (↑(⊤ : NonUnitalSubalgebra R A) : Set A) = Set.univ :=\n  rfl\n\n@[simp]\ntheorem mem_top {x : A} : x ∈ (⊤ : NonUnitalSubalgebra R A) :=\n  Set.mem_univ x\n\n@[simp]\ntheorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toNonUnitalSubsemiring : (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubsemiring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toSubring {R A : Type*} [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    [IsScalarTower R A A] [SMulCommClass R A A] :\n    (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem toSubmodule_eq_top {S : NonUnitalSubalgebra R A} : S.toSubmodule = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toSubmodule'.injective.eq_iff' top_toSubmodule\n\n@[simp]\ntheorem toNonUnitalSubsemiring_eq_top {S : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubsemiring_injective.eq_iff' top_toNonUnitalSubsemiring\n\n@[simp]\ntheorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubring_injective.eq_iff' top_toSubring\n\ntheorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_left\n\ntheorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_right\n\ntheorem mul_mem_sup {S T : NonUnitalSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) :\n    x * y ∈ S ⊔ T :=\n  mul_mem (mem_sup_left hx) (mem_sup_right hy)\n\ntheorem map_sup (f : F) (S T : NonUnitalSubalgebra R A) :\n    ((S ⊔ T).map f : NonUnitalSubalgebra R B) = S.map f ⊔ T.map f :=\n  (@NonUnitalSubalgebra.gc_map_comap F R A B _ _ _ _ _ _ f).l_sup\n\n@[simp, norm_cast]\ntheorem coe_inf (S T : NonUnitalSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T :=\n  rfl\n\n@[simp]\ntheorem mem_inf {S T : NonUnitalSubalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T :=\n  Iff.rfl\n\n@[simp]\ntheorem inf_toSubmodule (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toSubmodule = S.toSubmodule ⊓ T.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem inf_toNonUnitalSubsemiring (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring ⊓ T.toNonUnitalSubsemiring :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sInf (S : Set (NonUnitalSubalgebra R A)) : (↑(sInf S) : Set A) = ⋂ s ∈ S, ↑s :=\n  sInf_image\n\ntheorem mem_sInf {S : Set (NonUnitalSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p := by\n  simp only [← SetLike.mem_coe, coe_sInf, Set.mem_iInter₂]\n\n@[simp]\ntheorem sInf_toSubmodule (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toSubmodule = sInf (NonUnitalSubalgebra.toSubmodule '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp]\ntheorem sInf_toNonUnitalSubsemiring (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toNonUnitalSubsemiring = sInf (NonUnitalSubalgebra.toNonUnitalSubsemiring '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp, norm_cast]\ntheorem coe_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} :\n    (↑(⨅ i, S i) : Set A) = ⋂ i, S i := by simp [iInf]\n\ntheorem mem_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} {x : A} :\n    (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by simp only [iInf, mem_sInf, Set.forall_range_iff]\n\n@[simp]\ntheorem iInf_toSubmodule {ι : Sort*} (S : ι → NonUnitalSubalgebra R A) :\n    (⨅ i, S i).toSubmodule = ⨅ i, (S i).toSubmodule :=\n  SetLike.coe_injective <| by simp\n\ninstance : Inhabited (NonUnitalSubalgebra R A) :=\n  ⟨⊥⟩\n\ntheorem mem_bot {x : A} : x ∈ (⊥ : NonUnitalSubalgebra R A) ↔ x = 0 :=\n  show x ∈ Submodule.span R (NonUnitalSubsemiring.closure (∅ : Set A) : Set A) ↔ x = 0 by\n    rw [NonUnitalSubsemiring.closure_empty, NonUnitalSubsemiring.coe_bot,\n      Submodule.span_zero_singleton, Submodule.mem_bot]\n\ntheorem toSubmodule_bot : (⊥ : NonUnitalSubalgebra R A).toSubmodule = ⊥ := by\n  ext\n  simp only [mem_bot, NonUnitalSubalgebra.mem_toSubmodule, Submodule.mem_bot]\n\n@[simp]\ntheorem coe_bot : ((⊥ : NonUnitalSubalgebra R A) : Set A) = {0} := by\n  simp [Set.ext_iff, NonUnitalAlgebra.mem_bot]\n\ntheorem eq_top_iff {S : NonUnitalSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S :=\n  ⟨fun h x => by rw [h]; exact mem_top, fun h => by ext x; exact ⟨fun _ => mem_top, fun _ => h x⟩⟩\n\ntheorem range_top_iff_surjective (f : A →ₙₐ[R] B) :\n    NonUnitalAlgHom.range f = (⊤ : NonUnitalSubalgebra R B) ↔ Function.Surjective f :=\n  NonUnitalAlgebra.eq_top_iff\n\n@[simp]\ntheorem range_id : NonUnitalAlgHom.range (NonUnitalAlgHom.id R A) = ⊤ :=\n  SetLike.coe_injective Set.range_id\n\n@[simp]\ntheorem map_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R A).map f = NonUnitalAlgHom.range f :=\n  SetLike.coe_injective Set.image_univ\n\n@[simp]\ntheorem map_bot (f : A →ₙₐ[R] B) : (⊥ : NonUnitalSubalgebra R A).map f = ⊥ :=\n  SetLike.coe_injective <| by simp [NonUnitalAlgebra.coe_bot, NonUnitalSubalgebra.coe_map]\n\n@[simp]\ntheorem comap_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R B).comap f = ⊤ :=\n  eq_top_iff.2 fun _ => mem_top\n\n/-- `NonUnitalAlgHom` to `⊤ : NonUnitalSubalgebra R A`. -/\ndef toTop : A →ₙₐ[R] (⊤ : NonUnitalSubalgebra R A) :=\n  NonUnitalAlgHom.codRestrict (NonUnitalAlgHom.id R A) ⊤ fun _ => mem_top\n\nend NonUnitalAlgebra\n\nnamespace NonUnitalSubalgebra\n\nopen NonUnitalAlgebra\n\nsection NonAssoc\n\nvariable {R : Type u} {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable (S : NonUnitalSubalgebra R A)\n\ninstance subsingleton_of_subsingleton [Subsingleton A] : Subsingleton (NonUnitalSubalgebra R A) :=\n  ⟨fun B C => ext fun x => by simp only [Subsingleton.elim x 0, zero_mem B, zero_mem C]⟩\n\ninstance _root_.NonUnitalAlgHom.subsingleton [Subsingleton (NonUnitalSubalgebra R A)] :\n    Subsingleton (A →ₙₐ[R] B) :=\n  ⟨fun f g =>\n    NonUnitalAlgHom.ext fun a =>\n      have : a ∈ (⊥ : NonUnitalSubalgebra R A) :=\n        Subsingleton.elim (⊤ : NonUnitalSubalgebra R A) ⊥ ▸ mem_top\n      (mem_bot.mp this).symm ▸ (map_zero f).trans (map_zero g).symm⟩\n\ntheorem range_val : NonUnitalAlgHom.range (NonUnitalSubalgebraClass.subtype S) = S :=\n  ext <| Set.ext_iff.1 <| (NonUnitalSubalgebraClass.subtype S).coe_range.trans Subtype.range_val\n\n/-- The map `S → T` when `S` is a non-unital subalgebra contained in the non-unital subalgebra `T`.\n\nThis is the non-unital subalgebra version of `Submodule.inclusion`, or `Subring.inclusion`  -/\ndef inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) : S →ₙₐ[R] T\n    where\n  toFun := Set.inclusion h\n  map_add' _ _ := rfl\n  map_mul' _ _ := rfl\n  map_zero' := rfl\n  map_smul' _ _ := rfl\n\ntheorem inclusion_injective {S T : NonUnitalSubalgebra R A} (h : S ≤ T) :\n    Function.Injective (inclusion h) := fun _ _ => Subtype.ext ∘ Subtype.mk.inj\n\n@[simp]\ntheorem inclusion_self {S : NonUnitalSubalgebra R A} :\n    inclusion (le_refl S) = NonUnitalAlgHom.id R S :=\n  NonUnitalAlgHom.ext fun _ => Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_mk {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : A) (hx : x ∈ S) :\n    inclusion h ⟨x, hx⟩ = ⟨x, h hx⟩ :=\n  rfl\n\ntheorem inclusion_right {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : T) (m : (x : A) ∈ S) :\n    inclusion h ⟨x, m⟩ = x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_inclusion {S T U : NonUnitalSubalgebra R A} (hst : S ≤ T) (htu : T ≤ U) (x : S) :\n    inclusion htu (inclusion hst x) = inclusion (le_trans hst htu) x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem coe_inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (s : S) :\n    (inclusion h s : A) = s :=\n  rfl\n\nsection Prod\n\nvariable (S₁ : NonUnitalSubalgebra R B)\n\n/-- The product of two non-unital subalgebras is a non-unital subalgebra. -/\ndef prod : NonUnitalSubalgebra R (A × B) :=\n  { S.toNonUnitalSubsemiring.prod S₁.toNonUnitalSubsemiring with\n    carrier := S ×ˢ S₁\n    smul_mem' := fun r _x hx => ⟨SMulMemClass.smul_mem r hx.1, SMulMemClass.smul_mem r hx.2⟩ }\n\n@[simp]\ntheorem coe_prod : (prod S S₁ : Set (A × B)) = (S : Set A) ×ˢ S₁ :=\n  rfl\n\ntheorem prod_toSubmodule : (S.prod S₁).toSubmodule = S.toSubmodule.prod S₁.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem mem_prod {S : NonUnitalSubalgebra R A} {S₁ : NonUnitalSubalgebra R B} {x : A × B} :\n    x ∈ prod S S₁ ↔ x.1 ∈ S ∧ x.2 ∈ S₁ :=\n  Set.mem_prod\n\n@[simp]\ntheorem prod_top : (prod ⊤ ⊤ : NonUnitalSubalgebra R (A × B)) = ⊤ := by ext; simp\n\ntheorem prod_mono {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S ≤ T → S₁ ≤ T₁ → prod S S₁ ≤ prod T T₁ :=\n  Set.prod_mono\n\n@[simp]\ntheorem prod_inf_prod {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S.prod S₁ ⊓ T.prod T₁ = (S ⊓ T).prod (S₁ ⊓ T₁) :=\n  SetLike.coe_injective Set.prod_inter_prod\n\nend Prod\n\nsection SuprLift\n\nvariable {ι : Type*}\n\ntheorem coe_iSup_of_directed [Nonempty ι] {S : ι → NonUnitalSubalgebra R A}\n    (dir : Directed (· ≤ ·) S) : ↑(iSup S) = ⋃ i, (S i : Set A) :=\n  let K : NonUnitalSubalgebra R A :=\n    { __ := NonUnitalSubsemiring.copy _ _ (NonUnitalSubsemiring.coe_iSup_of_directed dir).symm\n      smul_mem' := fun r _x hx ↦\n        let ⟨i, hi⟩ := Set.mem_iUnion.1 hx\n        Set.mem_iUnion.2 ⟨i, (S i).smul_mem' r hi⟩ }\n  have : iSup S = K := le_antisymm\n    (iSup_le fun i ↦ le_iSup (fun i ↦ (S i : Set A)) i) (Set.iUnion_subset fun _ ↦ le_iSup S _)\n  this.symm ▸ rfl\n\n/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              rw [hf j k hjk]\n              rfl)\n            (↑(iSup K)) (by rw [coe_iSup_of_directed dir])\n        map_zero' := by\n          dsimp\n          exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl) _ (by simp)\n        map_mul' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· * ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_add' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· + ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_smul' := fun r => by\n          dsimp\n          apply Set.iUnionLift_unary (coe_iSup_of_directed dir) _ (fun _ x => r • x)\n            (fun _ _ => rfl)\n          ","nextTactic":"on_goal 2 => rw [coe_iSup_of_directed dir]","declUpToTactic":"/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              rw [hf j k hjk]\n              rfl)\n            (↑(iSup K)) (by rw [coe_iSup_of_directed dir])\n        map_zero' := by\n          dsimp\n          exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl) _ (by simp)\n        map_mul' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· * ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_add' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· + ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_smul' := fun r => by\n          dsimp\n          apply Set.iUnionLift_unary (coe_iSup_of_directed dir) _ (fun _ x => r • x)\n            (fun _ _ => rfl)\n          ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.892_0.jQvdAcOcyVirh23","decl":"/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B "}
-{"state":"R : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\nS : NonUnitalSubalgebra R A\nι : Type u_1\ninst✝ : Nonempty ι\nK : ι → NonUnitalSubalgebra R A\ndir : Directed (fun x x_1 => x ≤ x_1) K\nf : (i : ι) → ↥(K i) →ₙₐ[R] B\nhf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalAlgHom.comp (f j) (inclusion h)\nr : R\n⊢ ↑(iSup K) ⊆ ⋃ i, ↑(K i)","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact NonUnitalAlgebra.gc\n\n@[simp]\ntheorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=\n  eq_top_iff.2 fun _x hx => subset_adjoin R hx\n\nvariable {R A}\n\n@[simp]\ntheorem coe_top : (↑(⊤ : NonUnitalSubalgebra R A) : Set A) = Set.univ :=\n  rfl\n\n@[simp]\ntheorem mem_top {x : A} : x ∈ (⊤ : NonUnitalSubalgebra R A) :=\n  Set.mem_univ x\n\n@[simp]\ntheorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toNonUnitalSubsemiring : (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubsemiring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toSubring {R A : Type*} [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    [IsScalarTower R A A] [SMulCommClass R A A] :\n    (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem toSubmodule_eq_top {S : NonUnitalSubalgebra R A} : S.toSubmodule = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toSubmodule'.injective.eq_iff' top_toSubmodule\n\n@[simp]\ntheorem toNonUnitalSubsemiring_eq_top {S : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubsemiring_injective.eq_iff' top_toNonUnitalSubsemiring\n\n@[simp]\ntheorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubring_injective.eq_iff' top_toSubring\n\ntheorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_left\n\ntheorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_right\n\ntheorem mul_mem_sup {S T : NonUnitalSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) :\n    x * y ∈ S ⊔ T :=\n  mul_mem (mem_sup_left hx) (mem_sup_right hy)\n\ntheorem map_sup (f : F) (S T : NonUnitalSubalgebra R A) :\n    ((S ⊔ T).map f : NonUnitalSubalgebra R B) = S.map f ⊔ T.map f :=\n  (@NonUnitalSubalgebra.gc_map_comap F R A B _ _ _ _ _ _ f).l_sup\n\n@[simp, norm_cast]\ntheorem coe_inf (S T : NonUnitalSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T :=\n  rfl\n\n@[simp]\ntheorem mem_inf {S T : NonUnitalSubalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T :=\n  Iff.rfl\n\n@[simp]\ntheorem inf_toSubmodule (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toSubmodule = S.toSubmodule ⊓ T.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem inf_toNonUnitalSubsemiring (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring ⊓ T.toNonUnitalSubsemiring :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sInf (S : Set (NonUnitalSubalgebra R A)) : (↑(sInf S) : Set A) = ⋂ s ∈ S, ↑s :=\n  sInf_image\n\ntheorem mem_sInf {S : Set (NonUnitalSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p := by\n  simp only [← SetLike.mem_coe, coe_sInf, Set.mem_iInter₂]\n\n@[simp]\ntheorem sInf_toSubmodule (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toSubmodule = sInf (NonUnitalSubalgebra.toSubmodule '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp]\ntheorem sInf_toNonUnitalSubsemiring (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toNonUnitalSubsemiring = sInf (NonUnitalSubalgebra.toNonUnitalSubsemiring '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp, norm_cast]\ntheorem coe_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} :\n    (↑(⨅ i, S i) : Set A) = ⋂ i, S i := by simp [iInf]\n\ntheorem mem_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} {x : A} :\n    (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by simp only [iInf, mem_sInf, Set.forall_range_iff]\n\n@[simp]\ntheorem iInf_toSubmodule {ι : Sort*} (S : ι → NonUnitalSubalgebra R A) :\n    (⨅ i, S i).toSubmodule = ⨅ i, (S i).toSubmodule :=\n  SetLike.coe_injective <| by simp\n\ninstance : Inhabited (NonUnitalSubalgebra R A) :=\n  ⟨⊥⟩\n\ntheorem mem_bot {x : A} : x ∈ (⊥ : NonUnitalSubalgebra R A) ↔ x = 0 :=\n  show x ∈ Submodule.span R (NonUnitalSubsemiring.closure (∅ : Set A) : Set A) ↔ x = 0 by\n    rw [NonUnitalSubsemiring.closure_empty, NonUnitalSubsemiring.coe_bot,\n      Submodule.span_zero_singleton, Submodule.mem_bot]\n\ntheorem toSubmodule_bot : (⊥ : NonUnitalSubalgebra R A).toSubmodule = ⊥ := by\n  ext\n  simp only [mem_bot, NonUnitalSubalgebra.mem_toSubmodule, Submodule.mem_bot]\n\n@[simp]\ntheorem coe_bot : ((⊥ : NonUnitalSubalgebra R A) : Set A) = {0} := by\n  simp [Set.ext_iff, NonUnitalAlgebra.mem_bot]\n\ntheorem eq_top_iff {S : NonUnitalSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S :=\n  ⟨fun h x => by rw [h]; exact mem_top, fun h => by ext x; exact ⟨fun _ => mem_top, fun _ => h x⟩⟩\n\ntheorem range_top_iff_surjective (f : A →ₙₐ[R] B) :\n    NonUnitalAlgHom.range f = (⊤ : NonUnitalSubalgebra R B) ↔ Function.Surjective f :=\n  NonUnitalAlgebra.eq_top_iff\n\n@[simp]\ntheorem range_id : NonUnitalAlgHom.range (NonUnitalAlgHom.id R A) = ⊤ :=\n  SetLike.coe_injective Set.range_id\n\n@[simp]\ntheorem map_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R A).map f = NonUnitalAlgHom.range f :=\n  SetLike.coe_injective Set.image_univ\n\n@[simp]\ntheorem map_bot (f : A →ₙₐ[R] B) : (⊥ : NonUnitalSubalgebra R A).map f = ⊥ :=\n  SetLike.coe_injective <| by simp [NonUnitalAlgebra.coe_bot, NonUnitalSubalgebra.coe_map]\n\n@[simp]\ntheorem comap_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R B).comap f = ⊤ :=\n  eq_top_iff.2 fun _ => mem_top\n\n/-- `NonUnitalAlgHom` to `⊤ : NonUnitalSubalgebra R A`. -/\ndef toTop : A →ₙₐ[R] (⊤ : NonUnitalSubalgebra R A) :=\n  NonUnitalAlgHom.codRestrict (NonUnitalAlgHom.id R A) ⊤ fun _ => mem_top\n\nend NonUnitalAlgebra\n\nnamespace NonUnitalSubalgebra\n\nopen NonUnitalAlgebra\n\nsection NonAssoc\n\nvariable {R : Type u} {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable (S : NonUnitalSubalgebra R A)\n\ninstance subsingleton_of_subsingleton [Subsingleton A] : Subsingleton (NonUnitalSubalgebra R A) :=\n  ⟨fun B C => ext fun x => by simp only [Subsingleton.elim x 0, zero_mem B, zero_mem C]⟩\n\ninstance _root_.NonUnitalAlgHom.subsingleton [Subsingleton (NonUnitalSubalgebra R A)] :\n    Subsingleton (A →ₙₐ[R] B) :=\n  ⟨fun f g =>\n    NonUnitalAlgHom.ext fun a =>\n      have : a ∈ (⊥ : NonUnitalSubalgebra R A) :=\n        Subsingleton.elim (⊤ : NonUnitalSubalgebra R A) ⊥ ▸ mem_top\n      (mem_bot.mp this).symm ▸ (map_zero f).trans (map_zero g).symm⟩\n\ntheorem range_val : NonUnitalAlgHom.range (NonUnitalSubalgebraClass.subtype S) = S :=\n  ext <| Set.ext_iff.1 <| (NonUnitalSubalgebraClass.subtype S).coe_range.trans Subtype.range_val\n\n/-- The map `S → T` when `S` is a non-unital subalgebra contained in the non-unital subalgebra `T`.\n\nThis is the non-unital subalgebra version of `Submodule.inclusion`, or `Subring.inclusion`  -/\ndef inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) : S →ₙₐ[R] T\n    where\n  toFun := Set.inclusion h\n  map_add' _ _ := rfl\n  map_mul' _ _ := rfl\n  map_zero' := rfl\n  map_smul' _ _ := rfl\n\ntheorem inclusion_injective {S T : NonUnitalSubalgebra R A} (h : S ≤ T) :\n    Function.Injective (inclusion h) := fun _ _ => Subtype.ext ∘ Subtype.mk.inj\n\n@[simp]\ntheorem inclusion_self {S : NonUnitalSubalgebra R A} :\n    inclusion (le_refl S) = NonUnitalAlgHom.id R S :=\n  NonUnitalAlgHom.ext fun _ => Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_mk {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : A) (hx : x ∈ S) :\n    inclusion h ⟨x, hx⟩ = ⟨x, h hx⟩ :=\n  rfl\n\ntheorem inclusion_right {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : T) (m : (x : A) ∈ S) :\n    inclusion h ⟨x, m⟩ = x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_inclusion {S T U : NonUnitalSubalgebra R A} (hst : S ≤ T) (htu : T ≤ U) (x : S) :\n    inclusion htu (inclusion hst x) = inclusion (le_trans hst htu) x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem coe_inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (s : S) :\n    (inclusion h s : A) = s :=\n  rfl\n\nsection Prod\n\nvariable (S₁ : NonUnitalSubalgebra R B)\n\n/-- The product of two non-unital subalgebras is a non-unital subalgebra. -/\ndef prod : NonUnitalSubalgebra R (A × B) :=\n  { S.toNonUnitalSubsemiring.prod S₁.toNonUnitalSubsemiring with\n    carrier := S ×ˢ S₁\n    smul_mem' := fun r _x hx => ⟨SMulMemClass.smul_mem r hx.1, SMulMemClass.smul_mem r hx.2⟩ }\n\n@[simp]\ntheorem coe_prod : (prod S S₁ : Set (A × B)) = (S : Set A) ×ˢ S₁ :=\n  rfl\n\ntheorem prod_toSubmodule : (S.prod S₁).toSubmodule = S.toSubmodule.prod S₁.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem mem_prod {S : NonUnitalSubalgebra R A} {S₁ : NonUnitalSubalgebra R B} {x : A × B} :\n    x ∈ prod S S₁ ↔ x.1 ∈ S ∧ x.2 ∈ S₁ :=\n  Set.mem_prod\n\n@[simp]\ntheorem prod_top : (prod ⊤ ⊤ : NonUnitalSubalgebra R (A × B)) = ⊤ := by ext; simp\n\ntheorem prod_mono {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S ≤ T → S₁ ≤ T₁ → prod S S₁ ≤ prod T T₁ :=\n  Set.prod_mono\n\n@[simp]\ntheorem prod_inf_prod {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S.prod S₁ ⊓ T.prod T₁ = (S ⊓ T).prod (S₁ ⊓ T₁) :=\n  SetLike.coe_injective Set.prod_inter_prod\n\nend Prod\n\nsection SuprLift\n\nvariable {ι : Type*}\n\ntheorem coe_iSup_of_directed [Nonempty ι] {S : ι → NonUnitalSubalgebra R A}\n    (dir : Directed (· ≤ ·) S) : ↑(iSup S) = ⋃ i, (S i : Set A) :=\n  let K : NonUnitalSubalgebra R A :=\n    { __ := NonUnitalSubsemiring.copy _ _ (NonUnitalSubsemiring.coe_iSup_of_directed dir).symm\n      smul_mem' := fun r _x hx ↦\n        let ⟨i, hi⟩ := Set.mem_iUnion.1 hx\n        Set.mem_iUnion.2 ⟨i, (S i).smul_mem' r hi⟩ }\n  have : iSup S = K := le_antisymm\n    (iSup_le fun i ↦ le_iSup (fun i ↦ (S i : Set A)) i) (Set.iUnion_subset fun _ ↦ le_iSup S _)\n  this.symm ▸ rfl\n\n/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              rw [hf j k hjk]\n              rfl)\n            (↑(iSup K)) (by rw [coe_iSup_of_directed dir])\n        map_zero' := by\n          dsimp\n          exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl) _ (by simp)\n        map_mul' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· * ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_add' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· + ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_smul' := fun r => by\n          dsimp\n          apply Set.iUnionLift_unary (coe_iSup_of_directed dir) _ (fun _ x => r • x)\n            (fun _ _ => rfl)\n          on_goal 2 => ","nextTactic":"rw [coe_iSup_of_directed dir]","declUpToTactic":"/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              rw [hf j k hjk]\n              rfl)\n            (↑(iSup K)) (by rw [coe_iSup_of_directed dir])\n        map_zero' := by\n          dsimp\n          exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl) _ (by simp)\n        map_mul' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· * ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_add' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· + ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_smul' := fun r => by\n          dsimp\n          apply Set.iUnionLift_unary (coe_iSup_of_directed dir) _ (fun _ x => r • x)\n            (fun _ _ => rfl)\n          on_goal 2 => ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.892_0.jQvdAcOcyVirh23","decl":"/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B "}
-{"state":"case h\nR : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\nS : NonUnitalSubalgebra R A\nι : Type u_1\ninst✝ : Nonempty ι\nK : ι → NonUnitalSubalgebra R A\ndir : Directed (fun x x_1 => x ≤ x_1) K\nf : (i : ι) → ↥(K i) →ₙₐ[R] B\nhf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalAlgHom.comp (f j) (inclusion h)\nr : R\n⊢ ∀ (i : ι) (x : ↑↑(K i)), (f i) (r • x) = r • (f i) x","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact NonUnitalAlgebra.gc\n\n@[simp]\ntheorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=\n  eq_top_iff.2 fun _x hx => subset_adjoin R hx\n\nvariable {R A}\n\n@[simp]\ntheorem coe_top : (↑(⊤ : NonUnitalSubalgebra R A) : Set A) = Set.univ :=\n  rfl\n\n@[simp]\ntheorem mem_top {x : A} : x ∈ (⊤ : NonUnitalSubalgebra R A) :=\n  Set.mem_univ x\n\n@[simp]\ntheorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toNonUnitalSubsemiring : (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubsemiring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toSubring {R A : Type*} [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    [IsScalarTower R A A] [SMulCommClass R A A] :\n    (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem toSubmodule_eq_top {S : NonUnitalSubalgebra R A} : S.toSubmodule = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toSubmodule'.injective.eq_iff' top_toSubmodule\n\n@[simp]\ntheorem toNonUnitalSubsemiring_eq_top {S : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubsemiring_injective.eq_iff' top_toNonUnitalSubsemiring\n\n@[simp]\ntheorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubring_injective.eq_iff' top_toSubring\n\ntheorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_left\n\ntheorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_right\n\ntheorem mul_mem_sup {S T : NonUnitalSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) :\n    x * y ∈ S ⊔ T :=\n  mul_mem (mem_sup_left hx) (mem_sup_right hy)\n\ntheorem map_sup (f : F) (S T : NonUnitalSubalgebra R A) :\n    ((S ⊔ T).map f : NonUnitalSubalgebra R B) = S.map f ⊔ T.map f :=\n  (@NonUnitalSubalgebra.gc_map_comap F R A B _ _ _ _ _ _ f).l_sup\n\n@[simp, norm_cast]\ntheorem coe_inf (S T : NonUnitalSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T :=\n  rfl\n\n@[simp]\ntheorem mem_inf {S T : NonUnitalSubalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T :=\n  Iff.rfl\n\n@[simp]\ntheorem inf_toSubmodule (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toSubmodule = S.toSubmodule ⊓ T.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem inf_toNonUnitalSubsemiring (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring ⊓ T.toNonUnitalSubsemiring :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sInf (S : Set (NonUnitalSubalgebra R A)) : (↑(sInf S) : Set A) = ⋂ s ∈ S, ↑s :=\n  sInf_image\n\ntheorem mem_sInf {S : Set (NonUnitalSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p := by\n  simp only [← SetLike.mem_coe, coe_sInf, Set.mem_iInter₂]\n\n@[simp]\ntheorem sInf_toSubmodule (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toSubmodule = sInf (NonUnitalSubalgebra.toSubmodule '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp]\ntheorem sInf_toNonUnitalSubsemiring (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toNonUnitalSubsemiring = sInf (NonUnitalSubalgebra.toNonUnitalSubsemiring '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp, norm_cast]\ntheorem coe_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} :\n    (↑(⨅ i, S i) : Set A) = ⋂ i, S i := by simp [iInf]\n\ntheorem mem_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} {x : A} :\n    (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by simp only [iInf, mem_sInf, Set.forall_range_iff]\n\n@[simp]\ntheorem iInf_toSubmodule {ι : Sort*} (S : ι → NonUnitalSubalgebra R A) :\n    (⨅ i, S i).toSubmodule = ⨅ i, (S i).toSubmodule :=\n  SetLike.coe_injective <| by simp\n\ninstance : Inhabited (NonUnitalSubalgebra R A) :=\n  ⟨⊥⟩\n\ntheorem mem_bot {x : A} : x ∈ (⊥ : NonUnitalSubalgebra R A) ↔ x = 0 :=\n  show x ∈ Submodule.span R (NonUnitalSubsemiring.closure (∅ : Set A) : Set A) ↔ x = 0 by\n    rw [NonUnitalSubsemiring.closure_empty, NonUnitalSubsemiring.coe_bot,\n      Submodule.span_zero_singleton, Submodule.mem_bot]\n\ntheorem toSubmodule_bot : (⊥ : NonUnitalSubalgebra R A).toSubmodule = ⊥ := by\n  ext\n  simp only [mem_bot, NonUnitalSubalgebra.mem_toSubmodule, Submodule.mem_bot]\n\n@[simp]\ntheorem coe_bot : ((⊥ : NonUnitalSubalgebra R A) : Set A) = {0} := by\n  simp [Set.ext_iff, NonUnitalAlgebra.mem_bot]\n\ntheorem eq_top_iff {S : NonUnitalSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S :=\n  ⟨fun h x => by rw [h]; exact mem_top, fun h => by ext x; exact ⟨fun _ => mem_top, fun _ => h x⟩⟩\n\ntheorem range_top_iff_surjective (f : A →ₙₐ[R] B) :\n    NonUnitalAlgHom.range f = (⊤ : NonUnitalSubalgebra R B) ↔ Function.Surjective f :=\n  NonUnitalAlgebra.eq_top_iff\n\n@[simp]\ntheorem range_id : NonUnitalAlgHom.range (NonUnitalAlgHom.id R A) = ⊤ :=\n  SetLike.coe_injective Set.range_id\n\n@[simp]\ntheorem map_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R A).map f = NonUnitalAlgHom.range f :=\n  SetLike.coe_injective Set.image_univ\n\n@[simp]\ntheorem map_bot (f : A →ₙₐ[R] B) : (⊥ : NonUnitalSubalgebra R A).map f = ⊥ :=\n  SetLike.coe_injective <| by simp [NonUnitalAlgebra.coe_bot, NonUnitalSubalgebra.coe_map]\n\n@[simp]\ntheorem comap_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R B).comap f = ⊤ :=\n  eq_top_iff.2 fun _ => mem_top\n\n/-- `NonUnitalAlgHom` to `⊤ : NonUnitalSubalgebra R A`. -/\ndef toTop : A →ₙₐ[R] (⊤ : NonUnitalSubalgebra R A) :=\n  NonUnitalAlgHom.codRestrict (NonUnitalAlgHom.id R A) ⊤ fun _ => mem_top\n\nend NonUnitalAlgebra\n\nnamespace NonUnitalSubalgebra\n\nopen NonUnitalAlgebra\n\nsection NonAssoc\n\nvariable {R : Type u} {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable (S : NonUnitalSubalgebra R A)\n\ninstance subsingleton_of_subsingleton [Subsingleton A] : Subsingleton (NonUnitalSubalgebra R A) :=\n  ⟨fun B C => ext fun x => by simp only [Subsingleton.elim x 0, zero_mem B, zero_mem C]⟩\n\ninstance _root_.NonUnitalAlgHom.subsingleton [Subsingleton (NonUnitalSubalgebra R A)] :\n    Subsingleton (A →ₙₐ[R] B) :=\n  ⟨fun f g =>\n    NonUnitalAlgHom.ext fun a =>\n      have : a ∈ (⊥ : NonUnitalSubalgebra R A) :=\n        Subsingleton.elim (⊤ : NonUnitalSubalgebra R A) ⊥ ▸ mem_top\n      (mem_bot.mp this).symm ▸ (map_zero f).trans (map_zero g).symm⟩\n\ntheorem range_val : NonUnitalAlgHom.range (NonUnitalSubalgebraClass.subtype S) = S :=\n  ext <| Set.ext_iff.1 <| (NonUnitalSubalgebraClass.subtype S).coe_range.trans Subtype.range_val\n\n/-- The map `S → T` when `S` is a non-unital subalgebra contained in the non-unital subalgebra `T`.\n\nThis is the non-unital subalgebra version of `Submodule.inclusion`, or `Subring.inclusion`  -/\ndef inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) : S →ₙₐ[R] T\n    where\n  toFun := Set.inclusion h\n  map_add' _ _ := rfl\n  map_mul' _ _ := rfl\n  map_zero' := rfl\n  map_smul' _ _ := rfl\n\ntheorem inclusion_injective {S T : NonUnitalSubalgebra R A} (h : S ≤ T) :\n    Function.Injective (inclusion h) := fun _ _ => Subtype.ext ∘ Subtype.mk.inj\n\n@[simp]\ntheorem inclusion_self {S : NonUnitalSubalgebra R A} :\n    inclusion (le_refl S) = NonUnitalAlgHom.id R S :=\n  NonUnitalAlgHom.ext fun _ => Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_mk {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : A) (hx : x ∈ S) :\n    inclusion h ⟨x, hx⟩ = ⟨x, h hx⟩ :=\n  rfl\n\ntheorem inclusion_right {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : T) (m : (x : A) ∈ S) :\n    inclusion h ⟨x, m⟩ = x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_inclusion {S T U : NonUnitalSubalgebra R A} (hst : S ≤ T) (htu : T ≤ U) (x : S) :\n    inclusion htu (inclusion hst x) = inclusion (le_trans hst htu) x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem coe_inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (s : S) :\n    (inclusion h s : A) = s :=\n  rfl\n\nsection Prod\n\nvariable (S₁ : NonUnitalSubalgebra R B)\n\n/-- The product of two non-unital subalgebras is a non-unital subalgebra. -/\ndef prod : NonUnitalSubalgebra R (A × B) :=\n  { S.toNonUnitalSubsemiring.prod S₁.toNonUnitalSubsemiring with\n    carrier := S ×ˢ S₁\n    smul_mem' := fun r _x hx => ⟨SMulMemClass.smul_mem r hx.1, SMulMemClass.smul_mem r hx.2⟩ }\n\n@[simp]\ntheorem coe_prod : (prod S S₁ : Set (A × B)) = (S : Set A) ×ˢ S₁ :=\n  rfl\n\ntheorem prod_toSubmodule : (S.prod S₁).toSubmodule = S.toSubmodule.prod S₁.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem mem_prod {S : NonUnitalSubalgebra R A} {S₁ : NonUnitalSubalgebra R B} {x : A × B} :\n    x ∈ prod S S₁ ↔ x.1 ∈ S ∧ x.2 ∈ S₁ :=\n  Set.mem_prod\n\n@[simp]\ntheorem prod_top : (prod ⊤ ⊤ : NonUnitalSubalgebra R (A × B)) = ⊤ := by ext; simp\n\ntheorem prod_mono {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S ≤ T → S₁ ≤ T₁ → prod S S₁ ≤ prod T T₁ :=\n  Set.prod_mono\n\n@[simp]\ntheorem prod_inf_prod {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S.prod S₁ ⊓ T.prod T₁ = (S ⊓ T).prod (S₁ ⊓ T₁) :=\n  SetLike.coe_injective Set.prod_inter_prod\n\nend Prod\n\nsection SuprLift\n\nvariable {ι : Type*}\n\ntheorem coe_iSup_of_directed [Nonempty ι] {S : ι → NonUnitalSubalgebra R A}\n    (dir : Directed (· ≤ ·) S) : ↑(iSup S) = ⋃ i, (S i : Set A) :=\n  let K : NonUnitalSubalgebra R A :=\n    { __ := NonUnitalSubsemiring.copy _ _ (NonUnitalSubsemiring.coe_iSup_of_directed dir).symm\n      smul_mem' := fun r _x hx ↦\n        let ⟨i, hi⟩ := Set.mem_iUnion.1 hx\n        Set.mem_iUnion.2 ⟨i, (S i).smul_mem' r hi⟩ }\n  have : iSup S = K := le_antisymm\n    (iSup_le fun i ↦ le_iSup (fun i ↦ (S i : Set A)) i) (Set.iUnion_subset fun _ ↦ le_iSup S _)\n  this.symm ▸ rfl\n\n/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              rw [hf j k hjk]\n              rfl)\n            (↑(iSup K)) (by rw [coe_iSup_of_directed dir])\n        map_zero' := by\n          dsimp\n          exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl) _ (by simp)\n        map_mul' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· * ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_add' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· + ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_smul' := fun r => by\n          dsimp\n          apply Set.iUnionLift_unary (coe_iSup_of_directed dir) _ (fun _ x => r • x)\n            (fun _ _ => rfl)\n          on_goal 2 => rw [coe_iSup_of_directed dir]\n          ","nextTactic":"all_goals simp","declUpToTactic":"/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              rw [hf j k hjk]\n              rfl)\n            (↑(iSup K)) (by rw [coe_iSup_of_directed dir])\n        map_zero' := by\n          dsimp\n          exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl) _ (by simp)\n        map_mul' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· * ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_add' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· + ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_smul' := fun r => by\n          dsimp\n          apply Set.iUnionLift_unary (coe_iSup_of_directed dir) _ (fun _ x => r • x)\n            (fun _ _ => rfl)\n          on_goal 2 => rw [coe_iSup_of_directed dir]\n          ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.892_0.jQvdAcOcyVirh23","decl":"/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B "}
-{"state":"case h\nR : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\nS : NonUnitalSubalgebra R A\nι : Type u_1\ninst✝ : Nonempty ι\nK : ι → NonUnitalSubalgebra R A\ndir : Directed (fun x x_1 => x ≤ x_1) K\nf : (i : ι) → ↥(K i) →ₙₐ[R] B\nhf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalAlgHom.comp (f j) (inclusion h)\nr : R\n⊢ ∀ (i : ι) (x : ↑↑(K i)), (f i) (r • x) = r • (f i) x","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact NonUnitalAlgebra.gc\n\n@[simp]\ntheorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=\n  eq_top_iff.2 fun _x hx => subset_adjoin R hx\n\nvariable {R A}\n\n@[simp]\ntheorem coe_top : (↑(⊤ : NonUnitalSubalgebra R A) : Set A) = Set.univ :=\n  rfl\n\n@[simp]\ntheorem mem_top {x : A} : x ∈ (⊤ : NonUnitalSubalgebra R A) :=\n  Set.mem_univ x\n\n@[simp]\ntheorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toNonUnitalSubsemiring : (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubsemiring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toSubring {R A : Type*} [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    [IsScalarTower R A A] [SMulCommClass R A A] :\n    (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem toSubmodule_eq_top {S : NonUnitalSubalgebra R A} : S.toSubmodule = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toSubmodule'.injective.eq_iff' top_toSubmodule\n\n@[simp]\ntheorem toNonUnitalSubsemiring_eq_top {S : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubsemiring_injective.eq_iff' top_toNonUnitalSubsemiring\n\n@[simp]\ntheorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubring_injective.eq_iff' top_toSubring\n\ntheorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_left\n\ntheorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_right\n\ntheorem mul_mem_sup {S T : NonUnitalSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) :\n    x * y ∈ S ⊔ T :=\n  mul_mem (mem_sup_left hx) (mem_sup_right hy)\n\ntheorem map_sup (f : F) (S T : NonUnitalSubalgebra R A) :\n    ((S ⊔ T).map f : NonUnitalSubalgebra R B) = S.map f ⊔ T.map f :=\n  (@NonUnitalSubalgebra.gc_map_comap F R A B _ _ _ _ _ _ f).l_sup\n\n@[simp, norm_cast]\ntheorem coe_inf (S T : NonUnitalSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T :=\n  rfl\n\n@[simp]\ntheorem mem_inf {S T : NonUnitalSubalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T :=\n  Iff.rfl\n\n@[simp]\ntheorem inf_toSubmodule (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toSubmodule = S.toSubmodule ⊓ T.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem inf_toNonUnitalSubsemiring (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring ⊓ T.toNonUnitalSubsemiring :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sInf (S : Set (NonUnitalSubalgebra R A)) : (↑(sInf S) : Set A) = ⋂ s ∈ S, ↑s :=\n  sInf_image\n\ntheorem mem_sInf {S : Set (NonUnitalSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p := by\n  simp only [← SetLike.mem_coe, coe_sInf, Set.mem_iInter₂]\n\n@[simp]\ntheorem sInf_toSubmodule (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toSubmodule = sInf (NonUnitalSubalgebra.toSubmodule '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp]\ntheorem sInf_toNonUnitalSubsemiring (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toNonUnitalSubsemiring = sInf (NonUnitalSubalgebra.toNonUnitalSubsemiring '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp, norm_cast]\ntheorem coe_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} :\n    (↑(⨅ i, S i) : Set A) = ⋂ i, S i := by simp [iInf]\n\ntheorem mem_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} {x : A} :\n    (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by simp only [iInf, mem_sInf, Set.forall_range_iff]\n\n@[simp]\ntheorem iInf_toSubmodule {ι : Sort*} (S : ι → NonUnitalSubalgebra R A) :\n    (⨅ i, S i).toSubmodule = ⨅ i, (S i).toSubmodule :=\n  SetLike.coe_injective <| by simp\n\ninstance : Inhabited (NonUnitalSubalgebra R A) :=\n  ⟨⊥⟩\n\ntheorem mem_bot {x : A} : x ∈ (⊥ : NonUnitalSubalgebra R A) ↔ x = 0 :=\n  show x ∈ Submodule.span R (NonUnitalSubsemiring.closure (∅ : Set A) : Set A) ↔ x = 0 by\n    rw [NonUnitalSubsemiring.closure_empty, NonUnitalSubsemiring.coe_bot,\n      Submodule.span_zero_singleton, Submodule.mem_bot]\n\ntheorem toSubmodule_bot : (⊥ : NonUnitalSubalgebra R A).toSubmodule = ⊥ := by\n  ext\n  simp only [mem_bot, NonUnitalSubalgebra.mem_toSubmodule, Submodule.mem_bot]\n\n@[simp]\ntheorem coe_bot : ((⊥ : NonUnitalSubalgebra R A) : Set A) = {0} := by\n  simp [Set.ext_iff, NonUnitalAlgebra.mem_bot]\n\ntheorem eq_top_iff {S : NonUnitalSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S :=\n  ⟨fun h x => by rw [h]; exact mem_top, fun h => by ext x; exact ⟨fun _ => mem_top, fun _ => h x⟩⟩\n\ntheorem range_top_iff_surjective (f : A →ₙₐ[R] B) :\n    NonUnitalAlgHom.range f = (⊤ : NonUnitalSubalgebra R B) ↔ Function.Surjective f :=\n  NonUnitalAlgebra.eq_top_iff\n\n@[simp]\ntheorem range_id : NonUnitalAlgHom.range (NonUnitalAlgHom.id R A) = ⊤ :=\n  SetLike.coe_injective Set.range_id\n\n@[simp]\ntheorem map_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R A).map f = NonUnitalAlgHom.range f :=\n  SetLike.coe_injective Set.image_univ\n\n@[simp]\ntheorem map_bot (f : A →ₙₐ[R] B) : (⊥ : NonUnitalSubalgebra R A).map f = ⊥ :=\n  SetLike.coe_injective <| by simp [NonUnitalAlgebra.coe_bot, NonUnitalSubalgebra.coe_map]\n\n@[simp]\ntheorem comap_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R B).comap f = ⊤ :=\n  eq_top_iff.2 fun _ => mem_top\n\n/-- `NonUnitalAlgHom` to `⊤ : NonUnitalSubalgebra R A`. -/\ndef toTop : A →ₙₐ[R] (⊤ : NonUnitalSubalgebra R A) :=\n  NonUnitalAlgHom.codRestrict (NonUnitalAlgHom.id R A) ⊤ fun _ => mem_top\n\nend NonUnitalAlgebra\n\nnamespace NonUnitalSubalgebra\n\nopen NonUnitalAlgebra\n\nsection NonAssoc\n\nvariable {R : Type u} {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable (S : NonUnitalSubalgebra R A)\n\ninstance subsingleton_of_subsingleton [Subsingleton A] : Subsingleton (NonUnitalSubalgebra R A) :=\n  ⟨fun B C => ext fun x => by simp only [Subsingleton.elim x 0, zero_mem B, zero_mem C]⟩\n\ninstance _root_.NonUnitalAlgHom.subsingleton [Subsingleton (NonUnitalSubalgebra R A)] :\n    Subsingleton (A →ₙₐ[R] B) :=\n  ⟨fun f g =>\n    NonUnitalAlgHom.ext fun a =>\n      have : a ∈ (⊥ : NonUnitalSubalgebra R A) :=\n        Subsingleton.elim (⊤ : NonUnitalSubalgebra R A) ⊥ ▸ mem_top\n      (mem_bot.mp this).symm ▸ (map_zero f).trans (map_zero g).symm⟩\n\ntheorem range_val : NonUnitalAlgHom.range (NonUnitalSubalgebraClass.subtype S) = S :=\n  ext <| Set.ext_iff.1 <| (NonUnitalSubalgebraClass.subtype S).coe_range.trans Subtype.range_val\n\n/-- The map `S → T` when `S` is a non-unital subalgebra contained in the non-unital subalgebra `T`.\n\nThis is the non-unital subalgebra version of `Submodule.inclusion`, or `Subring.inclusion`  -/\ndef inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) : S →ₙₐ[R] T\n    where\n  toFun := Set.inclusion h\n  map_add' _ _ := rfl\n  map_mul' _ _ := rfl\n  map_zero' := rfl\n  map_smul' _ _ := rfl\n\ntheorem inclusion_injective {S T : NonUnitalSubalgebra R A} (h : S ≤ T) :\n    Function.Injective (inclusion h) := fun _ _ => Subtype.ext ∘ Subtype.mk.inj\n\n@[simp]\ntheorem inclusion_self {S : NonUnitalSubalgebra R A} :\n    inclusion (le_refl S) = NonUnitalAlgHom.id R S :=\n  NonUnitalAlgHom.ext fun _ => Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_mk {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : A) (hx : x ∈ S) :\n    inclusion h ⟨x, hx⟩ = ⟨x, h hx⟩ :=\n  rfl\n\ntheorem inclusion_right {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : T) (m : (x : A) ∈ S) :\n    inclusion h ⟨x, m⟩ = x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_inclusion {S T U : NonUnitalSubalgebra R A} (hst : S ≤ T) (htu : T ≤ U) (x : S) :\n    inclusion htu (inclusion hst x) = inclusion (le_trans hst htu) x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem coe_inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (s : S) :\n    (inclusion h s : A) = s :=\n  rfl\n\nsection Prod\n\nvariable (S₁ : NonUnitalSubalgebra R B)\n\n/-- The product of two non-unital subalgebras is a non-unital subalgebra. -/\ndef prod : NonUnitalSubalgebra R (A × B) :=\n  { S.toNonUnitalSubsemiring.prod S₁.toNonUnitalSubsemiring with\n    carrier := S ×ˢ S₁\n    smul_mem' := fun r _x hx => ⟨SMulMemClass.smul_mem r hx.1, SMulMemClass.smul_mem r hx.2⟩ }\n\n@[simp]\ntheorem coe_prod : (prod S S₁ : Set (A × B)) = (S : Set A) ×ˢ S₁ :=\n  rfl\n\ntheorem prod_toSubmodule : (S.prod S₁).toSubmodule = S.toSubmodule.prod S₁.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem mem_prod {S : NonUnitalSubalgebra R A} {S₁ : NonUnitalSubalgebra R B} {x : A × B} :\n    x ∈ prod S S₁ ↔ x.1 ∈ S ∧ x.2 ∈ S₁ :=\n  Set.mem_prod\n\n@[simp]\ntheorem prod_top : (prod ⊤ ⊤ : NonUnitalSubalgebra R (A × B)) = ⊤ := by ext; simp\n\ntheorem prod_mono {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S ≤ T → S₁ ≤ T₁ → prod S S₁ ≤ prod T T₁ :=\n  Set.prod_mono\n\n@[simp]\ntheorem prod_inf_prod {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S.prod S₁ ⊓ T.prod T₁ = (S ⊓ T).prod (S₁ ⊓ T₁) :=\n  SetLike.coe_injective Set.prod_inter_prod\n\nend Prod\n\nsection SuprLift\n\nvariable {ι : Type*}\n\ntheorem coe_iSup_of_directed [Nonempty ι] {S : ι → NonUnitalSubalgebra R A}\n    (dir : Directed (· ≤ ·) S) : ↑(iSup S) = ⋃ i, (S i : Set A) :=\n  let K : NonUnitalSubalgebra R A :=\n    { __ := NonUnitalSubsemiring.copy _ _ (NonUnitalSubsemiring.coe_iSup_of_directed dir).symm\n      smul_mem' := fun r _x hx ↦\n        let ⟨i, hi⟩ := Set.mem_iUnion.1 hx\n        Set.mem_iUnion.2 ⟨i, (S i).smul_mem' r hi⟩ }\n  have : iSup S = K := le_antisymm\n    (iSup_le fun i ↦ le_iSup (fun i ↦ (S i : Set A)) i) (Set.iUnion_subset fun _ ↦ le_iSup S _)\n  this.symm ▸ rfl\n\n/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              rw [hf j k hjk]\n              rfl)\n            (↑(iSup K)) (by rw [coe_iSup_of_directed dir])\n        map_zero' := by\n          dsimp\n          exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl) _ (by simp)\n        map_mul' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· * ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_add' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· + ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_smul' := fun r => by\n          dsimp\n          apply Set.iUnionLift_unary (coe_iSup_of_directed dir) _ (fun _ x => r • x)\n            (fun _ _ => rfl)\n          on_goal 2 => rw [coe_iSup_of_directed dir]\n          all_goals ","nextTactic":"simp","declUpToTactic":"/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              rw [hf j k hjk]\n              rfl)\n            (↑(iSup K)) (by rw [coe_iSup_of_directed dir])\n        map_zero' := by\n          dsimp\n          exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl) _ (by simp)\n        map_mul' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· * ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_add' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· + ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_smul' := fun r => by\n          dsimp\n          apply Set.iUnionLift_unary (coe_iSup_of_directed dir) _ (fun _ x => r • x)\n            (fun _ _ => rfl)\n          on_goal 2 => rw [coe_iSup_of_directed dir]\n          all_goals ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.892_0.jQvdAcOcyVirh23","decl":"/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B "}
-{"state":"R : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\nS : NonUnitalSubalgebra R A\nι : Type u_1\ninst✝ : Nonempty ι\nK : ι → NonUnitalSubalgebra R A\ndir : Directed (fun x x_1 => x ≤ x_1) K\nf : (i : ι) → ↥(K i) →ₙₐ[R] B\nhf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalAlgHom.comp (f j) (inclusion h)\n⊢ MulActionHom.toFun\n      {\n        toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x)\n            (_ :\n              ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),\n                (fun i x => (f i) x) i { val := x, property := hxi } =\n                  (fun i x => (f i) x) j { val := x, property := hxj })\n            ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ i, ↑(K i)),\n        map_smul' :=\n          (_ :\n            ∀ (r : R) (x : ↥(iSup K)),\n              Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x)\n                  (_ :\n                    ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),\n                      (fun i x => (f i) x) i { val := x, property := hxi } =\n                        (fun i x => (f i) x) j { val := x, property := hxj })\n                  ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ i, ↑(K i)) (r • x) =\n                r •\n                  Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x)\n                    (_ :\n                      ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),\n                        (fun i x => (f i) x) i { val := x, property := hxi } =\n                          (fun i x => (f i) x) j { val := x, property := hxj })\n                    ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ i, ↑(K i)) x) }\n      0 =\n    0","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact NonUnitalAlgebra.gc\n\n@[simp]\ntheorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=\n  eq_top_iff.2 fun _x hx => subset_adjoin R hx\n\nvariable {R A}\n\n@[simp]\ntheorem coe_top : (↑(⊤ : NonUnitalSubalgebra R A) : Set A) = Set.univ :=\n  rfl\n\n@[simp]\ntheorem mem_top {x : A} : x ∈ (⊤ : NonUnitalSubalgebra R A) :=\n  Set.mem_univ x\n\n@[simp]\ntheorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toNonUnitalSubsemiring : (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubsemiring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toSubring {R A : Type*} [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    [IsScalarTower R A A] [SMulCommClass R A A] :\n    (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem toSubmodule_eq_top {S : NonUnitalSubalgebra R A} : S.toSubmodule = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toSubmodule'.injective.eq_iff' top_toSubmodule\n\n@[simp]\ntheorem toNonUnitalSubsemiring_eq_top {S : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubsemiring_injective.eq_iff' top_toNonUnitalSubsemiring\n\n@[simp]\ntheorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubring_injective.eq_iff' top_toSubring\n\ntheorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_left\n\ntheorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_right\n\ntheorem mul_mem_sup {S T : NonUnitalSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) :\n    x * y ∈ S ⊔ T :=\n  mul_mem (mem_sup_left hx) (mem_sup_right hy)\n\ntheorem map_sup (f : F) (S T : NonUnitalSubalgebra R A) :\n    ((S ⊔ T).map f : NonUnitalSubalgebra R B) = S.map f ⊔ T.map f :=\n  (@NonUnitalSubalgebra.gc_map_comap F R A B _ _ _ _ _ _ f).l_sup\n\n@[simp, norm_cast]\ntheorem coe_inf (S T : NonUnitalSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T :=\n  rfl\n\n@[simp]\ntheorem mem_inf {S T : NonUnitalSubalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T :=\n  Iff.rfl\n\n@[simp]\ntheorem inf_toSubmodule (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toSubmodule = S.toSubmodule ⊓ T.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem inf_toNonUnitalSubsemiring (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring ⊓ T.toNonUnitalSubsemiring :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sInf (S : Set (NonUnitalSubalgebra R A)) : (↑(sInf S) : Set A) = ⋂ s ∈ S, ↑s :=\n  sInf_image\n\ntheorem mem_sInf {S : Set (NonUnitalSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p := by\n  simp only [← SetLike.mem_coe, coe_sInf, Set.mem_iInter₂]\n\n@[simp]\ntheorem sInf_toSubmodule (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toSubmodule = sInf (NonUnitalSubalgebra.toSubmodule '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp]\ntheorem sInf_toNonUnitalSubsemiring (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toNonUnitalSubsemiring = sInf (NonUnitalSubalgebra.toNonUnitalSubsemiring '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp, norm_cast]\ntheorem coe_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} :\n    (↑(⨅ i, S i) : Set A) = ⋂ i, S i := by simp [iInf]\n\ntheorem mem_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} {x : A} :\n    (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by simp only [iInf, mem_sInf, Set.forall_range_iff]\n\n@[simp]\ntheorem iInf_toSubmodule {ι : Sort*} (S : ι → NonUnitalSubalgebra R A) :\n    (⨅ i, S i).toSubmodule = ⨅ i, (S i).toSubmodule :=\n  SetLike.coe_injective <| by simp\n\ninstance : Inhabited (NonUnitalSubalgebra R A) :=\n  ⟨⊥⟩\n\ntheorem mem_bot {x : A} : x ∈ (⊥ : NonUnitalSubalgebra R A) ↔ x = 0 :=\n  show x ∈ Submodule.span R (NonUnitalSubsemiring.closure (∅ : Set A) : Set A) ↔ x = 0 by\n    rw [NonUnitalSubsemiring.closure_empty, NonUnitalSubsemiring.coe_bot,\n      Submodule.span_zero_singleton, Submodule.mem_bot]\n\ntheorem toSubmodule_bot : (⊥ : NonUnitalSubalgebra R A).toSubmodule = ⊥ := by\n  ext\n  simp only [mem_bot, NonUnitalSubalgebra.mem_toSubmodule, Submodule.mem_bot]\n\n@[simp]\ntheorem coe_bot : ((⊥ : NonUnitalSubalgebra R A) : Set A) = {0} := by\n  simp [Set.ext_iff, NonUnitalAlgebra.mem_bot]\n\ntheorem eq_top_iff {S : NonUnitalSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S :=\n  ⟨fun h x => by rw [h]; exact mem_top, fun h => by ext x; exact ⟨fun _ => mem_top, fun _ => h x⟩⟩\n\ntheorem range_top_iff_surjective (f : A →ₙₐ[R] B) :\n    NonUnitalAlgHom.range f = (⊤ : NonUnitalSubalgebra R B) ↔ Function.Surjective f :=\n  NonUnitalAlgebra.eq_top_iff\n\n@[simp]\ntheorem range_id : NonUnitalAlgHom.range (NonUnitalAlgHom.id R A) = ⊤ :=\n  SetLike.coe_injective Set.range_id\n\n@[simp]\ntheorem map_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R A).map f = NonUnitalAlgHom.range f :=\n  SetLike.coe_injective Set.image_univ\n\n@[simp]\ntheorem map_bot (f : A →ₙₐ[R] B) : (⊥ : NonUnitalSubalgebra R A).map f = ⊥ :=\n  SetLike.coe_injective <| by simp [NonUnitalAlgebra.coe_bot, NonUnitalSubalgebra.coe_map]\n\n@[simp]\ntheorem comap_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R B).comap f = ⊤ :=\n  eq_top_iff.2 fun _ => mem_top\n\n/-- `NonUnitalAlgHom` to `⊤ : NonUnitalSubalgebra R A`. -/\ndef toTop : A →ₙₐ[R] (⊤ : NonUnitalSubalgebra R A) :=\n  NonUnitalAlgHom.codRestrict (NonUnitalAlgHom.id R A) ⊤ fun _ => mem_top\n\nend NonUnitalAlgebra\n\nnamespace NonUnitalSubalgebra\n\nopen NonUnitalAlgebra\n\nsection NonAssoc\n\nvariable {R : Type u} {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable (S : NonUnitalSubalgebra R A)\n\ninstance subsingleton_of_subsingleton [Subsingleton A] : Subsingleton (NonUnitalSubalgebra R A) :=\n  ⟨fun B C => ext fun x => by simp only [Subsingleton.elim x 0, zero_mem B, zero_mem C]⟩\n\ninstance _root_.NonUnitalAlgHom.subsingleton [Subsingleton (NonUnitalSubalgebra R A)] :\n    Subsingleton (A →ₙₐ[R] B) :=\n  ⟨fun f g =>\n    NonUnitalAlgHom.ext fun a =>\n      have : a ∈ (⊥ : NonUnitalSubalgebra R A) :=\n        Subsingleton.elim (⊤ : NonUnitalSubalgebra R A) ⊥ ▸ mem_top\n      (mem_bot.mp this).symm ▸ (map_zero f).trans (map_zero g).symm⟩\n\ntheorem range_val : NonUnitalAlgHom.range (NonUnitalSubalgebraClass.subtype S) = S :=\n  ext <| Set.ext_iff.1 <| (NonUnitalSubalgebraClass.subtype S).coe_range.trans Subtype.range_val\n\n/-- The map `S → T` when `S` is a non-unital subalgebra contained in the non-unital subalgebra `T`.\n\nThis is the non-unital subalgebra version of `Submodule.inclusion`, or `Subring.inclusion`  -/\ndef inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) : S →ₙₐ[R] T\n    where\n  toFun := Set.inclusion h\n  map_add' _ _ := rfl\n  map_mul' _ _ := rfl\n  map_zero' := rfl\n  map_smul' _ _ := rfl\n\ntheorem inclusion_injective {S T : NonUnitalSubalgebra R A} (h : S ≤ T) :\n    Function.Injective (inclusion h) := fun _ _ => Subtype.ext ∘ Subtype.mk.inj\n\n@[simp]\ntheorem inclusion_self {S : NonUnitalSubalgebra R A} :\n    inclusion (le_refl S) = NonUnitalAlgHom.id R S :=\n  NonUnitalAlgHom.ext fun _ => Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_mk {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : A) (hx : x ∈ S) :\n    inclusion h ⟨x, hx⟩ = ⟨x, h hx⟩ :=\n  rfl\n\ntheorem inclusion_right {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : T) (m : (x : A) ∈ S) :\n    inclusion h ⟨x, m⟩ = x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_inclusion {S T U : NonUnitalSubalgebra R A} (hst : S ≤ T) (htu : T ≤ U) (x : S) :\n    inclusion htu (inclusion hst x) = inclusion (le_trans hst htu) x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem coe_inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (s : S) :\n    (inclusion h s : A) = s :=\n  rfl\n\nsection Prod\n\nvariable (S₁ : NonUnitalSubalgebra R B)\n\n/-- The product of two non-unital subalgebras is a non-unital subalgebra. -/\ndef prod : NonUnitalSubalgebra R (A × B) :=\n  { S.toNonUnitalSubsemiring.prod S₁.toNonUnitalSubsemiring with\n    carrier := S ×ˢ S₁\n    smul_mem' := fun r _x hx => ⟨SMulMemClass.smul_mem r hx.1, SMulMemClass.smul_mem r hx.2⟩ }\n\n@[simp]\ntheorem coe_prod : (prod S S₁ : Set (A × B)) = (S : Set A) ×ˢ S₁ :=\n  rfl\n\ntheorem prod_toSubmodule : (S.prod S₁).toSubmodule = S.toSubmodule.prod S₁.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem mem_prod {S : NonUnitalSubalgebra R A} {S₁ : NonUnitalSubalgebra R B} {x : A × B} :\n    x ∈ prod S S₁ ↔ x.1 ∈ S ∧ x.2 ∈ S₁ :=\n  Set.mem_prod\n\n@[simp]\ntheorem prod_top : (prod ⊤ ⊤ : NonUnitalSubalgebra R (A × B)) = ⊤ := by ext; simp\n\ntheorem prod_mono {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S ≤ T → S₁ ≤ T₁ → prod S S₁ ≤ prod T T₁ :=\n  Set.prod_mono\n\n@[simp]\ntheorem prod_inf_prod {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S.prod S₁ ⊓ T.prod T₁ = (S ⊓ T).prod (S₁ ⊓ T₁) :=\n  SetLike.coe_injective Set.prod_inter_prod\n\nend Prod\n\nsection SuprLift\n\nvariable {ι : Type*}\n\ntheorem coe_iSup_of_directed [Nonempty ι] {S : ι → NonUnitalSubalgebra R A}\n    (dir : Directed (· ≤ ·) S) : ↑(iSup S) = ⋃ i, (S i : Set A) :=\n  let K : NonUnitalSubalgebra R A :=\n    { __ := NonUnitalSubsemiring.copy _ _ (NonUnitalSubsemiring.coe_iSup_of_directed dir).symm\n      smul_mem' := fun r _x hx ↦\n        let ⟨i, hi⟩ := Set.mem_iUnion.1 hx\n        Set.mem_iUnion.2 ⟨i, (S i).smul_mem' r hi⟩ }\n  have : iSup S = K := le_antisymm\n    (iSup_le fun i ↦ le_iSup (fun i ↦ (S i : Set A)) i) (Set.iUnion_subset fun _ ↦ le_iSup S _)\n  this.symm ▸ rfl\n\n/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              rw [hf j k hjk]\n              rfl)\n            (↑(iSup K)) (by rw [coe_iSup_of_directed dir])\n        map_zero' := by\n          ","nextTactic":"dsimp","declUpToTactic":"/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              rw [hf j k hjk]\n              rfl)\n            (↑(iSup K)) (by rw [coe_iSup_of_directed dir])\n        map_zero' := by\n          ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.892_0.jQvdAcOcyVirh23","decl":"/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B "}
-{"state":"R : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\nS : NonUnitalSubalgebra R A\nι : Type u_1\ninst✝ : Nonempty ι\nK : ι → NonUnitalSubalgebra R A\ndir : Directed (fun x x_1 => x ≤ x_1) K\nf : (i : ι) → ↥(K i) →ₙₐ[R] B\nhf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalAlgHom.comp (f j) (inclusion h)\n⊢ Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x)\n      (_ :\n        ∀ (i j : ι) (x : A) (hxi : x ∈ ↑(K i)) (hxj : x ∈ ↑(K j)),\n          (f i) { val := x, property := hxi } = (f j) { val := x, property := hxj })\n      ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ i, ↑(K i)) 0 =\n    0","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact NonUnitalAlgebra.gc\n\n@[simp]\ntheorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=\n  eq_top_iff.2 fun _x hx => subset_adjoin R hx\n\nvariable {R A}\n\n@[simp]\ntheorem coe_top : (↑(⊤ : NonUnitalSubalgebra R A) : Set A) = Set.univ :=\n  rfl\n\n@[simp]\ntheorem mem_top {x : A} : x ∈ (⊤ : NonUnitalSubalgebra R A) :=\n  Set.mem_univ x\n\n@[simp]\ntheorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toNonUnitalSubsemiring : (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubsemiring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toSubring {R A : Type*} [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    [IsScalarTower R A A] [SMulCommClass R A A] :\n    (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem toSubmodule_eq_top {S : NonUnitalSubalgebra R A} : S.toSubmodule = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toSubmodule'.injective.eq_iff' top_toSubmodule\n\n@[simp]\ntheorem toNonUnitalSubsemiring_eq_top {S : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubsemiring_injective.eq_iff' top_toNonUnitalSubsemiring\n\n@[simp]\ntheorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubring_injective.eq_iff' top_toSubring\n\ntheorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_left\n\ntheorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_right\n\ntheorem mul_mem_sup {S T : NonUnitalSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) :\n    x * y ∈ S ⊔ T :=\n  mul_mem (mem_sup_left hx) (mem_sup_right hy)\n\ntheorem map_sup (f : F) (S T : NonUnitalSubalgebra R A) :\n    ((S ⊔ T).map f : NonUnitalSubalgebra R B) = S.map f ⊔ T.map f :=\n  (@NonUnitalSubalgebra.gc_map_comap F R A B _ _ _ _ _ _ f).l_sup\n\n@[simp, norm_cast]\ntheorem coe_inf (S T : NonUnitalSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T :=\n  rfl\n\n@[simp]\ntheorem mem_inf {S T : NonUnitalSubalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T :=\n  Iff.rfl\n\n@[simp]\ntheorem inf_toSubmodule (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toSubmodule = S.toSubmodule ⊓ T.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem inf_toNonUnitalSubsemiring (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring ⊓ T.toNonUnitalSubsemiring :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sInf (S : Set (NonUnitalSubalgebra R A)) : (↑(sInf S) : Set A) = ⋂ s ∈ S, ↑s :=\n  sInf_image\n\ntheorem mem_sInf {S : Set (NonUnitalSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p := by\n  simp only [← SetLike.mem_coe, coe_sInf, Set.mem_iInter₂]\n\n@[simp]\ntheorem sInf_toSubmodule (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toSubmodule = sInf (NonUnitalSubalgebra.toSubmodule '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp]\ntheorem sInf_toNonUnitalSubsemiring (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toNonUnitalSubsemiring = sInf (NonUnitalSubalgebra.toNonUnitalSubsemiring '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp, norm_cast]\ntheorem coe_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} :\n    (↑(⨅ i, S i) : Set A) = ⋂ i, S i := by simp [iInf]\n\ntheorem mem_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} {x : A} :\n    (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by simp only [iInf, mem_sInf, Set.forall_range_iff]\n\n@[simp]\ntheorem iInf_toSubmodule {ι : Sort*} (S : ι → NonUnitalSubalgebra R A) :\n    (⨅ i, S i).toSubmodule = ⨅ i, (S i).toSubmodule :=\n  SetLike.coe_injective <| by simp\n\ninstance : Inhabited (NonUnitalSubalgebra R A) :=\n  ⟨⊥⟩\n\ntheorem mem_bot {x : A} : x ∈ (⊥ : NonUnitalSubalgebra R A) ↔ x = 0 :=\n  show x ∈ Submodule.span R (NonUnitalSubsemiring.closure (∅ : Set A) : Set A) ↔ x = 0 by\n    rw [NonUnitalSubsemiring.closure_empty, NonUnitalSubsemiring.coe_bot,\n      Submodule.span_zero_singleton, Submodule.mem_bot]\n\ntheorem toSubmodule_bot : (⊥ : NonUnitalSubalgebra R A).toSubmodule = ⊥ := by\n  ext\n  simp only [mem_bot, NonUnitalSubalgebra.mem_toSubmodule, Submodule.mem_bot]\n\n@[simp]\ntheorem coe_bot : ((⊥ : NonUnitalSubalgebra R A) : Set A) = {0} := by\n  simp [Set.ext_iff, NonUnitalAlgebra.mem_bot]\n\ntheorem eq_top_iff {S : NonUnitalSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S :=\n  ⟨fun h x => by rw [h]; exact mem_top, fun h => by ext x; exact ⟨fun _ => mem_top, fun _ => h x⟩⟩\n\ntheorem range_top_iff_surjective (f : A →ₙₐ[R] B) :\n    NonUnitalAlgHom.range f = (⊤ : NonUnitalSubalgebra R B) ↔ Function.Surjective f :=\n  NonUnitalAlgebra.eq_top_iff\n\n@[simp]\ntheorem range_id : NonUnitalAlgHom.range (NonUnitalAlgHom.id R A) = ⊤ :=\n  SetLike.coe_injective Set.range_id\n\n@[simp]\ntheorem map_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R A).map f = NonUnitalAlgHom.range f :=\n  SetLike.coe_injective Set.image_univ\n\n@[simp]\ntheorem map_bot (f : A →ₙₐ[R] B) : (⊥ : NonUnitalSubalgebra R A).map f = ⊥ :=\n  SetLike.coe_injective <| by simp [NonUnitalAlgebra.coe_bot, NonUnitalSubalgebra.coe_map]\n\n@[simp]\ntheorem comap_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R B).comap f = ⊤ :=\n  eq_top_iff.2 fun _ => mem_top\n\n/-- `NonUnitalAlgHom` to `⊤ : NonUnitalSubalgebra R A`. -/\ndef toTop : A →ₙₐ[R] (⊤ : NonUnitalSubalgebra R A) :=\n  NonUnitalAlgHom.codRestrict (NonUnitalAlgHom.id R A) ⊤ fun _ => mem_top\n\nend NonUnitalAlgebra\n\nnamespace NonUnitalSubalgebra\n\nopen NonUnitalAlgebra\n\nsection NonAssoc\n\nvariable {R : Type u} {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable (S : NonUnitalSubalgebra R A)\n\ninstance subsingleton_of_subsingleton [Subsingleton A] : Subsingleton (NonUnitalSubalgebra R A) :=\n  ⟨fun B C => ext fun x => by simp only [Subsingleton.elim x 0, zero_mem B, zero_mem C]⟩\n\ninstance _root_.NonUnitalAlgHom.subsingleton [Subsingleton (NonUnitalSubalgebra R A)] :\n    Subsingleton (A →ₙₐ[R] B) :=\n  ⟨fun f g =>\n    NonUnitalAlgHom.ext fun a =>\n      have : a ∈ (⊥ : NonUnitalSubalgebra R A) :=\n        Subsingleton.elim (⊤ : NonUnitalSubalgebra R A) ⊥ ▸ mem_top\n      (mem_bot.mp this).symm ▸ (map_zero f).trans (map_zero g).symm⟩\n\ntheorem range_val : NonUnitalAlgHom.range (NonUnitalSubalgebraClass.subtype S) = S :=\n  ext <| Set.ext_iff.1 <| (NonUnitalSubalgebraClass.subtype S).coe_range.trans Subtype.range_val\n\n/-- The map `S → T` when `S` is a non-unital subalgebra contained in the non-unital subalgebra `T`.\n\nThis is the non-unital subalgebra version of `Submodule.inclusion`, or `Subring.inclusion`  -/\ndef inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) : S →ₙₐ[R] T\n    where\n  toFun := Set.inclusion h\n  map_add' _ _ := rfl\n  map_mul' _ _ := rfl\n  map_zero' := rfl\n  map_smul' _ _ := rfl\n\ntheorem inclusion_injective {S T : NonUnitalSubalgebra R A} (h : S ≤ T) :\n    Function.Injective (inclusion h) := fun _ _ => Subtype.ext ∘ Subtype.mk.inj\n\n@[simp]\ntheorem inclusion_self {S : NonUnitalSubalgebra R A} :\n    inclusion (le_refl S) = NonUnitalAlgHom.id R S :=\n  NonUnitalAlgHom.ext fun _ => Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_mk {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : A) (hx : x ∈ S) :\n    inclusion h ⟨x, hx⟩ = ⟨x, h hx⟩ :=\n  rfl\n\ntheorem inclusion_right {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : T) (m : (x : A) ∈ S) :\n    inclusion h ⟨x, m⟩ = x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_inclusion {S T U : NonUnitalSubalgebra R A} (hst : S ≤ T) (htu : T ≤ U) (x : S) :\n    inclusion htu (inclusion hst x) = inclusion (le_trans hst htu) x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem coe_inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (s : S) :\n    (inclusion h s : A) = s :=\n  rfl\n\nsection Prod\n\nvariable (S₁ : NonUnitalSubalgebra R B)\n\n/-- The product of two non-unital subalgebras is a non-unital subalgebra. -/\ndef prod : NonUnitalSubalgebra R (A × B) :=\n  { S.toNonUnitalSubsemiring.prod S₁.toNonUnitalSubsemiring with\n    carrier := S ×ˢ S₁\n    smul_mem' := fun r _x hx => ⟨SMulMemClass.smul_mem r hx.1, SMulMemClass.smul_mem r hx.2⟩ }\n\n@[simp]\ntheorem coe_prod : (prod S S₁ : Set (A × B)) = (S : Set A) ×ˢ S₁ :=\n  rfl\n\ntheorem prod_toSubmodule : (S.prod S₁).toSubmodule = S.toSubmodule.prod S₁.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem mem_prod {S : NonUnitalSubalgebra R A} {S₁ : NonUnitalSubalgebra R B} {x : A × B} :\n    x ∈ prod S S₁ ↔ x.1 ∈ S ∧ x.2 ∈ S₁ :=\n  Set.mem_prod\n\n@[simp]\ntheorem prod_top : (prod ⊤ ⊤ : NonUnitalSubalgebra R (A × B)) = ⊤ := by ext; simp\n\ntheorem prod_mono {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S ≤ T → S₁ ≤ T₁ → prod S S₁ ≤ prod T T₁ :=\n  Set.prod_mono\n\n@[simp]\ntheorem prod_inf_prod {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S.prod S₁ ⊓ T.prod T₁ = (S ⊓ T).prod (S₁ ⊓ T₁) :=\n  SetLike.coe_injective Set.prod_inter_prod\n\nend Prod\n\nsection SuprLift\n\nvariable {ι : Type*}\n\ntheorem coe_iSup_of_directed [Nonempty ι] {S : ι → NonUnitalSubalgebra R A}\n    (dir : Directed (· ≤ ·) S) : ↑(iSup S) = ⋃ i, (S i : Set A) :=\n  let K : NonUnitalSubalgebra R A :=\n    { __ := NonUnitalSubsemiring.copy _ _ (NonUnitalSubsemiring.coe_iSup_of_directed dir).symm\n      smul_mem' := fun r _x hx ↦\n        let ⟨i, hi⟩ := Set.mem_iUnion.1 hx\n        Set.mem_iUnion.2 ⟨i, (S i).smul_mem' r hi⟩ }\n  have : iSup S = K := le_antisymm\n    (iSup_le fun i ↦ le_iSup (fun i ↦ (S i : Set A)) i) (Set.iUnion_subset fun _ ↦ le_iSup S _)\n  this.symm ▸ rfl\n\n/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              rw [hf j k hjk]\n              rfl)\n            (↑(iSup K)) (by rw [coe_iSup_of_directed dir])\n        map_zero' := by\n          dsimp\n          ","nextTactic":"exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl) _ (by simp)","declUpToTactic":"/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              rw [hf j k hjk]\n              rfl)\n            (↑(iSup K)) (by rw [coe_iSup_of_directed dir])\n        map_zero' := by\n          dsimp\n          ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.892_0.jQvdAcOcyVirh23","decl":"/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B "}
-{"state":"R : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\nS : NonUnitalSubalgebra R A\nι : Type u_1\ninst✝ : Nonempty ι\nK : ι → NonUnitalSubalgebra R A\ndir : Directed (fun x x_1 => x ≤ x_1) K\nf : (i : ι) → ↥(K i) →ₙₐ[R] B\nhf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalAlgHom.comp (f j) (inclusion h)\n⊢ ∀ (i : ι), (f i) ((fun i => 0) i) = 0","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact NonUnitalAlgebra.gc\n\n@[simp]\ntheorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=\n  eq_top_iff.2 fun _x hx => subset_adjoin R hx\n\nvariable {R A}\n\n@[simp]\ntheorem coe_top : (↑(⊤ : NonUnitalSubalgebra R A) : Set A) = Set.univ :=\n  rfl\n\n@[simp]\ntheorem mem_top {x : A} : x ∈ (⊤ : NonUnitalSubalgebra R A) :=\n  Set.mem_univ x\n\n@[simp]\ntheorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toNonUnitalSubsemiring : (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubsemiring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toSubring {R A : Type*} [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    [IsScalarTower R A A] [SMulCommClass R A A] :\n    (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem toSubmodule_eq_top {S : NonUnitalSubalgebra R A} : S.toSubmodule = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toSubmodule'.injective.eq_iff' top_toSubmodule\n\n@[simp]\ntheorem toNonUnitalSubsemiring_eq_top {S : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubsemiring_injective.eq_iff' top_toNonUnitalSubsemiring\n\n@[simp]\ntheorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubring_injective.eq_iff' top_toSubring\n\ntheorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_left\n\ntheorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_right\n\ntheorem mul_mem_sup {S T : NonUnitalSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) :\n    x * y ∈ S ⊔ T :=\n  mul_mem (mem_sup_left hx) (mem_sup_right hy)\n\ntheorem map_sup (f : F) (S T : NonUnitalSubalgebra R A) :\n    ((S ⊔ T).map f : NonUnitalSubalgebra R B) = S.map f ⊔ T.map f :=\n  (@NonUnitalSubalgebra.gc_map_comap F R A B _ _ _ _ _ _ f).l_sup\n\n@[simp, norm_cast]\ntheorem coe_inf (S T : NonUnitalSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T :=\n  rfl\n\n@[simp]\ntheorem mem_inf {S T : NonUnitalSubalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T :=\n  Iff.rfl\n\n@[simp]\ntheorem inf_toSubmodule (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toSubmodule = S.toSubmodule ⊓ T.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem inf_toNonUnitalSubsemiring (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring ⊓ T.toNonUnitalSubsemiring :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sInf (S : Set (NonUnitalSubalgebra R A)) : (↑(sInf S) : Set A) = ⋂ s ∈ S, ↑s :=\n  sInf_image\n\ntheorem mem_sInf {S : Set (NonUnitalSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p := by\n  simp only [← SetLike.mem_coe, coe_sInf, Set.mem_iInter₂]\n\n@[simp]\ntheorem sInf_toSubmodule (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toSubmodule = sInf (NonUnitalSubalgebra.toSubmodule '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp]\ntheorem sInf_toNonUnitalSubsemiring (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toNonUnitalSubsemiring = sInf (NonUnitalSubalgebra.toNonUnitalSubsemiring '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp, norm_cast]\ntheorem coe_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} :\n    (↑(⨅ i, S i) : Set A) = ⋂ i, S i := by simp [iInf]\n\ntheorem mem_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} {x : A} :\n    (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by simp only [iInf, mem_sInf, Set.forall_range_iff]\n\n@[simp]\ntheorem iInf_toSubmodule {ι : Sort*} (S : ι → NonUnitalSubalgebra R A) :\n    (⨅ i, S i).toSubmodule = ⨅ i, (S i).toSubmodule :=\n  SetLike.coe_injective <| by simp\n\ninstance : Inhabited (NonUnitalSubalgebra R A) :=\n  ⟨⊥⟩\n\ntheorem mem_bot {x : A} : x ∈ (⊥ : NonUnitalSubalgebra R A) ↔ x = 0 :=\n  show x ∈ Submodule.span R (NonUnitalSubsemiring.closure (∅ : Set A) : Set A) ↔ x = 0 by\n    rw [NonUnitalSubsemiring.closure_empty, NonUnitalSubsemiring.coe_bot,\n      Submodule.span_zero_singleton, Submodule.mem_bot]\n\ntheorem toSubmodule_bot : (⊥ : NonUnitalSubalgebra R A).toSubmodule = ⊥ := by\n  ext\n  simp only [mem_bot, NonUnitalSubalgebra.mem_toSubmodule, Submodule.mem_bot]\n\n@[simp]\ntheorem coe_bot : ((⊥ : NonUnitalSubalgebra R A) : Set A) = {0} := by\n  simp [Set.ext_iff, NonUnitalAlgebra.mem_bot]\n\ntheorem eq_top_iff {S : NonUnitalSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S :=\n  ⟨fun h x => by rw [h]; exact mem_top, fun h => by ext x; exact ⟨fun _ => mem_top, fun _ => h x⟩⟩\n\ntheorem range_top_iff_surjective (f : A →ₙₐ[R] B) :\n    NonUnitalAlgHom.range f = (⊤ : NonUnitalSubalgebra R B) ↔ Function.Surjective f :=\n  NonUnitalAlgebra.eq_top_iff\n\n@[simp]\ntheorem range_id : NonUnitalAlgHom.range (NonUnitalAlgHom.id R A) = ⊤ :=\n  SetLike.coe_injective Set.range_id\n\n@[simp]\ntheorem map_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R A).map f = NonUnitalAlgHom.range f :=\n  SetLike.coe_injective Set.image_univ\n\n@[simp]\ntheorem map_bot (f : A →ₙₐ[R] B) : (⊥ : NonUnitalSubalgebra R A).map f = ⊥ :=\n  SetLike.coe_injective <| by simp [NonUnitalAlgebra.coe_bot, NonUnitalSubalgebra.coe_map]\n\n@[simp]\ntheorem comap_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R B).comap f = ⊤ :=\n  eq_top_iff.2 fun _ => mem_top\n\n/-- `NonUnitalAlgHom` to `⊤ : NonUnitalSubalgebra R A`. -/\ndef toTop : A →ₙₐ[R] (⊤ : NonUnitalSubalgebra R A) :=\n  NonUnitalAlgHom.codRestrict (NonUnitalAlgHom.id R A) ⊤ fun _ => mem_top\n\nend NonUnitalAlgebra\n\nnamespace NonUnitalSubalgebra\n\nopen NonUnitalAlgebra\n\nsection NonAssoc\n\nvariable {R : Type u} {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable (S : NonUnitalSubalgebra R A)\n\ninstance subsingleton_of_subsingleton [Subsingleton A] : Subsingleton (NonUnitalSubalgebra R A) :=\n  ⟨fun B C => ext fun x => by simp only [Subsingleton.elim x 0, zero_mem B, zero_mem C]⟩\n\ninstance _root_.NonUnitalAlgHom.subsingleton [Subsingleton (NonUnitalSubalgebra R A)] :\n    Subsingleton (A →ₙₐ[R] B) :=\n  ⟨fun f g =>\n    NonUnitalAlgHom.ext fun a =>\n      have : a ∈ (⊥ : NonUnitalSubalgebra R A) :=\n        Subsingleton.elim (⊤ : NonUnitalSubalgebra R A) ⊥ ▸ mem_top\n      (mem_bot.mp this).symm ▸ (map_zero f).trans (map_zero g).symm⟩\n\ntheorem range_val : NonUnitalAlgHom.range (NonUnitalSubalgebraClass.subtype S) = S :=\n  ext <| Set.ext_iff.1 <| (NonUnitalSubalgebraClass.subtype S).coe_range.trans Subtype.range_val\n\n/-- The map `S → T` when `S` is a non-unital subalgebra contained in the non-unital subalgebra `T`.\n\nThis is the non-unital subalgebra version of `Submodule.inclusion`, or `Subring.inclusion`  -/\ndef inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) : S →ₙₐ[R] T\n    where\n  toFun := Set.inclusion h\n  map_add' _ _ := rfl\n  map_mul' _ _ := rfl\n  map_zero' := rfl\n  map_smul' _ _ := rfl\n\ntheorem inclusion_injective {S T : NonUnitalSubalgebra R A} (h : S ≤ T) :\n    Function.Injective (inclusion h) := fun _ _ => Subtype.ext ∘ Subtype.mk.inj\n\n@[simp]\ntheorem inclusion_self {S : NonUnitalSubalgebra R A} :\n    inclusion (le_refl S) = NonUnitalAlgHom.id R S :=\n  NonUnitalAlgHom.ext fun _ => Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_mk {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : A) (hx : x ∈ S) :\n    inclusion h ⟨x, hx⟩ = ⟨x, h hx⟩ :=\n  rfl\n\ntheorem inclusion_right {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : T) (m : (x : A) ∈ S) :\n    inclusion h ⟨x, m⟩ = x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_inclusion {S T U : NonUnitalSubalgebra R A} (hst : S ≤ T) (htu : T ≤ U) (x : S) :\n    inclusion htu (inclusion hst x) = inclusion (le_trans hst htu) x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem coe_inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (s : S) :\n    (inclusion h s : A) = s :=\n  rfl\n\nsection Prod\n\nvariable (S₁ : NonUnitalSubalgebra R B)\n\n/-- The product of two non-unital subalgebras is a non-unital subalgebra. -/\ndef prod : NonUnitalSubalgebra R (A × B) :=\n  { S.toNonUnitalSubsemiring.prod S₁.toNonUnitalSubsemiring with\n    carrier := S ×ˢ S₁\n    smul_mem' := fun r _x hx => ⟨SMulMemClass.smul_mem r hx.1, SMulMemClass.smul_mem r hx.2⟩ }\n\n@[simp]\ntheorem coe_prod : (prod S S₁ : Set (A × B)) = (S : Set A) ×ˢ S₁ :=\n  rfl\n\ntheorem prod_toSubmodule : (S.prod S₁).toSubmodule = S.toSubmodule.prod S₁.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem mem_prod {S : NonUnitalSubalgebra R A} {S₁ : NonUnitalSubalgebra R B} {x : A × B} :\n    x ∈ prod S S₁ ↔ x.1 ∈ S ∧ x.2 ∈ S₁ :=\n  Set.mem_prod\n\n@[simp]\ntheorem prod_top : (prod ⊤ ⊤ : NonUnitalSubalgebra R (A × B)) = ⊤ := by ext; simp\n\ntheorem prod_mono {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S ≤ T → S₁ ≤ T₁ → prod S S₁ ≤ prod T T₁ :=\n  Set.prod_mono\n\n@[simp]\ntheorem prod_inf_prod {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S.prod S₁ ⊓ T.prod T₁ = (S ⊓ T).prod (S₁ ⊓ T₁) :=\n  SetLike.coe_injective Set.prod_inter_prod\n\nend Prod\n\nsection SuprLift\n\nvariable {ι : Type*}\n\ntheorem coe_iSup_of_directed [Nonempty ι] {S : ι → NonUnitalSubalgebra R A}\n    (dir : Directed (· ≤ ·) S) : ↑(iSup S) = ⋃ i, (S i : Set A) :=\n  let K : NonUnitalSubalgebra R A :=\n    { __ := NonUnitalSubsemiring.copy _ _ (NonUnitalSubsemiring.coe_iSup_of_directed dir).symm\n      smul_mem' := fun r _x hx ↦\n        let ⟨i, hi⟩ := Set.mem_iUnion.1 hx\n        Set.mem_iUnion.2 ⟨i, (S i).smul_mem' r hi⟩ }\n  have : iSup S = K := le_antisymm\n    (iSup_le fun i ↦ le_iSup (fun i ↦ (S i : Set A)) i) (Set.iUnion_subset fun _ ↦ le_iSup S _)\n  this.symm ▸ rfl\n\n/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              rw [hf j k hjk]\n              rfl)\n            (↑(iSup K)) (by rw [coe_iSup_of_directed dir])\n        map_zero' := by\n          dsimp\n          exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl) _ (by ","nextTactic":"simp","declUpToTactic":"/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              rw [hf j k hjk]\n              rfl)\n            (↑(iSup K)) (by rw [coe_iSup_of_directed dir])\n        map_zero' := by\n          dsimp\n          exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl) _ (by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.892_0.jQvdAcOcyVirh23","decl":"/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B "}
-{"state":"R : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\nS : NonUnitalSubalgebra R A\nι : Type u_1\ninst✝ : Nonempty ι\nK : ι → NonUnitalSubalgebra R A\ndir : Directed (fun x x_1 => x ≤ x_1) K\nf : (i : ι) → ↥(K i) →ₙₐ[R] B\nhf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalAlgHom.comp (f j) (inclusion h)\n⊢ ∀ (x y : ↥(iSup K)),\n    MulActionHom.toFun\n        {\n          toFun :=\n            Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x)\n              (_ :\n                ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),\n                  (fun i x => (f i) x) i { val := x, property := hxi } =\n                    (fun i x => (f i) x) j { val := x, property := hxj })\n              ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ i, ↑(K i)),\n          map_smul' :=\n            (_ :\n              ∀ (r : R) (x : ↥(iSup K)),\n                Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x)\n                    (_ :\n                      ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),\n                        (fun i x => (f i) x) i { val := x, property := hxi } =\n                          (fun i x => (f i) x) j { val := x, property := hxj })\n                    ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ i, ↑(K i)) (r • x) =\n                  r •\n                    Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x)\n                      (_ :\n                        ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),\n                          (fun i x => (f i) x) i { val := x, property := hxi } =\n                            (fun i x => (f i) x) j { val := x, property := hxj })\n                      ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ i, ↑(K i)) x) }\n        (x + y) =\n      MulActionHom.toFun\n          {\n            toFun :=\n              Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x)\n                (_ :\n                  ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),\n                    (fun i x => (f i) x) i { val := x, property := hxi } =\n                      (fun i x => (f i) x) j { val := x, property := hxj })\n                ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ i, ↑(K i)),\n            map_smul' :=\n              (_ :\n                ∀ (r : R) (x : ↥(iSup K)),\n                  Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x)\n                      (_ :\n                        ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),\n                          (fun i x => (f i) x) i { val := x, property := hxi } =\n                            (fun i x => (f i) x) j { val := x, property := hxj })\n                      ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ i, ↑(K i)) (r • x) =\n                    r •\n                      Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x)\n                        (_ :\n                          ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),\n                            (fun i x => (f i) x) i { val := x, property := hxi } =\n                              (fun i x => (f i) x) j { val := x, property := hxj })\n                        ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ i, ↑(K i)) x) }\n          x +\n        MulActionHom.toFun\n          {\n            toFun :=\n              Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x)\n                (_ :\n                  ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),\n                    (fun i x => (f i) x) i { val := x, property := hxi } =\n                      (fun i x => (f i) x) j { val := x, property := hxj })\n                ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ i, ↑(K i)),\n            map_smul' :=\n              (_ :\n                ∀ (r : R) (x : ↥(iSup K)),\n                  Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x)\n                      (_ :\n                        ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),\n                          (fun i x => (f i) x) i { val := x, property := hxi } =\n                            (fun i x => (f i) x) j { val := x, property := hxj })\n                      ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ i, ↑(K i)) (r • x) =\n                    r •\n                      Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x)\n                        (_ :\n                          ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),\n                            (fun i x => (f i) x) i { val := x, property := hxi } =\n                              (fun i x => (f i) x) j { val := x, property := hxj })\n                        ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ i, ↑(K i)) x) }\n          y","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact NonUnitalAlgebra.gc\n\n@[simp]\ntheorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=\n  eq_top_iff.2 fun _x hx => subset_adjoin R hx\n\nvariable {R A}\n\n@[simp]\ntheorem coe_top : (↑(⊤ : NonUnitalSubalgebra R A) : Set A) = Set.univ :=\n  rfl\n\n@[simp]\ntheorem mem_top {x : A} : x ∈ (⊤ : NonUnitalSubalgebra R A) :=\n  Set.mem_univ x\n\n@[simp]\ntheorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toNonUnitalSubsemiring : (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubsemiring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toSubring {R A : Type*} [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    [IsScalarTower R A A] [SMulCommClass R A A] :\n    (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem toSubmodule_eq_top {S : NonUnitalSubalgebra R A} : S.toSubmodule = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toSubmodule'.injective.eq_iff' top_toSubmodule\n\n@[simp]\ntheorem toNonUnitalSubsemiring_eq_top {S : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubsemiring_injective.eq_iff' top_toNonUnitalSubsemiring\n\n@[simp]\ntheorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubring_injective.eq_iff' top_toSubring\n\ntheorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_left\n\ntheorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_right\n\ntheorem mul_mem_sup {S T : NonUnitalSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) :\n    x * y ∈ S ⊔ T :=\n  mul_mem (mem_sup_left hx) (mem_sup_right hy)\n\ntheorem map_sup (f : F) (S T : NonUnitalSubalgebra R A) :\n    ((S ⊔ T).map f : NonUnitalSubalgebra R B) = S.map f ⊔ T.map f :=\n  (@NonUnitalSubalgebra.gc_map_comap F R A B _ _ _ _ _ _ f).l_sup\n\n@[simp, norm_cast]\ntheorem coe_inf (S T : NonUnitalSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T :=\n  rfl\n\n@[simp]\ntheorem mem_inf {S T : NonUnitalSubalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T :=\n  Iff.rfl\n\n@[simp]\ntheorem inf_toSubmodule (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toSubmodule = S.toSubmodule ⊓ T.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem inf_toNonUnitalSubsemiring (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring ⊓ T.toNonUnitalSubsemiring :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sInf (S : Set (NonUnitalSubalgebra R A)) : (↑(sInf S) : Set A) = ⋂ s ∈ S, ↑s :=\n  sInf_image\n\ntheorem mem_sInf {S : Set (NonUnitalSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p := by\n  simp only [← SetLike.mem_coe, coe_sInf, Set.mem_iInter₂]\n\n@[simp]\ntheorem sInf_toSubmodule (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toSubmodule = sInf (NonUnitalSubalgebra.toSubmodule '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp]\ntheorem sInf_toNonUnitalSubsemiring (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toNonUnitalSubsemiring = sInf (NonUnitalSubalgebra.toNonUnitalSubsemiring '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp, norm_cast]\ntheorem coe_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} :\n    (↑(⨅ i, S i) : Set A) = ⋂ i, S i := by simp [iInf]\n\ntheorem mem_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} {x : A} :\n    (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by simp only [iInf, mem_sInf, Set.forall_range_iff]\n\n@[simp]\ntheorem iInf_toSubmodule {ι : Sort*} (S : ι → NonUnitalSubalgebra R A) :\n    (⨅ i, S i).toSubmodule = ⨅ i, (S i).toSubmodule :=\n  SetLike.coe_injective <| by simp\n\ninstance : Inhabited (NonUnitalSubalgebra R A) :=\n  ⟨⊥⟩\n\ntheorem mem_bot {x : A} : x ∈ (⊥ : NonUnitalSubalgebra R A) ↔ x = 0 :=\n  show x ∈ Submodule.span R (NonUnitalSubsemiring.closure (∅ : Set A) : Set A) ↔ x = 0 by\n    rw [NonUnitalSubsemiring.closure_empty, NonUnitalSubsemiring.coe_bot,\n      Submodule.span_zero_singleton, Submodule.mem_bot]\n\ntheorem toSubmodule_bot : (⊥ : NonUnitalSubalgebra R A).toSubmodule = ⊥ := by\n  ext\n  simp only [mem_bot, NonUnitalSubalgebra.mem_toSubmodule, Submodule.mem_bot]\n\n@[simp]\ntheorem coe_bot : ((⊥ : NonUnitalSubalgebra R A) : Set A) = {0} := by\n  simp [Set.ext_iff, NonUnitalAlgebra.mem_bot]\n\ntheorem eq_top_iff {S : NonUnitalSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S :=\n  ⟨fun h x => by rw [h]; exact mem_top, fun h => by ext x; exact ⟨fun _ => mem_top, fun _ => h x⟩⟩\n\ntheorem range_top_iff_surjective (f : A →ₙₐ[R] B) :\n    NonUnitalAlgHom.range f = (⊤ : NonUnitalSubalgebra R B) ↔ Function.Surjective f :=\n  NonUnitalAlgebra.eq_top_iff\n\n@[simp]\ntheorem range_id : NonUnitalAlgHom.range (NonUnitalAlgHom.id R A) = ⊤ :=\n  SetLike.coe_injective Set.range_id\n\n@[simp]\ntheorem map_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R A).map f = NonUnitalAlgHom.range f :=\n  SetLike.coe_injective Set.image_univ\n\n@[simp]\ntheorem map_bot (f : A →ₙₐ[R] B) : (⊥ : NonUnitalSubalgebra R A).map f = ⊥ :=\n  SetLike.coe_injective <| by simp [NonUnitalAlgebra.coe_bot, NonUnitalSubalgebra.coe_map]\n\n@[simp]\ntheorem comap_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R B).comap f = ⊤ :=\n  eq_top_iff.2 fun _ => mem_top\n\n/-- `NonUnitalAlgHom` to `⊤ : NonUnitalSubalgebra R A`. -/\ndef toTop : A →ₙₐ[R] (⊤ : NonUnitalSubalgebra R A) :=\n  NonUnitalAlgHom.codRestrict (NonUnitalAlgHom.id R A) ⊤ fun _ => mem_top\n\nend NonUnitalAlgebra\n\nnamespace NonUnitalSubalgebra\n\nopen NonUnitalAlgebra\n\nsection NonAssoc\n\nvariable {R : Type u} {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable (S : NonUnitalSubalgebra R A)\n\ninstance subsingleton_of_subsingleton [Subsingleton A] : Subsingleton (NonUnitalSubalgebra R A) :=\n  ⟨fun B C => ext fun x => by simp only [Subsingleton.elim x 0, zero_mem B, zero_mem C]⟩\n\ninstance _root_.NonUnitalAlgHom.subsingleton [Subsingleton (NonUnitalSubalgebra R A)] :\n    Subsingleton (A →ₙₐ[R] B) :=\n  ⟨fun f g =>\n    NonUnitalAlgHom.ext fun a =>\n      have : a ∈ (⊥ : NonUnitalSubalgebra R A) :=\n        Subsingleton.elim (⊤ : NonUnitalSubalgebra R A) ⊥ ▸ mem_top\n      (mem_bot.mp this).symm ▸ (map_zero f).trans (map_zero g).symm⟩\n\ntheorem range_val : NonUnitalAlgHom.range (NonUnitalSubalgebraClass.subtype S) = S :=\n  ext <| Set.ext_iff.1 <| (NonUnitalSubalgebraClass.subtype S).coe_range.trans Subtype.range_val\n\n/-- The map `S → T` when `S` is a non-unital subalgebra contained in the non-unital subalgebra `T`.\n\nThis is the non-unital subalgebra version of `Submodule.inclusion`, or `Subring.inclusion`  -/\ndef inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) : S →ₙₐ[R] T\n    where\n  toFun := Set.inclusion h\n  map_add' _ _ := rfl\n  map_mul' _ _ := rfl\n  map_zero' := rfl\n  map_smul' _ _ := rfl\n\ntheorem inclusion_injective {S T : NonUnitalSubalgebra R A} (h : S ≤ T) :\n    Function.Injective (inclusion h) := fun _ _ => Subtype.ext ∘ Subtype.mk.inj\n\n@[simp]\ntheorem inclusion_self {S : NonUnitalSubalgebra R A} :\n    inclusion (le_refl S) = NonUnitalAlgHom.id R S :=\n  NonUnitalAlgHom.ext fun _ => Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_mk {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : A) (hx : x ∈ S) :\n    inclusion h ⟨x, hx⟩ = ⟨x, h hx⟩ :=\n  rfl\n\ntheorem inclusion_right {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : T) (m : (x : A) ∈ S) :\n    inclusion h ⟨x, m⟩ = x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_inclusion {S T U : NonUnitalSubalgebra R A} (hst : S ≤ T) (htu : T ≤ U) (x : S) :\n    inclusion htu (inclusion hst x) = inclusion (le_trans hst htu) x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem coe_inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (s : S) :\n    (inclusion h s : A) = s :=\n  rfl\n\nsection Prod\n\nvariable (S₁ : NonUnitalSubalgebra R B)\n\n/-- The product of two non-unital subalgebras is a non-unital subalgebra. -/\ndef prod : NonUnitalSubalgebra R (A × B) :=\n  { S.toNonUnitalSubsemiring.prod S₁.toNonUnitalSubsemiring with\n    carrier := S ×ˢ S₁\n    smul_mem' := fun r _x hx => ⟨SMulMemClass.smul_mem r hx.1, SMulMemClass.smul_mem r hx.2⟩ }\n\n@[simp]\ntheorem coe_prod : (prod S S₁ : Set (A × B)) = (S : Set A) ×ˢ S₁ :=\n  rfl\n\ntheorem prod_toSubmodule : (S.prod S₁).toSubmodule = S.toSubmodule.prod S₁.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem mem_prod {S : NonUnitalSubalgebra R A} {S₁ : NonUnitalSubalgebra R B} {x : A × B} :\n    x ∈ prod S S₁ ↔ x.1 ∈ S ∧ x.2 ∈ S₁ :=\n  Set.mem_prod\n\n@[simp]\ntheorem prod_top : (prod ⊤ ⊤ : NonUnitalSubalgebra R (A × B)) = ⊤ := by ext; simp\n\ntheorem prod_mono {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S ≤ T → S₁ ≤ T₁ → prod S S₁ ≤ prod T T₁ :=\n  Set.prod_mono\n\n@[simp]\ntheorem prod_inf_prod {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S.prod S₁ ⊓ T.prod T₁ = (S ⊓ T).prod (S₁ ⊓ T₁) :=\n  SetLike.coe_injective Set.prod_inter_prod\n\nend Prod\n\nsection SuprLift\n\nvariable {ι : Type*}\n\ntheorem coe_iSup_of_directed [Nonempty ι] {S : ι → NonUnitalSubalgebra R A}\n    (dir : Directed (· ≤ ·) S) : ↑(iSup S) = ⋃ i, (S i : Set A) :=\n  let K : NonUnitalSubalgebra R A :=\n    { __ := NonUnitalSubsemiring.copy _ _ (NonUnitalSubsemiring.coe_iSup_of_directed dir).symm\n      smul_mem' := fun r _x hx ↦\n        let ⟨i, hi⟩ := Set.mem_iUnion.1 hx\n        Set.mem_iUnion.2 ⟨i, (S i).smul_mem' r hi⟩ }\n  have : iSup S = K := le_antisymm\n    (iSup_le fun i ↦ le_iSup (fun i ↦ (S i : Set A)) i) (Set.iUnion_subset fun _ ↦ le_iSup S _)\n  this.symm ▸ rfl\n\n/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              rw [hf j k hjk]\n              rfl)\n            (↑(iSup K)) (by rw [coe_iSup_of_directed dir])\n        map_zero' := by\n          dsimp\n          exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl) _ (by simp)\n        map_mul' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· * ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_add' := by\n          ","nextTactic":"dsimp","declUpToTactic":"/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T ���ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              rw [hf j k hjk]\n              rfl)\n            (↑(iSup K)) (by rw [coe_iSup_of_directed dir])\n        map_zero' := by\n          dsimp\n          exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl) _ (by simp)\n        map_mul' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· * ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_add' := by\n          ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.892_0.jQvdAcOcyVirh23","decl":"/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B "}
-{"state":"R : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\nS : NonUnitalSubalgebra R A\nι : Type u_1\ninst✝ : Nonempty ι\nK : ι → NonUnitalSubalgebra R A\ndir : Directed (fun x x_1 => x ≤ x_1) K\nf : (i : ι) → ↥(K i) →ₙₐ[R] B\nhf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalAlgHom.comp (f j) (inclusion h)\n⊢ ∀ (x y : ↥(iSup K)),\n    Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x)\n        (_ :\n          ∀ (i j : ι) (x : A) (hxi : x ∈ ↑(K i)) (hxj : x ∈ ↑(K j)),\n            (f i) { val := x, property := hxi } = (f j) { val := x, property := hxj })\n        ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ i, ↑(K i)) (x + y) =\n      Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x)\n          (_ :\n            ∀ (i j : ι) (x : A) (hxi : x ∈ ↑(K i)) (hxj : x ∈ ↑(K j)),\n              (f i) { val := x, property := hxi } = (f j) { val := x, property := hxj })\n          ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ i, ↑(K i)) x +\n        Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x)\n          (_ :\n            ∀ (i j : ι) (x : A) (hxi : x ∈ ↑(K i)) (hxj : x ∈ ↑(K j)),\n              (f i) { val := x, property := hxi } = (f j) { val := x, property := hxj })\n          ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ i, ↑(K i)) y","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact NonUnitalAlgebra.gc\n\n@[simp]\ntheorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=\n  eq_top_iff.2 fun _x hx => subset_adjoin R hx\n\nvariable {R A}\n\n@[simp]\ntheorem coe_top : (↑(⊤ : NonUnitalSubalgebra R A) : Set A) = Set.univ :=\n  rfl\n\n@[simp]\ntheorem mem_top {x : A} : x ∈ (⊤ : NonUnitalSubalgebra R A) :=\n  Set.mem_univ x\n\n@[simp]\ntheorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toNonUnitalSubsemiring : (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubsemiring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toSubring {R A : Type*} [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    [IsScalarTower R A A] [SMulCommClass R A A] :\n    (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem toSubmodule_eq_top {S : NonUnitalSubalgebra R A} : S.toSubmodule = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toSubmodule'.injective.eq_iff' top_toSubmodule\n\n@[simp]\ntheorem toNonUnitalSubsemiring_eq_top {S : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubsemiring_injective.eq_iff' top_toNonUnitalSubsemiring\n\n@[simp]\ntheorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubring_injective.eq_iff' top_toSubring\n\ntheorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_left\n\ntheorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_right\n\ntheorem mul_mem_sup {S T : NonUnitalSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) :\n    x * y ∈ S ⊔ T :=\n  mul_mem (mem_sup_left hx) (mem_sup_right hy)\n\ntheorem map_sup (f : F) (S T : NonUnitalSubalgebra R A) :\n    ((S ⊔ T).map f : NonUnitalSubalgebra R B) = S.map f ⊔ T.map f :=\n  (@NonUnitalSubalgebra.gc_map_comap F R A B _ _ _ _ _ _ f).l_sup\n\n@[simp, norm_cast]\ntheorem coe_inf (S T : NonUnitalSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T :=\n  rfl\n\n@[simp]\ntheorem mem_inf {S T : NonUnitalSubalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T :=\n  Iff.rfl\n\n@[simp]\ntheorem inf_toSubmodule (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toSubmodule = S.toSubmodule ⊓ T.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem inf_toNonUnitalSubsemiring (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring ⊓ T.toNonUnitalSubsemiring :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sInf (S : Set (NonUnitalSubalgebra R A)) : (↑(sInf S) : Set A) = ⋂ s ∈ S, ↑s :=\n  sInf_image\n\ntheorem mem_sInf {S : Set (NonUnitalSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p := by\n  simp only [← SetLike.mem_coe, coe_sInf, Set.mem_iInter₂]\n\n@[simp]\ntheorem sInf_toSubmodule (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toSubmodule = sInf (NonUnitalSubalgebra.toSubmodule '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp]\ntheorem sInf_toNonUnitalSubsemiring (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toNonUnitalSubsemiring = sInf (NonUnitalSubalgebra.toNonUnitalSubsemiring '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp, norm_cast]\ntheorem coe_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} :\n    (↑(⨅ i, S i) : Set A) = ⋂ i, S i := by simp [iInf]\n\ntheorem mem_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} {x : A} :\n    (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by simp only [iInf, mem_sInf, Set.forall_range_iff]\n\n@[simp]\ntheorem iInf_toSubmodule {ι : Sort*} (S : ι → NonUnitalSubalgebra R A) :\n    (⨅ i, S i).toSubmodule = ⨅ i, (S i).toSubmodule :=\n  SetLike.coe_injective <| by simp\n\ninstance : Inhabited (NonUnitalSubalgebra R A) :=\n  ⟨⊥⟩\n\ntheorem mem_bot {x : A} : x ∈ (⊥ : NonUnitalSubalgebra R A) ↔ x = 0 :=\n  show x ∈ Submodule.span R (NonUnitalSubsemiring.closure (∅ : Set A) : Set A) ↔ x = 0 by\n    rw [NonUnitalSubsemiring.closure_empty, NonUnitalSubsemiring.coe_bot,\n      Submodule.span_zero_singleton, Submodule.mem_bot]\n\ntheorem toSubmodule_bot : (⊥ : NonUnitalSubalgebra R A).toSubmodule = ⊥ := by\n  ext\n  simp only [mem_bot, NonUnitalSubalgebra.mem_toSubmodule, Submodule.mem_bot]\n\n@[simp]\ntheorem coe_bot : ((⊥ : NonUnitalSubalgebra R A) : Set A) = {0} := by\n  simp [Set.ext_iff, NonUnitalAlgebra.mem_bot]\n\ntheorem eq_top_iff {S : NonUnitalSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S :=\n  ⟨fun h x => by rw [h]; exact mem_top, fun h => by ext x; exact ⟨fun _ => mem_top, fun _ => h x⟩⟩\n\ntheorem range_top_iff_surjective (f : A →ₙₐ[R] B) :\n    NonUnitalAlgHom.range f = (⊤ : NonUnitalSubalgebra R B) ↔ Function.Surjective f :=\n  NonUnitalAlgebra.eq_top_iff\n\n@[simp]\ntheorem range_id : NonUnitalAlgHom.range (NonUnitalAlgHom.id R A) = ⊤ :=\n  SetLike.coe_injective Set.range_id\n\n@[simp]\ntheorem map_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R A).map f = NonUnitalAlgHom.range f :=\n  SetLike.coe_injective Set.image_univ\n\n@[simp]\ntheorem map_bot (f : A →ₙₐ[R] B) : (⊥ : NonUnitalSubalgebra R A).map f = ⊥ :=\n  SetLike.coe_injective <| by simp [NonUnitalAlgebra.coe_bot, NonUnitalSubalgebra.coe_map]\n\n@[simp]\ntheorem comap_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R B).comap f = ⊤ :=\n  eq_top_iff.2 fun _ => mem_top\n\n/-- `NonUnitalAlgHom` to `⊤ : NonUnitalSubalgebra R A`. -/\ndef toTop : A →ₙₐ[R] (⊤ : NonUnitalSubalgebra R A) :=\n  NonUnitalAlgHom.codRestrict (NonUnitalAlgHom.id R A) ⊤ fun _ => mem_top\n\nend NonUnitalAlgebra\n\nnamespace NonUnitalSubalgebra\n\nopen NonUnitalAlgebra\n\nsection NonAssoc\n\nvariable {R : Type u} {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable (S : NonUnitalSubalgebra R A)\n\ninstance subsingleton_of_subsingleton [Subsingleton A] : Subsingleton (NonUnitalSubalgebra R A) :=\n  ⟨fun B C => ext fun x => by simp only [Subsingleton.elim x 0, zero_mem B, zero_mem C]⟩\n\ninstance _root_.NonUnitalAlgHom.subsingleton [Subsingleton (NonUnitalSubalgebra R A)] :\n    Subsingleton (A →ₙₐ[R] B) :=\n  ⟨fun f g =>\n    NonUnitalAlgHom.ext fun a =>\n      have : a ∈ (⊥ : NonUnitalSubalgebra R A) :=\n        Subsingleton.elim (⊤ : NonUnitalSubalgebra R A) ⊥ ▸ mem_top\n      (mem_bot.mp this).symm ▸ (map_zero f).trans (map_zero g).symm⟩\n\ntheorem range_val : NonUnitalAlgHom.range (NonUnitalSubalgebraClass.subtype S) = S :=\n  ext <| Set.ext_iff.1 <| (NonUnitalSubalgebraClass.subtype S).coe_range.trans Subtype.range_val\n\n/-- The map `S → T` when `S` is a non-unital subalgebra contained in the non-unital subalgebra `T`.\n\nThis is the non-unital subalgebra version of `Submodule.inclusion`, or `Subring.inclusion`  -/\ndef inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) : S →ₙₐ[R] T\n    where\n  toFun := Set.inclusion h\n  map_add' _ _ := rfl\n  map_mul' _ _ := rfl\n  map_zero' := rfl\n  map_smul' _ _ := rfl\n\ntheorem inclusion_injective {S T : NonUnitalSubalgebra R A} (h : S ≤ T) :\n    Function.Injective (inclusion h) := fun _ _ => Subtype.ext ∘ Subtype.mk.inj\n\n@[simp]\ntheorem inclusion_self {S : NonUnitalSubalgebra R A} :\n    inclusion (le_refl S) = NonUnitalAlgHom.id R S :=\n  NonUnitalAlgHom.ext fun _ => Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_mk {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : A) (hx : x ∈ S) :\n    inclusion h ⟨x, hx⟩ = ⟨x, h hx⟩ :=\n  rfl\n\ntheorem inclusion_right {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : T) (m : (x : A) ∈ S) :\n    inclusion h ⟨x, m⟩ = x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_inclusion {S T U : NonUnitalSubalgebra R A} (hst : S ≤ T) (htu : T ≤ U) (x : S) :\n    inclusion htu (inclusion hst x) = inclusion (le_trans hst htu) x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem coe_inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (s : S) :\n    (inclusion h s : A) = s :=\n  rfl\n\nsection Prod\n\nvariable (S₁ : NonUnitalSubalgebra R B)\n\n/-- The product of two non-unital subalgebras is a non-unital subalgebra. -/\ndef prod : NonUnitalSubalgebra R (A × B) :=\n  { S.toNonUnitalSubsemiring.prod S₁.toNonUnitalSubsemiring with\n    carrier := S ×ˢ S₁\n    smul_mem' := fun r _x hx => ⟨SMulMemClass.smul_mem r hx.1, SMulMemClass.smul_mem r hx.2⟩ }\n\n@[simp]\ntheorem coe_prod : (prod S S₁ : Set (A × B)) = (S : Set A) ×ˢ S₁ :=\n  rfl\n\ntheorem prod_toSubmodule : (S.prod S₁).toSubmodule = S.toSubmodule.prod S₁.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem mem_prod {S : NonUnitalSubalgebra R A} {S₁ : NonUnitalSubalgebra R B} {x : A × B} :\n    x ∈ prod S S₁ ↔ x.1 ∈ S ∧ x.2 ∈ S₁ :=\n  Set.mem_prod\n\n@[simp]\ntheorem prod_top : (prod ⊤ ⊤ : NonUnitalSubalgebra R (A × B)) = ⊤ := by ext; simp\n\ntheorem prod_mono {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S ≤ T → S₁ ≤ T₁ → prod S S₁ ≤ prod T T₁ :=\n  Set.prod_mono\n\n@[simp]\ntheorem prod_inf_prod {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S.prod S₁ ⊓ T.prod T₁ = (S ⊓ T).prod (S₁ ⊓ T₁) :=\n  SetLike.coe_injective Set.prod_inter_prod\n\nend Prod\n\nsection SuprLift\n\nvariable {ι : Type*}\n\ntheorem coe_iSup_of_directed [Nonempty ι] {S : ι → NonUnitalSubalgebra R A}\n    (dir : Directed (· ≤ ·) S) : ↑(iSup S) = ⋃ i, (S i : Set A) :=\n  let K : NonUnitalSubalgebra R A :=\n    { __ := NonUnitalSubsemiring.copy _ _ (NonUnitalSubsemiring.coe_iSup_of_directed dir).symm\n      smul_mem' := fun r _x hx ↦\n        let ⟨i, hi⟩ := Set.mem_iUnion.1 hx\n        Set.mem_iUnion.2 ⟨i, (S i).smul_mem' r hi⟩ }\n  have : iSup S = K := le_antisymm\n    (iSup_le fun i ↦ le_iSup (fun i ↦ (S i : Set A)) i) (Set.iUnion_subset fun _ ↦ le_iSup S _)\n  this.symm ▸ rfl\n\n/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              rw [hf j k hjk]\n              rfl)\n            (↑(iSup K)) (by rw [coe_iSup_of_directed dir])\n        map_zero' := by\n          dsimp\n          exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl) _ (by simp)\n        map_mul' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· * ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_add' := by\n          dsimp\n          ","nextTactic":"apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· + ·))","declUpToTactic":"/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              rw [hf j k hjk]\n              rfl)\n            (↑(iSup K)) (by rw [coe_iSup_of_directed dir])\n        map_zero' := by\n          dsimp\n          exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl) _ (by simp)\n        map_mul' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· * ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_add' := by\n          dsimp\n          ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.892_0.jQvdAcOcyVirh23","decl":"/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B "}
-{"state":"case hopi\nR : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\nS : NonUnitalSubalgebra R A\nι : Type u_1\ninst✝ : Nonempty ι\nK : ι → NonUnitalSubalgebra R A\ndir : Directed (fun x x_1 => x ≤ x_1) K\nf : (i : ι) → ↥(K i) →ₙₐ[R] B\nhf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalAlgHom.comp (f j) (inclusion h)\n⊢ ∀ (i : ι) (x y : ↑↑(K i)),\n    Set.inclusion (_ : ↑(K i) ⊆ ↑(iSup K)) (x + y) =\n      Set.inclusion (_ : ↑(K i) ⊆ ↑(iSup K)) x + Set.inclusion (_ : ↑(K i) ⊆ ↑(iSup K)) y\ncase h\nR : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\nS : NonUnitalSubalgebra R A\nι : Type u_1\ninst✝ : Nonempty ι\nK : ι → NonUnitalSubalgebra R A\ndir : Directed (fun x x_1 => x ≤ x_1) K\nf : (i : ι) → ↥(K i) →ₙₐ[R] B\nhf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalAlgHom.comp (f j) (inclusion h)\n⊢ ∀ (i : ι) (x y : ↑↑(K i)), (f i) (x + y) = (f i) x + (f i) y\nR : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\nS : NonUnitalSubalgebra R A\nι : Type u_1\ninst✝ : Nonempty ι\nK : ι → NonUnitalSubalgebra R A\ndir : Directed (fun x x_1 => x ≤ x_1) K\nf : (i : ι) → ↥(K i) →ₙₐ[R] B\nhf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalAlgHom.comp (f j) (inclusion h)\n⊢ ↑(iSup K) ⊆ ⋃ i, ↑(K i)","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ���a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact NonUnitalAlgebra.gc\n\n@[simp]\ntheorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=\n  eq_top_iff.2 fun _x hx => subset_adjoin R hx\n\nvariable {R A}\n\n@[simp]\ntheorem coe_top : (↑(⊤ : NonUnitalSubalgebra R A) : Set A) = Set.univ :=\n  rfl\n\n@[simp]\ntheorem mem_top {x : A} : x ∈ (⊤ : NonUnitalSubalgebra R A) :=\n  Set.mem_univ x\n\n@[simp]\ntheorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toNonUnitalSubsemiring : (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubsemiring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toSubring {R A : Type*} [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    [IsScalarTower R A A] [SMulCommClass R A A] :\n    (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem toSubmodule_eq_top {S : NonUnitalSubalgebra R A} : S.toSubmodule = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toSubmodule'.injective.eq_iff' top_toSubmodule\n\n@[simp]\ntheorem toNonUnitalSubsemiring_eq_top {S : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubsemiring_injective.eq_iff' top_toNonUnitalSubsemiring\n\n@[simp]\ntheorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubring_injective.eq_iff' top_toSubring\n\ntheorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_left\n\ntheorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_right\n\ntheorem mul_mem_sup {S T : NonUnitalSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) :\n    x * y ∈ S ⊔ T :=\n  mul_mem (mem_sup_left hx) (mem_sup_right hy)\n\ntheorem map_sup (f : F) (S T : NonUnitalSubalgebra R A) :\n    ((S ⊔ T).map f : NonUnitalSubalgebra R B) = S.map f ⊔ T.map f :=\n  (@NonUnitalSubalgebra.gc_map_comap F R A B _ _ _ _ _ _ f).l_sup\n\n@[simp, norm_cast]\ntheorem coe_inf (S T : NonUnitalSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T :=\n  rfl\n\n@[simp]\ntheorem mem_inf {S T : NonUnitalSubalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T :=\n  Iff.rfl\n\n@[simp]\ntheorem inf_toSubmodule (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toSubmodule = S.toSubmodule ⊓ T.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem inf_toNonUnitalSubsemiring (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring ⊓ T.toNonUnitalSubsemiring :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sInf (S : Set (NonUnitalSubalgebra R A)) : (↑(sInf S) : Set A) = ⋂ s ∈ S, ↑s :=\n  sInf_image\n\ntheorem mem_sInf {S : Set (NonUnitalSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p := by\n  simp only [← SetLike.mem_coe, coe_sInf, Set.mem_iInter₂]\n\n@[simp]\ntheorem sInf_toSubmodule (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toSubmodule = sInf (NonUnitalSubalgebra.toSubmodule '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp]\ntheorem sInf_toNonUnitalSubsemiring (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toNonUnitalSubsemiring = sInf (NonUnitalSubalgebra.toNonUnitalSubsemiring '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp, norm_cast]\ntheorem coe_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} :\n    (↑(⨅ i, S i) : Set A) = ⋂ i, S i := by simp [iInf]\n\ntheorem mem_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} {x : A} :\n    (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by simp only [iInf, mem_sInf, Set.forall_range_iff]\n\n@[simp]\ntheorem iInf_toSubmodule {ι : Sort*} (S : ι → NonUnitalSubalgebra R A) :\n    (⨅ i, S i).toSubmodule = ⨅ i, (S i).toSubmodule :=\n  SetLike.coe_injective <| by simp\n\ninstance : Inhabited (NonUnitalSubalgebra R A) :=\n  ⟨⊥⟩\n\ntheorem mem_bot {x : A} : x ∈ (⊥ : NonUnitalSubalgebra R A) ↔ x = 0 :=\n  show x ∈ Submodule.span R (NonUnitalSubsemiring.closure (∅ : Set A) : Set A) ↔ x = 0 by\n    rw [NonUnitalSubsemiring.closure_empty, NonUnitalSubsemiring.coe_bot,\n      Submodule.span_zero_singleton, Submodule.mem_bot]\n\ntheorem toSubmodule_bot : (⊥ : NonUnitalSubalgebra R A).toSubmodule = ⊥ := by\n  ext\n  simp only [mem_bot, NonUnitalSubalgebra.mem_toSubmodule, Submodule.mem_bot]\n\n@[simp]\ntheorem coe_bot : ((⊥ : NonUnitalSubalgebra R A) : Set A) = {0} := by\n  simp [Set.ext_iff, NonUnitalAlgebra.mem_bot]\n\ntheorem eq_top_iff {S : NonUnitalSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S :=\n  ⟨fun h x => by rw [h]; exact mem_top, fun h => by ext x; exact ⟨fun _ => mem_top, fun _ => h x⟩⟩\n\ntheorem range_top_iff_surjective (f : A →ₙₐ[R] B) :\n    NonUnitalAlgHom.range f = (⊤ : NonUnitalSubalgebra R B) ↔ Function.Surjective f :=\n  NonUnitalAlgebra.eq_top_iff\n\n@[simp]\ntheorem range_id : NonUnitalAlgHom.range (NonUnitalAlgHom.id R A) = ⊤ :=\n  SetLike.coe_injective Set.range_id\n\n@[simp]\ntheorem map_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R A).map f = NonUnitalAlgHom.range f :=\n  SetLike.coe_injective Set.image_univ\n\n@[simp]\ntheorem map_bot (f : A →ₙₐ[R] B) : (⊥ : NonUnitalSubalgebra R A).map f = ⊥ :=\n  SetLike.coe_injective <| by simp [NonUnitalAlgebra.coe_bot, NonUnitalSubalgebra.coe_map]\n\n@[simp]\ntheorem comap_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R B).comap f = ⊤ :=\n  eq_top_iff.2 fun _ => mem_top\n\n/-- `NonUnitalAlgHom` to `⊤ : NonUnitalSubalgebra R A`. -/\ndef toTop : A →ₙₐ[R] (⊤ : NonUnitalSubalgebra R A) :=\n  NonUnitalAlgHom.codRestrict (NonUnitalAlgHom.id R A) ⊤ fun _ => mem_top\n\nend NonUnitalAlgebra\n\nnamespace NonUnitalSubalgebra\n\nopen NonUnitalAlgebra\n\nsection NonAssoc\n\nvariable {R : Type u} {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable (S : NonUnitalSubalgebra R A)\n\ninstance subsingleton_of_subsingleton [Subsingleton A] : Subsingleton (NonUnitalSubalgebra R A) :=\n  ⟨fun B C => ext fun x => by simp only [Subsingleton.elim x 0, zero_mem B, zero_mem C]⟩\n\ninstance _root_.NonUnitalAlgHom.subsingleton [Subsingleton (NonUnitalSubalgebra R A)] :\n    Subsingleton (A →ₙₐ[R] B) :=\n  ⟨fun f g =>\n    NonUnitalAlgHom.ext fun a =>\n      have : a ∈ (⊥ : NonUnitalSubalgebra R A) :=\n        Subsingleton.elim (⊤ : NonUnitalSubalgebra R A) ⊥ ▸ mem_top\n      (mem_bot.mp this).symm ▸ (map_zero f).trans (map_zero g).symm⟩\n\ntheorem range_val : NonUnitalAlgHom.range (NonUnitalSubalgebraClass.subtype S) = S :=\n  ext <| Set.ext_iff.1 <| (NonUnitalSubalgebraClass.subtype S).coe_range.trans Subtype.range_val\n\n/-- The map `S → T` when `S` is a non-unital subalgebra contained in the non-unital subalgebra `T`.\n\nThis is the non-unital subalgebra version of `Submodule.inclusion`, or `Subring.inclusion`  -/\ndef inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) : S →ₙₐ[R] T\n    where\n  toFun := Set.inclusion h\n  map_add' _ _ := rfl\n  map_mul' _ _ := rfl\n  map_zero' := rfl\n  map_smul' _ _ := rfl\n\ntheorem inclusion_injective {S T : NonUnitalSubalgebra R A} (h : S ≤ T) :\n    Function.Injective (inclusion h) := fun _ _ => Subtype.ext ∘ Subtype.mk.inj\n\n@[simp]\ntheorem inclusion_self {S : NonUnitalSubalgebra R A} :\n    inclusion (le_refl S) = NonUnitalAlgHom.id R S :=\n  NonUnitalAlgHom.ext fun _ => Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_mk {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : A) (hx : x ∈ S) :\n    inclusion h ⟨x, hx⟩ = ⟨x, h hx⟩ :=\n  rfl\n\ntheorem inclusion_right {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : T) (m : (x : A) ∈ S) :\n    inclusion h ⟨x, m⟩ = x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_inclusion {S T U : NonUnitalSubalgebra R A} (hst : S ≤ T) (htu : T ≤ U) (x : S) :\n    inclusion htu (inclusion hst x) = inclusion (le_trans hst htu) x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem coe_inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (s : S) :\n    (inclusion h s : A) = s :=\n  rfl\n\nsection Prod\n\nvariable (S₁ : NonUnitalSubalgebra R B)\n\n/-- The product of two non-unital subalgebras is a non-unital subalgebra. -/\ndef prod : NonUnitalSubalgebra R (A × B) :=\n  { S.toNonUnitalSubsemiring.prod S₁.toNonUnitalSubsemiring with\n    carrier := S ×ˢ S₁\n    smul_mem' := fun r _x hx => ⟨SMulMemClass.smul_mem r hx.1, SMulMemClass.smul_mem r hx.2⟩ }\n\n@[simp]\ntheorem coe_prod : (prod S S₁ : Set (A × B)) = (S : Set A) ×ˢ S₁ :=\n  rfl\n\ntheorem prod_toSubmodule : (S.prod S₁).toSubmodule = S.toSubmodule.prod S₁.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem mem_prod {S : NonUnitalSubalgebra R A} {S₁ : NonUnitalSubalgebra R B} {x : A × B} :\n    x ∈ prod S S₁ ↔ x.1 ∈ S ∧ x.2 ∈ S₁ :=\n  Set.mem_prod\n\n@[simp]\ntheorem prod_top : (prod ⊤ ⊤ : NonUnitalSubalgebra R (A × B)) = ⊤ := by ext; simp\n\ntheorem prod_mono {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S ≤ T → S₁ ≤ T₁ → prod S S₁ ≤ prod T T₁ :=\n  Set.prod_mono\n\n@[simp]\ntheorem prod_inf_prod {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S.prod S₁ ⊓ T.prod T₁ = (S ⊓ T).prod (S₁ ⊓ T₁) :=\n  SetLike.coe_injective Set.prod_inter_prod\n\nend Prod\n\nsection SuprLift\n\nvariable {ι : Type*}\n\ntheorem coe_iSup_of_directed [Nonempty ι] {S : ι → NonUnitalSubalgebra R A}\n    (dir : Directed (· ≤ ·) S) : ↑(iSup S) = ⋃ i, (S i : Set A) :=\n  let K : NonUnitalSubalgebra R A :=\n    { __ := NonUnitalSubsemiring.copy _ _ (NonUnitalSubsemiring.coe_iSup_of_directed dir).symm\n      smul_mem' := fun r _x hx ↦\n        let ⟨i, hi⟩ := Set.mem_iUnion.1 hx\n        Set.mem_iUnion.2 ⟨i, (S i).smul_mem' r hi⟩ }\n  have : iSup S = K := le_antisymm\n    (iSup_le fun i ↦ le_iSup (fun i ↦ (S i : Set A)) i) (Set.iUnion_subset fun _ ↦ le_iSup S _)\n  this.symm ▸ rfl\n\n/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              rw [hf j k hjk]\n              rfl)\n            (↑(iSup K)) (by rw [coe_iSup_of_directed dir])\n        map_zero' := by\n          dsimp\n          exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl) _ (by simp)\n        map_mul' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· * ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_add' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· + ·))\n          ","nextTactic":"on_goal 3 => rw [coe_iSup_of_directed dir]","declUpToTactic":"/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              rw [hf j k hjk]\n              rfl)\n            (↑(iSup K)) (by rw [coe_iSup_of_directed dir])\n        map_zero' := by\n          dsimp\n          exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl) _ (by simp)\n        map_mul' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· * ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_add' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· + ·))\n          ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.892_0.jQvdAcOcyVirh23","decl":"/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B "}
-{"state":"case hopi\nR : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\nS : NonUnitalSubalgebra R A\nι : Type u_1\ninst✝ : Nonempty ι\nK : ι → NonUnitalSubalgebra R A\ndir : Directed (fun x x_1 => x ≤ x_1) K\nf : (i : ι) → ↥(K i) →ₙₐ[R] B\nhf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalAlgHom.comp (f j) (inclusion h)\n⊢ ∀ (i : ι) (x y : ↑↑(K i)),\n    Set.inclusion (_ : ↑(K i) ⊆ ↑(iSup K)) (x + y) =\n      Set.inclusion (_ : ↑(K i) ⊆ ↑(iSup K)) x + Set.inclusion (_ : ↑(K i) ⊆ ↑(iSup K)) y\ncase h\nR : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\nS : NonUnitalSubalgebra R A\nι : Type u_1\ninst✝ : Nonempty ι\nK : ι → NonUnitalSubalgebra R A\ndir : Directed (fun x x_1 => x ≤ x_1) K\nf : (i : ι) → ↥(K i) →ₙₐ[R] B\nhf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalAlgHom.comp (f j) (inclusion h)\n⊢ ∀ (i : ι) (x y : ↑↑(K i)), (f i) (x + y) = (f i) x + (f i) y\nR : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\nS : NonUnitalSubalgebra R A\nι : Type u_1\ninst✝ : Nonempty ι\nK : ι → NonUnitalSubalgebra R A\ndir : Directed (fun x x_1 => x ≤ x_1) K\nf : (i : ι) → ↥(K i) →ₙₐ[R] B\nhf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalAlgHom.comp (f j) (inclusion h)\n⊢ ↑(iSup K) ⊆ ⋃ i, ↑(K i)","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact NonUnitalAlgebra.gc\n\n@[simp]\ntheorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=\n  eq_top_iff.2 fun _x hx => subset_adjoin R hx\n\nvariable {R A}\n\n@[simp]\ntheorem coe_top : (↑(⊤ : NonUnitalSubalgebra R A) : Set A) = Set.univ :=\n  rfl\n\n@[simp]\ntheorem mem_top {x : A} : x ∈ (⊤ : NonUnitalSubalgebra R A) :=\n  Set.mem_univ x\n\n@[simp]\ntheorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toNonUnitalSubsemiring : (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubsemiring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toSubring {R A : Type*} [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    [IsScalarTower R A A] [SMulCommClass R A A] :\n    (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem toSubmodule_eq_top {S : NonUnitalSubalgebra R A} : S.toSubmodule = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toSubmodule'.injective.eq_iff' top_toSubmodule\n\n@[simp]\ntheorem toNonUnitalSubsemiring_eq_top {S : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubsemiring_injective.eq_iff' top_toNonUnitalSubsemiring\n\n@[simp]\ntheorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubring_injective.eq_iff' top_toSubring\n\ntheorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_left\n\ntheorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_right\n\ntheorem mul_mem_sup {S T : NonUnitalSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) :\n    x * y ∈ S ⊔ T :=\n  mul_mem (mem_sup_left hx) (mem_sup_right hy)\n\ntheorem map_sup (f : F) (S T : NonUnitalSubalgebra R A) :\n    ((S ⊔ T).map f : NonUnitalSubalgebra R B) = S.map f ⊔ T.map f :=\n  (@NonUnitalSubalgebra.gc_map_comap F R A B _ _ _ _ _ _ f).l_sup\n\n@[simp, norm_cast]\ntheorem coe_inf (S T : NonUnitalSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T :=\n  rfl\n\n@[simp]\ntheorem mem_inf {S T : NonUnitalSubalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T :=\n  Iff.rfl\n\n@[simp]\ntheorem inf_toSubmodule (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toSubmodule = S.toSubmodule ⊓ T.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem inf_toNonUnitalSubsemiring (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring ⊓ T.toNonUnitalSubsemiring :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sInf (S : Set (NonUnitalSubalgebra R A)) : (↑(sInf S) : Set A) = ⋂ s ∈ S, ↑s :=\n  sInf_image\n\ntheorem mem_sInf {S : Set (NonUnitalSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p := by\n  simp only [← SetLike.mem_coe, coe_sInf, Set.mem_iInter₂]\n\n@[simp]\ntheorem sInf_toSubmodule (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toSubmodule = sInf (NonUnitalSubalgebra.toSubmodule '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp]\ntheorem sInf_toNonUnitalSubsemiring (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toNonUnitalSubsemiring = sInf (NonUnitalSubalgebra.toNonUnitalSubsemiring '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp, norm_cast]\ntheorem coe_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} :\n    (↑(⨅ i, S i) : Set A) = ⋂ i, S i := by simp [iInf]\n\ntheorem mem_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} {x : A} :\n    (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by simp only [iInf, mem_sInf, Set.forall_range_iff]\n\n@[simp]\ntheorem iInf_toSubmodule {ι : Sort*} (S : ι → NonUnitalSubalgebra R A) :\n    (⨅ i, S i).toSubmodule = ⨅ i, (S i).toSubmodule :=\n  SetLike.coe_injective <| by simp\n\ninstance : Inhabited (NonUnitalSubalgebra R A) :=\n  ⟨⊥⟩\n\ntheorem mem_bot {x : A} : x ∈ (⊥ : NonUnitalSubalgebra R A) ↔ x = 0 :=\n  show x ∈ Submodule.span R (NonUnitalSubsemiring.closure (∅ : Set A) : Set A) ↔ x = 0 by\n    rw [NonUnitalSubsemiring.closure_empty, NonUnitalSubsemiring.coe_bot,\n      Submodule.span_zero_singleton, Submodule.mem_bot]\n\ntheorem toSubmodule_bot : (⊥ : NonUnitalSubalgebra R A).toSubmodule = ⊥ := by\n  ext\n  simp only [mem_bot, NonUnitalSubalgebra.mem_toSubmodule, Submodule.mem_bot]\n\n@[simp]\ntheorem coe_bot : ((⊥ : NonUnitalSubalgebra R A) : Set A) = {0} := by\n  simp [Set.ext_iff, NonUnitalAlgebra.mem_bot]\n\ntheorem eq_top_iff {S : NonUnitalSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S :=\n  ⟨fun h x => by rw [h]; exact mem_top, fun h => by ext x; exact ⟨fun _ => mem_top, fun _ => h x⟩⟩\n\ntheorem range_top_iff_surjective (f : A →ₙₐ[R] B) :\n    NonUnitalAlgHom.range f = (⊤ : NonUnitalSubalgebra R B) ↔ Function.Surjective f :=\n  NonUnitalAlgebra.eq_top_iff\n\n@[simp]\ntheorem range_id : NonUnitalAlgHom.range (NonUnitalAlgHom.id R A) = ⊤ :=\n  SetLike.coe_injective Set.range_id\n\n@[simp]\ntheorem map_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R A).map f = NonUnitalAlgHom.range f :=\n  SetLike.coe_injective Set.image_univ\n\n@[simp]\ntheorem map_bot (f : A →ₙₐ[R] B) : (⊥ : NonUnitalSubalgebra R A).map f = ⊥ :=\n  SetLike.coe_injective <| by simp [NonUnitalAlgebra.coe_bot, NonUnitalSubalgebra.coe_map]\n\n@[simp]\ntheorem comap_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R B).comap f = ⊤ :=\n  eq_top_iff.2 fun _ => mem_top\n\n/-- `NonUnitalAlgHom` to `⊤ : NonUnitalSubalgebra R A`. -/\ndef toTop : A →ₙₐ[R] (⊤ : NonUnitalSubalgebra R A) :=\n  NonUnitalAlgHom.codRestrict (NonUnitalAlgHom.id R A) ⊤ fun _ => mem_top\n\nend NonUnitalAlgebra\n\nnamespace NonUnitalSubalgebra\n\nopen NonUnitalAlgebra\n\nsection NonAssoc\n\nvariable {R : Type u} {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable (S : NonUnitalSubalgebra R A)\n\ninstance subsingleton_of_subsingleton [Subsingleton A] : Subsingleton (NonUnitalSubalgebra R A) :=\n  ⟨fun B C => ext fun x => by simp only [Subsingleton.elim x 0, zero_mem B, zero_mem C]⟩\n\ninstance _root_.NonUnitalAlgHom.subsingleton [Subsingleton (NonUnitalSubalgebra R A)] :\n    Subsingleton (A →ₙₐ[R] B) :=\n  ⟨fun f g =>\n    NonUnitalAlgHom.ext fun a =>\n      have : a ∈ (⊥ : NonUnitalSubalgebra R A) :=\n        Subsingleton.elim (⊤ : NonUnitalSubalgebra R A) ⊥ ▸ mem_top\n      (mem_bot.mp this).symm ▸ (map_zero f).trans (map_zero g).symm⟩\n\ntheorem range_val : NonUnitalAlgHom.range (NonUnitalSubalgebraClass.subtype S) = S :=\n  ext <| Set.ext_iff.1 <| (NonUnitalSubalgebraClass.subtype S).coe_range.trans Subtype.range_val\n\n/-- The map `S → T` when `S` is a non-unital subalgebra contained in the non-unital subalgebra `T`.\n\nThis is the non-unital subalgebra version of `Submodule.inclusion`, or `Subring.inclusion`  -/\ndef inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) : S →ₙₐ[R] T\n    where\n  toFun := Set.inclusion h\n  map_add' _ _ := rfl\n  map_mul' _ _ := rfl\n  map_zero' := rfl\n  map_smul' _ _ := rfl\n\ntheorem inclusion_injective {S T : NonUnitalSubalgebra R A} (h : S ≤ T) :\n    Function.Injective (inclusion h) := fun _ _ => Subtype.ext ∘ Subtype.mk.inj\n\n@[simp]\ntheorem inclusion_self {S : NonUnitalSubalgebra R A} :\n    inclusion (le_refl S) = NonUnitalAlgHom.id R S :=\n  NonUnitalAlgHom.ext fun _ => Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_mk {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : A) (hx : x ∈ S) :\n    inclusion h ⟨x, hx⟩ = ⟨x, h hx⟩ :=\n  rfl\n\ntheorem inclusion_right {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : T) (m : (x : A) ∈ S) :\n    inclusion h ⟨x, m⟩ = x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_inclusion {S T U : NonUnitalSubalgebra R A} (hst : S ≤ T) (htu : T ≤ U) (x : S) :\n    inclusion htu (inclusion hst x) = inclusion (le_trans hst htu) x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem coe_inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (s : S) :\n    (inclusion h s : A) = s :=\n  rfl\n\nsection Prod\n\nvariable (S₁ : NonUnitalSubalgebra R B)\n\n/-- The product of two non-unital subalgebras is a non-unital subalgebra. -/\ndef prod : NonUnitalSubalgebra R (A × B) :=\n  { S.toNonUnitalSubsemiring.prod S₁.toNonUnitalSubsemiring with\n    carrier := S ×ˢ S₁\n    smul_mem' := fun r _x hx => ⟨SMulMemClass.smul_mem r hx.1, SMulMemClass.smul_mem r hx.2⟩ }\n\n@[simp]\ntheorem coe_prod : (prod S S₁ : Set (A × B)) = (S : Set A) ×ˢ S₁ :=\n  rfl\n\ntheorem prod_toSubmodule : (S.prod S₁).toSubmodule = S.toSubmodule.prod S₁.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem mem_prod {S : NonUnitalSubalgebra R A} {S₁ : NonUnitalSubalgebra R B} {x : A × B} :\n    x ∈ prod S S₁ ↔ x.1 ∈ S ∧ x.2 ∈ S₁ :=\n  Set.mem_prod\n\n@[simp]\ntheorem prod_top : (prod ⊤ ⊤ : NonUnitalSubalgebra R (A × B)) = ⊤ := by ext; simp\n\ntheorem prod_mono {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S ≤ T → S₁ ≤ T₁ → prod S S₁ ≤ prod T T₁ :=\n  Set.prod_mono\n\n@[simp]\ntheorem prod_inf_prod {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S.prod S₁ ⊓ T.prod T₁ = (S ⊓ T).prod (S₁ ⊓ T₁) :=\n  SetLike.coe_injective Set.prod_inter_prod\n\nend Prod\n\nsection SuprLift\n\nvariable {ι : Type*}\n\ntheorem coe_iSup_of_directed [Nonempty ι] {S : ι → NonUnitalSubalgebra R A}\n    (dir : Directed (· ≤ ·) S) : ↑(iSup S) = ⋃ i, (S i : Set A) :=\n  let K : NonUnitalSubalgebra R A :=\n    { __ := NonUnitalSubsemiring.copy _ _ (NonUnitalSubsemiring.coe_iSup_of_directed dir).symm\n      smul_mem' := fun r _x hx ↦\n        let ⟨i, hi⟩ := Set.mem_iUnion.1 hx\n        Set.mem_iUnion.2 ⟨i, (S i).smul_mem' r hi⟩ }\n  have : iSup S = K := le_antisymm\n    (iSup_le fun i ↦ le_iSup (fun i ↦ (S i : Set A)) i) (Set.iUnion_subset fun _ ↦ le_iSup S _)\n  this.symm ▸ rfl\n\n/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              rw [hf j k hjk]\n              rfl)\n            (↑(iSup K)) (by rw [coe_iSup_of_directed dir])\n        map_zero' := by\n          dsimp\n          exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl) _ (by simp)\n        map_mul' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· * ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_add' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· + ·))\n          ","nextTactic":"on_goal 3 => rw [coe_iSup_of_directed dir]","declUpToTactic":"/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              rw [hf j k hjk]\n              rfl)\n            (↑(iSup K)) (by rw [coe_iSup_of_directed dir])\n        map_zero' := by\n          dsimp\n          exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl) _ (by simp)\n        map_mul' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· * ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_add' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· + ·))\n          ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.892_0.jQvdAcOcyVirh23","decl":"/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B "}
-{"state":"R : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\nS : NonUnitalSubalgebra R A\nι : Type u_1\ninst✝ : Nonempty ι\nK : ι → NonUnitalSubalgebra R A\ndir : Directed (fun x x_1 => x ≤ x_1) K\nf : (i : ι) → ↥(K i) →ₙₐ[R] B\nhf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalAlgHom.comp (f j) (inclusion h)\n⊢ ↑(iSup K) ⊆ ⋃ i, ↑(K i)","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact NonUnitalAlgebra.gc\n\n@[simp]\ntheorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=\n  eq_top_iff.2 fun _x hx => subset_adjoin R hx\n\nvariable {R A}\n\n@[simp]\ntheorem coe_top : (↑(⊤ : NonUnitalSubalgebra R A) : Set A) = Set.univ :=\n  rfl\n\n@[simp]\ntheorem mem_top {x : A} : x ∈ (⊤ : NonUnitalSubalgebra R A) :=\n  Set.mem_univ x\n\n@[simp]\ntheorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toNonUnitalSubsemiring : (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubsemiring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toSubring {R A : Type*} [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    [IsScalarTower R A A] [SMulCommClass R A A] :\n    (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem toSubmodule_eq_top {S : NonUnitalSubalgebra R A} : S.toSubmodule = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toSubmodule'.injective.eq_iff' top_toSubmodule\n\n@[simp]\ntheorem toNonUnitalSubsemiring_eq_top {S : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubsemiring_injective.eq_iff' top_toNonUnitalSubsemiring\n\n@[simp]\ntheorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubring_injective.eq_iff' top_toSubring\n\ntheorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_left\n\ntheorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_right\n\ntheorem mul_mem_sup {S T : NonUnitalSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) :\n    x * y ∈ S ⊔ T :=\n  mul_mem (mem_sup_left hx) (mem_sup_right hy)\n\ntheorem map_sup (f : F) (S T : NonUnitalSubalgebra R A) :\n    ((S ⊔ T).map f : NonUnitalSubalgebra R B) = S.map f ⊔ T.map f :=\n  (@NonUnitalSubalgebra.gc_map_comap F R A B _ _ _ _ _ _ f).l_sup\n\n@[simp, norm_cast]\ntheorem coe_inf (S T : NonUnitalSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T :=\n  rfl\n\n@[simp]\ntheorem mem_inf {S T : NonUnitalSubalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T :=\n  Iff.rfl\n\n@[simp]\ntheorem inf_toSubmodule (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toSubmodule = S.toSubmodule ⊓ T.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem inf_toNonUnitalSubsemiring (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring ⊓ T.toNonUnitalSubsemiring :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sInf (S : Set (NonUnitalSubalgebra R A)) : (↑(sInf S) : Set A) = ⋂ s ∈ S, ↑s :=\n  sInf_image\n\ntheorem mem_sInf {S : Set (NonUnitalSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p := by\n  simp only [← SetLike.mem_coe, coe_sInf, Set.mem_iInter₂]\n\n@[simp]\ntheorem sInf_toSubmodule (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toSubmodule = sInf (NonUnitalSubalgebra.toSubmodule '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp]\ntheorem sInf_toNonUnitalSubsemiring (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toNonUnitalSubsemiring = sInf (NonUnitalSubalgebra.toNonUnitalSubsemiring '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp, norm_cast]\ntheorem coe_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} :\n    (↑(⨅ i, S i) : Set A) = ⋂ i, S i := by simp [iInf]\n\ntheorem mem_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} {x : A} :\n    (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by simp only [iInf, mem_sInf, Set.forall_range_iff]\n\n@[simp]\ntheorem iInf_toSubmodule {ι : Sort*} (S : ι → NonUnitalSubalgebra R A) :\n    (⨅ i, S i).toSubmodule = ⨅ i, (S i).toSubmodule :=\n  SetLike.coe_injective <| by simp\n\ninstance : Inhabited (NonUnitalSubalgebra R A) :=\n  ⟨⊥⟩\n\ntheorem mem_bot {x : A} : x ∈ (⊥ : NonUnitalSubalgebra R A) ↔ x = 0 :=\n  show x ∈ Submodule.span R (NonUnitalSubsemiring.closure (∅ : Set A) : Set A) ↔ x = 0 by\n    rw [NonUnitalSubsemiring.closure_empty, NonUnitalSubsemiring.coe_bot,\n      Submodule.span_zero_singleton, Submodule.mem_bot]\n\ntheorem toSubmodule_bot : (⊥ : NonUnitalSubalgebra R A).toSubmodule = ⊥ := by\n  ext\n  simp only [mem_bot, NonUnitalSubalgebra.mem_toSubmodule, Submodule.mem_bot]\n\n@[simp]\ntheorem coe_bot : ((⊥ : NonUnitalSubalgebra R A) : Set A) = {0} := by\n  simp [Set.ext_iff, NonUnitalAlgebra.mem_bot]\n\ntheorem eq_top_iff {S : NonUnitalSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S :=\n  ⟨fun h x => by rw [h]; exact mem_top, fun h => by ext x; exact ⟨fun _ => mem_top, fun _ => h x⟩⟩\n\ntheorem range_top_iff_surjective (f : A →ₙₐ[R] B) :\n    NonUnitalAlgHom.range f = (⊤ : NonUnitalSubalgebra R B) ↔ Function.Surjective f :=\n  NonUnitalAlgebra.eq_top_iff\n\n@[simp]\ntheorem range_id : NonUnitalAlgHom.range (NonUnitalAlgHom.id R A) = ⊤ :=\n  SetLike.coe_injective Set.range_id\n\n@[simp]\ntheorem map_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R A).map f = NonUnitalAlgHom.range f :=\n  SetLike.coe_injective Set.image_univ\n\n@[simp]\ntheorem map_bot (f : A →ₙₐ[R] B) : (⊥ : NonUnitalSubalgebra R A).map f = ⊥ :=\n  SetLike.coe_injective <| by simp [NonUnitalAlgebra.coe_bot, NonUnitalSubalgebra.coe_map]\n\n@[simp]\ntheorem comap_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R B).comap f = ⊤ :=\n  eq_top_iff.2 fun _ => mem_top\n\n/-- `NonUnitalAlgHom` to `⊤ : NonUnitalSubalgebra R A`. -/\ndef toTop : A →ₙₐ[R] (⊤ : NonUnitalSubalgebra R A) :=\n  NonUnitalAlgHom.codRestrict (NonUnitalAlgHom.id R A) ⊤ fun _ => mem_top\n\nend NonUnitalAlgebra\n\nnamespace NonUnitalSubalgebra\n\nopen NonUnitalAlgebra\n\nsection NonAssoc\n\nvariable {R : Type u} {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable (S : NonUnitalSubalgebra R A)\n\ninstance subsingleton_of_subsingleton [Subsingleton A] : Subsingleton (NonUnitalSubalgebra R A) :=\n  ⟨fun B C => ext fun x => by simp only [Subsingleton.elim x 0, zero_mem B, zero_mem C]⟩\n\ninstance _root_.NonUnitalAlgHom.subsingleton [Subsingleton (NonUnitalSubalgebra R A)] :\n    Subsingleton (A →ₙₐ[R] B) :=\n  ⟨fun f g =>\n    NonUnitalAlgHom.ext fun a =>\n      have : a ∈ (⊥ : NonUnitalSubalgebra R A) :=\n        Subsingleton.elim (⊤ : NonUnitalSubalgebra R A) ⊥ ▸ mem_top\n      (mem_bot.mp this).symm ▸ (map_zero f).trans (map_zero g).symm⟩\n\ntheorem range_val : NonUnitalAlgHom.range (NonUnitalSubalgebraClass.subtype S) = S :=\n  ext <| Set.ext_iff.1 <| (NonUnitalSubalgebraClass.subtype S).coe_range.trans Subtype.range_val\n\n/-- The map `S → T` when `S` is a non-unital subalgebra contained in the non-unital subalgebra `T`.\n\nThis is the non-unital subalgebra version of `Submodule.inclusion`, or `Subring.inclusion`  -/\ndef inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) : S →ₙₐ[R] T\n    where\n  toFun := Set.inclusion h\n  map_add' _ _ := rfl\n  map_mul' _ _ := rfl\n  map_zero' := rfl\n  map_smul' _ _ := rfl\n\ntheorem inclusion_injective {S T : NonUnitalSubalgebra R A} (h : S ≤ T) :\n    Function.Injective (inclusion h) := fun _ _ => Subtype.ext ∘ Subtype.mk.inj\n\n@[simp]\ntheorem inclusion_self {S : NonUnitalSubalgebra R A} :\n    inclusion (le_refl S) = NonUnitalAlgHom.id R S :=\n  NonUnitalAlgHom.ext fun _ => Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_mk {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : A) (hx : x ∈ S) :\n    inclusion h ⟨x, hx⟩ = ⟨x, h hx⟩ :=\n  rfl\n\ntheorem inclusion_right {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : T) (m : (x : A) ∈ S) :\n    inclusion h ⟨x, m⟩ = x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_inclusion {S T U : NonUnitalSubalgebra R A} (hst : S ≤ T) (htu : T ≤ U) (x : S) :\n    inclusion htu (inclusion hst x) = inclusion (le_trans hst htu) x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem coe_inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (s : S) :\n    (inclusion h s : A) = s :=\n  rfl\n\nsection Prod\n\nvariable (S₁ : NonUnitalSubalgebra R B)\n\n/-- The product of two non-unital subalgebras is a non-unital subalgebra. -/\ndef prod : NonUnitalSubalgebra R (A × B) :=\n  { S.toNonUnitalSubsemiring.prod S₁.toNonUnitalSubsemiring with\n    carrier := S ×ˢ S₁\n    smul_mem' := fun r _x hx => ⟨SMulMemClass.smul_mem r hx.1, SMulMemClass.smul_mem r hx.2⟩ }\n\n@[simp]\ntheorem coe_prod : (prod S S₁ : Set (A × B)) = (S : Set A) ×ˢ S₁ :=\n  rfl\n\ntheorem prod_toSubmodule : (S.prod S₁).toSubmodule = S.toSubmodule.prod S₁.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem mem_prod {S : NonUnitalSubalgebra R A} {S₁ : NonUnitalSubalgebra R B} {x : A × B} :\n    x ∈ prod S S₁ ↔ x.1 ∈ S ∧ x.2 ∈ S₁ :=\n  Set.mem_prod\n\n@[simp]\ntheorem prod_top : (prod ⊤ ⊤ : NonUnitalSubalgebra R (A × B)) = ⊤ := by ext; simp\n\ntheorem prod_mono {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S ≤ T → S₁ ≤ T₁ → prod S S₁ ≤ prod T T₁ :=\n  Set.prod_mono\n\n@[simp]\ntheorem prod_inf_prod {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S.prod S₁ ⊓ T.prod T₁ = (S ⊓ T).prod (S₁ ⊓ T₁) :=\n  SetLike.coe_injective Set.prod_inter_prod\n\nend Prod\n\nsection SuprLift\n\nvariable {ι : Type*}\n\ntheorem coe_iSup_of_directed [Nonempty ι] {S : ι → NonUnitalSubalgebra R A}\n    (dir : Directed (· ≤ ·) S) : ↑(iSup S) = ⋃ i, (S i : Set A) :=\n  let K : NonUnitalSubalgebra R A :=\n    { __ := NonUnitalSubsemiring.copy _ _ (NonUnitalSubsemiring.coe_iSup_of_directed dir).symm\n      smul_mem' := fun r _x hx ↦\n        let ⟨i, hi⟩ := Set.mem_iUnion.1 hx\n        Set.mem_iUnion.2 ⟨i, (S i).smul_mem' r hi⟩ }\n  have : iSup S = K := le_antisymm\n    (iSup_le fun i ↦ le_iSup (fun i ↦ (S i : Set A)) i) (Set.iUnion_subset fun _ ↦ le_iSup S _)\n  this.symm ▸ rfl\n\n/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              rw [hf j k hjk]\n              rfl)\n            (↑(iSup K)) (by rw [coe_iSup_of_directed dir])\n        map_zero' := by\n          dsimp\n          exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl) _ (by simp)\n        map_mul' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· * ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_add' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· + ·))\n          on_goal 3 => ","nextTactic":"rw [coe_iSup_of_directed dir]","declUpToTactic":"/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              rw [hf j k hjk]\n              rfl)\n            (↑(iSup K)) (by rw [coe_iSup_of_directed dir])\n        map_zero' := by\n          dsimp\n          exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl) _ (by simp)\n        map_mul' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· * ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_add' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· + ·))\n          on_goal 3 => ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.892_0.jQvdAcOcyVirh23","decl":"/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B "}
-{"state":"case hopi\nR : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\nS : NonUnitalSubalgebra R A\nι : Type u_1\ninst✝ : Nonempty ι\nK : ι → NonUnitalSubalgebra R A\ndir : Directed (fun x x_1 => x ≤ x_1) K\nf : (i : ι) → ↥(K i) →ₙₐ[R] B\nhf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalAlgHom.comp (f j) (inclusion h)\n⊢ ∀ (i : ι) (x y : ↑↑(K i)),\n    Set.inclusion (_ : ↑(K i) ⊆ ↑(iSup K)) (x + y) =\n      Set.inclusion (_ : ↑(K i) ⊆ ↑(iSup K)) x + Set.inclusion (_ : ↑(K i) ⊆ ↑(iSup K)) y\ncase h\nR : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\nS : NonUnitalSubalgebra R A\nι : Type u_1\ninst✝ : Nonempty ι\nK : ι → NonUnitalSubalgebra R A\ndir : Directed (fun x x_1 => x ≤ x_1) K\nf : (i : ι) → ↥(K i) →ₙₐ[R] B\nhf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalAlgHom.comp (f j) (inclusion h)\n⊢ ∀ (i : ι) (x y : ↑↑(K i)), (f i) (x + y) = (f i) x + (f i) y","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact NonUnitalAlgebra.gc\n\n@[simp]\ntheorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=\n  eq_top_iff.2 fun _x hx => subset_adjoin R hx\n\nvariable {R A}\n\n@[simp]\ntheorem coe_top : (↑(⊤ : NonUnitalSubalgebra R A) : Set A) = Set.univ :=\n  rfl\n\n@[simp]\ntheorem mem_top {x : A} : x ∈ (⊤ : NonUnitalSubalgebra R A) :=\n  Set.mem_univ x\n\n@[simp]\ntheorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toNonUnitalSubsemiring : (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubsemiring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toSubring {R A : Type*} [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    [IsScalarTower R A A] [SMulCommClass R A A] :\n    (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem toSubmodule_eq_top {S : NonUnitalSubalgebra R A} : S.toSubmodule = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toSubmodule'.injective.eq_iff' top_toSubmodule\n\n@[simp]\ntheorem toNonUnitalSubsemiring_eq_top {S : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubsemiring_injective.eq_iff' top_toNonUnitalSubsemiring\n\n@[simp]\ntheorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubring_injective.eq_iff' top_toSubring\n\ntheorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_left\n\ntheorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_right\n\ntheorem mul_mem_sup {S T : NonUnitalSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) :\n    x * y ∈ S ⊔ T :=\n  mul_mem (mem_sup_left hx) (mem_sup_right hy)\n\ntheorem map_sup (f : F) (S T : NonUnitalSubalgebra R A) :\n    ((S ⊔ T).map f : NonUnitalSubalgebra R B) = S.map f ⊔ T.map f :=\n  (@NonUnitalSubalgebra.gc_map_comap F R A B _ _ _ _ _ _ f).l_sup\n\n@[simp, norm_cast]\ntheorem coe_inf (S T : NonUnitalSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T :=\n  rfl\n\n@[simp]\ntheorem mem_inf {S T : NonUnitalSubalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T :=\n  Iff.rfl\n\n@[simp]\ntheorem inf_toSubmodule (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toSubmodule = S.toSubmodule ⊓ T.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem inf_toNonUnitalSubsemiring (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring ⊓ T.toNonUnitalSubsemiring :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sInf (S : Set (NonUnitalSubalgebra R A)) : (↑(sInf S) : Set A) = ⋂ s ∈ S, ↑s :=\n  sInf_image\n\ntheorem mem_sInf {S : Set (NonUnitalSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p := by\n  simp only [← SetLike.mem_coe, coe_sInf, Set.mem_iInter₂]\n\n@[simp]\ntheorem sInf_toSubmodule (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toSubmodule = sInf (NonUnitalSubalgebra.toSubmodule '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp]\ntheorem sInf_toNonUnitalSubsemiring (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toNonUnitalSubsemiring = sInf (NonUnitalSubalgebra.toNonUnitalSubsemiring '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp, norm_cast]\ntheorem coe_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} :\n    (↑(⨅ i, S i) : Set A) = ⋂ i, S i := by simp [iInf]\n\ntheorem mem_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} {x : A} :\n    (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by simp only [iInf, mem_sInf, Set.forall_range_iff]\n\n@[simp]\ntheorem iInf_toSubmodule {ι : Sort*} (S : ι → NonUnitalSubalgebra R A) :\n    (⨅ i, S i).toSubmodule = ⨅ i, (S i).toSubmodule :=\n  SetLike.coe_injective <| by simp\n\ninstance : Inhabited (NonUnitalSubalgebra R A) :=\n  ⟨⊥⟩\n\ntheorem mem_bot {x : A} : x ∈ (⊥ : NonUnitalSubalgebra R A) ↔ x = 0 :=\n  show x ∈ Submodule.span R (NonUnitalSubsemiring.closure (∅ : Set A) : Set A) ↔ x = 0 by\n    rw [NonUnitalSubsemiring.closure_empty, NonUnitalSubsemiring.coe_bot,\n      Submodule.span_zero_singleton, Submodule.mem_bot]\n\ntheorem toSubmodule_bot : (⊥ : NonUnitalSubalgebra R A).toSubmodule = ⊥ := by\n  ext\n  simp only [mem_bot, NonUnitalSubalgebra.mem_toSubmodule, Submodule.mem_bot]\n\n@[simp]\ntheorem coe_bot : ((⊥ : NonUnitalSubalgebra R A) : Set A) = {0} := by\n  simp [Set.ext_iff, NonUnitalAlgebra.mem_bot]\n\ntheorem eq_top_iff {S : NonUnitalSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S :=\n  ⟨fun h x => by rw [h]; exact mem_top, fun h => by ext x; exact ⟨fun _ => mem_top, fun _ => h x⟩⟩\n\ntheorem range_top_iff_surjective (f : A →ₙₐ[R] B) :\n    NonUnitalAlgHom.range f = (⊤ : NonUnitalSubalgebra R B) ↔ Function.Surjective f :=\n  NonUnitalAlgebra.eq_top_iff\n\n@[simp]\ntheorem range_id : NonUnitalAlgHom.range (NonUnitalAlgHom.id R A) = ⊤ :=\n  SetLike.coe_injective Set.range_id\n\n@[simp]\ntheorem map_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R A).map f = NonUnitalAlgHom.range f :=\n  SetLike.coe_injective Set.image_univ\n\n@[simp]\ntheorem map_bot (f : A →ₙₐ[R] B) : (⊥ : NonUnitalSubalgebra R A).map f = ⊥ :=\n  SetLike.coe_injective <| by simp [NonUnitalAlgebra.coe_bot, NonUnitalSubalgebra.coe_map]\n\n@[simp]\ntheorem comap_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R B).comap f = ⊤ :=\n  eq_top_iff.2 fun _ => mem_top\n\n/-- `NonUnitalAlgHom` to `⊤ : NonUnitalSubalgebra R A`. -/\ndef toTop : A →ₙₐ[R] (⊤ : NonUnitalSubalgebra R A) :=\n  NonUnitalAlgHom.codRestrict (NonUnitalAlgHom.id R A) ⊤ fun _ => mem_top\n\nend NonUnitalAlgebra\n\nnamespace NonUnitalSubalgebra\n\nopen NonUnitalAlgebra\n\nsection NonAssoc\n\nvariable {R : Type u} {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable (S : NonUnitalSubalgebra R A)\n\ninstance subsingleton_of_subsingleton [Subsingleton A] : Subsingleton (NonUnitalSubalgebra R A) :=\n  ⟨fun B C => ext fun x => by simp only [Subsingleton.elim x 0, zero_mem B, zero_mem C]⟩\n\ninstance _root_.NonUnitalAlgHom.subsingleton [Subsingleton (NonUnitalSubalgebra R A)] :\n    Subsingleton (A →ₙₐ[R] B) :=\n  ⟨fun f g =>\n    NonUnitalAlgHom.ext fun a =>\n      have : a ∈ (⊥ : NonUnitalSubalgebra R A) :=\n        Subsingleton.elim (⊤ : NonUnitalSubalgebra R A) ⊥ ▸ mem_top\n      (mem_bot.mp this).symm ▸ (map_zero f).trans (map_zero g).symm⟩\n\ntheorem range_val : NonUnitalAlgHom.range (NonUnitalSubalgebraClass.subtype S) = S :=\n  ext <| Set.ext_iff.1 <| (NonUnitalSubalgebraClass.subtype S).coe_range.trans Subtype.range_val\n\n/-- The map `S → T` when `S` is a non-unital subalgebra contained in the non-unital subalgebra `T`.\n\nThis is the non-unital subalgebra version of `Submodule.inclusion`, or `Subring.inclusion`  -/\ndef inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) : S →ₙₐ[R] T\n    where\n  toFun := Set.inclusion h\n  map_add' _ _ := rfl\n  map_mul' _ _ := rfl\n  map_zero' := rfl\n  map_smul' _ _ := rfl\n\ntheorem inclusion_injective {S T : NonUnitalSubalgebra R A} (h : S ≤ T) :\n    Function.Injective (inclusion h) := fun _ _ => Subtype.ext ∘ Subtype.mk.inj\n\n@[simp]\ntheorem inclusion_self {S : NonUnitalSubalgebra R A} :\n    inclusion (le_refl S) = NonUnitalAlgHom.id R S :=\n  NonUnitalAlgHom.ext fun _ => Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_mk {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : A) (hx : x ∈ S) :\n    inclusion h ⟨x, hx⟩ = ⟨x, h hx⟩ :=\n  rfl\n\ntheorem inclusion_right {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : T) (m : (x : A) ∈ S) :\n    inclusion h ⟨x, m⟩ = x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_inclusion {S T U : NonUnitalSubalgebra R A} (hst : S ≤ T) (htu : T ≤ U) (x : S) :\n    inclusion htu (inclusion hst x) = inclusion (le_trans hst htu) x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem coe_inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (s : S) :\n    (inclusion h s : A) = s :=\n  rfl\n\nsection Prod\n\nvariable (S₁ : NonUnitalSubalgebra R B)\n\n/-- The product of two non-unital subalgebras is a non-unital subalgebra. -/\ndef prod : NonUnitalSubalgebra R (A × B) :=\n  { S.toNonUnitalSubsemiring.prod S₁.toNonUnitalSubsemiring with\n    carrier := S ×ˢ S₁\n    smul_mem' := fun r _x hx => ⟨SMulMemClass.smul_mem r hx.1, SMulMemClass.smul_mem r hx.2⟩ }\n\n@[simp]\ntheorem coe_prod : (prod S S₁ : Set (A × B)) = (S : Set A) ×ˢ S₁ :=\n  rfl\n\ntheorem prod_toSubmodule : (S.prod S₁).toSubmodule = S.toSubmodule.prod S₁.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem mem_prod {S : NonUnitalSubalgebra R A} {S₁ : NonUnitalSubalgebra R B} {x : A × B} :\n    x ∈ prod S S₁ ↔ x.1 ∈ S ∧ x.2 ∈ S₁ :=\n  Set.mem_prod\n\n@[simp]\ntheorem prod_top : (prod ⊤ ⊤ : NonUnitalSubalgebra R (A × B)) = ⊤ := by ext; simp\n\ntheorem prod_mono {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S ≤ T → S₁ ≤ T₁ → prod S S₁ ≤ prod T T₁ :=\n  Set.prod_mono\n\n@[simp]\ntheorem prod_inf_prod {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S.prod S₁ ⊓ T.prod T₁ = (S ⊓ T).prod (S₁ ⊓ T₁) :=\n  SetLike.coe_injective Set.prod_inter_prod\n\nend Prod\n\nsection SuprLift\n\nvariable {ι : Type*}\n\ntheorem coe_iSup_of_directed [Nonempty ι] {S : ι → NonUnitalSubalgebra R A}\n    (dir : Directed (· ≤ ·) S) : ↑(iSup S) = ⋃ i, (S i : Set A) :=\n  let K : NonUnitalSubalgebra R A :=\n    { __ := NonUnitalSubsemiring.copy _ _ (NonUnitalSubsemiring.coe_iSup_of_directed dir).symm\n      smul_mem' := fun r _x hx ↦\n        let ⟨i, hi⟩ := Set.mem_iUnion.1 hx\n        Set.mem_iUnion.2 ⟨i, (S i).smul_mem' r hi⟩ }\n  have : iSup S = K := le_antisymm\n    (iSup_le fun i ↦ le_iSup (fun i ↦ (S i : Set A)) i) (Set.iUnion_subset fun _ ↦ le_iSup S _)\n  this.symm ▸ rfl\n\n/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              rw [hf j k hjk]\n              rfl)\n            (↑(iSup K)) (by rw [coe_iSup_of_directed dir])\n        map_zero' := by\n          dsimp\n          exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl) _ (by simp)\n        map_mul' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· * ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_add' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· + ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          ","nextTactic":"all_goals simp","declUpToTactic":"/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              rw [hf j k hjk]\n              rfl)\n            (↑(iSup K)) (by rw [coe_iSup_of_directed dir])\n        map_zero' := by\n          dsimp\n          exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl) _ (by simp)\n        map_mul' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· * ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_add' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· + ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.892_0.jQvdAcOcyVirh23","decl":"/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B "}
-{"state":"case hopi\nR : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\nS : NonUnitalSubalgebra R A\nι : Type u_1\ninst✝ : Nonempty ι\nK : ι → NonUnitalSubalgebra R A\ndir : Directed (fun x x_1 => x ≤ x_1) K\nf : (i : ι) → ↥(K i) →ₙₐ[R] B\nhf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalAlgHom.comp (f j) (inclusion h)\n⊢ ∀ (i : ι) (x y : ↑↑(K i)),\n    Set.inclusion (_ : ↑(K i) ⊆ ↑(iSup K)) (x + y) =\n      Set.inclusion (_ : ↑(K i) ⊆ ↑(iSup K)) x + Set.inclusion (_ : ↑(K i) ⊆ ↑(iSup K)) y","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact NonUnitalAlgebra.gc\n\n@[simp]\ntheorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=\n  eq_top_iff.2 fun _x hx => subset_adjoin R hx\n\nvariable {R A}\n\n@[simp]\ntheorem coe_top : (↑(⊤ : NonUnitalSubalgebra R A) : Set A) = Set.univ :=\n  rfl\n\n@[simp]\ntheorem mem_top {x : A} : x ∈ (⊤ : NonUnitalSubalgebra R A) :=\n  Set.mem_univ x\n\n@[simp]\ntheorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toNonUnitalSubsemiring : (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubsemiring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toSubring {R A : Type*} [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    [IsScalarTower R A A] [SMulCommClass R A A] :\n    (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem toSubmodule_eq_top {S : NonUnitalSubalgebra R A} : S.toSubmodule = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toSubmodule'.injective.eq_iff' top_toSubmodule\n\n@[simp]\ntheorem toNonUnitalSubsemiring_eq_top {S : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubsemiring_injective.eq_iff' top_toNonUnitalSubsemiring\n\n@[simp]\ntheorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubring_injective.eq_iff' top_toSubring\n\ntheorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_left\n\ntheorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_right\n\ntheorem mul_mem_sup {S T : NonUnitalSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) :\n    x * y ∈ S ⊔ T :=\n  mul_mem (mem_sup_left hx) (mem_sup_right hy)\n\ntheorem map_sup (f : F) (S T : NonUnitalSubalgebra R A) :\n    ((S ⊔ T).map f : NonUnitalSubalgebra R B) = S.map f ⊔ T.map f :=\n  (@NonUnitalSubalgebra.gc_map_comap F R A B _ _ _ _ _ _ f).l_sup\n\n@[simp, norm_cast]\ntheorem coe_inf (S T : NonUnitalSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T :=\n  rfl\n\n@[simp]\ntheorem mem_inf {S T : NonUnitalSubalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T :=\n  Iff.rfl\n\n@[simp]\ntheorem inf_toSubmodule (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toSubmodule = S.toSubmodule ⊓ T.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem inf_toNonUnitalSubsemiring (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring ⊓ T.toNonUnitalSubsemiring :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sInf (S : Set (NonUnitalSubalgebra R A)) : (↑(sInf S) : Set A) = ⋂ s ∈ S, ↑s :=\n  sInf_image\n\ntheorem mem_sInf {S : Set (NonUnitalSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p := by\n  simp only [← SetLike.mem_coe, coe_sInf, Set.mem_iInter₂]\n\n@[simp]\ntheorem sInf_toSubmodule (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toSubmodule = sInf (NonUnitalSubalgebra.toSubmodule '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp]\ntheorem sInf_toNonUnitalSubsemiring (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toNonUnitalSubsemiring = sInf (NonUnitalSubalgebra.toNonUnitalSubsemiring '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp, norm_cast]\ntheorem coe_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} :\n    (↑(⨅ i, S i) : Set A) = ⋂ i, S i := by simp [iInf]\n\ntheorem mem_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} {x : A} :\n    (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by simp only [iInf, mem_sInf, Set.forall_range_iff]\n\n@[simp]\ntheorem iInf_toSubmodule {ι : Sort*} (S : ι → NonUnitalSubalgebra R A) :\n    (⨅ i, S i).toSubmodule = ⨅ i, (S i).toSubmodule :=\n  SetLike.coe_injective <| by simp\n\ninstance : Inhabited (NonUnitalSubalgebra R A) :=\n  ⟨⊥⟩\n\ntheorem mem_bot {x : A} : x ∈ (⊥ : NonUnitalSubalgebra R A) ↔ x = 0 :=\n  show x ∈ Submodule.span R (NonUnitalSubsemiring.closure (∅ : Set A) : Set A) ↔ x = 0 by\n    rw [NonUnitalSubsemiring.closure_empty, NonUnitalSubsemiring.coe_bot,\n      Submodule.span_zero_singleton, Submodule.mem_bot]\n\ntheorem toSubmodule_bot : (⊥ : NonUnitalSubalgebra R A).toSubmodule = ⊥ := by\n  ext\n  simp only [mem_bot, NonUnitalSubalgebra.mem_toSubmodule, Submodule.mem_bot]\n\n@[simp]\ntheorem coe_bot : ((⊥ : NonUnitalSubalgebra R A) : Set A) = {0} := by\n  simp [Set.ext_iff, NonUnitalAlgebra.mem_bot]\n\ntheorem eq_top_iff {S : NonUnitalSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S :=\n  ⟨fun h x => by rw [h]; exact mem_top, fun h => by ext x; exact ⟨fun _ => mem_top, fun _ => h x⟩⟩\n\ntheorem range_top_iff_surjective (f : A →ₙₐ[R] B) :\n    NonUnitalAlgHom.range f = (⊤ : NonUnitalSubalgebra R B) ↔ Function.Surjective f :=\n  NonUnitalAlgebra.eq_top_iff\n\n@[simp]\ntheorem range_id : NonUnitalAlgHom.range (NonUnitalAlgHom.id R A) = ⊤ :=\n  SetLike.coe_injective Set.range_id\n\n@[simp]\ntheorem map_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R A).map f = NonUnitalAlgHom.range f :=\n  SetLike.coe_injective Set.image_univ\n\n@[simp]\ntheorem map_bot (f : A →ₙₐ[R] B) : (⊥ : NonUnitalSubalgebra R A).map f = ⊥ :=\n  SetLike.coe_injective <| by simp [NonUnitalAlgebra.coe_bot, NonUnitalSubalgebra.coe_map]\n\n@[simp]\ntheorem comap_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R B).comap f = ⊤ :=\n  eq_top_iff.2 fun _ => mem_top\n\n/-- `NonUnitalAlgHom` to `⊤ : NonUnitalSubalgebra R A`. -/\ndef toTop : A →ₙₐ[R] (⊤ : NonUnitalSubalgebra R A) :=\n  NonUnitalAlgHom.codRestrict (NonUnitalAlgHom.id R A) ⊤ fun _ => mem_top\n\nend NonUnitalAlgebra\n\nnamespace NonUnitalSubalgebra\n\nopen NonUnitalAlgebra\n\nsection NonAssoc\n\nvariable {R : Type u} {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable (S : NonUnitalSubalgebra R A)\n\ninstance subsingleton_of_subsingleton [Subsingleton A] : Subsingleton (NonUnitalSubalgebra R A) :=\n  ⟨fun B C => ext fun x => by simp only [Subsingleton.elim x 0, zero_mem B, zero_mem C]⟩\n\ninstance _root_.NonUnitalAlgHom.subsingleton [Subsingleton (NonUnitalSubalgebra R A)] :\n    Subsingleton (A →ₙₐ[R] B) :=\n  ⟨fun f g =>\n    NonUnitalAlgHom.ext fun a =>\n      have : a ∈ (⊥ : NonUnitalSubalgebra R A) :=\n        Subsingleton.elim (⊤ : NonUnitalSubalgebra R A) ⊥ ▸ mem_top\n      (mem_bot.mp this).symm ▸ (map_zero f).trans (map_zero g).symm⟩\n\ntheorem range_val : NonUnitalAlgHom.range (NonUnitalSubalgebraClass.subtype S) = S :=\n  ext <| Set.ext_iff.1 <| (NonUnitalSubalgebraClass.subtype S).coe_range.trans Subtype.range_val\n\n/-- The map `S → T` when `S` is a non-unital subalgebra contained in the non-unital subalgebra `T`.\n\nThis is the non-unital subalgebra version of `Submodule.inclusion`, or `Subring.inclusion`  -/\ndef inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) : S →ₙₐ[R] T\n    where\n  toFun := Set.inclusion h\n  map_add' _ _ := rfl\n  map_mul' _ _ := rfl\n  map_zero' := rfl\n  map_smul' _ _ := rfl\n\ntheorem inclusion_injective {S T : NonUnitalSubalgebra R A} (h : S ≤ T) :\n    Function.Injective (inclusion h) := fun _ _ => Subtype.ext ∘ Subtype.mk.inj\n\n@[simp]\ntheorem inclusion_self {S : NonUnitalSubalgebra R A} :\n    inclusion (le_refl S) = NonUnitalAlgHom.id R S :=\n  NonUnitalAlgHom.ext fun _ => Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_mk {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : A) (hx : x ∈ S) :\n    inclusion h ⟨x, hx⟩ = ⟨x, h hx⟩ :=\n  rfl\n\ntheorem inclusion_right {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : T) (m : (x : A) ∈ S) :\n    inclusion h ⟨x, m⟩ = x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_inclusion {S T U : NonUnitalSubalgebra R A} (hst : S ≤ T) (htu : T ≤ U) (x : S) :\n    inclusion htu (inclusion hst x) = inclusion (le_trans hst htu) x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem coe_inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (s : S) :\n    (inclusion h s : A) = s :=\n  rfl\n\nsection Prod\n\nvariable (S₁ : NonUnitalSubalgebra R B)\n\n/-- The product of two non-unital subalgebras is a non-unital subalgebra. -/\ndef prod : NonUnitalSubalgebra R (A × B) :=\n  { S.toNonUnitalSubsemiring.prod S₁.toNonUnitalSubsemiring with\n    carrier := S ×ˢ S₁\n    smul_mem' := fun r _x hx => ⟨SMulMemClass.smul_mem r hx.1, SMulMemClass.smul_mem r hx.2⟩ }\n\n@[simp]\ntheorem coe_prod : (prod S S₁ : Set (A × B)) = (S : Set A) ×ˢ S₁ :=\n  rfl\n\ntheorem prod_toSubmodule : (S.prod S₁).toSubmodule = S.toSubmodule.prod S₁.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem mem_prod {S : NonUnitalSubalgebra R A} {S₁ : NonUnitalSubalgebra R B} {x : A × B} :\n    x ∈ prod S S₁ ↔ x.1 ∈ S ∧ x.2 ∈ S₁ :=\n  Set.mem_prod\n\n@[simp]\ntheorem prod_top : (prod ⊤ ⊤ : NonUnitalSubalgebra R (A × B)) = ⊤ := by ext; simp\n\ntheorem prod_mono {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S ≤ T → S₁ ≤ T₁ → prod S S₁ ≤ prod T T₁ :=\n  Set.prod_mono\n\n@[simp]\ntheorem prod_inf_prod {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S.prod S₁ ⊓ T.prod T₁ = (S ⊓ T).prod (S₁ ⊓ T₁) :=\n  SetLike.coe_injective Set.prod_inter_prod\n\nend Prod\n\nsection SuprLift\n\nvariable {ι : Type*}\n\ntheorem coe_iSup_of_directed [Nonempty ι] {S : ι → NonUnitalSubalgebra R A}\n    (dir : Directed (· ≤ ·) S) : ↑(iSup S) = ⋃ i, (S i : Set A) :=\n  let K : NonUnitalSubalgebra R A :=\n    { __ := NonUnitalSubsemiring.copy _ _ (NonUnitalSubsemiring.coe_iSup_of_directed dir).symm\n      smul_mem' := fun r _x hx ↦\n        let ⟨i, hi⟩ := Set.mem_iUnion.1 hx\n        Set.mem_iUnion.2 ⟨i, (S i).smul_mem' r hi⟩ }\n  have : iSup S = K := le_antisymm\n    (iSup_le fun i ↦ le_iSup (fun i ↦ (S i : Set A)) i) (Set.iUnion_subset fun _ ↦ le_iSup S _)\n  this.symm ▸ rfl\n\n/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              rw [hf j k hjk]\n              rfl)\n            (↑(iSup K)) (by rw [coe_iSup_of_directed dir])\n        map_zero' := by\n          dsimp\n          exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl) _ (by simp)\n        map_mul' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· * ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_add' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· + ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals ","nextTactic":"simp","declUpToTactic":"/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              rw [hf j k hjk]\n              rfl)\n            (↑(iSup K)) (by rw [coe_iSup_of_directed dir])\n        map_zero' := by\n          dsimp\n          exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl) _ (by simp)\n        map_mul' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· * ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_add' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· + ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.892_0.jQvdAcOcyVirh23","decl":"/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B "}
-{"state":"case h\nR : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\nS : NonUnitalSubalgebra R A\nι : Type u_1\ninst✝ : Nonempty ι\nK : ι → NonUnitalSubalgebra R A\ndir : Directed (fun x x_1 => x ≤ x_1) K\nf : (i : ι) → ↥(K i) →ₙₐ[R] B\nhf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalAlgHom.comp (f j) (inclusion h)\n⊢ ∀ (i : ι) (x y : ↑↑(K i)), (f i) (x + y) = (f i) x + (f i) y","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact NonUnitalAlgebra.gc\n\n@[simp]\ntheorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=\n  eq_top_iff.2 fun _x hx => subset_adjoin R hx\n\nvariable {R A}\n\n@[simp]\ntheorem coe_top : (↑(⊤ : NonUnitalSubalgebra R A) : Set A) = Set.univ :=\n  rfl\n\n@[simp]\ntheorem mem_top {x : A} : x ∈ (⊤ : NonUnitalSubalgebra R A) :=\n  Set.mem_univ x\n\n@[simp]\ntheorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toNonUnitalSubsemiring : (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubsemiring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toSubring {R A : Type*} [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    [IsScalarTower R A A] [SMulCommClass R A A] :\n    (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem toSubmodule_eq_top {S : NonUnitalSubalgebra R A} : S.toSubmodule = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toSubmodule'.injective.eq_iff' top_toSubmodule\n\n@[simp]\ntheorem toNonUnitalSubsemiring_eq_top {S : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubsemiring_injective.eq_iff' top_toNonUnitalSubsemiring\n\n@[simp]\ntheorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubring_injective.eq_iff' top_toSubring\n\ntheorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_left\n\ntheorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_right\n\ntheorem mul_mem_sup {S T : NonUnitalSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) :\n    x * y ∈ S ⊔ T :=\n  mul_mem (mem_sup_left hx) (mem_sup_right hy)\n\ntheorem map_sup (f : F) (S T : NonUnitalSubalgebra R A) :\n    ((S ⊔ T).map f : NonUnitalSubalgebra R B) = S.map f ⊔ T.map f :=\n  (@NonUnitalSubalgebra.gc_map_comap F R A B _ _ _ _ _ _ f).l_sup\n\n@[simp, norm_cast]\ntheorem coe_inf (S T : NonUnitalSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T :=\n  rfl\n\n@[simp]\ntheorem mem_inf {S T : NonUnitalSubalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T :=\n  Iff.rfl\n\n@[simp]\ntheorem inf_toSubmodule (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toSubmodule = S.toSubmodule ⊓ T.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem inf_toNonUnitalSubsemiring (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring ⊓ T.toNonUnitalSubsemiring :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sInf (S : Set (NonUnitalSubalgebra R A)) : (↑(sInf S) : Set A) = ⋂ s ∈ S, ↑s :=\n  sInf_image\n\ntheorem mem_sInf {S : Set (NonUnitalSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p := by\n  simp only [← SetLike.mem_coe, coe_sInf, Set.mem_iInter₂]\n\n@[simp]\ntheorem sInf_toSubmodule (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toSubmodule = sInf (NonUnitalSubalgebra.toSubmodule '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp]\ntheorem sInf_toNonUnitalSubsemiring (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toNonUnitalSubsemiring = sInf (NonUnitalSubalgebra.toNonUnitalSubsemiring '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp, norm_cast]\ntheorem coe_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} :\n    (↑(⨅ i, S i) : Set A) = ⋂ i, S i := by simp [iInf]\n\ntheorem mem_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} {x : A} :\n    (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by simp only [iInf, mem_sInf, Set.forall_range_iff]\n\n@[simp]\ntheorem iInf_toSubmodule {ι : Sort*} (S : ι → NonUnitalSubalgebra R A) :\n    (⨅ i, S i).toSubmodule = ⨅ i, (S i).toSubmodule :=\n  SetLike.coe_injective <| by simp\n\ninstance : Inhabited (NonUnitalSubalgebra R A) :=\n  ⟨⊥⟩\n\ntheorem mem_bot {x : A} : x ∈ (⊥ : NonUnitalSubalgebra R A) ↔ x = 0 :=\n  show x ∈ Submodule.span R (NonUnitalSubsemiring.closure (∅ : Set A) : Set A) ↔ x = 0 by\n    rw [NonUnitalSubsemiring.closure_empty, NonUnitalSubsemiring.coe_bot,\n      Submodule.span_zero_singleton, Submodule.mem_bot]\n\ntheorem toSubmodule_bot : (⊥ : NonUnitalSubalgebra R A).toSubmodule = ⊥ := by\n  ext\n  simp only [mem_bot, NonUnitalSubalgebra.mem_toSubmodule, Submodule.mem_bot]\n\n@[simp]\ntheorem coe_bot : ((⊥ : NonUnitalSubalgebra R A) : Set A) = {0} := by\n  simp [Set.ext_iff, NonUnitalAlgebra.mem_bot]\n\ntheorem eq_top_iff {S : NonUnitalSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S :=\n  ⟨fun h x => by rw [h]; exact mem_top, fun h => by ext x; exact ⟨fun _ => mem_top, fun _ => h x⟩⟩\n\ntheorem range_top_iff_surjective (f : A →ₙₐ[R] B) :\n    NonUnitalAlgHom.range f = (⊤ : NonUnitalSubalgebra R B) ↔ Function.Surjective f :=\n  NonUnitalAlgebra.eq_top_iff\n\n@[simp]\ntheorem range_id : NonUnitalAlgHom.range (NonUnitalAlgHom.id R A) = ⊤ :=\n  SetLike.coe_injective Set.range_id\n\n@[simp]\ntheorem map_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R A).map f = NonUnitalAlgHom.range f :=\n  SetLike.coe_injective Set.image_univ\n\n@[simp]\ntheorem map_bot (f : A →ₙₐ[R] B) : (⊥ : NonUnitalSubalgebra R A).map f = ⊥ :=\n  SetLike.coe_injective <| by simp [NonUnitalAlgebra.coe_bot, NonUnitalSubalgebra.coe_map]\n\n@[simp]\ntheorem comap_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R B).comap f = ⊤ :=\n  eq_top_iff.2 fun _ => mem_top\n\n/-- `NonUnitalAlgHom` to `⊤ : NonUnitalSubalgebra R A`. -/\ndef toTop : A →ₙₐ[R] (⊤ : NonUnitalSubalgebra R A) :=\n  NonUnitalAlgHom.codRestrict (NonUnitalAlgHom.id R A) ⊤ fun _ => mem_top\n\nend NonUnitalAlgebra\n\nnamespace NonUnitalSubalgebra\n\nopen NonUnitalAlgebra\n\nsection NonAssoc\n\nvariable {R : Type u} {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable (S : NonUnitalSubalgebra R A)\n\ninstance subsingleton_of_subsingleton [Subsingleton A] : Subsingleton (NonUnitalSubalgebra R A) :=\n  ⟨fun B C => ext fun x => by simp only [Subsingleton.elim x 0, zero_mem B, zero_mem C]⟩\n\ninstance _root_.NonUnitalAlgHom.subsingleton [Subsingleton (NonUnitalSubalgebra R A)] :\n    Subsingleton (A →ₙₐ[R] B) :=\n  ⟨fun f g =>\n    NonUnitalAlgHom.ext fun a =>\n      have : a ∈ (⊥ : NonUnitalSubalgebra R A) :=\n        Subsingleton.elim (⊤ : NonUnitalSubalgebra R A) ⊥ ▸ mem_top\n      (mem_bot.mp this).symm ▸ (map_zero f).trans (map_zero g).symm⟩\n\ntheorem range_val : NonUnitalAlgHom.range (NonUnitalSubalgebraClass.subtype S) = S :=\n  ext <| Set.ext_iff.1 <| (NonUnitalSubalgebraClass.subtype S).coe_range.trans Subtype.range_val\n\n/-- The map `S → T` when `S` is a non-unital subalgebra contained in the non-unital subalgebra `T`.\n\nThis is the non-unital subalgebra version of `Submodule.inclusion`, or `Subring.inclusion`  -/\ndef inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) : S →ₙₐ[R] T\n    where\n  toFun := Set.inclusion h\n  map_add' _ _ := rfl\n  map_mul' _ _ := rfl\n  map_zero' := rfl\n  map_smul' _ _ := rfl\n\ntheorem inclusion_injective {S T : NonUnitalSubalgebra R A} (h : S ≤ T) :\n    Function.Injective (inclusion h) := fun _ _ => Subtype.ext ∘ Subtype.mk.inj\n\n@[simp]\ntheorem inclusion_self {S : NonUnitalSubalgebra R A} :\n    inclusion (le_refl S) = NonUnitalAlgHom.id R S :=\n  NonUnitalAlgHom.ext fun _ => Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_mk {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : A) (hx : x ∈ S) :\n    inclusion h ⟨x, hx⟩ = ⟨x, h hx⟩ :=\n  rfl\n\ntheorem inclusion_right {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : T) (m : (x : A) ∈ S) :\n    inclusion h ⟨x, m⟩ = x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_inclusion {S T U : NonUnitalSubalgebra R A} (hst : S ≤ T) (htu : T ≤ U) (x : S) :\n    inclusion htu (inclusion hst x) = inclusion (le_trans hst htu) x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem coe_inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (s : S) :\n    (inclusion h s : A) = s :=\n  rfl\n\nsection Prod\n\nvariable (S₁ : NonUnitalSubalgebra R B)\n\n/-- The product of two non-unital subalgebras is a non-unital subalgebra. -/\ndef prod : NonUnitalSubalgebra R (A × B) :=\n  { S.toNonUnitalSubsemiring.prod S₁.toNonUnitalSubsemiring with\n    carrier := S ×ˢ S₁\n    smul_mem' := fun r _x hx => ⟨SMulMemClass.smul_mem r hx.1, SMulMemClass.smul_mem r hx.2⟩ }\n\n@[simp]\ntheorem coe_prod : (prod S S₁ : Set (A × B)) = (S : Set A) ×ˢ S₁ :=\n  rfl\n\ntheorem prod_toSubmodule : (S.prod S₁).toSubmodule = S.toSubmodule.prod S₁.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem mem_prod {S : NonUnitalSubalgebra R A} {S₁ : NonUnitalSubalgebra R B} {x : A × B} :\n    x ∈ prod S S₁ ↔ x.1 ∈ S ∧ x.2 ∈ S₁ :=\n  Set.mem_prod\n\n@[simp]\ntheorem prod_top : (prod ⊤ ⊤ : NonUnitalSubalgebra R (A × B)) = ⊤ := by ext; simp\n\ntheorem prod_mono {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S ≤ T → S₁ ≤ T₁ → prod S S₁ ≤ prod T T₁ :=\n  Set.prod_mono\n\n@[simp]\ntheorem prod_inf_prod {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S.prod S₁ ⊓ T.prod T₁ = (S ⊓ T).prod (S₁ ⊓ T₁) :=\n  SetLike.coe_injective Set.prod_inter_prod\n\nend Prod\n\nsection SuprLift\n\nvariable {ι : Type*}\n\ntheorem coe_iSup_of_directed [Nonempty ι] {S : ι → NonUnitalSubalgebra R A}\n    (dir : Directed (· ≤ ·) S) : ↑(iSup S) = ⋃ i, (S i : Set A) :=\n  let K : NonUnitalSubalgebra R A :=\n    { __ := NonUnitalSubsemiring.copy _ _ (NonUnitalSubsemiring.coe_iSup_of_directed dir).symm\n      smul_mem' := fun r _x hx ↦\n        let ⟨i, hi⟩ := Set.mem_iUnion.1 hx\n        Set.mem_iUnion.2 ⟨i, (S i).smul_mem' r hi⟩ }\n  have : iSup S = K := le_antisymm\n    (iSup_le fun i ↦ le_iSup (fun i ↦ (S i : Set A)) i) (Set.iUnion_subset fun _ ↦ le_iSup S _)\n  this.symm ▸ rfl\n\n/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              rw [hf j k hjk]\n              rfl)\n            (↑(iSup K)) (by rw [coe_iSup_of_directed dir])\n        map_zero' := by\n          dsimp\n          exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl) _ (by simp)\n        map_mul' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· * ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_add' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· + ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals ","nextTactic":"simp","declUpToTactic":"/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              rw [hf j k hjk]\n              rfl)\n            (↑(iSup K)) (by rw [coe_iSup_of_directed dir])\n        map_zero' := by\n          dsimp\n          exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl) _ (by simp)\n        map_mul' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· * ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_add' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· + ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.892_0.jQvdAcOcyVirh23","decl":"/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B "}
-{"state":"R : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\nS : NonUnitalSubalgebra R A\nι : Type u_1\ninst✝ : Nonempty ι\nK : ι → NonUnitalSubalgebra R A\ndir : Directed (fun x x_1 => x ≤ x_1) K\nf : (i : ι) → ↥(K i) →ₙₐ[R] B\nhf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalAlgHom.comp (f j) (inclusion h)\n⊢ ∀ (x y : ↥(iSup K)),\n    MulActionHom.toFun\n        {\n            toMulActionHom :=\n              {\n                toFun :=\n                  Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x)\n                    (_ :\n                      ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),\n                        (fun i x => (f i) x) i { val := x, property := hxi } =\n                          (fun i x => (f i) x) j { val := x, property := hxj })\n                    ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ i, ↑(K i)),\n                map_smul' :=\n                  (_ :\n                    ∀ (r : R) (x : ↥(iSup K)),\n                      Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x)\n                          (_ :\n                            ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),\n                              (fun i x => (f i) x) i { val := x, property := hxi } =\n                                (fun i x => (f i) x) j { val := x, property := hxj })\n                          ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ i, ↑(K i)) (r • x) =\n                        r •\n                          Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x)\n                            (_ :\n                              ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),\n                                (fun i x => (f i) x) i { val := x, property := hxi } =\n                                  (fun i x => (f i) x) j { val := x, property := hxj })\n                            ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ i, ↑(K i)) x) },\n            map_zero' :=\n              (_ :\n                MulActionHom.toFun\n                    {\n                      toFun :=\n                        Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x)\n                          (_ :\n                            ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),\n                              (fun i x => (f i) x) i { val := x, property := hxi } =\n                                (fun i x => (f i) x) j { val := x, property := hxj })\n                          ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ i, ↑(K i)),\n                      map_smul' :=\n                        (_ :\n                          ∀ (r : R) (x : ↥(iSup K)),\n                            Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x)\n                                (_ :\n                                  ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),\n                                    (fun i x => (f i) x) i { val := x, property := hxi } =\n                                      (fun i x => (f i) x) j { val := x, property := hxj })\n                                ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ i, ↑(K i)) (r • x) =\n                              r •\n                                Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x)\n                                  (_ :\n                                    ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),\n                                      (fun i x => (f i) x) i { val := x, property := hxi } =\n                                        (fun i x => (f i) x) j { val := x, property := hxj })\n                                  ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ i, ↑(K i)) x) }\n                    0 =\n                  0),\n            map_add' :=\n              (_ :\n                ∀ (x y : ↥(iSup K)),\n                  MulActionHom.toFun\n                      {\n                        toFun :=\n                          Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x)\n                            (_ :\n                              ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),\n                                (fun i x => (f i) x) i { val := x, property := hxi } =\n                                  (fun i x => (f i) x) j { val := x, property := hxj })\n                            ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ i, ↑(K i)),\n                        map_smul' :=\n                          (_ :\n                            ∀ (r : R) (x : ↥(iSup K)),\n                              Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x)\n                                  (_ :\n                                    ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),\n                                      (fun i x => (f i) x) i { val := x, property := hxi } =\n                                        (fun i x => (f i) x) j { val := x, property := hxj })\n                                  ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ i, ↑(K i)) (r • x) =\n                                r •\n                                  Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x)\n                                    (_ :\n                                      ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)\n                                        (hxj : x ∈ (fun i => ↑(K i)) j),\n                                        (fun i x => (f i) x) i { val := x, property := hxi } =\n                                          (fun i x => (f i) x) j { val := x, property := hxj })\n                                    ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ i, ↑(K i)) x) }\n                      (x + y) =\n                    MulActionHom.toFun\n                        {\n                          toFun :=\n                            Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x)\n                              (_ :\n                                ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),\n                                  (fun i x => (f i) x) i { val := x, property := hxi } =\n                                    (fun i x => (f i) x) j { val := x, property := hxj })\n                              ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ i, ↑(K i)),\n                          map_smul' :=\n                            (_ :\n                              ∀ (r : R) (x : ↥(iSup K)),\n                                Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x)\n                                    (_ :\n                                      ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)\n                                        (hxj : x ∈ (fun i => ↑(K i)) j),\n                                        (fun i x => (f i) x) i { val := x, property := hxi } =\n                                          (fun i x => (f i) x) j { val := x, property := hxj })\n                                    ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ i, ↑(K i)) (r • x) =\n                                  r •\n                                    Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x)\n                                      (_ :\n                                        ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)\n                                          (hxj : x ∈ (fun i => ↑(K i)) j),\n                                          (fun i x => (f i) x) i { val := x, property := hxi } =\n                                            (fun i x => (f i) x) j { val := x, property := hxj })\n                                      ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ i, ↑(K i)) x) }\n                        x +\n                      MulActionHom.toFun\n                        {\n                          toFun :=\n                            Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x)\n                              (_ :\n                                ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),\n                                  (fun i x => (f i) x) i { val := x, property := hxi } =\n                                    (fun i x => (f i) x) j { val := x, property := hxj })\n                              ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ i, ↑(K i)),\n                          map_smul' :=\n                            (_ :\n                              ∀ (r : R) (x : ↥(iSup K)),\n                                Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x)\n                                    (_ :\n                                      ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)\n                                        (hxj : x ∈ (fun i => ↑(K i)) j),\n                                        (fun i x => (f i) x) i { val := x, property := hxi } =\n                                          (fun i x => (f i) x) j { val := x, property := hxj })\n                                    ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ i, ↑(K i)) (r • x) =\n                                  r •\n                                    Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x)\n                                      (_ :\n                                        ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)\n                                          (hxj : x ∈ (fun i => ↑(K i)) j),\n                                          (fun i x => (f i) x) i { val := x, property := hxi } =\n                                            (fun i x => (f i) x) j { val := x, property := hxj })\n                                      ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ i, ↑(K i)) x) }\n                        y) }.toMulActionHom\n        (x * y) =\n      MulActionHom.toFun\n          {\n              toMulActionHom :=\n                {\n                  toFun :=\n                    Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x)\n                      (_ :\n                        ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),\n                          (fun i x => (f i) x) i { val := x, property := hxi } =\n                            (fun i x => (f i) x) j { val := x, property := hxj })\n                      ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ i, ↑(K i)),\n                  map_smul' :=\n                    (_ :\n                      ∀ (r : R) (x : ↥(iSup K)),\n                        Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x)\n                            (_ :\n                              ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),\n                                (fun i x => (f i) x) i { val := x, property := hxi } =\n                                  (fun i x => (f i) x) j { val := x, property := hxj })\n                            ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ i, ↑(K i)) (r • x) =\n                          r •\n                            Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x)\n                              (_ :\n                                ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),\n                                  (fun i x => (f i) x) i { val := x, property := hxi } =\n                                    (fun i x => (f i) x) j { val := x, property := hxj })\n                              ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ i, ↑(K i)) x) },\n              map_zero' :=\n                (_ :\n                  MulActionHom.toFun\n                      {\n                        toFun :=\n                          Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x)\n                            (_ :\n                              ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),\n                                (fun i x => (f i) x) i { val := x, property := hxi } =\n                                  (fun i x => (f i) x) j { val := x, property := hxj })\n                            ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ i, ↑(K i)),\n                        map_smul' :=\n                          (_ :\n                            ∀ (r : R) (x : ↥(iSup K)),\n                              Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x)\n                                  (_ :\n                                    ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),\n                                      (fun i x => (f i) x) i { val := x, property := hxi } =\n                                        (fun i x => (f i) x) j { val := x, property := hxj })\n                                  ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ i, ↑(K i)) (r • x) =\n                                r •\n                                  Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x)\n                                    (_ :\n                                      ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)\n                                        (hxj : x ∈ (fun i => ↑(K i)) j),\n                                        (fun i x => (f i) x) i { val := x, property := hxi } =\n                                          (fun i x => (f i) x) j { val := x, property := hxj })\n                                    ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ i, ↑(K i)) x) }\n                      0 =\n                    0),\n              map_add' :=\n                (_ :\n                  ∀ (x y : ↥(iSup K)),\n                    MulActionHom.toFun\n                        {\n                          toFun :=\n                            Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x)\n                              (_ :\n                                ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),\n                                  (fun i x => (f i) x) i { val := x, property := hxi } =\n                                    (fun i x => (f i) x) j { val := x, property := hxj })\n                              ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ i, ↑(K i)),\n                          map_smul' :=\n                            (_ :\n                              ∀ (r : R) (x : ↥(iSup K)),\n                                Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x)\n                                    (_ :\n                                      ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)\n                                        (hxj : x ∈ (fun i => ↑(K i)) j),\n                                        (fun i x => (f i) x) i { val := x, property := hxi } =\n                                          (fun i x => (f i) x) j { val := x, property := hxj })\n                                    ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ i, ↑(K i)) (r • x) =\n                                  r •\n                                    Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x)\n                                      (_ :\n                                        ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)\n                                          (hxj : x ∈ (fun i => ↑(K i)) j),\n                                          (fun i x => (f i) x) i { val := x, property := hxi } =\n                                            (fun i x => (f i) x) j { val := x, property := hxj })\n                                      ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ i, ↑(K i)) x) }\n                        (x + y) =\n                      MulActionHom.toFun\n                          {\n                            toFun :=\n                              Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x)\n                                (_ :\n                                  ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),\n                                    (fun i x => (f i) x) i { val := x, property := hxi } =\n                                      (fun i x => (f i) x) j { val := x, property := hxj })\n                                ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ i, ↑(K i)),\n                            map_smul' :=\n                              (_ :\n                                ∀ (r : R) (x : ↥(iSup K)),\n                                  Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x)\n                                      (_ :\n                                        ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)\n                                          (hxj : x ∈ (fun i => ↑(K i)) j),\n                                          (fun i x => (f i) x) i { val := x, property := hxi } =\n                                            (fun i x => (f i) x) j { val := x, property := hxj })\n                                      ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ i, ↑(K i)) (r • x) =\n                                    r •\n                                      Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x)\n                                        (_ :\n                                          ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)\n                                            (hxj : x ∈ (fun i => ↑(K i)) j),\n                                            (fun i x => (f i) x) i { val := x, property := hxi } =\n                                              (fun i x => (f i) x) j { val := x, property := hxj })\n                                        ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ i, ↑(K i)) x) }\n                          x +\n                        MulActionHom.toFun\n                          {\n                            toFun :=\n                              Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x)\n                                (_ :\n                                  ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),\n                                    (fun i x => (f i) x) i { val := x, property := hxi } =\n                                      (fun i x => (f i) x) j { val := x, property := hxj })\n                                ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ i, ↑(K i)),\n                            map_smul' :=\n                              (_ :\n                                ∀ (r : R) (x : ↥(iSup K)),\n                                  Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x)\n                                      (_ :\n                                        ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)\n                                          (hxj : x ∈ (fun i => ↑(K i)) j),\n                                          (fun i x => (f i) x) i { val := x, property := hxi } =\n                                            (fun i x => (f i) x) j { val := x, property := hxj })\n                                      ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ i, ↑(K i)) (r • x) =\n                                    r •\n                                      Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x)\n                                        (_ :\n                                          ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)\n                                            (hxj : x ∈ (fun i => ↑(K i)) j),\n                                            (fun i x => (f i) x) i { val := x, property := hxi } =\n                                              (fun i x => (f i) x) j { val := x, property := hxj })\n                                        ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ i, ↑(K i)) x) }\n                          y) }.toMulActionHom\n          x *\n        MulActionHom.toFun\n          {\n              toMulActionHom :=\n                {\n                  toFun :=\n                    Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x)\n                      (_ :\n                        ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),\n                          (fun i x => (f i) x) i { val := x, property := hxi } =\n                            (fun i x => (f i) x) j { val := x, property := hxj })\n                      ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ i, ↑(K i)),\n                  map_smul' :=\n                    (_ :\n                      ∀ (r : R) (x : ↥(iSup K)),\n                        Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x)\n                            (_ :\n                              ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),\n                                (fun i x => (f i) x) i { val := x, property := hxi } =\n                                  (fun i x => (f i) x) j { val := x, property := hxj })\n                            ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ i, ↑(K i)) (r • x) =\n                          r •\n                            Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x)\n                              (_ :\n                                ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),\n                                  (fun i x => (f i) x) i { val := x, property := hxi } =\n                                    (fun i x => (f i) x) j { val := x, property := hxj })\n                              ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ i, ↑(K i)) x) },\n              map_zero' :=\n                (_ :\n                  MulActionHom.toFun\n                      {\n                        toFun :=\n                          Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x)\n                            (_ :\n                              ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),\n                                (fun i x => (f i) x) i { val := x, property := hxi } =\n                                  (fun i x => (f i) x) j { val := x, property := hxj })\n                            ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ i, ↑(K i)),\n                        map_smul' :=\n                          (_ :\n                            ∀ (r : R) (x : ↥(iSup K)),\n                              Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x)\n                                  (_ :\n                                    ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),\n                                      (fun i x => (f i) x) i { val := x, property := hxi } =\n                                        (fun i x => (f i) x) j { val := x, property := hxj })\n                                  ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ i, ↑(K i)) (r • x) =\n                                r •\n                                  Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x)\n                                    (_ :\n                                      ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)\n                                        (hxj : x ∈ (fun i => ↑(K i)) j),\n                                        (fun i x => (f i) x) i { val := x, property := hxi } =\n                                          (fun i x => (f i) x) j { val := x, property := hxj })\n                                    ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ i, ↑(K i)) x) }\n                      0 =\n                    0),\n              map_add' :=\n                (_ :\n                  ∀ (x y : ↥(iSup K)),\n                    MulActionHom.toFun\n                        {\n                          toFun :=\n                            Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x)\n                              (_ :\n                                ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),\n                                  (fun i x => (f i) x) i { val := x, property := hxi } =\n                                    (fun i x => (f i) x) j { val := x, property := hxj })\n                              ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ i, ↑(K i)),\n                          map_smul' :=\n                            (_ :\n                              ∀ (r : R) (x : ↥(iSup K)),\n                                Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x)\n                                    (_ :\n                                      ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)\n                                        (hxj : x ∈ (fun i => ↑(K i)) j),\n                                        (fun i x => (f i) x) i { val := x, property := hxi } =\n                                          (fun i x => (f i) x) j { val := x, property := hxj })\n                                    ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ i, ↑(K i)) (r • x) =\n                                  r •\n                                    Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x)\n                                      (_ :\n                                        ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)\n                                          (hxj : x ∈ (fun i => ↑(K i)) j),\n                                          (fun i x => (f i) x) i { val := x, property := hxi } =\n                                            (fun i x => (f i) x) j { val := x, property := hxj })\n                                      ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ i, ↑(K i)) x) }\n                        (x + y) =\n                      MulActionHom.toFun\n                          {\n                            toFun :=\n                              Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x)\n                                (_ :\n                                  ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),\n                                    (fun i x => (f i) x) i { val := x, property := hxi } =\n                                      (fun i x => (f i) x) j { val := x, property := hxj })\n                                ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ i, ↑(K i)),\n                            map_smul' :=\n                              (_ :\n                                ∀ (r : R) (x : ↥(iSup K)),\n                                  Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x)\n                                      (_ :\n                                        ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)\n                                          (hxj : x ∈ (fun i => ↑(K i)) j),\n                                          (fun i x => (f i) x) i { val := x, property := hxi } =\n                                            (fun i x => (f i) x) j { val := x, property := hxj })\n                                      ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ i, ↑(K i)) (r • x) =\n                                    r •\n                                      Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x)\n                                        (_ :\n                                          ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)\n                                            (hxj : x ∈ (fun i => ↑(K i)) j),\n                                            (fun i x => (f i) x) i { val := x, property := hxi } =\n                                              (fun i x => (f i) x) j { val := x, property := hxj })\n                                        ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ i, ↑(K i)) x) }\n                          x +\n                        MulActionHom.toFun\n                          {\n                            toFun :=\n                              Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x)\n                                (_ :\n                                  ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),\n                                    (fun i x => (f i) x) i { val := x, property := hxi } =\n                                      (fun i x => (f i) x) j { val := x, property := hxj })\n                                ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ i, ↑(K i)),\n                            map_smul' :=\n                              (_ :\n                                ∀ (r : R) (x : ↥(iSup K)),\n                                  Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x)\n                                      (_ :\n                                        ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)\n                                          (hxj : x ∈ (fun i => ↑(K i)) j),\n                                          (fun i x => (f i) x) i { val := x, property := hxi } =\n                                            (fun i x => (f i) x) j { val := x, property := hxj })\n                                      ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ i, ↑(K i)) (r • x) =\n                                    r •\n                                      Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x)\n                                        (_ :\n                                          ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i)\n                                            (hxj : x ∈ (fun i => ↑(K i)) j),\n                                            (fun i x => (f i) x) i { val := x, property := hxi } =\n                                              (fun i x => (f i) x) j { val := x, property := hxj })\n                                        ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ i, ↑(K i)) x) }\n                          y) }.toMulActionHom\n          y","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact NonUnitalAlgebra.gc\n\n@[simp]\ntheorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=\n  eq_top_iff.2 fun _x hx => subset_adjoin R hx\n\nvariable {R A}\n\n@[simp]\ntheorem coe_top : (↑(⊤ : NonUnitalSubalgebra R A) : Set A) = Set.univ :=\n  rfl\n\n@[simp]\ntheorem mem_top {x : A} : x ∈ (⊤ : NonUnitalSubalgebra R A) :=\n  Set.mem_univ x\n\n@[simp]\ntheorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toNonUnitalSubsemiring : (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubsemiring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toSubring {R A : Type*} [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    [IsScalarTower R A A] [SMulCommClass R A A] :\n    (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem toSubmodule_eq_top {S : NonUnitalSubalgebra R A} : S.toSubmodule = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toSubmodule'.injective.eq_iff' top_toSubmodule\n\n@[simp]\ntheorem toNonUnitalSubsemiring_eq_top {S : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubsemiring_injective.eq_iff' top_toNonUnitalSubsemiring\n\n@[simp]\ntheorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubring_injective.eq_iff' top_toSubring\n\ntheorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_left\n\ntheorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_right\n\ntheorem mul_mem_sup {S T : NonUnitalSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) :\n    x * y ∈ S ⊔ T :=\n  mul_mem (mem_sup_left hx) (mem_sup_right hy)\n\ntheorem map_sup (f : F) (S T : NonUnitalSubalgebra R A) :\n    ((S ⊔ T).map f : NonUnitalSubalgebra R B) = S.map f ⊔ T.map f :=\n  (@NonUnitalSubalgebra.gc_map_comap F R A B _ _ _ _ _ _ f).l_sup\n\n@[simp, norm_cast]\ntheorem coe_inf (S T : NonUnitalSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T :=\n  rfl\n\n@[simp]\ntheorem mem_inf {S T : NonUnitalSubalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T :=\n  Iff.rfl\n\n@[simp]\ntheorem inf_toSubmodule (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toSubmodule = S.toSubmodule ⊓ T.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem inf_toNonUnitalSubsemiring (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring ⊓ T.toNonUnitalSubsemiring :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sInf (S : Set (NonUnitalSubalgebra R A)) : (↑(sInf S) : Set A) = ⋂ s ∈ S, ↑s :=\n  sInf_image\n\ntheorem mem_sInf {S : Set (NonUnitalSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p := by\n  simp only [← SetLike.mem_coe, coe_sInf, Set.mem_iInter₂]\n\n@[simp]\ntheorem sInf_toSubmodule (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toSubmodule = sInf (NonUnitalSubalgebra.toSubmodule '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp]\ntheorem sInf_toNonUnitalSubsemiring (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toNonUnitalSubsemiring = sInf (NonUnitalSubalgebra.toNonUnitalSubsemiring '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp, norm_cast]\ntheorem coe_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} :\n    (↑(⨅ i, S i) : Set A) = ⋂ i, S i := by simp [iInf]\n\ntheorem mem_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} {x : A} :\n    (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by simp only [iInf, mem_sInf, Set.forall_range_iff]\n\n@[simp]\ntheorem iInf_toSubmodule {ι : Sort*} (S : ι → NonUnitalSubalgebra R A) :\n    (⨅ i, S i).toSubmodule = ⨅ i, (S i).toSubmodule :=\n  SetLike.coe_injective <| by simp\n\ninstance : Inhabited (NonUnitalSubalgebra R A) :=\n  ⟨⊥⟩\n\ntheorem mem_bot {x : A} : x ∈ (⊥ : NonUnitalSubalgebra R A) ↔ x = 0 :=\n  show x ∈ Submodule.span R (NonUnitalSubsemiring.closure (∅ : Set A) : Set A) ↔ x = 0 by\n    rw [NonUnitalSubsemiring.closure_empty, NonUnitalSubsemiring.coe_bot,\n      Submodule.span_zero_singleton, Submodule.mem_bot]\n\ntheorem toSubmodule_bot : (⊥ : NonUnitalSubalgebra R A).toSubmodule = ⊥ := by\n  ext\n  simp only [mem_bot, NonUnitalSubalgebra.mem_toSubmodule, Submodule.mem_bot]\n\n@[simp]\ntheorem coe_bot : ((⊥ : NonUnitalSubalgebra R A) : Set A) = {0} := by\n  simp [Set.ext_iff, NonUnitalAlgebra.mem_bot]\n\ntheorem eq_top_iff {S : NonUnitalSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S :=\n  ⟨fun h x => by rw [h]; exact mem_top, fun h => by ext x; exact ⟨fun _ => mem_top, fun _ => h x⟩⟩\n\ntheorem range_top_iff_surjective (f : A →ₙₐ[R] B) :\n    NonUnitalAlgHom.range f = (⊤ : NonUnitalSubalgebra R B) ↔ Function.Surjective f :=\n  NonUnitalAlgebra.eq_top_iff\n\n@[simp]\ntheorem range_id : NonUnitalAlgHom.range (NonUnitalAlgHom.id R A) = ⊤ :=\n  SetLike.coe_injective Set.range_id\n\n@[simp]\ntheorem map_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R A).map f = NonUnitalAlgHom.range f :=\n  SetLike.coe_injective Set.image_univ\n\n@[simp]\ntheorem map_bot (f : A →ₙₐ[R] B) : (⊥ : NonUnitalSubalgebra R A).map f = ⊥ :=\n  SetLike.coe_injective <| by simp [NonUnitalAlgebra.coe_bot, NonUnitalSubalgebra.coe_map]\n\n@[simp]\ntheorem comap_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R B).comap f = ⊤ :=\n  eq_top_iff.2 fun _ => mem_top\n\n/-- `NonUnitalAlgHom` to `⊤ : NonUnitalSubalgebra R A`. -/\ndef toTop : A →ₙₐ[R] (⊤ : NonUnitalSubalgebra R A) :=\n  NonUnitalAlgHom.codRestrict (NonUnitalAlgHom.id R A) ⊤ fun _ => mem_top\n\nend NonUnitalAlgebra\n\nnamespace NonUnitalSubalgebra\n\nopen NonUnitalAlgebra\n\nsection NonAssoc\n\nvariable {R : Type u} {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable (S : NonUnitalSubalgebra R A)\n\ninstance subsingleton_of_subsingleton [Subsingleton A] : Subsingleton (NonUnitalSubalgebra R A) :=\n  ⟨fun B C => ext fun x => by simp only [Subsingleton.elim x 0, zero_mem B, zero_mem C]⟩\n\ninstance _root_.NonUnitalAlgHom.subsingleton [Subsingleton (NonUnitalSubalgebra R A)] :\n    Subsingleton (A →ₙₐ[R] B) :=\n  ⟨fun f g =>\n    NonUnitalAlgHom.ext fun a =>\n      have : a ∈ (⊥ : NonUnitalSubalgebra R A) :=\n        Subsingleton.elim (⊤ : NonUnitalSubalgebra R A) ⊥ ▸ mem_top\n      (mem_bot.mp this).symm ▸ (map_zero f).trans (map_zero g).symm⟩\n\ntheorem range_val : NonUnitalAlgHom.range (NonUnitalSubalgebraClass.subtype S) = S :=\n  ext <| Set.ext_iff.1 <| (NonUnitalSubalgebraClass.subtype S).coe_range.trans Subtype.range_val\n\n/-- The map `S → T` when `S` is a non-unital subalgebra contained in the non-unital subalgebra `T`.\n\nThis is the non-unital subalgebra version of `Submodule.inclusion`, or `Subring.inclusion`  -/\ndef inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) : S →ₙₐ[R] T\n    where\n  toFun := Set.inclusion h\n  map_add' _ _ := rfl\n  map_mul' _ _ := rfl\n  map_zero' := rfl\n  map_smul' _ _ := rfl\n\ntheorem inclusion_injective {S T : NonUnitalSubalgebra R A} (h : S ≤ T) :\n    Function.Injective (inclusion h) := fun _ _ => Subtype.ext ∘ Subtype.mk.inj\n\n@[simp]\ntheorem inclusion_self {S : NonUnitalSubalgebra R A} :\n    inclusion (le_refl S) = NonUnitalAlgHom.id R S :=\n  NonUnitalAlgHom.ext fun _ => Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_mk {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : A) (hx : x ∈ S) :\n    inclusion h ⟨x, hx⟩ = ⟨x, h hx⟩ :=\n  rfl\n\ntheorem inclusion_right {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : T) (m : (x : A) ∈ S) :\n    inclusion h ⟨x, m⟩ = x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_inclusion {S T U : NonUnitalSubalgebra R A} (hst : S ≤ T) (htu : T ≤ U) (x : S) :\n    inclusion htu (inclusion hst x) = inclusion (le_trans hst htu) x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem coe_inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (s : S) :\n    (inclusion h s : A) = s :=\n  rfl\n\nsection Prod\n\nvariable (S₁ : NonUnitalSubalgebra R B)\n\n/-- The product of two non-unital subalgebras is a non-unital subalgebra. -/\ndef prod : NonUnitalSubalgebra R (A × B) :=\n  { S.toNonUnitalSubsemiring.prod S₁.toNonUnitalSubsemiring with\n    carrier := S ×ˢ S₁\n    smul_mem' := fun r _x hx => ⟨SMulMemClass.smul_mem r hx.1, SMulMemClass.smul_mem r hx.2⟩ }\n\n@[simp]\ntheorem coe_prod : (prod S S₁ : Set (A × B)) = (S : Set A) ×ˢ S₁ :=\n  rfl\n\ntheorem prod_toSubmodule : (S.prod S₁).toSubmodule = S.toSubmodule.prod S₁.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem mem_prod {S : NonUnitalSubalgebra R A} {S₁ : NonUnitalSubalgebra R B} {x : A × B} :\n    x ∈ prod S S₁ ↔ x.1 ∈ S ∧ x.2 ∈ S₁ :=\n  Set.mem_prod\n\n@[simp]\ntheorem prod_top : (prod ⊤ ⊤ : NonUnitalSubalgebra R (A × B)) = ⊤ := by ext; simp\n\ntheorem prod_mono {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S ≤ T → S₁ ≤ T₁ → prod S S₁ ≤ prod T T₁ :=\n  Set.prod_mono\n\n@[simp]\ntheorem prod_inf_prod {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S.prod S₁ ⊓ T.prod T₁ = (S ⊓ T).prod (S₁ ⊓ T₁) :=\n  SetLike.coe_injective Set.prod_inter_prod\n\nend Prod\n\nsection SuprLift\n\nvariable {ι : Type*}\n\ntheorem coe_iSup_of_directed [Nonempty ι] {S : ι → NonUnitalSubalgebra R A}\n    (dir : Directed (· ≤ ·) S) : ↑(iSup S) = ⋃ i, (S i : Set A) :=\n  let K : NonUnitalSubalgebra R A :=\n    { __ := NonUnitalSubsemiring.copy _ _ (NonUnitalSubsemiring.coe_iSup_of_directed dir).symm\n      smul_mem' := fun r _x hx ↦\n        let ⟨i, hi⟩ := Set.mem_iUnion.1 hx\n        Set.mem_iUnion.2 ⟨i, (S i).smul_mem' r hi⟩ }\n  have : iSup S = K := le_antisymm\n    (iSup_le fun i ↦ le_iSup (fun i ↦ (S i : Set A)) i) (Set.iUnion_subset fun _ ↦ le_iSup S _)\n  this.symm ▸ rfl\n\n/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              rw [hf j k hjk]\n              rfl)\n            (↑(iSup K)) (by rw [coe_iSup_of_directed dir])\n        map_zero' := by\n          dsimp\n          exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl) _ (by simp)\n        map_mul' := by\n          ","nextTactic":"dsimp","declUpToTactic":"/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              rw [hf j k hjk]\n              rfl)\n            (↑(iSup K)) (by rw [coe_iSup_of_directed dir])\n        map_zero' := by\n          dsimp\n          exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl) _ (by simp)\n        map_mul' := by\n          ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.892_0.jQvdAcOcyVirh23","decl":"/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B "}
-{"state":"R : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\nS : NonUnitalSubalgebra R A\nι : Type u_1\ninst✝ : Nonempty ι\nK : ι → NonUnitalSubalgebra R A\ndir : Directed (fun x x_1 => x ≤ x_1) K\nf : (i : ι) → ↥(K i) →ₙₐ[R] B\nhf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalAlgHom.comp (f j) (inclusion h)\n⊢ ∀ (x y : ↥(iSup K)),\n    Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x)\n        (_ :\n          ∀ (i j : ι) (x : A) (hxi : x ∈ ↑(K i)) (hxj : x ∈ ↑(K j)),\n            (f i) { val := x, property := hxi } = (f j) { val := x, property := hxj })\n        ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ i, ↑(K i)) (x * y) =\n      Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x)\n          (_ :\n            ∀ (i j : ι) (x : A) (hxi : x ∈ ↑(K i)) (hxj : x ∈ ↑(K j)),\n              (f i) { val := x, property := hxi } = (f j) { val := x, property := hxj })\n          ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ i, ↑(K i)) x *\n        Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x)\n          (_ :\n            ∀ (i j : ι) (x : A) (hxi : x ∈ ↑(K i)) (hxj : x ∈ ↑(K j)),\n              (f i) { val := x, property := hxi } = (f j) { val := x, property := hxj })\n          ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ i, ↑(K i)) y","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact NonUnitalAlgebra.gc\n\n@[simp]\ntheorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=\n  eq_top_iff.2 fun _x hx => subset_adjoin R hx\n\nvariable {R A}\n\n@[simp]\ntheorem coe_top : (↑(⊤ : NonUnitalSubalgebra R A) : Set A) = Set.univ :=\n  rfl\n\n@[simp]\ntheorem mem_top {x : A} : x ∈ (⊤ : NonUnitalSubalgebra R A) :=\n  Set.mem_univ x\n\n@[simp]\ntheorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toNonUnitalSubsemiring : (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubsemiring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toSubring {R A : Type*} [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    [IsScalarTower R A A] [SMulCommClass R A A] :\n    (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem toSubmodule_eq_top {S : NonUnitalSubalgebra R A} : S.toSubmodule = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toSubmodule'.injective.eq_iff' top_toSubmodule\n\n@[simp]\ntheorem toNonUnitalSubsemiring_eq_top {S : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubsemiring_injective.eq_iff' top_toNonUnitalSubsemiring\n\n@[simp]\ntheorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubring_injective.eq_iff' top_toSubring\n\ntheorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_left\n\ntheorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_right\n\ntheorem mul_mem_sup {S T : NonUnitalSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) :\n    x * y ∈ S ⊔ T :=\n  mul_mem (mem_sup_left hx) (mem_sup_right hy)\n\ntheorem map_sup (f : F) (S T : NonUnitalSubalgebra R A) :\n    ((S ⊔ T).map f : NonUnitalSubalgebra R B) = S.map f ⊔ T.map f :=\n  (@NonUnitalSubalgebra.gc_map_comap F R A B _ _ _ _ _ _ f).l_sup\n\n@[simp, norm_cast]\ntheorem coe_inf (S T : NonUnitalSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T :=\n  rfl\n\n@[simp]\ntheorem mem_inf {S T : NonUnitalSubalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T :=\n  Iff.rfl\n\n@[simp]\ntheorem inf_toSubmodule (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toSubmodule = S.toSubmodule ⊓ T.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem inf_toNonUnitalSubsemiring (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring ⊓ T.toNonUnitalSubsemiring :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sInf (S : Set (NonUnitalSubalgebra R A)) : (↑(sInf S) : Set A) = ⋂ s ∈ S, ↑s :=\n  sInf_image\n\ntheorem mem_sInf {S : Set (NonUnitalSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p := by\n  simp only [← SetLike.mem_coe, coe_sInf, Set.mem_iInter₂]\n\n@[simp]\ntheorem sInf_toSubmodule (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toSubmodule = sInf (NonUnitalSubalgebra.toSubmodule '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp]\ntheorem sInf_toNonUnitalSubsemiring (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toNonUnitalSubsemiring = sInf (NonUnitalSubalgebra.toNonUnitalSubsemiring '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp, norm_cast]\ntheorem coe_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} :\n    (↑(⨅ i, S i) : Set A) = ⋂ i, S i := by simp [iInf]\n\ntheorem mem_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} {x : A} :\n    (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by simp only [iInf, mem_sInf, Set.forall_range_iff]\n\n@[simp]\ntheorem iInf_toSubmodule {ι : Sort*} (S : ι → NonUnitalSubalgebra R A) :\n    (⨅ i, S i).toSubmodule = ⨅ i, (S i).toSubmodule :=\n  SetLike.coe_injective <| by simp\n\ninstance : Inhabited (NonUnitalSubalgebra R A) :=\n  ⟨⊥⟩\n\ntheorem mem_bot {x : A} : x ∈ (⊥ : NonUnitalSubalgebra R A) ↔ x = 0 :=\n  show x ∈ Submodule.span R (NonUnitalSubsemiring.closure (∅ : Set A) : Set A) ↔ x = 0 by\n    rw [NonUnitalSubsemiring.closure_empty, NonUnitalSubsemiring.coe_bot,\n      Submodule.span_zero_singleton, Submodule.mem_bot]\n\ntheorem toSubmodule_bot : (⊥ : NonUnitalSubalgebra R A).toSubmodule = ⊥ := by\n  ext\n  simp only [mem_bot, NonUnitalSubalgebra.mem_toSubmodule, Submodule.mem_bot]\n\n@[simp]\ntheorem coe_bot : ((⊥ : NonUnitalSubalgebra R A) : Set A) = {0} := by\n  simp [Set.ext_iff, NonUnitalAlgebra.mem_bot]\n\ntheorem eq_top_iff {S : NonUnitalSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S :=\n  ⟨fun h x => by rw [h]; exact mem_top, fun h => by ext x; exact ⟨fun _ => mem_top, fun _ => h x⟩⟩\n\ntheorem range_top_iff_surjective (f : A →ₙₐ[R] B) :\n    NonUnitalAlgHom.range f = (⊤ : NonUnitalSubalgebra R B) ↔ Function.Surjective f :=\n  NonUnitalAlgebra.eq_top_iff\n\n@[simp]\ntheorem range_id : NonUnitalAlgHom.range (NonUnitalAlgHom.id R A) = ⊤ :=\n  SetLike.coe_injective Set.range_id\n\n@[simp]\ntheorem map_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R A).map f = NonUnitalAlgHom.range f :=\n  SetLike.coe_injective Set.image_univ\n\n@[simp]\ntheorem map_bot (f : A →ₙₐ[R] B) : (⊥ : NonUnitalSubalgebra R A).map f = ⊥ :=\n  SetLike.coe_injective <| by simp [NonUnitalAlgebra.coe_bot, NonUnitalSubalgebra.coe_map]\n\n@[simp]\ntheorem comap_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R B).comap f = ⊤ :=\n  eq_top_iff.2 fun _ => mem_top\n\n/-- `NonUnitalAlgHom` to `⊤ : NonUnitalSubalgebra R A`. -/\ndef toTop : A →ₙₐ[R] (⊤ : NonUnitalSubalgebra R A) :=\n  NonUnitalAlgHom.codRestrict (NonUnitalAlgHom.id R A) ⊤ fun _ => mem_top\n\nend NonUnitalAlgebra\n\nnamespace NonUnitalSubalgebra\n\nopen NonUnitalAlgebra\n\nsection NonAssoc\n\nvariable {R : Type u} {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable (S : NonUnitalSubalgebra R A)\n\ninstance subsingleton_of_subsingleton [Subsingleton A] : Subsingleton (NonUnitalSubalgebra R A) :=\n  ⟨fun B C => ext fun x => by simp only [Subsingleton.elim x 0, zero_mem B, zero_mem C]⟩\n\ninstance _root_.NonUnitalAlgHom.subsingleton [Subsingleton (NonUnitalSubalgebra R A)] :\n    Subsingleton (A →ₙₐ[R] B) :=\n  ⟨fun f g =>\n    NonUnitalAlgHom.ext fun a =>\n      have : a ∈ (⊥ : NonUnitalSubalgebra R A) :=\n        Subsingleton.elim (⊤ : NonUnitalSubalgebra R A) ⊥ ▸ mem_top\n      (mem_bot.mp this).symm ▸ (map_zero f).trans (map_zero g).symm⟩\n\ntheorem range_val : NonUnitalAlgHom.range (NonUnitalSubalgebraClass.subtype S) = S :=\n  ext <| Set.ext_iff.1 <| (NonUnitalSubalgebraClass.subtype S).coe_range.trans Subtype.range_val\n\n/-- The map `S → T` when `S` is a non-unital subalgebra contained in the non-unital subalgebra `T`.\n\nThis is the non-unital subalgebra version of `Submodule.inclusion`, or `Subring.inclusion`  -/\ndef inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) : S →ₙₐ[R] T\n    where\n  toFun := Set.inclusion h\n  map_add' _ _ := rfl\n  map_mul' _ _ := rfl\n  map_zero' := rfl\n  map_smul' _ _ := rfl\n\ntheorem inclusion_injective {S T : NonUnitalSubalgebra R A} (h : S ≤ T) :\n    Function.Injective (inclusion h) := fun _ _ => Subtype.ext ∘ Subtype.mk.inj\n\n@[simp]\ntheorem inclusion_self {S : NonUnitalSubalgebra R A} :\n    inclusion (le_refl S) = NonUnitalAlgHom.id R S :=\n  NonUnitalAlgHom.ext fun _ => Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_mk {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : A) (hx : x ∈ S) :\n    inclusion h ⟨x, hx⟩ = ⟨x, h hx⟩ :=\n  rfl\n\ntheorem inclusion_right {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : T) (m : (x : A) ∈ S) :\n    inclusion h ⟨x, m⟩ = x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_inclusion {S T U : NonUnitalSubalgebra R A} (hst : S ≤ T) (htu : T ≤ U) (x : S) :\n    inclusion htu (inclusion hst x) = inclusion (le_trans hst htu) x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem coe_inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (s : S) :\n    (inclusion h s : A) = s :=\n  rfl\n\nsection Prod\n\nvariable (S₁ : NonUnitalSubalgebra R B)\n\n/-- The product of two non-unital subalgebras is a non-unital subalgebra. -/\ndef prod : NonUnitalSubalgebra R (A × B) :=\n  { S.toNonUnitalSubsemiring.prod S₁.toNonUnitalSubsemiring with\n    carrier := S ×ˢ S₁\n    smul_mem' := fun r _x hx => ⟨SMulMemClass.smul_mem r hx.1, SMulMemClass.smul_mem r hx.2⟩ }\n\n@[simp]\ntheorem coe_prod : (prod S S₁ : Set (A × B)) = (S : Set A) ×ˢ S₁ :=\n  rfl\n\ntheorem prod_toSubmodule : (S.prod S₁).toSubmodule = S.toSubmodule.prod S₁.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem mem_prod {S : NonUnitalSubalgebra R A} {S₁ : NonUnitalSubalgebra R B} {x : A × B} :\n    x ∈ prod S S₁ ↔ x.1 ∈ S ∧ x.2 ∈ S₁ :=\n  Set.mem_prod\n\n@[simp]\ntheorem prod_top : (prod ⊤ ⊤ : NonUnitalSubalgebra R (A × B)) = ⊤ := by ext; simp\n\ntheorem prod_mono {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S ≤ T → S₁ ≤ T₁ → prod S S₁ ≤ prod T T₁ :=\n  Set.prod_mono\n\n@[simp]\ntheorem prod_inf_prod {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S.prod S₁ ⊓ T.prod T₁ = (S ⊓ T).prod (S₁ ⊓ T₁) :=\n  SetLike.coe_injective Set.prod_inter_prod\n\nend Prod\n\nsection SuprLift\n\nvariable {ι : Type*}\n\ntheorem coe_iSup_of_directed [Nonempty ι] {S : ι → NonUnitalSubalgebra R A}\n    (dir : Directed (· ≤ ·) S) : ↑(iSup S) = ⋃ i, (S i : Set A) :=\n  let K : NonUnitalSubalgebra R A :=\n    { __ := NonUnitalSubsemiring.copy _ _ (NonUnitalSubsemiring.coe_iSup_of_directed dir).symm\n      smul_mem' := fun r _x hx ↦\n        let ⟨i, hi⟩ := Set.mem_iUnion.1 hx\n        Set.mem_iUnion.2 ⟨i, (S i).smul_mem' r hi⟩ }\n  have : iSup S = K := le_antisymm\n    (iSup_le fun i ↦ le_iSup (fun i ↦ (S i : Set A)) i) (Set.iUnion_subset fun _ ↦ le_iSup S _)\n  this.symm ▸ rfl\n\n/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              rw [hf j k hjk]\n              rfl)\n            (↑(iSup K)) (by rw [coe_iSup_of_directed dir])\n        map_zero' := by\n          dsimp\n          exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl) _ (by simp)\n        map_mul' := by\n          dsimp\n          ","nextTactic":"apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· * ·))","declUpToTactic":"/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              rw [hf j k hjk]\n              rfl)\n            (↑(iSup K)) (by rw [coe_iSup_of_directed dir])\n        map_zero' := by\n          dsimp\n          exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl) _ (by simp)\n        map_mul' := by\n          dsimp\n          ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.892_0.jQvdAcOcyVirh23","decl":"/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B "}
-{"state":"case hopi\nR : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\nS : NonUnitalSubalgebra R A\nι : Type u_1\ninst✝ : Nonempty ι\nK : ι → NonUnitalSubalgebra R A\ndir : Directed (fun x x_1 => x ≤ x_1) K\nf : (i : ι) → ↥(K i) →ₙₐ[R] B\nhf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalAlgHom.comp (f j) (inclusion h)\n⊢ ∀ (i : ι) (x y : ↑↑(K i)),\n    Set.inclusion (_ : ↑(K i) ⊆ ↑(iSup K)) (x * y) =\n      Set.inclusion (_ : ↑(K i) ⊆ ↑(iSup K)) x * Set.inclusion (_ : ↑(K i) ⊆ ↑(iSup K)) y\ncase h\nR : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\nS : NonUnitalSubalgebra R A\nι : Type u_1\ninst✝ : Nonempty ι\nK : ι → NonUnitalSubalgebra R A\ndir : Directed (fun x x_1 => x ≤ x_1) K\nf : (i : ι) → ↥(K i) →ₙₐ[R] B\nhf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalAlgHom.comp (f j) (inclusion h)\n⊢ ∀ (i : ι) (x y : ↑↑(K i)), (f i) (x * y) = (f i) x * (f i) y\nR : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\nS : NonUnitalSubalgebra R A\nι : Type u_1\ninst✝ : Nonempty ι\nK : ι → NonUnitalSubalgebra R A\ndir : Directed (fun x x_1 => x ≤ x_1) K\nf : (i : ι) → ↥(K i) →ₙₐ[R] B\nhf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalAlgHom.comp (f j) (inclusion h)\n⊢ ↑(iSup K) ⊆ ⋃ i, ↑(K i)","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact NonUnitalAlgebra.gc\n\n@[simp]\ntheorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=\n  eq_top_iff.2 fun _x hx => subset_adjoin R hx\n\nvariable {R A}\n\n@[simp]\ntheorem coe_top : (↑(⊤ : NonUnitalSubalgebra R A) : Set A) = Set.univ :=\n  rfl\n\n@[simp]\ntheorem mem_top {x : A} : x ∈ (⊤ : NonUnitalSubalgebra R A) :=\n  Set.mem_univ x\n\n@[simp]\ntheorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toNonUnitalSubsemiring : (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubsemiring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toSubring {R A : Type*} [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    [IsScalarTower R A A] [SMulCommClass R A A] :\n    (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem toSubmodule_eq_top {S : NonUnitalSubalgebra R A} : S.toSubmodule = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toSubmodule'.injective.eq_iff' top_toSubmodule\n\n@[simp]\ntheorem toNonUnitalSubsemiring_eq_top {S : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubsemiring_injective.eq_iff' top_toNonUnitalSubsemiring\n\n@[simp]\ntheorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubring_injective.eq_iff' top_toSubring\n\ntheorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_left\n\ntheorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_right\n\ntheorem mul_mem_sup {S T : NonUnitalSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) :\n    x * y ∈ S ⊔ T :=\n  mul_mem (mem_sup_left hx) (mem_sup_right hy)\n\ntheorem map_sup (f : F) (S T : NonUnitalSubalgebra R A) :\n    ((S ⊔ T).map f : NonUnitalSubalgebra R B) = S.map f ⊔ T.map f :=\n  (@NonUnitalSubalgebra.gc_map_comap F R A B _ _ _ _ _ _ f).l_sup\n\n@[simp, norm_cast]\ntheorem coe_inf (S T : NonUnitalSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T :=\n  rfl\n\n@[simp]\ntheorem mem_inf {S T : NonUnitalSubalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T :=\n  Iff.rfl\n\n@[simp]\ntheorem inf_toSubmodule (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toSubmodule = S.toSubmodule ⊓ T.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem inf_toNonUnitalSubsemiring (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring ⊓ T.toNonUnitalSubsemiring :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sInf (S : Set (NonUnitalSubalgebra R A)) : (↑(sInf S) : Set A) = ⋂ s ∈ S, ↑s :=\n  sInf_image\n\ntheorem mem_sInf {S : Set (NonUnitalSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p := by\n  simp only [← SetLike.mem_coe, coe_sInf, Set.mem_iInter₂]\n\n@[simp]\ntheorem sInf_toSubmodule (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toSubmodule = sInf (NonUnitalSubalgebra.toSubmodule '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp]\ntheorem sInf_toNonUnitalSubsemiring (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toNonUnitalSubsemiring = sInf (NonUnitalSubalgebra.toNonUnitalSubsemiring '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp, norm_cast]\ntheorem coe_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} :\n    (↑(⨅ i, S i) : Set A) = ⋂ i, S i := by simp [iInf]\n\ntheorem mem_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} {x : A} :\n    (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by simp only [iInf, mem_sInf, Set.forall_range_iff]\n\n@[simp]\ntheorem iInf_toSubmodule {ι : Sort*} (S : ι → NonUnitalSubalgebra R A) :\n    (⨅ i, S i).toSubmodule = ⨅ i, (S i).toSubmodule :=\n  SetLike.coe_injective <| by simp\n\ninstance : Inhabited (NonUnitalSubalgebra R A) :=\n  ⟨⊥⟩\n\ntheorem mem_bot {x : A} : x ∈ (⊥ : NonUnitalSubalgebra R A) ↔ x = 0 :=\n  show x ∈ Submodule.span R (NonUnitalSubsemiring.closure (∅ : Set A) : Set A) ↔ x = 0 by\n    rw [NonUnitalSubsemiring.closure_empty, NonUnitalSubsemiring.coe_bot,\n      Submodule.span_zero_singleton, Submodule.mem_bot]\n\ntheorem toSubmodule_bot : (⊥ : NonUnitalSubalgebra R A).toSubmodule = ⊥ := by\n  ext\n  simp only [mem_bot, NonUnitalSubalgebra.mem_toSubmodule, Submodule.mem_bot]\n\n@[simp]\ntheorem coe_bot : ((⊥ : NonUnitalSubalgebra R A) : Set A) = {0} := by\n  simp [Set.ext_iff, NonUnitalAlgebra.mem_bot]\n\ntheorem eq_top_iff {S : NonUnitalSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S :=\n  ⟨fun h x => by rw [h]; exact mem_top, fun h => by ext x; exact ⟨fun _ => mem_top, fun _ => h x⟩⟩\n\ntheorem range_top_iff_surjective (f : A →ₙₐ[R] B) :\n    NonUnitalAlgHom.range f = (⊤ : NonUnitalSubalgebra R B) ↔ Function.Surjective f :=\n  NonUnitalAlgebra.eq_top_iff\n\n@[simp]\ntheorem range_id : NonUnitalAlgHom.range (NonUnitalAlgHom.id R A) = ⊤ :=\n  SetLike.coe_injective Set.range_id\n\n@[simp]\ntheorem map_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R A).map f = NonUnitalAlgHom.range f :=\n  SetLike.coe_injective Set.image_univ\n\n@[simp]\ntheorem map_bot (f : A →ₙₐ[R] B) : (⊥ : NonUnitalSubalgebra R A).map f = ⊥ :=\n  SetLike.coe_injective <| by simp [NonUnitalAlgebra.coe_bot, NonUnitalSubalgebra.coe_map]\n\n@[simp]\ntheorem comap_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R B).comap f = ⊤ :=\n  eq_top_iff.2 fun _ => mem_top\n\n/-- `NonUnitalAlgHom` to `⊤ : NonUnitalSubalgebra R A`. -/\ndef toTop : A →ₙₐ[R] (⊤ : NonUnitalSubalgebra R A) :=\n  NonUnitalAlgHom.codRestrict (NonUnitalAlgHom.id R A) ⊤ fun _ => mem_top\n\nend NonUnitalAlgebra\n\nnamespace NonUnitalSubalgebra\n\nopen NonUnitalAlgebra\n\nsection NonAssoc\n\nvariable {R : Type u} {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable (S : NonUnitalSubalgebra R A)\n\ninstance subsingleton_of_subsingleton [Subsingleton A] : Subsingleton (NonUnitalSubalgebra R A) :=\n  ⟨fun B C => ext fun x => by simp only [Subsingleton.elim x 0, zero_mem B, zero_mem C]⟩\n\ninstance _root_.NonUnitalAlgHom.subsingleton [Subsingleton (NonUnitalSubalgebra R A)] :\n    Subsingleton (A →ₙₐ[R] B) :=\n  ⟨fun f g =>\n    NonUnitalAlgHom.ext fun a =>\n      have : a ∈ (⊥ : NonUnitalSubalgebra R A) :=\n        Subsingleton.elim (⊤ : NonUnitalSubalgebra R A) ⊥ ▸ mem_top\n      (mem_bot.mp this).symm ▸ (map_zero f).trans (map_zero g).symm⟩\n\ntheorem range_val : NonUnitalAlgHom.range (NonUnitalSubalgebraClass.subtype S) = S :=\n  ext <| Set.ext_iff.1 <| (NonUnitalSubalgebraClass.subtype S).coe_range.trans Subtype.range_val\n\n/-- The map `S → T` when `S` is a non-unital subalgebra contained in the non-unital subalgebra `T`.\n\nThis is the non-unital subalgebra version of `Submodule.inclusion`, or `Subring.inclusion`  -/\ndef inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) : S →ₙₐ[R] T\n    where\n  toFun := Set.inclusion h\n  map_add' _ _ := rfl\n  map_mul' _ _ := rfl\n  map_zero' := rfl\n  map_smul' _ _ := rfl\n\ntheorem inclusion_injective {S T : NonUnitalSubalgebra R A} (h : S ≤ T) :\n    Function.Injective (inclusion h) := fun _ _ => Subtype.ext ∘ Subtype.mk.inj\n\n@[simp]\ntheorem inclusion_self {S : NonUnitalSubalgebra R A} :\n    inclusion (le_refl S) = NonUnitalAlgHom.id R S :=\n  NonUnitalAlgHom.ext fun _ => Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_mk {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : A) (hx : x ∈ S) :\n    inclusion h ⟨x, hx⟩ = ⟨x, h hx⟩ :=\n  rfl\n\ntheorem inclusion_right {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : T) (m : (x : A) ∈ S) :\n    inclusion h ⟨x, m⟩ = x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_inclusion {S T U : NonUnitalSubalgebra R A} (hst : S ≤ T) (htu : T ≤ U) (x : S) :\n    inclusion htu (inclusion hst x) = inclusion (le_trans hst htu) x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem coe_inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (s : S) :\n    (inclusion h s : A) = s :=\n  rfl\n\nsection Prod\n\nvariable (S₁ : NonUnitalSubalgebra R B)\n\n/-- The product of two non-unital subalgebras is a non-unital subalgebra. -/\ndef prod : NonUnitalSubalgebra R (A × B) :=\n  { S.toNonUnitalSubsemiring.prod S₁.toNonUnitalSubsemiring with\n    carrier := S ×ˢ S₁\n    smul_mem' := fun r _x hx => ⟨SMulMemClass.smul_mem r hx.1, SMulMemClass.smul_mem r hx.2⟩ }\n\n@[simp]\ntheorem coe_prod : (prod S S₁ : Set (A × B)) = (S : Set A) ×ˢ S₁ :=\n  rfl\n\ntheorem prod_toSubmodule : (S.prod S₁).toSubmodule = S.toSubmodule.prod S₁.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem mem_prod {S : NonUnitalSubalgebra R A} {S₁ : NonUnitalSubalgebra R B} {x : A × B} :\n    x ∈ prod S S₁ ↔ x.1 ∈ S ∧ x.2 ∈ S₁ :=\n  Set.mem_prod\n\n@[simp]\ntheorem prod_top : (prod ⊤ ⊤ : NonUnitalSubalgebra R (A × B)) = ⊤ := by ext; simp\n\ntheorem prod_mono {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S ≤ T → S₁ ≤ T₁ → prod S S₁ ≤ prod T T₁ :=\n  Set.prod_mono\n\n@[simp]\ntheorem prod_inf_prod {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S.prod S₁ ⊓ T.prod T₁ = (S ⊓ T).prod (S₁ ⊓ T₁) :=\n  SetLike.coe_injective Set.prod_inter_prod\n\nend Prod\n\nsection SuprLift\n\nvariable {ι : Type*}\n\ntheorem coe_iSup_of_directed [Nonempty ι] {S : ι → NonUnitalSubalgebra R A}\n    (dir : Directed (· ≤ ·) S) : ↑(iSup S) = ⋃ i, (S i : Set A) :=\n  let K : NonUnitalSubalgebra R A :=\n    { __ := NonUnitalSubsemiring.copy _ _ (NonUnitalSubsemiring.coe_iSup_of_directed dir).symm\n      smul_mem' := fun r _x hx ↦\n        let ⟨i, hi⟩ := Set.mem_iUnion.1 hx\n        Set.mem_iUnion.2 ⟨i, (S i).smul_mem' r hi⟩ }\n  have : iSup S = K := le_antisymm\n    (iSup_le fun i ↦ le_iSup (fun i ↦ (S i : Set A)) i) (Set.iUnion_subset fun _ ↦ le_iSup S _)\n  this.symm ▸ rfl\n\n/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              rw [hf j k hjk]\n              rfl)\n            (↑(iSup K)) (by rw [coe_iSup_of_directed dir])\n        map_zero' := by\n          dsimp\n          exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl) _ (by simp)\n        map_mul' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· * ·))\n          ","nextTactic":"on_goal 3 => rw [coe_iSup_of_directed dir]","declUpToTactic":"/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              rw [hf j k hjk]\n              rfl)\n            (↑(iSup K)) (by rw [coe_iSup_of_directed dir])\n        map_zero' := by\n          dsimp\n          exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl) _ (by simp)\n        map_mul' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· * ·))\n          ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.892_0.jQvdAcOcyVirh23","decl":"/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B "}
-{"state":"case hopi\nR : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\nS : NonUnitalSubalgebra R A\nι : Type u_1\ninst✝ : Nonempty ι\nK : ι → NonUnitalSubalgebra R A\ndir : Directed (fun x x_1 => x ≤ x_1) K\nf : (i : ι) → ↥(K i) →ₙₐ[R] B\nhf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalAlgHom.comp (f j) (inclusion h)\n⊢ ∀ (i : ι) (x y : ↑↑(K i)),\n    Set.inclusion (_ : ↑(K i) ⊆ ↑(iSup K)) (x * y) =\n      Set.inclusion (_ : ↑(K i) ⊆ ↑(iSup K)) x * Set.inclusion (_ : ↑(K i) ⊆ ↑(iSup K)) y\ncase h\nR : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\nS : NonUnitalSubalgebra R A\nι : Type u_1\ninst✝ : Nonempty ι\nK : ι → NonUnitalSubalgebra R A\ndir : Directed (fun x x_1 => x ≤ x_1) K\nf : (i : ι) → ↥(K i) →ₙₐ[R] B\nhf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalAlgHom.comp (f j) (inclusion h)\n⊢ ∀ (i : ι) (x y : ↑↑(K i)), (f i) (x * y) = (f i) x * (f i) y\nR : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\nS : NonUnitalSubalgebra R A\nι : Type u_1\ninst✝ : Nonempty ι\nK : ι → NonUnitalSubalgebra R A\ndir : Directed (fun x x_1 => x ≤ x_1) K\nf : (i : ι) → ↥(K i) →ₙₐ[R] B\nhf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalAlgHom.comp (f j) (inclusion h)\n⊢ ↑(iSup K) ⊆ ⋃ i, ↑(K i)","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact NonUnitalAlgebra.gc\n\n@[simp]\ntheorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=\n  eq_top_iff.2 fun _x hx => subset_adjoin R hx\n\nvariable {R A}\n\n@[simp]\ntheorem coe_top : (↑(⊤ : NonUnitalSubalgebra R A) : Set A) = Set.univ :=\n  rfl\n\n@[simp]\ntheorem mem_top {x : A} : x ∈ (⊤ : NonUnitalSubalgebra R A) :=\n  Set.mem_univ x\n\n@[simp]\ntheorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toNonUnitalSubsemiring : (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubsemiring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toSubring {R A : Type*} [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    [IsScalarTower R A A] [SMulCommClass R A A] :\n    (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem toSubmodule_eq_top {S : NonUnitalSubalgebra R A} : S.toSubmodule = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toSubmodule'.injective.eq_iff' top_toSubmodule\n\n@[simp]\ntheorem toNonUnitalSubsemiring_eq_top {S : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubsemiring_injective.eq_iff' top_toNonUnitalSubsemiring\n\n@[simp]\ntheorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubring_injective.eq_iff' top_toSubring\n\ntheorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_left\n\ntheorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_right\n\ntheorem mul_mem_sup {S T : NonUnitalSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) :\n    x * y ∈ S ⊔ T :=\n  mul_mem (mem_sup_left hx) (mem_sup_right hy)\n\ntheorem map_sup (f : F) (S T : NonUnitalSubalgebra R A) :\n    ((S ⊔ T).map f : NonUnitalSubalgebra R B) = S.map f ⊔ T.map f :=\n  (@NonUnitalSubalgebra.gc_map_comap F R A B _ _ _ _ _ _ f).l_sup\n\n@[simp, norm_cast]\ntheorem coe_inf (S T : NonUnitalSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T :=\n  rfl\n\n@[simp]\ntheorem mem_inf {S T : NonUnitalSubalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T :=\n  Iff.rfl\n\n@[simp]\ntheorem inf_toSubmodule (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toSubmodule = S.toSubmodule ⊓ T.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem inf_toNonUnitalSubsemiring (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring ⊓ T.toNonUnitalSubsemiring :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sInf (S : Set (NonUnitalSubalgebra R A)) : (↑(sInf S) : Set A) = ⋂ s ∈ S, ↑s :=\n  sInf_image\n\ntheorem mem_sInf {S : Set (NonUnitalSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p := by\n  simp only [← SetLike.mem_coe, coe_sInf, Set.mem_iInter₂]\n\n@[simp]\ntheorem sInf_toSubmodule (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toSubmodule = sInf (NonUnitalSubalgebra.toSubmodule '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp]\ntheorem sInf_toNonUnitalSubsemiring (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toNonUnitalSubsemiring = sInf (NonUnitalSubalgebra.toNonUnitalSubsemiring '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp, norm_cast]\ntheorem coe_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} :\n    (↑(⨅ i, S i) : Set A) = ⋂ i, S i := by simp [iInf]\n\ntheorem mem_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} {x : A} :\n    (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by simp only [iInf, mem_sInf, Set.forall_range_iff]\n\n@[simp]\ntheorem iInf_toSubmodule {ι : Sort*} (S : ι → NonUnitalSubalgebra R A) :\n    (⨅ i, S i).toSubmodule = ⨅ i, (S i).toSubmodule :=\n  SetLike.coe_injective <| by simp\n\ninstance : Inhabited (NonUnitalSubalgebra R A) :=\n  ⟨⊥⟩\n\ntheorem mem_bot {x : A} : x ∈ (⊥ : NonUnitalSubalgebra R A) ↔ x = 0 :=\n  show x ∈ Submodule.span R (NonUnitalSubsemiring.closure (∅ : Set A) : Set A) ↔ x = 0 by\n    rw [NonUnitalSubsemiring.closure_empty, NonUnitalSubsemiring.coe_bot,\n      Submodule.span_zero_singleton, Submodule.mem_bot]\n\ntheorem toSubmodule_bot : (⊥ : NonUnitalSubalgebra R A).toSubmodule = ⊥ := by\n  ext\n  simp only [mem_bot, NonUnitalSubalgebra.mem_toSubmodule, Submodule.mem_bot]\n\n@[simp]\ntheorem coe_bot : ((⊥ : NonUnitalSubalgebra R A) : Set A) = {0} := by\n  simp [Set.ext_iff, NonUnitalAlgebra.mem_bot]\n\ntheorem eq_top_iff {S : NonUnitalSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S :=\n  ⟨fun h x => by rw [h]; exact mem_top, fun h => by ext x; exact ⟨fun _ => mem_top, fun _ => h x⟩⟩\n\ntheorem range_top_iff_surjective (f : A →ₙₐ[R] B) :\n    NonUnitalAlgHom.range f = (⊤ : NonUnitalSubalgebra R B) ↔ Function.Surjective f :=\n  NonUnitalAlgebra.eq_top_iff\n\n@[simp]\ntheorem range_id : NonUnitalAlgHom.range (NonUnitalAlgHom.id R A) = ⊤ :=\n  SetLike.coe_injective Set.range_id\n\n@[simp]\ntheorem map_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R A).map f = NonUnitalAlgHom.range f :=\n  SetLike.coe_injective Set.image_univ\n\n@[simp]\ntheorem map_bot (f : A →ₙₐ[R] B) : (⊥ : NonUnitalSubalgebra R A).map f = ⊥ :=\n  SetLike.coe_injective <| by simp [NonUnitalAlgebra.coe_bot, NonUnitalSubalgebra.coe_map]\n\n@[simp]\ntheorem comap_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R B).comap f = ⊤ :=\n  eq_top_iff.2 fun _ => mem_top\n\n/-- `NonUnitalAlgHom` to `⊤ : NonUnitalSubalgebra R A`. -/\ndef toTop : A →ₙₐ[R] (⊤ : NonUnitalSubalgebra R A) :=\n  NonUnitalAlgHom.codRestrict (NonUnitalAlgHom.id R A) ⊤ fun _ => mem_top\n\nend NonUnitalAlgebra\n\nnamespace NonUnitalSubalgebra\n\nopen NonUnitalAlgebra\n\nsection NonAssoc\n\nvariable {R : Type u} {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable (S : NonUnitalSubalgebra R A)\n\ninstance subsingleton_of_subsingleton [Subsingleton A] : Subsingleton (NonUnitalSubalgebra R A) :=\n  ⟨fun B C => ext fun x => by simp only [Subsingleton.elim x 0, zero_mem B, zero_mem C]⟩\n\ninstance _root_.NonUnitalAlgHom.subsingleton [Subsingleton (NonUnitalSubalgebra R A)] :\n    Subsingleton (A →ₙₐ[R] B) :=\n  ⟨fun f g =>\n    NonUnitalAlgHom.ext fun a =>\n      have : a ∈ (⊥ : NonUnitalSubalgebra R A) :=\n        Subsingleton.elim (⊤ : NonUnitalSubalgebra R A) ⊥ ▸ mem_top\n      (mem_bot.mp this).symm ▸ (map_zero f).trans (map_zero g).symm⟩\n\ntheorem range_val : NonUnitalAlgHom.range (NonUnitalSubalgebraClass.subtype S) = S :=\n  ext <| Set.ext_iff.1 <| (NonUnitalSubalgebraClass.subtype S).coe_range.trans Subtype.range_val\n\n/-- The map `S → T` when `S` is a non-unital subalgebra contained in the non-unital subalgebra `T`.\n\nThis is the non-unital subalgebra version of `Submodule.inclusion`, or `Subring.inclusion`  -/\ndef inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) : S →ₙₐ[R] T\n    where\n  toFun := Set.inclusion h\n  map_add' _ _ := rfl\n  map_mul' _ _ := rfl\n  map_zero' := rfl\n  map_smul' _ _ := rfl\n\ntheorem inclusion_injective {S T : NonUnitalSubalgebra R A} (h : S ≤ T) :\n    Function.Injective (inclusion h) := fun _ _ => Subtype.ext ∘ Subtype.mk.inj\n\n@[simp]\ntheorem inclusion_self {S : NonUnitalSubalgebra R A} :\n    inclusion (le_refl S) = NonUnitalAlgHom.id R S :=\n  NonUnitalAlgHom.ext fun _ => Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_mk {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : A) (hx : x ∈ S) :\n    inclusion h ⟨x, hx⟩ = ⟨x, h hx⟩ :=\n  rfl\n\ntheorem inclusion_right {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : T) (m : (x : A) ∈ S) :\n    inclusion h ⟨x, m⟩ = x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_inclusion {S T U : NonUnitalSubalgebra R A} (hst : S ≤ T) (htu : T ≤ U) (x : S) :\n    inclusion htu (inclusion hst x) = inclusion (le_trans hst htu) x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem coe_inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (s : S) :\n    (inclusion h s : A) = s :=\n  rfl\n\nsection Prod\n\nvariable (S₁ : NonUnitalSubalgebra R B)\n\n/-- The product of two non-unital subalgebras is a non-unital subalgebra. -/\ndef prod : NonUnitalSubalgebra R (A × B) :=\n  { S.toNonUnitalSubsemiring.prod S₁.toNonUnitalSubsemiring with\n    carrier := S ×ˢ S₁\n    smul_mem' := fun r _x hx => ⟨SMulMemClass.smul_mem r hx.1, SMulMemClass.smul_mem r hx.2⟩ }\n\n@[simp]\ntheorem coe_prod : (prod S S₁ : Set (A × B)) = (S : Set A) ×ˢ S₁ :=\n  rfl\n\ntheorem prod_toSubmodule : (S.prod S₁).toSubmodule = S.toSubmodule.prod S₁.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem mem_prod {S : NonUnitalSubalgebra R A} {S₁ : NonUnitalSubalgebra R B} {x : A × B} :\n    x ∈ prod S S₁ ↔ x.1 ∈ S ∧ x.2 ∈ S₁ :=\n  Set.mem_prod\n\n@[simp]\ntheorem prod_top : (prod ⊤ ⊤ : NonUnitalSubalgebra R (A × B)) = ⊤ := by ext; simp\n\ntheorem prod_mono {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S ≤ T → S₁ ≤ T₁ → prod S S₁ ≤ prod T T₁ :=\n  Set.prod_mono\n\n@[simp]\ntheorem prod_inf_prod {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S.prod S₁ ⊓ T.prod T₁ = (S ⊓ T).prod (S₁ ⊓ T₁) :=\n  SetLike.coe_injective Set.prod_inter_prod\n\nend Prod\n\nsection SuprLift\n\nvariable {ι : Type*}\n\ntheorem coe_iSup_of_directed [Nonempty ι] {S : ι → NonUnitalSubalgebra R A}\n    (dir : Directed (· ≤ ·) S) : ↑(iSup S) = ⋃ i, (S i : Set A) :=\n  let K : NonUnitalSubalgebra R A :=\n    { __ := NonUnitalSubsemiring.copy _ _ (NonUnitalSubsemiring.coe_iSup_of_directed dir).symm\n      smul_mem' := fun r _x hx ↦\n        let ⟨i, hi⟩ := Set.mem_iUnion.1 hx\n        Set.mem_iUnion.2 ⟨i, (S i).smul_mem' r hi⟩ }\n  have : iSup S = K := le_antisymm\n    (iSup_le fun i ↦ le_iSup (fun i ↦ (S i : Set A)) i) (Set.iUnion_subset fun _ ↦ le_iSup S _)\n  this.symm ▸ rfl\n\n/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              rw [hf j k hjk]\n              rfl)\n            (↑(iSup K)) (by rw [coe_iSup_of_directed dir])\n        map_zero' := by\n          dsimp\n          exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl) _ (by simp)\n        map_mul' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· * ·))\n          ","nextTactic":"on_goal 3 => rw [coe_iSup_of_directed dir]","declUpToTactic":"/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              rw [hf j k hjk]\n              rfl)\n            (↑(iSup K)) (by rw [coe_iSup_of_directed dir])\n        map_zero' := by\n          dsimp\n          exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl) _ (by simp)\n        map_mul' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· * ·))\n          ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.892_0.jQvdAcOcyVirh23","decl":"/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B "}
-{"state":"R : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\nS : NonUnitalSubalgebra R A\nι : Type u_1\ninst✝ : Nonempty ι\nK : ι → NonUnitalSubalgebra R A\ndir : Directed (fun x x_1 => x ≤ x_1) K\nf : (i : ι) → ↥(K i) →ₙₐ[R] B\nhf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalAlgHom.comp (f j) (inclusion h)\n⊢ ↑(iSup K) ⊆ ⋃ i, ↑(K i)","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact NonUnitalAlgebra.gc\n\n@[simp]\ntheorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=\n  eq_top_iff.2 fun _x hx => subset_adjoin R hx\n\nvariable {R A}\n\n@[simp]\ntheorem coe_top : (↑(⊤ : NonUnitalSubalgebra R A) : Set A) = Set.univ :=\n  rfl\n\n@[simp]\ntheorem mem_top {x : A} : x ∈ (⊤ : NonUnitalSubalgebra R A) :=\n  Set.mem_univ x\n\n@[simp]\ntheorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toNonUnitalSubsemiring : (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubsemiring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toSubring {R A : Type*} [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    [IsScalarTower R A A] [SMulCommClass R A A] :\n    (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem toSubmodule_eq_top {S : NonUnitalSubalgebra R A} : S.toSubmodule = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toSubmodule'.injective.eq_iff' top_toSubmodule\n\n@[simp]\ntheorem toNonUnitalSubsemiring_eq_top {S : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubsemiring_injective.eq_iff' top_toNonUnitalSubsemiring\n\n@[simp]\ntheorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubring_injective.eq_iff' top_toSubring\n\ntheorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_left\n\ntheorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_right\n\ntheorem mul_mem_sup {S T : NonUnitalSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) :\n    x * y ∈ S ⊔ T :=\n  mul_mem (mem_sup_left hx) (mem_sup_right hy)\n\ntheorem map_sup (f : F) (S T : NonUnitalSubalgebra R A) :\n    ((S ⊔ T).map f : NonUnitalSubalgebra R B) = S.map f ⊔ T.map f :=\n  (@NonUnitalSubalgebra.gc_map_comap F R A B _ _ _ _ _ _ f).l_sup\n\n@[simp, norm_cast]\ntheorem coe_inf (S T : NonUnitalSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T :=\n  rfl\n\n@[simp]\ntheorem mem_inf {S T : NonUnitalSubalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T :=\n  Iff.rfl\n\n@[simp]\ntheorem inf_toSubmodule (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toSubmodule = S.toSubmodule ⊓ T.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem inf_toNonUnitalSubsemiring (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring ⊓ T.toNonUnitalSubsemiring :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sInf (S : Set (NonUnitalSubalgebra R A)) : (↑(sInf S) : Set A) = ⋂ s ∈ S, ↑s :=\n  sInf_image\n\ntheorem mem_sInf {S : Set (NonUnitalSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p := by\n  simp only [← SetLike.mem_coe, coe_sInf, Set.mem_iInter₂]\n\n@[simp]\ntheorem sInf_toSubmodule (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toSubmodule = sInf (NonUnitalSubalgebra.toSubmodule '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp]\ntheorem sInf_toNonUnitalSubsemiring (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toNonUnitalSubsemiring = sInf (NonUnitalSubalgebra.toNonUnitalSubsemiring '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp, norm_cast]\ntheorem coe_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} :\n    (↑(⨅ i, S i) : Set A) = ⋂ i, S i := by simp [iInf]\n\ntheorem mem_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} {x : A} :\n    (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by simp only [iInf, mem_sInf, Set.forall_range_iff]\n\n@[simp]\ntheorem iInf_toSubmodule {ι : Sort*} (S : ι → NonUnitalSubalgebra R A) :\n    (⨅ i, S i).toSubmodule = ⨅ i, (S i).toSubmodule :=\n  SetLike.coe_injective <| by simp\n\ninstance : Inhabited (NonUnitalSubalgebra R A) :=\n  ⟨⊥⟩\n\ntheorem mem_bot {x : A} : x ∈ (⊥ : NonUnitalSubalgebra R A) ↔ x = 0 :=\n  show x ∈ Submodule.span R (NonUnitalSubsemiring.closure (∅ : Set A) : Set A) ↔ x = 0 by\n    rw [NonUnitalSubsemiring.closure_empty, NonUnitalSubsemiring.coe_bot,\n      Submodule.span_zero_singleton, Submodule.mem_bot]\n\ntheorem toSubmodule_bot : (⊥ : NonUnitalSubalgebra R A).toSubmodule = ⊥ := by\n  ext\n  simp only [mem_bot, NonUnitalSubalgebra.mem_toSubmodule, Submodule.mem_bot]\n\n@[simp]\ntheorem coe_bot : ((⊥ : NonUnitalSubalgebra R A) : Set A) = {0} := by\n  simp [Set.ext_iff, NonUnitalAlgebra.mem_bot]\n\ntheorem eq_top_iff {S : NonUnitalSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S :=\n  ⟨fun h x => by rw [h]; exact mem_top, fun h => by ext x; exact ⟨fun _ => mem_top, fun _ => h x⟩⟩\n\ntheorem range_top_iff_surjective (f : A →ₙₐ[R] B) :\n    NonUnitalAlgHom.range f = (⊤ : NonUnitalSubalgebra R B) ↔ Function.Surjective f :=\n  NonUnitalAlgebra.eq_top_iff\n\n@[simp]\ntheorem range_id : NonUnitalAlgHom.range (NonUnitalAlgHom.id R A) = ⊤ :=\n  SetLike.coe_injective Set.range_id\n\n@[simp]\ntheorem map_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R A).map f = NonUnitalAlgHom.range f :=\n  SetLike.coe_injective Set.image_univ\n\n@[simp]\ntheorem map_bot (f : A →ₙₐ[R] B) : (⊥ : NonUnitalSubalgebra R A).map f = ⊥ :=\n  SetLike.coe_injective <| by simp [NonUnitalAlgebra.coe_bot, NonUnitalSubalgebra.coe_map]\n\n@[simp]\ntheorem comap_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R B).comap f = ⊤ :=\n  eq_top_iff.2 fun _ => mem_top\n\n/-- `NonUnitalAlgHom` to `⊤ : NonUnitalSubalgebra R A`. -/\ndef toTop : A →ₙₐ[R] (⊤ : NonUnitalSubalgebra R A) :=\n  NonUnitalAlgHom.codRestrict (NonUnitalAlgHom.id R A) ⊤ fun _ => mem_top\n\nend NonUnitalAlgebra\n\nnamespace NonUnitalSubalgebra\n\nopen NonUnitalAlgebra\n\nsection NonAssoc\n\nvariable {R : Type u} {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable (S : NonUnitalSubalgebra R A)\n\ninstance subsingleton_of_subsingleton [Subsingleton A] : Subsingleton (NonUnitalSubalgebra R A) :=\n  ⟨fun B C => ext fun x => by simp only [Subsingleton.elim x 0, zero_mem B, zero_mem C]⟩\n\ninstance _root_.NonUnitalAlgHom.subsingleton [Subsingleton (NonUnitalSubalgebra R A)] :\n    Subsingleton (A →ₙₐ[R] B) :=\n  ⟨fun f g =>\n    NonUnitalAlgHom.ext fun a =>\n      have : a ∈ (⊥ : NonUnitalSubalgebra R A) :=\n        Subsingleton.elim (⊤ : NonUnitalSubalgebra R A) ⊥ ▸ mem_top\n      (mem_bot.mp this).symm ▸ (map_zero f).trans (map_zero g).symm⟩\n\ntheorem range_val : NonUnitalAlgHom.range (NonUnitalSubalgebraClass.subtype S) = S :=\n  ext <| Set.ext_iff.1 <| (NonUnitalSubalgebraClass.subtype S).coe_range.trans Subtype.range_val\n\n/-- The map `S → T` when `S` is a non-unital subalgebra contained in the non-unital subalgebra `T`.\n\nThis is the non-unital subalgebra version of `Submodule.inclusion`, or `Subring.inclusion`  -/\ndef inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) : S →ₙₐ[R] T\n    where\n  toFun := Set.inclusion h\n  map_add' _ _ := rfl\n  map_mul' _ _ := rfl\n  map_zero' := rfl\n  map_smul' _ _ := rfl\n\ntheorem inclusion_injective {S T : NonUnitalSubalgebra R A} (h : S ≤ T) :\n    Function.Injective (inclusion h) := fun _ _ => Subtype.ext ∘ Subtype.mk.inj\n\n@[simp]\ntheorem inclusion_self {S : NonUnitalSubalgebra R A} :\n    inclusion (le_refl S) = NonUnitalAlgHom.id R S :=\n  NonUnitalAlgHom.ext fun _ => Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_mk {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : A) (hx : x ∈ S) :\n    inclusion h ⟨x, hx⟩ = ⟨x, h hx⟩ :=\n  rfl\n\ntheorem inclusion_right {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : T) (m : (x : A) ∈ S) :\n    inclusion h ⟨x, m⟩ = x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_inclusion {S T U : NonUnitalSubalgebra R A} (hst : S ≤ T) (htu : T ≤ U) (x : S) :\n    inclusion htu (inclusion hst x) = inclusion (le_trans hst htu) x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem coe_inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (s : S) :\n    (inclusion h s : A) = s :=\n  rfl\n\nsection Prod\n\nvariable (S₁ : NonUnitalSubalgebra R B)\n\n/-- The product of two non-unital subalgebras is a non-unital subalgebra. -/\ndef prod : NonUnitalSubalgebra R (A × B) :=\n  { S.toNonUnitalSubsemiring.prod S₁.toNonUnitalSubsemiring with\n    carrier := S ×ˢ S₁\n    smul_mem' := fun r _x hx => ⟨SMulMemClass.smul_mem r hx.1, SMulMemClass.smul_mem r hx.2⟩ }\n\n@[simp]\ntheorem coe_prod : (prod S S₁ : Set (A × B)) = (S : Set A) ×ˢ S₁ :=\n  rfl\n\ntheorem prod_toSubmodule : (S.prod S₁).toSubmodule = S.toSubmodule.prod S₁.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem mem_prod {S : NonUnitalSubalgebra R A} {S₁ : NonUnitalSubalgebra R B} {x : A × B} :\n    x ∈ prod S S₁ ↔ x.1 ∈ S ∧ x.2 ∈ S₁ :=\n  Set.mem_prod\n\n@[simp]\ntheorem prod_top : (prod ⊤ ⊤ : NonUnitalSubalgebra R (A × B)) = ⊤ := by ext; simp\n\ntheorem prod_mono {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S ≤ T → S₁ ≤ T₁ → prod S S₁ ≤ prod T T₁ :=\n  Set.prod_mono\n\n@[simp]\ntheorem prod_inf_prod {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S.prod S₁ ⊓ T.prod T₁ = (S ⊓ T).prod (S₁ ⊓ T₁) :=\n  SetLike.coe_injective Set.prod_inter_prod\n\nend Prod\n\nsection SuprLift\n\nvariable {ι : Type*}\n\ntheorem coe_iSup_of_directed [Nonempty ι] {S : ι → NonUnitalSubalgebra R A}\n    (dir : Directed (· ≤ ·) S) : ↑(iSup S) = ⋃ i, (S i : Set A) :=\n  let K : NonUnitalSubalgebra R A :=\n    { __ := NonUnitalSubsemiring.copy _ _ (NonUnitalSubsemiring.coe_iSup_of_directed dir).symm\n      smul_mem' := fun r _x hx ↦\n        let ⟨i, hi⟩ := Set.mem_iUnion.1 hx\n        Set.mem_iUnion.2 ⟨i, (S i).smul_mem' r hi⟩ }\n  have : iSup S = K := le_antisymm\n    (iSup_le fun i ↦ le_iSup (fun i ↦ (S i : Set A)) i) (Set.iUnion_subset fun _ ↦ le_iSup S _)\n  this.symm ▸ rfl\n\n/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              rw [hf j k hjk]\n              rfl)\n            (↑(iSup K)) (by rw [coe_iSup_of_directed dir])\n        map_zero' := by\n          dsimp\n          exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl) _ (by simp)\n        map_mul' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· * ·))\n          on_goal 3 => ","nextTactic":"rw [coe_iSup_of_directed dir]","declUpToTactic":"/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              rw [hf j k hjk]\n              rfl)\n            (↑(iSup K)) (by rw [coe_iSup_of_directed dir])\n        map_zero' := by\n          dsimp\n          exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl) _ (by simp)\n        map_mul' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· * ·))\n          on_goal 3 => ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.892_0.jQvdAcOcyVirh23","decl":"/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B "}
-{"state":"case hopi\nR : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\nS : NonUnitalSubalgebra R A\nι : Type u_1\ninst✝ : Nonempty ι\nK : ι → NonUnitalSubalgebra R A\ndir : Directed (fun x x_1 => x ≤ x_1) K\nf : (i : ι) → ↥(K i) →ₙₐ[R] B\nhf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalAlgHom.comp (f j) (inclusion h)\n⊢ ∀ (i : ι) (x y : ↑↑(K i)),\n    Set.inclusion (_ : ↑(K i) ⊆ ↑(iSup K)) (x * y) =\n      Set.inclusion (_ : ↑(K i) ⊆ ↑(iSup K)) x * Set.inclusion (_ : ↑(K i) ⊆ ↑(iSup K)) y\ncase h\nR : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\nS : NonUnitalSubalgebra R A\nι : Type u_1\ninst✝ : Nonempty ι\nK : ι → NonUnitalSubalgebra R A\ndir : Directed (fun x x_1 => x ≤ x_1) K\nf : (i : ι) → ↥(K i) →ₙₐ[R] B\nhf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalAlgHom.comp (f j) (inclusion h)\n⊢ ∀ (i : ι) (x y : ↑↑(K i)), (f i) (x * y) = (f i) x * (f i) y","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact NonUnitalAlgebra.gc\n\n@[simp]\ntheorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=\n  eq_top_iff.2 fun _x hx => subset_adjoin R hx\n\nvariable {R A}\n\n@[simp]\ntheorem coe_top : (↑(⊤ : NonUnitalSubalgebra R A) : Set A) = Set.univ :=\n  rfl\n\n@[simp]\ntheorem mem_top {x : A} : x ∈ (⊤ : NonUnitalSubalgebra R A) :=\n  Set.mem_univ x\n\n@[simp]\ntheorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toNonUnitalSubsemiring : (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubsemiring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toSubring {R A : Type*} [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    [IsScalarTower R A A] [SMulCommClass R A A] :\n    (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem toSubmodule_eq_top {S : NonUnitalSubalgebra R A} : S.toSubmodule = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toSubmodule'.injective.eq_iff' top_toSubmodule\n\n@[simp]\ntheorem toNonUnitalSubsemiring_eq_top {S : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubsemiring_injective.eq_iff' top_toNonUnitalSubsemiring\n\n@[simp]\ntheorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubring_injective.eq_iff' top_toSubring\n\ntheorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_left\n\ntheorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_right\n\ntheorem mul_mem_sup {S T : NonUnitalSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) :\n    x * y ∈ S ⊔ T :=\n  mul_mem (mem_sup_left hx) (mem_sup_right hy)\n\ntheorem map_sup (f : F) (S T : NonUnitalSubalgebra R A) :\n    ((S ⊔ T).map f : NonUnitalSubalgebra R B) = S.map f ⊔ T.map f :=\n  (@NonUnitalSubalgebra.gc_map_comap F R A B _ _ _ _ _ _ f).l_sup\n\n@[simp, norm_cast]\ntheorem coe_inf (S T : NonUnitalSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T :=\n  rfl\n\n@[simp]\ntheorem mem_inf {S T : NonUnitalSubalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T :=\n  Iff.rfl\n\n@[simp]\ntheorem inf_toSubmodule (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toSubmodule = S.toSubmodule ⊓ T.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem inf_toNonUnitalSubsemiring (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring ⊓ T.toNonUnitalSubsemiring :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sInf (S : Set (NonUnitalSubalgebra R A)) : (↑(sInf S) : Set A) = ⋂ s ∈ S, ↑s :=\n  sInf_image\n\ntheorem mem_sInf {S : Set (NonUnitalSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p := by\n  simp only [← SetLike.mem_coe, coe_sInf, Set.mem_iInter₂]\n\n@[simp]\ntheorem sInf_toSubmodule (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toSubmodule = sInf (NonUnitalSubalgebra.toSubmodule '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp]\ntheorem sInf_toNonUnitalSubsemiring (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toNonUnitalSubsemiring = sInf (NonUnitalSubalgebra.toNonUnitalSubsemiring '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp, norm_cast]\ntheorem coe_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} :\n    (↑(⨅ i, S i) : Set A) = ⋂ i, S i := by simp [iInf]\n\ntheorem mem_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} {x : A} :\n    (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by simp only [iInf, mem_sInf, Set.forall_range_iff]\n\n@[simp]\ntheorem iInf_toSubmodule {ι : Sort*} (S : ι → NonUnitalSubalgebra R A) :\n    (⨅ i, S i).toSubmodule = ⨅ i, (S i).toSubmodule :=\n  SetLike.coe_injective <| by simp\n\ninstance : Inhabited (NonUnitalSubalgebra R A) :=\n  ⟨⊥⟩\n\ntheorem mem_bot {x : A} : x ∈ (⊥ : NonUnitalSubalgebra R A) ↔ x = 0 :=\n  show x ∈ Submodule.span R (NonUnitalSubsemiring.closure (∅ : Set A) : Set A) ↔ x = 0 by\n    rw [NonUnitalSubsemiring.closure_empty, NonUnitalSubsemiring.coe_bot,\n      Submodule.span_zero_singleton, Submodule.mem_bot]\n\ntheorem toSubmodule_bot : (⊥ : NonUnitalSubalgebra R A).toSubmodule = ⊥ := by\n  ext\n  simp only [mem_bot, NonUnitalSubalgebra.mem_toSubmodule, Submodule.mem_bot]\n\n@[simp]\ntheorem coe_bot : ((⊥ : NonUnitalSubalgebra R A) : Set A) = {0} := by\n  simp [Set.ext_iff, NonUnitalAlgebra.mem_bot]\n\ntheorem eq_top_iff {S : NonUnitalSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S :=\n  ⟨fun h x => by rw [h]; exact mem_top, fun h => by ext x; exact ⟨fun _ => mem_top, fun _ => h x⟩⟩\n\ntheorem range_top_iff_surjective (f : A →ₙₐ[R] B) :\n    NonUnitalAlgHom.range f = (⊤ : NonUnitalSubalgebra R B) ↔ Function.Surjective f :=\n  NonUnitalAlgebra.eq_top_iff\n\n@[simp]\ntheorem range_id : NonUnitalAlgHom.range (NonUnitalAlgHom.id R A) = ⊤ :=\n  SetLike.coe_injective Set.range_id\n\n@[simp]\ntheorem map_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R A).map f = NonUnitalAlgHom.range f :=\n  SetLike.coe_injective Set.image_univ\n\n@[simp]\ntheorem map_bot (f : A →ₙₐ[R] B) : (⊥ : NonUnitalSubalgebra R A).map f = ⊥ :=\n  SetLike.coe_injective <| by simp [NonUnitalAlgebra.coe_bot, NonUnitalSubalgebra.coe_map]\n\n@[simp]\ntheorem comap_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R B).comap f = ⊤ :=\n  eq_top_iff.2 fun _ => mem_top\n\n/-- `NonUnitalAlgHom` to `⊤ : NonUnitalSubalgebra R A`. -/\ndef toTop : A →ₙₐ[R] (⊤ : NonUnitalSubalgebra R A) :=\n  NonUnitalAlgHom.codRestrict (NonUnitalAlgHom.id R A) ⊤ fun _ => mem_top\n\nend NonUnitalAlgebra\n\nnamespace NonUnitalSubalgebra\n\nopen NonUnitalAlgebra\n\nsection NonAssoc\n\nvariable {R : Type u} {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable (S : NonUnitalSubalgebra R A)\n\ninstance subsingleton_of_subsingleton [Subsingleton A] : Subsingleton (NonUnitalSubalgebra R A) :=\n  ⟨fun B C => ext fun x => by simp only [Subsingleton.elim x 0, zero_mem B, zero_mem C]⟩\n\ninstance _root_.NonUnitalAlgHom.subsingleton [Subsingleton (NonUnitalSubalgebra R A)] :\n    Subsingleton (A →ₙₐ[R] B) :=\n  ⟨fun f g =>\n    NonUnitalAlgHom.ext fun a =>\n      have : a ∈ (⊥ : NonUnitalSubalgebra R A) :=\n        Subsingleton.elim (⊤ : NonUnitalSubalgebra R A) ⊥ ▸ mem_top\n      (mem_bot.mp this).symm ▸ (map_zero f).trans (map_zero g).symm⟩\n\ntheorem range_val : NonUnitalAlgHom.range (NonUnitalSubalgebraClass.subtype S) = S :=\n  ext <| Set.ext_iff.1 <| (NonUnitalSubalgebraClass.subtype S).coe_range.trans Subtype.range_val\n\n/-- The map `S → T` when `S` is a non-unital subalgebra contained in the non-unital subalgebra `T`.\n\nThis is the non-unital subalgebra version of `Submodule.inclusion`, or `Subring.inclusion`  -/\ndef inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) : S →ₙₐ[R] T\n    where\n  toFun := Set.inclusion h\n  map_add' _ _ := rfl\n  map_mul' _ _ := rfl\n  map_zero' := rfl\n  map_smul' _ _ := rfl\n\ntheorem inclusion_injective {S T : NonUnitalSubalgebra R A} (h : S ≤ T) :\n    Function.Injective (inclusion h) := fun _ _ => Subtype.ext ∘ Subtype.mk.inj\n\n@[simp]\ntheorem inclusion_self {S : NonUnitalSubalgebra R A} :\n    inclusion (le_refl S) = NonUnitalAlgHom.id R S :=\n  NonUnitalAlgHom.ext fun _ => Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_mk {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : A) (hx : x ∈ S) :\n    inclusion h ⟨x, hx⟩ = ⟨x, h hx⟩ :=\n  rfl\n\ntheorem inclusion_right {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : T) (m : (x : A) ∈ S) :\n    inclusion h ⟨x, m⟩ = x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_inclusion {S T U : NonUnitalSubalgebra R A} (hst : S ≤ T) (htu : T ≤ U) (x : S) :\n    inclusion htu (inclusion hst x) = inclusion (le_trans hst htu) x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem coe_inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (s : S) :\n    (inclusion h s : A) = s :=\n  rfl\n\nsection Prod\n\nvariable (S₁ : NonUnitalSubalgebra R B)\n\n/-- The product of two non-unital subalgebras is a non-unital subalgebra. -/\ndef prod : NonUnitalSubalgebra R (A × B) :=\n  { S.toNonUnitalSubsemiring.prod S₁.toNonUnitalSubsemiring with\n    carrier := S ×ˢ S₁\n    smul_mem' := fun r _x hx => ⟨SMulMemClass.smul_mem r hx.1, SMulMemClass.smul_mem r hx.2⟩ }\n\n@[simp]\ntheorem coe_prod : (prod S S₁ : Set (A × B)) = (S : Set A) ×ˢ S₁ :=\n  rfl\n\ntheorem prod_toSubmodule : (S.prod S₁).toSubmodule = S.toSubmodule.prod S₁.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem mem_prod {S : NonUnitalSubalgebra R A} {S₁ : NonUnitalSubalgebra R B} {x : A × B} :\n    x ∈ prod S S₁ ↔ x.1 ∈ S ∧ x.2 ∈ S₁ :=\n  Set.mem_prod\n\n@[simp]\ntheorem prod_top : (prod ⊤ ⊤ : NonUnitalSubalgebra R (A × B)) = ⊤ := by ext; simp\n\ntheorem prod_mono {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S ≤ T → S₁ ≤ T₁ → prod S S₁ ≤ prod T T₁ :=\n  Set.prod_mono\n\n@[simp]\ntheorem prod_inf_prod {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S.prod S₁ ⊓ T.prod T₁ = (S ⊓ T).prod (S₁ ⊓ T₁) :=\n  SetLike.coe_injective Set.prod_inter_prod\n\nend Prod\n\nsection SuprLift\n\nvariable {ι : Type*}\n\ntheorem coe_iSup_of_directed [Nonempty ι] {S : ι → NonUnitalSubalgebra R A}\n    (dir : Directed (· ≤ ·) S) : ↑(iSup S) = ⋃ i, (S i : Set A) :=\n  let K : NonUnitalSubalgebra R A :=\n    { __ := NonUnitalSubsemiring.copy _ _ (NonUnitalSubsemiring.coe_iSup_of_directed dir).symm\n      smul_mem' := fun r _x hx ↦\n        let ⟨i, hi⟩ := Set.mem_iUnion.1 hx\n        Set.mem_iUnion.2 ⟨i, (S i).smul_mem' r hi⟩ }\n  have : iSup S = K := le_antisymm\n    (iSup_le fun i ↦ le_iSup (fun i ↦ (S i : Set A)) i) (Set.iUnion_subset fun _ ↦ le_iSup S _)\n  this.symm ▸ rfl\n\n/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              rw [hf j k hjk]\n              rfl)\n            (↑(iSup K)) (by rw [coe_iSup_of_directed dir])\n        map_zero' := by\n          dsimp\n          exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl) _ (by simp)\n        map_mul' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· * ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          ","nextTactic":"all_goals simp","declUpToTactic":"/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              rw [hf j k hjk]\n              rfl)\n            (↑(iSup K)) (by rw [coe_iSup_of_directed dir])\n        map_zero' := by\n          dsimp\n          exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl) _ (by simp)\n        map_mul' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· * ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.892_0.jQvdAcOcyVirh23","decl":"/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B "}
-{"state":"case hopi\nR : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\nS : NonUnitalSubalgebra R A\nι : Type u_1\ninst✝ : Nonempty ι\nK : ι → NonUnitalSubalgebra R A\ndir : Directed (fun x x_1 => x ≤ x_1) K\nf : (i : ι) → ↥(K i) →ₙₐ[R] B\nhf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalAlgHom.comp (f j) (inclusion h)\n⊢ ∀ (i : ι) (x y : ↑↑(K i)),\n    Set.inclusion (_ : ↑(K i) ⊆ ↑(iSup K)) (x * y) =\n      Set.inclusion (_ : ↑(K i) ⊆ ↑(iSup K)) x * Set.inclusion (_ : ↑(K i) ⊆ ↑(iSup K)) y","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact NonUnitalAlgebra.gc\n\n@[simp]\ntheorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=\n  eq_top_iff.2 fun _x hx => subset_adjoin R hx\n\nvariable {R A}\n\n@[simp]\ntheorem coe_top : (↑(⊤ : NonUnitalSubalgebra R A) : Set A) = Set.univ :=\n  rfl\n\n@[simp]\ntheorem mem_top {x : A} : x ∈ (⊤ : NonUnitalSubalgebra R A) :=\n  Set.mem_univ x\n\n@[simp]\ntheorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toNonUnitalSubsemiring : (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubsemiring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toSubring {R A : Type*} [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    [IsScalarTower R A A] [SMulCommClass R A A] :\n    (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem toSubmodule_eq_top {S : NonUnitalSubalgebra R A} : S.toSubmodule = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toSubmodule'.injective.eq_iff' top_toSubmodule\n\n@[simp]\ntheorem toNonUnitalSubsemiring_eq_top {S : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubsemiring_injective.eq_iff' top_toNonUnitalSubsemiring\n\n@[simp]\ntheorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubring_injective.eq_iff' top_toSubring\n\ntheorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_left\n\ntheorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_right\n\ntheorem mul_mem_sup {S T : NonUnitalSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) :\n    x * y ∈ S ⊔ T :=\n  mul_mem (mem_sup_left hx) (mem_sup_right hy)\n\ntheorem map_sup (f : F) (S T : NonUnitalSubalgebra R A) :\n    ((S ⊔ T).map f : NonUnitalSubalgebra R B) = S.map f ⊔ T.map f :=\n  (@NonUnitalSubalgebra.gc_map_comap F R A B _ _ _ _ _ _ f).l_sup\n\n@[simp, norm_cast]\ntheorem coe_inf (S T : NonUnitalSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T :=\n  rfl\n\n@[simp]\ntheorem mem_inf {S T : NonUnitalSubalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T :=\n  Iff.rfl\n\n@[simp]\ntheorem inf_toSubmodule (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toSubmodule = S.toSubmodule ⊓ T.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem inf_toNonUnitalSubsemiring (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring ⊓ T.toNonUnitalSubsemiring :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sInf (S : Set (NonUnitalSubalgebra R A)) : (↑(sInf S) : Set A) = ⋂ s ∈ S, ↑s :=\n  sInf_image\n\ntheorem mem_sInf {S : Set (NonUnitalSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p := by\n  simp only [← SetLike.mem_coe, coe_sInf, Set.mem_iInter₂]\n\n@[simp]\ntheorem sInf_toSubmodule (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toSubmodule = sInf (NonUnitalSubalgebra.toSubmodule '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp]\ntheorem sInf_toNonUnitalSubsemiring (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toNonUnitalSubsemiring = sInf (NonUnitalSubalgebra.toNonUnitalSubsemiring '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp, norm_cast]\ntheorem coe_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} :\n    (↑(⨅ i, S i) : Set A) = ⋂ i, S i := by simp [iInf]\n\ntheorem mem_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} {x : A} :\n    (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by simp only [iInf, mem_sInf, Set.forall_range_iff]\n\n@[simp]\ntheorem iInf_toSubmodule {ι : Sort*} (S : ι → NonUnitalSubalgebra R A) :\n    (⨅ i, S i).toSubmodule = ⨅ i, (S i).toSubmodule :=\n  SetLike.coe_injective <| by simp\n\ninstance : Inhabited (NonUnitalSubalgebra R A) :=\n  ⟨⊥⟩\n\ntheorem mem_bot {x : A} : x ∈ (⊥ : NonUnitalSubalgebra R A) ↔ x = 0 :=\n  show x ∈ Submodule.span R (NonUnitalSubsemiring.closure (∅ : Set A) : Set A) ↔ x = 0 by\n    rw [NonUnitalSubsemiring.closure_empty, NonUnitalSubsemiring.coe_bot,\n      Submodule.span_zero_singleton, Submodule.mem_bot]\n\ntheorem toSubmodule_bot : (⊥ : NonUnitalSubalgebra R A).toSubmodule = ⊥ := by\n  ext\n  simp only [mem_bot, NonUnitalSubalgebra.mem_toSubmodule, Submodule.mem_bot]\n\n@[simp]\ntheorem coe_bot : ((⊥ : NonUnitalSubalgebra R A) : Set A) = {0} := by\n  simp [Set.ext_iff, NonUnitalAlgebra.mem_bot]\n\ntheorem eq_top_iff {S : NonUnitalSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S :=\n  ⟨fun h x => by rw [h]; exact mem_top, fun h => by ext x; exact ⟨fun _ => mem_top, fun _ => h x⟩⟩\n\ntheorem range_top_iff_surjective (f : A →ₙₐ[R] B) :\n    NonUnitalAlgHom.range f = (⊤ : NonUnitalSubalgebra R B) ↔ Function.Surjective f :=\n  NonUnitalAlgebra.eq_top_iff\n\n@[simp]\ntheorem range_id : NonUnitalAlgHom.range (NonUnitalAlgHom.id R A) = ⊤ :=\n  SetLike.coe_injective Set.range_id\n\n@[simp]\ntheorem map_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R A).map f = NonUnitalAlgHom.range f :=\n  SetLike.coe_injective Set.image_univ\n\n@[simp]\ntheorem map_bot (f : A →ₙₐ[R] B) : (⊥ : NonUnitalSubalgebra R A).map f = ⊥ :=\n  SetLike.coe_injective <| by simp [NonUnitalAlgebra.coe_bot, NonUnitalSubalgebra.coe_map]\n\n@[simp]\ntheorem comap_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R B).comap f = ⊤ :=\n  eq_top_iff.2 fun _ => mem_top\n\n/-- `NonUnitalAlgHom` to `⊤ : NonUnitalSubalgebra R A`. -/\ndef toTop : A →ₙₐ[R] (⊤ : NonUnitalSubalgebra R A) :=\n  NonUnitalAlgHom.codRestrict (NonUnitalAlgHom.id R A) ⊤ fun _ => mem_top\n\nend NonUnitalAlgebra\n\nnamespace NonUnitalSubalgebra\n\nopen NonUnitalAlgebra\n\nsection NonAssoc\n\nvariable {R : Type u} {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable (S : NonUnitalSubalgebra R A)\n\ninstance subsingleton_of_subsingleton [Subsingleton A] : Subsingleton (NonUnitalSubalgebra R A) :=\n  ⟨fun B C => ext fun x => by simp only [Subsingleton.elim x 0, zero_mem B, zero_mem C]⟩\n\ninstance _root_.NonUnitalAlgHom.subsingleton [Subsingleton (NonUnitalSubalgebra R A)] :\n    Subsingleton (A →ₙₐ[R] B) :=\n  ⟨fun f g =>\n    NonUnitalAlgHom.ext fun a =>\n      have : a ∈ (⊥ : NonUnitalSubalgebra R A) :=\n        Subsingleton.elim (⊤ : NonUnitalSubalgebra R A) ⊥ ▸ mem_top\n      (mem_bot.mp this).symm ▸ (map_zero f).trans (map_zero g).symm⟩\n\ntheorem range_val : NonUnitalAlgHom.range (NonUnitalSubalgebraClass.subtype S) = S :=\n  ext <| Set.ext_iff.1 <| (NonUnitalSubalgebraClass.subtype S).coe_range.trans Subtype.range_val\n\n/-- The map `S → T` when `S` is a non-unital subalgebra contained in the non-unital subalgebra `T`.\n\nThis is the non-unital subalgebra version of `Submodule.inclusion`, or `Subring.inclusion`  -/\ndef inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) : S →ₙₐ[R] T\n    where\n  toFun := Set.inclusion h\n  map_add' _ _ := rfl\n  map_mul' _ _ := rfl\n  map_zero' := rfl\n  map_smul' _ _ := rfl\n\ntheorem inclusion_injective {S T : NonUnitalSubalgebra R A} (h : S ≤ T) :\n    Function.Injective (inclusion h) := fun _ _ => Subtype.ext ∘ Subtype.mk.inj\n\n@[simp]\ntheorem inclusion_self {S : NonUnitalSubalgebra R A} :\n    inclusion (le_refl S) = NonUnitalAlgHom.id R S :=\n  NonUnitalAlgHom.ext fun _ => Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_mk {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : A) (hx : x ∈ S) :\n    inclusion h ⟨x, hx⟩ = ⟨x, h hx⟩ :=\n  rfl\n\ntheorem inclusion_right {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : T) (m : (x : A) ∈ S) :\n    inclusion h ⟨x, m⟩ = x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_inclusion {S T U : NonUnitalSubalgebra R A} (hst : S ≤ T) (htu : T ≤ U) (x : S) :\n    inclusion htu (inclusion hst x) = inclusion (le_trans hst htu) x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem coe_inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (s : S) :\n    (inclusion h s : A) = s :=\n  rfl\n\nsection Prod\n\nvariable (S₁ : NonUnitalSubalgebra R B)\n\n/-- The product of two non-unital subalgebras is a non-unital subalgebra. -/\ndef prod : NonUnitalSubalgebra R (A × B) :=\n  { S.toNonUnitalSubsemiring.prod S₁.toNonUnitalSubsemiring with\n    carrier := S ×ˢ S₁\n    smul_mem' := fun r _x hx => ⟨SMulMemClass.smul_mem r hx.1, SMulMemClass.smul_mem r hx.2⟩ }\n\n@[simp]\ntheorem coe_prod : (prod S S₁ : Set (A × B)) = (S : Set A) ×ˢ S₁ :=\n  rfl\n\ntheorem prod_toSubmodule : (S.prod S₁).toSubmodule = S.toSubmodule.prod S₁.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem mem_prod {S : NonUnitalSubalgebra R A} {S₁ : NonUnitalSubalgebra R B} {x : A × B} :\n    x ∈ prod S S₁ ↔ x.1 ∈ S ∧ x.2 ∈ S₁ :=\n  Set.mem_prod\n\n@[simp]\ntheorem prod_top : (prod ⊤ ⊤ : NonUnitalSubalgebra R (A × B)) = ⊤ := by ext; simp\n\ntheorem prod_mono {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S ≤ T → S₁ ≤ T₁ → prod S S₁ ≤ prod T T₁ :=\n  Set.prod_mono\n\n@[simp]\ntheorem prod_inf_prod {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S.prod S₁ ⊓ T.prod T₁ = (S ⊓ T).prod (S₁ ⊓ T₁) :=\n  SetLike.coe_injective Set.prod_inter_prod\n\nend Prod\n\nsection SuprLift\n\nvariable {ι : Type*}\n\ntheorem coe_iSup_of_directed [Nonempty ι] {S : ι → NonUnitalSubalgebra R A}\n    (dir : Directed (· ≤ ·) S) : ↑(iSup S) = ⋃ i, (S i : Set A) :=\n  let K : NonUnitalSubalgebra R A :=\n    { __ := NonUnitalSubsemiring.copy _ _ (NonUnitalSubsemiring.coe_iSup_of_directed dir).symm\n      smul_mem' := fun r _x hx ↦\n        let ⟨i, hi⟩ := Set.mem_iUnion.1 hx\n        Set.mem_iUnion.2 ⟨i, (S i).smul_mem' r hi⟩ }\n  have : iSup S = K := le_antisymm\n    (iSup_le fun i ↦ le_iSup (fun i ↦ (S i : Set A)) i) (Set.iUnion_subset fun _ ↦ le_iSup S _)\n  this.symm ▸ rfl\n\n/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              rw [hf j k hjk]\n              rfl)\n            (↑(iSup K)) (by rw [coe_iSup_of_directed dir])\n        map_zero' := by\n          dsimp\n          exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl) _ (by simp)\n        map_mul' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· * ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals ","nextTactic":"simp","declUpToTactic":"/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              rw [hf j k hjk]\n              rfl)\n            (↑(iSup K)) (by rw [coe_iSup_of_directed dir])\n        map_zero' := by\n          dsimp\n          exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl) _ (by simp)\n        map_mul' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· * ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.892_0.jQvdAcOcyVirh23","decl":"/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B "}
-{"state":"case h\nR : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\nS : NonUnitalSubalgebra R A\nι : Type u_1\ninst✝ : Nonempty ι\nK : ι → NonUnitalSubalgebra R A\ndir : Directed (fun x x_1 => x ≤ x_1) K\nf : (i : ι) → ↥(K i) →ₙₐ[R] B\nhf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalAlgHom.comp (f j) (inclusion h)\n⊢ ∀ (i : ι) (x y : ↑↑(K i)), (f i) (x * y) = (f i) x * (f i) y","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact NonUnitalAlgebra.gc\n\n@[simp]\ntheorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=\n  eq_top_iff.2 fun _x hx => subset_adjoin R hx\n\nvariable {R A}\n\n@[simp]\ntheorem coe_top : (↑(⊤ : NonUnitalSubalgebra R A) : Set A) = Set.univ :=\n  rfl\n\n@[simp]\ntheorem mem_top {x : A} : x ∈ (⊤ : NonUnitalSubalgebra R A) :=\n  Set.mem_univ x\n\n@[simp]\ntheorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toNonUnitalSubsemiring : (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubsemiring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toSubring {R A : Type*} [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    [IsScalarTower R A A] [SMulCommClass R A A] :\n    (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem toSubmodule_eq_top {S : NonUnitalSubalgebra R A} : S.toSubmodule = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toSubmodule'.injective.eq_iff' top_toSubmodule\n\n@[simp]\ntheorem toNonUnitalSubsemiring_eq_top {S : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubsemiring_injective.eq_iff' top_toNonUnitalSubsemiring\n\n@[simp]\ntheorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubring_injective.eq_iff' top_toSubring\n\ntheorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_left\n\ntheorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_right\n\ntheorem mul_mem_sup {S T : NonUnitalSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) :\n    x * y ∈ S ⊔ T :=\n  mul_mem (mem_sup_left hx) (mem_sup_right hy)\n\ntheorem map_sup (f : F) (S T : NonUnitalSubalgebra R A) :\n    ((S ⊔ T).map f : NonUnitalSubalgebra R B) = S.map f ⊔ T.map f :=\n  (@NonUnitalSubalgebra.gc_map_comap F R A B _ _ _ _ _ _ f).l_sup\n\n@[simp, norm_cast]\ntheorem coe_inf (S T : NonUnitalSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T :=\n  rfl\n\n@[simp]\ntheorem mem_inf {S T : NonUnitalSubalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T :=\n  Iff.rfl\n\n@[simp]\ntheorem inf_toSubmodule (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toSubmodule = S.toSubmodule ⊓ T.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem inf_toNonUnitalSubsemiring (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring ⊓ T.toNonUnitalSubsemiring :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sInf (S : Set (NonUnitalSubalgebra R A)) : (↑(sInf S) : Set A) = ⋂ s ∈ S, ↑s :=\n  sInf_image\n\ntheorem mem_sInf {S : Set (NonUnitalSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p := by\n  simp only [← SetLike.mem_coe, coe_sInf, Set.mem_iInter₂]\n\n@[simp]\ntheorem sInf_toSubmodule (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toSubmodule = sInf (NonUnitalSubalgebra.toSubmodule '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp]\ntheorem sInf_toNonUnitalSubsemiring (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toNonUnitalSubsemiring = sInf (NonUnitalSubalgebra.toNonUnitalSubsemiring '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp, norm_cast]\ntheorem coe_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} :\n    (↑(⨅ i, S i) : Set A) = ⋂ i, S i := by simp [iInf]\n\ntheorem mem_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} {x : A} :\n    (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by simp only [iInf, mem_sInf, Set.forall_range_iff]\n\n@[simp]\ntheorem iInf_toSubmodule {ι : Sort*} (S : ι → NonUnitalSubalgebra R A) :\n    (⨅ i, S i).toSubmodule = ⨅ i, (S i).toSubmodule :=\n  SetLike.coe_injective <| by simp\n\ninstance : Inhabited (NonUnitalSubalgebra R A) :=\n  ⟨⊥⟩\n\ntheorem mem_bot {x : A} : x ∈ (⊥ : NonUnitalSubalgebra R A) ↔ x = 0 :=\n  show x ∈ Submodule.span R (NonUnitalSubsemiring.closure (∅ : Set A) : Set A) ↔ x = 0 by\n    rw [NonUnitalSubsemiring.closure_empty, NonUnitalSubsemiring.coe_bot,\n      Submodule.span_zero_singleton, Submodule.mem_bot]\n\ntheorem toSubmodule_bot : (⊥ : NonUnitalSubalgebra R A).toSubmodule = ⊥ := by\n  ext\n  simp only [mem_bot, NonUnitalSubalgebra.mem_toSubmodule, Submodule.mem_bot]\n\n@[simp]\ntheorem coe_bot : ((⊥ : NonUnitalSubalgebra R A) : Set A) = {0} := by\n  simp [Set.ext_iff, NonUnitalAlgebra.mem_bot]\n\ntheorem eq_top_iff {S : NonUnitalSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S :=\n  ⟨fun h x => by rw [h]; exact mem_top, fun h => by ext x; exact ⟨fun _ => mem_top, fun _ => h x⟩⟩\n\ntheorem range_top_iff_surjective (f : A →ₙₐ[R] B) :\n    NonUnitalAlgHom.range f = (⊤ : NonUnitalSubalgebra R B) ↔ Function.Surjective f :=\n  NonUnitalAlgebra.eq_top_iff\n\n@[simp]\ntheorem range_id : NonUnitalAlgHom.range (NonUnitalAlgHom.id R A) = ⊤ :=\n  SetLike.coe_injective Set.range_id\n\n@[simp]\ntheorem map_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R A).map f = NonUnitalAlgHom.range f :=\n  SetLike.coe_injective Set.image_univ\n\n@[simp]\ntheorem map_bot (f : A →ₙₐ[R] B) : (⊥ : NonUnitalSubalgebra R A).map f = ⊥ :=\n  SetLike.coe_injective <| by simp [NonUnitalAlgebra.coe_bot, NonUnitalSubalgebra.coe_map]\n\n@[simp]\ntheorem comap_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R B).comap f = ⊤ :=\n  eq_top_iff.2 fun _ => mem_top\n\n/-- `NonUnitalAlgHom` to `⊤ : NonUnitalSubalgebra R A`. -/\ndef toTop : A →ₙₐ[R] (⊤ : NonUnitalSubalgebra R A) :=\n  NonUnitalAlgHom.codRestrict (NonUnitalAlgHom.id R A) ⊤ fun _ => mem_top\n\nend NonUnitalAlgebra\n\nnamespace NonUnitalSubalgebra\n\nopen NonUnitalAlgebra\n\nsection NonAssoc\n\nvariable {R : Type u} {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable (S : NonUnitalSubalgebra R A)\n\ninstance subsingleton_of_subsingleton [Subsingleton A] : Subsingleton (NonUnitalSubalgebra R A) :=\n  ⟨fun B C => ext fun x => by simp only [Subsingleton.elim x 0, zero_mem B, zero_mem C]⟩\n\ninstance _root_.NonUnitalAlgHom.subsingleton [Subsingleton (NonUnitalSubalgebra R A)] :\n    Subsingleton (A →ₙₐ[R] B) :=\n  ⟨fun f g =>\n    NonUnitalAlgHom.ext fun a =>\n      have : a ∈ (⊥ : NonUnitalSubalgebra R A) :=\n        Subsingleton.elim (⊤ : NonUnitalSubalgebra R A) ⊥ ▸ mem_top\n      (mem_bot.mp this).symm ▸ (map_zero f).trans (map_zero g).symm⟩\n\ntheorem range_val : NonUnitalAlgHom.range (NonUnitalSubalgebraClass.subtype S) = S :=\n  ext <| Set.ext_iff.1 <| (NonUnitalSubalgebraClass.subtype S).coe_range.trans Subtype.range_val\n\n/-- The map `S → T` when `S` is a non-unital subalgebra contained in the non-unital subalgebra `T`.\n\nThis is the non-unital subalgebra version of `Submodule.inclusion`, or `Subring.inclusion`  -/\ndef inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) : S →ₙₐ[R] T\n    where\n  toFun := Set.inclusion h\n  map_add' _ _ := rfl\n  map_mul' _ _ := rfl\n  map_zero' := rfl\n  map_smul' _ _ := rfl\n\ntheorem inclusion_injective {S T : NonUnitalSubalgebra R A} (h : S ≤ T) :\n    Function.Injective (inclusion h) := fun _ _ => Subtype.ext ∘ Subtype.mk.inj\n\n@[simp]\ntheorem inclusion_self {S : NonUnitalSubalgebra R A} :\n    inclusion (le_refl S) = NonUnitalAlgHom.id R S :=\n  NonUnitalAlgHom.ext fun _ => Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_mk {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : A) (hx : x ∈ S) :\n    inclusion h ⟨x, hx⟩ = ⟨x, h hx⟩ :=\n  rfl\n\ntheorem inclusion_right {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : T) (m : (x : A) ∈ S) :\n    inclusion h ⟨x, m⟩ = x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_inclusion {S T U : NonUnitalSubalgebra R A} (hst : S ≤ T) (htu : T ≤ U) (x : S) :\n    inclusion htu (inclusion hst x) = inclusion (le_trans hst htu) x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem coe_inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (s : S) :\n    (inclusion h s : A) = s :=\n  rfl\n\nsection Prod\n\nvariable (S₁ : NonUnitalSubalgebra R B)\n\n/-- The product of two non-unital subalgebras is a non-unital subalgebra. -/\ndef prod : NonUnitalSubalgebra R (A × B) :=\n  { S.toNonUnitalSubsemiring.prod S₁.toNonUnitalSubsemiring with\n    carrier := S ×ˢ S₁\n    smul_mem' := fun r _x hx => ⟨SMulMemClass.smul_mem r hx.1, SMulMemClass.smul_mem r hx.2⟩ }\n\n@[simp]\ntheorem coe_prod : (prod S S₁ : Set (A × B)) = (S : Set A) ×ˢ S₁ :=\n  rfl\n\ntheorem prod_toSubmodule : (S.prod S₁).toSubmodule = S.toSubmodule.prod S₁.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem mem_prod {S : NonUnitalSubalgebra R A} {S₁ : NonUnitalSubalgebra R B} {x : A × B} :\n    x ∈ prod S S₁ ↔ x.1 ∈ S ∧ x.2 ∈ S₁ :=\n  Set.mem_prod\n\n@[simp]\ntheorem prod_top : (prod ⊤ ⊤ : NonUnitalSubalgebra R (A × B)) = ⊤ := by ext; simp\n\ntheorem prod_mono {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S ≤ T → S₁ ≤ T₁ → prod S S₁ ≤ prod T T₁ :=\n  Set.prod_mono\n\n@[simp]\ntheorem prod_inf_prod {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S.prod S₁ ⊓ T.prod T₁ = (S ⊓ T).prod (S₁ ⊓ T₁) :=\n  SetLike.coe_injective Set.prod_inter_prod\n\nend Prod\n\nsection SuprLift\n\nvariable {ι : Type*}\n\ntheorem coe_iSup_of_directed [Nonempty ι] {S : ι → NonUnitalSubalgebra R A}\n    (dir : Directed (· ≤ ·) S) : ↑(iSup S) = ⋃ i, (S i : Set A) :=\n  let K : NonUnitalSubalgebra R A :=\n    { __ := NonUnitalSubsemiring.copy _ _ (NonUnitalSubsemiring.coe_iSup_of_directed dir).symm\n      smul_mem' := fun r _x hx ↦\n        let ⟨i, hi⟩ := Set.mem_iUnion.1 hx\n        Set.mem_iUnion.2 ⟨i, (S i).smul_mem' r hi⟩ }\n  have : iSup S = K := le_antisymm\n    (iSup_le fun i ↦ le_iSup (fun i ↦ (S i : Set A)) i) (Set.iUnion_subset fun _ ↦ le_iSup S _)\n  this.symm ▸ rfl\n\n/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              rw [hf j k hjk]\n              rfl)\n            (↑(iSup K)) (by rw [coe_iSup_of_directed dir])\n        map_zero' := by\n          dsimp\n          exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl) _ (by simp)\n        map_mul' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· * ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals ","nextTactic":"simp","declUpToTactic":"/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              rw [hf j k hjk]\n              rfl)\n            (↑(iSup K)) (by rw [coe_iSup_of_directed dir])\n        map_zero' := by\n          dsimp\n          exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl) _ (by simp)\n        map_mul' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· * ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.892_0.jQvdAcOcyVirh23","decl":"/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B "}
-{"state":"R : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\nS : NonUnitalSubalgebra R A\nι : Type u_1\ninst✝ : Nonempty ι\nK : ι → NonUnitalSubalgebra R A\ndir : Directed (fun x x_1 => x ≤ x_1) K\nf : (i : ι) → ↥(K i) →ₙₐ[R] B\nhf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalAlgHom.comp (f j) (inclusion h)\nT : NonUnitalSubalgebra R A\nhT : T = iSup K\ni : ι\nx : ↥(K i)\nh : K i ≤ T\n⊢ (iSupLift K dir f hf T hT) ((inclusion h) x) = (f i) x","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact NonUnitalAlgebra.gc\n\n@[simp]\ntheorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=\n  eq_top_iff.2 fun _x hx => subset_adjoin R hx\n\nvariable {R A}\n\n@[simp]\ntheorem coe_top : (↑(⊤ : NonUnitalSubalgebra R A) : Set A) = Set.univ :=\n  rfl\n\n@[simp]\ntheorem mem_top {x : A} : x ∈ (⊤ : NonUnitalSubalgebra R A) :=\n  Set.mem_univ x\n\n@[simp]\ntheorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toNonUnitalSubsemiring : (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubsemiring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toSubring {R A : Type*} [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    [IsScalarTower R A A] [SMulCommClass R A A] :\n    (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem toSubmodule_eq_top {S : NonUnitalSubalgebra R A} : S.toSubmodule = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toSubmodule'.injective.eq_iff' top_toSubmodule\n\n@[simp]\ntheorem toNonUnitalSubsemiring_eq_top {S : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubsemiring_injective.eq_iff' top_toNonUnitalSubsemiring\n\n@[simp]\ntheorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubring_injective.eq_iff' top_toSubring\n\ntheorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_left\n\ntheorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_right\n\ntheorem mul_mem_sup {S T : NonUnitalSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) :\n    x * y ∈ S ⊔ T :=\n  mul_mem (mem_sup_left hx) (mem_sup_right hy)\n\ntheorem map_sup (f : F) (S T : NonUnitalSubalgebra R A) :\n    ((S ⊔ T).map f : NonUnitalSubalgebra R B) = S.map f ⊔ T.map f :=\n  (@NonUnitalSubalgebra.gc_map_comap F R A B _ _ _ _ _ _ f).l_sup\n\n@[simp, norm_cast]\ntheorem coe_inf (S T : NonUnitalSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T :=\n  rfl\n\n@[simp]\ntheorem mem_inf {S T : NonUnitalSubalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T :=\n  Iff.rfl\n\n@[simp]\ntheorem inf_toSubmodule (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toSubmodule = S.toSubmodule ⊓ T.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem inf_toNonUnitalSubsemiring (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring ⊓ T.toNonUnitalSubsemiring :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sInf (S : Set (NonUnitalSubalgebra R A)) : (↑(sInf S) : Set A) = ⋂ s ∈ S, ↑s :=\n  sInf_image\n\ntheorem mem_sInf {S : Set (NonUnitalSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p := by\n  simp only [← SetLike.mem_coe, coe_sInf, Set.mem_iInter₂]\n\n@[simp]\ntheorem sInf_toSubmodule (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toSubmodule = sInf (NonUnitalSubalgebra.toSubmodule '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp]\ntheorem sInf_toNonUnitalSubsemiring (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toNonUnitalSubsemiring = sInf (NonUnitalSubalgebra.toNonUnitalSubsemiring '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp, norm_cast]\ntheorem coe_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} :\n    (↑(⨅ i, S i) : Set A) = ⋂ i, S i := by simp [iInf]\n\ntheorem mem_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} {x : A} :\n    (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by simp only [iInf, mem_sInf, Set.forall_range_iff]\n\n@[simp]\ntheorem iInf_toSubmodule {ι : Sort*} (S : ι → NonUnitalSubalgebra R A) :\n    (⨅ i, S i).toSubmodule = ⨅ i, (S i).toSubmodule :=\n  SetLike.coe_injective <| by simp\n\ninstance : Inhabited (NonUnitalSubalgebra R A) :=\n  ⟨⊥⟩\n\ntheorem mem_bot {x : A} : x ∈ (⊥ : NonUnitalSubalgebra R A) ↔ x = 0 :=\n  show x ∈ Submodule.span R (NonUnitalSubsemiring.closure (∅ : Set A) : Set A) ↔ x = 0 by\n    rw [NonUnitalSubsemiring.closure_empty, NonUnitalSubsemiring.coe_bot,\n      Submodule.span_zero_singleton, Submodule.mem_bot]\n\ntheorem toSubmodule_bot : (⊥ : NonUnitalSubalgebra R A).toSubmodule = ⊥ := by\n  ext\n  simp only [mem_bot, NonUnitalSubalgebra.mem_toSubmodule, Submodule.mem_bot]\n\n@[simp]\ntheorem coe_bot : ((⊥ : NonUnitalSubalgebra R A) : Set A) = {0} := by\n  simp [Set.ext_iff, NonUnitalAlgebra.mem_bot]\n\ntheorem eq_top_iff {S : NonUnitalSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S :=\n  ⟨fun h x => by rw [h]; exact mem_top, fun h => by ext x; exact ⟨fun _ => mem_top, fun _ => h x⟩⟩\n\ntheorem range_top_iff_surjective (f : A →ₙₐ[R] B) :\n    NonUnitalAlgHom.range f = (⊤ : NonUnitalSubalgebra R B) ↔ Function.Surjective f :=\n  NonUnitalAlgebra.eq_top_iff\n\n@[simp]\ntheorem range_id : NonUnitalAlgHom.range (NonUnitalAlgHom.id R A) = ⊤ :=\n  SetLike.coe_injective Set.range_id\n\n@[simp]\ntheorem map_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R A).map f = NonUnitalAlgHom.range f :=\n  SetLike.coe_injective Set.image_univ\n\n@[simp]\ntheorem map_bot (f : A →ₙₐ[R] B) : (⊥ : NonUnitalSubalgebra R A).map f = ⊥ :=\n  SetLike.coe_injective <| by simp [NonUnitalAlgebra.coe_bot, NonUnitalSubalgebra.coe_map]\n\n@[simp]\ntheorem comap_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R B).comap f = ⊤ :=\n  eq_top_iff.2 fun _ => mem_top\n\n/-- `NonUnitalAlgHom` to `⊤ : NonUnitalSubalgebra R A`. -/\ndef toTop : A →ₙₐ[R] (⊤ : NonUnitalSubalgebra R A) :=\n  NonUnitalAlgHom.codRestrict (NonUnitalAlgHom.id R A) ⊤ fun _ => mem_top\n\nend NonUnitalAlgebra\n\nnamespace NonUnitalSubalgebra\n\nopen NonUnitalAlgebra\n\nsection NonAssoc\n\nvariable {R : Type u} {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable (S : NonUnitalSubalgebra R A)\n\ninstance subsingleton_of_subsingleton [Subsingleton A] : Subsingleton (NonUnitalSubalgebra R A) :=\n  ⟨fun B C => ext fun x => by simp only [Subsingleton.elim x 0, zero_mem B, zero_mem C]⟩\n\ninstance _root_.NonUnitalAlgHom.subsingleton [Subsingleton (NonUnitalSubalgebra R A)] :\n    Subsingleton (A →ₙₐ[R] B) :=\n  ⟨fun f g =>\n    NonUnitalAlgHom.ext fun a =>\n      have : a ∈ (⊥ : NonUnitalSubalgebra R A) :=\n        Subsingleton.elim (⊤ : NonUnitalSubalgebra R A) ⊥ ▸ mem_top\n      (mem_bot.mp this).symm ▸ (map_zero f).trans (map_zero g).symm⟩\n\ntheorem range_val : NonUnitalAlgHom.range (NonUnitalSubalgebraClass.subtype S) = S :=\n  ext <| Set.ext_iff.1 <| (NonUnitalSubalgebraClass.subtype S).coe_range.trans Subtype.range_val\n\n/-- The map `S → T` when `S` is a non-unital subalgebra contained in the non-unital subalgebra `T`.\n\nThis is the non-unital subalgebra version of `Submodule.inclusion`, or `Subring.inclusion`  -/\ndef inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) : S →ₙₐ[R] T\n    where\n  toFun := Set.inclusion h\n  map_add' _ _ := rfl\n  map_mul' _ _ := rfl\n  map_zero' := rfl\n  map_smul' _ _ := rfl\n\ntheorem inclusion_injective {S T : NonUnitalSubalgebra R A} (h : S ≤ T) :\n    Function.Injective (inclusion h) := fun _ _ => Subtype.ext ∘ Subtype.mk.inj\n\n@[simp]\ntheorem inclusion_self {S : NonUnitalSubalgebra R A} :\n    inclusion (le_refl S) = NonUnitalAlgHom.id R S :=\n  NonUnitalAlgHom.ext fun _ => Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_mk {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : A) (hx : x ∈ S) :\n    inclusion h ⟨x, hx⟩ = ⟨x, h hx⟩ :=\n  rfl\n\ntheorem inclusion_right {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : T) (m : (x : A) ∈ S) :\n    inclusion h ⟨x, m⟩ = x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_inclusion {S T U : NonUnitalSubalgebra R A} (hst : S ≤ T) (htu : T ≤ U) (x : S) :\n    inclusion htu (inclusion hst x) = inclusion (le_trans hst htu) x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem coe_inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (s : S) :\n    (inclusion h s : A) = s :=\n  rfl\n\nsection Prod\n\nvariable (S₁ : NonUnitalSubalgebra R B)\n\n/-- The product of two non-unital subalgebras is a non-unital subalgebra. -/\ndef prod : NonUnitalSubalgebra R (A × B) :=\n  { S.toNonUnitalSubsemiring.prod S₁.toNonUnitalSubsemiring with\n    carrier := S ×ˢ S₁\n    smul_mem' := fun r _x hx => ⟨SMulMemClass.smul_mem r hx.1, SMulMemClass.smul_mem r hx.2⟩ }\n\n@[simp]\ntheorem coe_prod : (prod S S₁ : Set (A × B)) = (S : Set A) ×ˢ S₁ :=\n  rfl\n\ntheorem prod_toSubmodule : (S.prod S₁).toSubmodule = S.toSubmodule.prod S₁.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem mem_prod {S : NonUnitalSubalgebra R A} {S₁ : NonUnitalSubalgebra R B} {x : A × B} :\n    x ∈ prod S S₁ ↔ x.1 ∈ S ∧ x.2 ∈ S₁ :=\n  Set.mem_prod\n\n@[simp]\ntheorem prod_top : (prod ⊤ ⊤ : NonUnitalSubalgebra R (A × B)) = ⊤ := by ext; simp\n\ntheorem prod_mono {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S ≤ T → S₁ ≤ T₁ → prod S S₁ ≤ prod T T₁ :=\n  Set.prod_mono\n\n@[simp]\ntheorem prod_inf_prod {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S.prod S₁ ⊓ T.prod T₁ = (S ⊓ T).prod (S₁ ⊓ T₁) :=\n  SetLike.coe_injective Set.prod_inter_prod\n\nend Prod\n\nsection SuprLift\n\nvariable {ι : Type*}\n\ntheorem coe_iSup_of_directed [Nonempty ι] {S : ι → NonUnitalSubalgebra R A}\n    (dir : Directed (· ≤ ·) S) : ↑(iSup S) = ⋃ i, (S i : Set A) :=\n  let K : NonUnitalSubalgebra R A :=\n    { __ := NonUnitalSubsemiring.copy _ _ (NonUnitalSubsemiring.coe_iSup_of_directed dir).symm\n      smul_mem' := fun r _x hx ↦\n        let ⟨i, hi⟩ := Set.mem_iUnion.1 hx\n        Set.mem_iUnion.2 ⟨i, (S i).smul_mem' r hi⟩ }\n  have : iSup S = K := le_antisymm\n    (iSup_le fun i ↦ le_iSup (fun i ↦ (S i : Set A)) i) (Set.iUnion_subset fun _ ↦ le_iSup S _)\n  this.symm ▸ rfl\n\n/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              rw [hf j k hjk]\n              rfl)\n            (↑(iSup K)) (by rw [coe_iSup_of_directed dir])\n        map_zero' := by\n          dsimp\n          exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl) _ (by simp)\n        map_mul' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· * ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_add' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· + ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_smul' := fun r => by\n          dsimp\n          apply Set.iUnionLift_unary (coe_iSup_of_directed dir) _ (fun _ x => r • x)\n            (fun _ _ => rfl)\n          on_goal 2 => rw [coe_iSup_of_directed dir]\n          all_goals simp }\n\nvariable [Nonempty ι] {K : ι → NonUnitalSubalgebra R A} {dir : Directed (· ≤ ·) K}\n  {f : ∀ i, K i →ₙₐ[R] B} {hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h)}\n  {T : NonUnitalSubalgebra R A} {hT : T = iSup K}\n\n@[simp]\ntheorem iSupLift_inclusion {i : ι} (x : K i) (h : K i ≤ T) :\n    iSupLift K dir f hf T hT (inclusion h x) = f i x := by\n  ","nextTactic":"subst T","declUpToTactic":"@[simp]\ntheorem iSupLift_inclusion {i : ι} (x : K i) (h : K i ≤ T) :\n    iSupLift K dir f hf T hT (inclusion h x) = f i x := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.933_0.jQvdAcOcyVirh23","decl":"@[simp]\ntheorem iSupLift_inclusion {i : ι} (x : K i) (h : K i ≤ T) :\n    iSupLift K dir f hf T hT (inclusion h x) = f i x "}
-{"state":"R : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\nS : NonUnitalSubalgebra R A\nι : Type u_1\ninst✝ : Nonempty ι\nK : ι → NonUnitalSubalgebra R A\ndir : Directed (fun x x_1 => x ≤ x_1) K\nf : (i : ι) → ↥(K i) →ₙₐ[R] B\nhf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalAlgHom.comp (f j) (inclusion h)\ni : ι\nx : ↥(K i)\nh : K i ≤ iSup K\n⊢ (iSupLift K dir f hf (iSup K) (_ : iSup K = iSup K)) ((inclusion h) x) = (f i) x","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact NonUnitalAlgebra.gc\n\n@[simp]\ntheorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=\n  eq_top_iff.2 fun _x hx => subset_adjoin R hx\n\nvariable {R A}\n\n@[simp]\ntheorem coe_top : (↑(⊤ : NonUnitalSubalgebra R A) : Set A) = Set.univ :=\n  rfl\n\n@[simp]\ntheorem mem_top {x : A} : x ∈ (⊤ : NonUnitalSubalgebra R A) :=\n  Set.mem_univ x\n\n@[simp]\ntheorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toNonUnitalSubsemiring : (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubsemiring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toSubring {R A : Type*} [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    [IsScalarTower R A A] [SMulCommClass R A A] :\n    (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem toSubmodule_eq_top {S : NonUnitalSubalgebra R A} : S.toSubmodule = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toSubmodule'.injective.eq_iff' top_toSubmodule\n\n@[simp]\ntheorem toNonUnitalSubsemiring_eq_top {S : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubsemiring_injective.eq_iff' top_toNonUnitalSubsemiring\n\n@[simp]\ntheorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubring_injective.eq_iff' top_toSubring\n\ntheorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_left\n\ntheorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_right\n\ntheorem mul_mem_sup {S T : NonUnitalSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) :\n    x * y ∈ S ⊔ T :=\n  mul_mem (mem_sup_left hx) (mem_sup_right hy)\n\ntheorem map_sup (f : F) (S T : NonUnitalSubalgebra R A) :\n    ((S ⊔ T).map f : NonUnitalSubalgebra R B) = S.map f ⊔ T.map f :=\n  (@NonUnitalSubalgebra.gc_map_comap F R A B _ _ _ _ _ _ f).l_sup\n\n@[simp, norm_cast]\ntheorem coe_inf (S T : NonUnitalSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T :=\n  rfl\n\n@[simp]\ntheorem mem_inf {S T : NonUnitalSubalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T :=\n  Iff.rfl\n\n@[simp]\ntheorem inf_toSubmodule (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toSubmodule = S.toSubmodule ⊓ T.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem inf_toNonUnitalSubsemiring (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring ⊓ T.toNonUnitalSubsemiring :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sInf (S : Set (NonUnitalSubalgebra R A)) : (↑(sInf S) : Set A) = ⋂ s ∈ S, ↑s :=\n  sInf_image\n\ntheorem mem_sInf {S : Set (NonUnitalSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p := by\n  simp only [← SetLike.mem_coe, coe_sInf, Set.mem_iInter₂]\n\n@[simp]\ntheorem sInf_toSubmodule (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toSubmodule = sInf (NonUnitalSubalgebra.toSubmodule '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp]\ntheorem sInf_toNonUnitalSubsemiring (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toNonUnitalSubsemiring = sInf (NonUnitalSubalgebra.toNonUnitalSubsemiring '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp, norm_cast]\ntheorem coe_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} :\n    (↑(⨅ i, S i) : Set A) = ⋂ i, S i := by simp [iInf]\n\ntheorem mem_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} {x : A} :\n    (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by simp only [iInf, mem_sInf, Set.forall_range_iff]\n\n@[simp]\ntheorem iInf_toSubmodule {ι : Sort*} (S : ι → NonUnitalSubalgebra R A) :\n    (⨅ i, S i).toSubmodule = ⨅ i, (S i).toSubmodule :=\n  SetLike.coe_injective <| by simp\n\ninstance : Inhabited (NonUnitalSubalgebra R A) :=\n  ⟨⊥⟩\n\ntheorem mem_bot {x : A} : x ∈ (⊥ : NonUnitalSubalgebra R A) ↔ x = 0 :=\n  show x ∈ Submodule.span R (NonUnitalSubsemiring.closure (∅ : Set A) : Set A) ↔ x = 0 by\n    rw [NonUnitalSubsemiring.closure_empty, NonUnitalSubsemiring.coe_bot,\n      Submodule.span_zero_singleton, Submodule.mem_bot]\n\ntheorem toSubmodule_bot : (⊥ : NonUnitalSubalgebra R A).toSubmodule = ⊥ := by\n  ext\n  simp only [mem_bot, NonUnitalSubalgebra.mem_toSubmodule, Submodule.mem_bot]\n\n@[simp]\ntheorem coe_bot : ((⊥ : NonUnitalSubalgebra R A) : Set A) = {0} := by\n  simp [Set.ext_iff, NonUnitalAlgebra.mem_bot]\n\ntheorem eq_top_iff {S : NonUnitalSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S :=\n  ⟨fun h x => by rw [h]; exact mem_top, fun h => by ext x; exact ⟨fun _ => mem_top, fun _ => h x⟩⟩\n\ntheorem range_top_iff_surjective (f : A →ₙₐ[R] B) :\n    NonUnitalAlgHom.range f = (⊤ : NonUnitalSubalgebra R B) ↔ Function.Surjective f :=\n  NonUnitalAlgebra.eq_top_iff\n\n@[simp]\ntheorem range_id : NonUnitalAlgHom.range (NonUnitalAlgHom.id R A) = ⊤ :=\n  SetLike.coe_injective Set.range_id\n\n@[simp]\ntheorem map_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R A).map f = NonUnitalAlgHom.range f :=\n  SetLike.coe_injective Set.image_univ\n\n@[simp]\ntheorem map_bot (f : A →ₙₐ[R] B) : (⊥ : NonUnitalSubalgebra R A).map f = ⊥ :=\n  SetLike.coe_injective <| by simp [NonUnitalAlgebra.coe_bot, NonUnitalSubalgebra.coe_map]\n\n@[simp]\ntheorem comap_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R B).comap f = ⊤ :=\n  eq_top_iff.2 fun _ => mem_top\n\n/-- `NonUnitalAlgHom` to `⊤ : NonUnitalSubalgebra R A`. -/\ndef toTop : A →ₙₐ[R] (⊤ : NonUnitalSubalgebra R A) :=\n  NonUnitalAlgHom.codRestrict (NonUnitalAlgHom.id R A) ⊤ fun _ => mem_top\n\nend NonUnitalAlgebra\n\nnamespace NonUnitalSubalgebra\n\nopen NonUnitalAlgebra\n\nsection NonAssoc\n\nvariable {R : Type u} {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable (S : NonUnitalSubalgebra R A)\n\ninstance subsingleton_of_subsingleton [Subsingleton A] : Subsingleton (NonUnitalSubalgebra R A) :=\n  ⟨fun B C => ext fun x => by simp only [Subsingleton.elim x 0, zero_mem B, zero_mem C]⟩\n\ninstance _root_.NonUnitalAlgHom.subsingleton [Subsingleton (NonUnitalSubalgebra R A)] :\n    Subsingleton (A →ₙₐ[R] B) :=\n  ⟨fun f g =>\n    NonUnitalAlgHom.ext fun a =>\n      have : a ∈ (⊥ : NonUnitalSubalgebra R A) :=\n        Subsingleton.elim (⊤ : NonUnitalSubalgebra R A) ⊥ ▸ mem_top\n      (mem_bot.mp this).symm ▸ (map_zero f).trans (map_zero g).symm⟩\n\ntheorem range_val : NonUnitalAlgHom.range (NonUnitalSubalgebraClass.subtype S) = S :=\n  ext <| Set.ext_iff.1 <| (NonUnitalSubalgebraClass.subtype S).coe_range.trans Subtype.range_val\n\n/-- The map `S → T` when `S` is a non-unital subalgebra contained in the non-unital subalgebra `T`.\n\nThis is the non-unital subalgebra version of `Submodule.inclusion`, or `Subring.inclusion`  -/\ndef inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) : S →ₙₐ[R] T\n    where\n  toFun := Set.inclusion h\n  map_add' _ _ := rfl\n  map_mul' _ _ := rfl\n  map_zero' := rfl\n  map_smul' _ _ := rfl\n\ntheorem inclusion_injective {S T : NonUnitalSubalgebra R A} (h : S ≤ T) :\n    Function.Injective (inclusion h) := fun _ _ => Subtype.ext ∘ Subtype.mk.inj\n\n@[simp]\ntheorem inclusion_self {S : NonUnitalSubalgebra R A} :\n    inclusion (le_refl S) = NonUnitalAlgHom.id R S :=\n  NonUnitalAlgHom.ext fun _ => Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_mk {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : A) (hx : x ∈ S) :\n    inclusion h ⟨x, hx⟩ = ⟨x, h hx⟩ :=\n  rfl\n\ntheorem inclusion_right {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : T) (m : (x : A) ∈ S) :\n    inclusion h ⟨x, m⟩ = x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_inclusion {S T U : NonUnitalSubalgebra R A} (hst : S ≤ T) (htu : T ≤ U) (x : S) :\n    inclusion htu (inclusion hst x) = inclusion (le_trans hst htu) x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem coe_inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (s : S) :\n    (inclusion h s : A) = s :=\n  rfl\n\nsection Prod\n\nvariable (S₁ : NonUnitalSubalgebra R B)\n\n/-- The product of two non-unital subalgebras is a non-unital subalgebra. -/\ndef prod : NonUnitalSubalgebra R (A × B) :=\n  { S.toNonUnitalSubsemiring.prod S₁.toNonUnitalSubsemiring with\n    carrier := S ×ˢ S₁\n    smul_mem' := fun r _x hx => ⟨SMulMemClass.smul_mem r hx.1, SMulMemClass.smul_mem r hx.2⟩ }\n\n@[simp]\ntheorem coe_prod : (prod S S₁ : Set (A × B)) = (S : Set A) ×ˢ S₁ :=\n  rfl\n\ntheorem prod_toSubmodule : (S.prod S₁).toSubmodule = S.toSubmodule.prod S₁.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem mem_prod {S : NonUnitalSubalgebra R A} {S₁ : NonUnitalSubalgebra R B} {x : A × B} :\n    x ∈ prod S S₁ ↔ x.1 ∈ S ∧ x.2 ∈ S₁ :=\n  Set.mem_prod\n\n@[simp]\ntheorem prod_top : (prod ⊤ ⊤ : NonUnitalSubalgebra R (A × B)) = ⊤ := by ext; simp\n\ntheorem prod_mono {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S ≤ T → S₁ ≤ T₁ → prod S S₁ ≤ prod T T₁ :=\n  Set.prod_mono\n\n@[simp]\ntheorem prod_inf_prod {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S.prod S₁ ⊓ T.prod T₁ = (S ⊓ T).prod (S₁ ⊓ T₁) :=\n  SetLike.coe_injective Set.prod_inter_prod\n\nend Prod\n\nsection SuprLift\n\nvariable {ι : Type*}\n\ntheorem coe_iSup_of_directed [Nonempty ι] {S : ι → NonUnitalSubalgebra R A}\n    (dir : Directed (· ≤ ·) S) : ↑(iSup S) = ⋃ i, (S i : Set A) :=\n  let K : NonUnitalSubalgebra R A :=\n    { __ := NonUnitalSubsemiring.copy _ _ (NonUnitalSubsemiring.coe_iSup_of_directed dir).symm\n      smul_mem' := fun r _x hx ↦\n        let ⟨i, hi⟩ := Set.mem_iUnion.1 hx\n        Set.mem_iUnion.2 ⟨i, (S i).smul_mem' r hi⟩ }\n  have : iSup S = K := le_antisymm\n    (iSup_le fun i ↦ le_iSup (fun i ↦ (S i : Set A)) i) (Set.iUnion_subset fun _ ↦ le_iSup S _)\n  this.symm ▸ rfl\n\n/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              rw [hf j k hjk]\n              rfl)\n            (↑(iSup K)) (by rw [coe_iSup_of_directed dir])\n        map_zero' := by\n          dsimp\n          exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl) _ (by simp)\n        map_mul' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· * ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_add' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· + ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_smul' := fun r => by\n          dsimp\n          apply Set.iUnionLift_unary (coe_iSup_of_directed dir) _ (fun _ x => r • x)\n            (fun _ _ => rfl)\n          on_goal 2 => rw [coe_iSup_of_directed dir]\n          all_goals simp }\n\nvariable [Nonempty ι] {K : ι → NonUnitalSubalgebra R A} {dir : Directed (· ≤ ·) K}\n  {f : ∀ i, K i →ₙₐ[R] B} {hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h)}\n  {T : NonUnitalSubalgebra R A} {hT : T = iSup K}\n\n@[simp]\ntheorem iSupLift_inclusion {i : ι} (x : K i) (h : K i ≤ T) :\n    iSupLift K dir f hf T hT (inclusion h x) = f i x := by\n  subst T\n  ","nextTactic":"dsimp [iSupLift]","declUpToTactic":"@[simp]\ntheorem iSupLift_inclusion {i : ι} (x : K i) (h : K i ≤ T) :\n    iSupLift K dir f hf T hT (inclusion h x) = f i x := by\n  subst T\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.933_0.jQvdAcOcyVirh23","decl":"@[simp]\ntheorem iSupLift_inclusion {i : ι} (x : K i) (h : K i ≤ T) :\n    iSupLift K dir f hf T hT (inclusion h x) = f i x "}
-{"state":"R : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\nS : NonUnitalSubalgebra R A\nι : Type u_1\ninst✝ : Nonempty ι\nK : ι → NonUnitalSubalgebra R A\ndir : Directed (fun x x_1 => x ≤ x_1) K\nf : (i : ι) → ↥(K i) →ₙₐ[R] B\nhf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalAlgHom.comp (f j) (inclusion h)\ni : ι\nx : ↥(K i)\nh : K i ≤ iSup K\n⊢ Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x)\n      (_ :\n        ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),\n          (fun i x => (f i) x) i { val := x, property := hxi } = (fun i x => (f i) x) j { val := x, property := hxj })\n      ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ i, ↑(K i)) ((inclusion h) x) =\n    (f i) x","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact NonUnitalAlgebra.gc\n\n@[simp]\ntheorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=\n  eq_top_iff.2 fun _x hx => subset_adjoin R hx\n\nvariable {R A}\n\n@[simp]\ntheorem coe_top : (↑(⊤ : NonUnitalSubalgebra R A) : Set A) = Set.univ :=\n  rfl\n\n@[simp]\ntheorem mem_top {x : A} : x ∈ (⊤ : NonUnitalSubalgebra R A) :=\n  Set.mem_univ x\n\n@[simp]\ntheorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toNonUnitalSubsemiring : (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubsemiring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toSubring {R A : Type*} [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    [IsScalarTower R A A] [SMulCommClass R A A] :\n    (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem toSubmodule_eq_top {S : NonUnitalSubalgebra R A} : S.toSubmodule = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toSubmodule'.injective.eq_iff' top_toSubmodule\n\n@[simp]\ntheorem toNonUnitalSubsemiring_eq_top {S : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubsemiring_injective.eq_iff' top_toNonUnitalSubsemiring\n\n@[simp]\ntheorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubring_injective.eq_iff' top_toSubring\n\ntheorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_left\n\ntheorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_right\n\ntheorem mul_mem_sup {S T : NonUnitalSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) :\n    x * y ∈ S ⊔ T :=\n  mul_mem (mem_sup_left hx) (mem_sup_right hy)\n\ntheorem map_sup (f : F) (S T : NonUnitalSubalgebra R A) :\n    ((S ⊔ T).map f : NonUnitalSubalgebra R B) = S.map f ⊔ T.map f :=\n  (@NonUnitalSubalgebra.gc_map_comap F R A B _ _ _ _ _ _ f).l_sup\n\n@[simp, norm_cast]\ntheorem coe_inf (S T : NonUnitalSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T :=\n  rfl\n\n@[simp]\ntheorem mem_inf {S T : NonUnitalSubalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T :=\n  Iff.rfl\n\n@[simp]\ntheorem inf_toSubmodule (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toSubmodule = S.toSubmodule ⊓ T.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem inf_toNonUnitalSubsemiring (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring ⊓ T.toNonUnitalSubsemiring :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sInf (S : Set (NonUnitalSubalgebra R A)) : (↑(sInf S) : Set A) = ⋂ s ∈ S, ↑s :=\n  sInf_image\n\ntheorem mem_sInf {S : Set (NonUnitalSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p := by\n  simp only [← SetLike.mem_coe, coe_sInf, Set.mem_iInter₂]\n\n@[simp]\ntheorem sInf_toSubmodule (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toSubmodule = sInf (NonUnitalSubalgebra.toSubmodule '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp]\ntheorem sInf_toNonUnitalSubsemiring (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toNonUnitalSubsemiring = sInf (NonUnitalSubalgebra.toNonUnitalSubsemiring '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp, norm_cast]\ntheorem coe_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} :\n    (↑(⨅ i, S i) : Set A) = ⋂ i, S i := by simp [iInf]\n\ntheorem mem_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} {x : A} :\n    (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by simp only [iInf, mem_sInf, Set.forall_range_iff]\n\n@[simp]\ntheorem iInf_toSubmodule {ι : Sort*} (S : ι → NonUnitalSubalgebra R A) :\n    (⨅ i, S i).toSubmodule = ⨅ i, (S i).toSubmodule :=\n  SetLike.coe_injective <| by simp\n\ninstance : Inhabited (NonUnitalSubalgebra R A) :=\n  ⟨⊥⟩\n\ntheorem mem_bot {x : A} : x ∈ (⊥ : NonUnitalSubalgebra R A) ↔ x = 0 :=\n  show x ∈ Submodule.span R (NonUnitalSubsemiring.closure (∅ : Set A) : Set A) ↔ x = 0 by\n    rw [NonUnitalSubsemiring.closure_empty, NonUnitalSubsemiring.coe_bot,\n      Submodule.span_zero_singleton, Submodule.mem_bot]\n\ntheorem toSubmodule_bot : (⊥ : NonUnitalSubalgebra R A).toSubmodule = ⊥ := by\n  ext\n  simp only [mem_bot, NonUnitalSubalgebra.mem_toSubmodule, Submodule.mem_bot]\n\n@[simp]\ntheorem coe_bot : ((⊥ : NonUnitalSubalgebra R A) : Set A) = {0} := by\n  simp [Set.ext_iff, NonUnitalAlgebra.mem_bot]\n\ntheorem eq_top_iff {S : NonUnitalSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S :=\n  ⟨fun h x => by rw [h]; exact mem_top, fun h => by ext x; exact ⟨fun _ => mem_top, fun _ => h x⟩⟩\n\ntheorem range_top_iff_surjective (f : A →ₙₐ[R] B) :\n    NonUnitalAlgHom.range f = (⊤ : NonUnitalSubalgebra R B) ↔ Function.Surjective f :=\n  NonUnitalAlgebra.eq_top_iff\n\n@[simp]\ntheorem range_id : NonUnitalAlgHom.range (NonUnitalAlgHom.id R A) = ⊤ :=\n  SetLike.coe_injective Set.range_id\n\n@[simp]\ntheorem map_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R A).map f = NonUnitalAlgHom.range f :=\n  SetLike.coe_injective Set.image_univ\n\n@[simp]\ntheorem map_bot (f : A →ₙₐ[R] B) : (⊥ : NonUnitalSubalgebra R A).map f = ⊥ :=\n  SetLike.coe_injective <| by simp [NonUnitalAlgebra.coe_bot, NonUnitalSubalgebra.coe_map]\n\n@[simp]\ntheorem comap_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R B).comap f = ⊤ :=\n  eq_top_iff.2 fun _ => mem_top\n\n/-- `NonUnitalAlgHom` to `⊤ : NonUnitalSubalgebra R A`. -/\ndef toTop : A →ₙₐ[R] (⊤ : NonUnitalSubalgebra R A) :=\n  NonUnitalAlgHom.codRestrict (NonUnitalAlgHom.id R A) ⊤ fun _ => mem_top\n\nend NonUnitalAlgebra\n\nnamespace NonUnitalSubalgebra\n\nopen NonUnitalAlgebra\n\nsection NonAssoc\n\nvariable {R : Type u} {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable (S : NonUnitalSubalgebra R A)\n\ninstance subsingleton_of_subsingleton [Subsingleton A] : Subsingleton (NonUnitalSubalgebra R A) :=\n  ⟨fun B C => ext fun x => by simp only [Subsingleton.elim x 0, zero_mem B, zero_mem C]⟩\n\ninstance _root_.NonUnitalAlgHom.subsingleton [Subsingleton (NonUnitalSubalgebra R A)] :\n    Subsingleton (A →ₙₐ[R] B) :=\n  ⟨fun f g =>\n    NonUnitalAlgHom.ext fun a =>\n      have : a ∈ (⊥ : NonUnitalSubalgebra R A) :=\n        Subsingleton.elim (⊤ : NonUnitalSubalgebra R A) ⊥ ▸ mem_top\n      (mem_bot.mp this).symm ▸ (map_zero f).trans (map_zero g).symm⟩\n\ntheorem range_val : NonUnitalAlgHom.range (NonUnitalSubalgebraClass.subtype S) = S :=\n  ext <| Set.ext_iff.1 <| (NonUnitalSubalgebraClass.subtype S).coe_range.trans Subtype.range_val\n\n/-- The map `S → T` when `S` is a non-unital subalgebra contained in the non-unital subalgebra `T`.\n\nThis is the non-unital subalgebra version of `Submodule.inclusion`, or `Subring.inclusion`  -/\ndef inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) : S →ₙₐ[R] T\n    where\n  toFun := Set.inclusion h\n  map_add' _ _ := rfl\n  map_mul' _ _ := rfl\n  map_zero' := rfl\n  map_smul' _ _ := rfl\n\ntheorem inclusion_injective {S T : NonUnitalSubalgebra R A} (h : S ≤ T) :\n    Function.Injective (inclusion h) := fun _ _ => Subtype.ext ∘ Subtype.mk.inj\n\n@[simp]\ntheorem inclusion_self {S : NonUnitalSubalgebra R A} :\n    inclusion (le_refl S) = NonUnitalAlgHom.id R S :=\n  NonUnitalAlgHom.ext fun _ => Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_mk {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : A) (hx : x ∈ S) :\n    inclusion h ⟨x, hx⟩ = ⟨x, h hx⟩ :=\n  rfl\n\ntheorem inclusion_right {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : T) (m : (x : A) ∈ S) :\n    inclusion h ⟨x, m⟩ = x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_inclusion {S T U : NonUnitalSubalgebra R A} (hst : S ≤ T) (htu : T ≤ U) (x : S) :\n    inclusion htu (inclusion hst x) = inclusion (le_trans hst htu) x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem coe_inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (s : S) :\n    (inclusion h s : A) = s :=\n  rfl\n\nsection Prod\n\nvariable (S₁ : NonUnitalSubalgebra R B)\n\n/-- The product of two non-unital subalgebras is a non-unital subalgebra. -/\ndef prod : NonUnitalSubalgebra R (A × B) :=\n  { S.toNonUnitalSubsemiring.prod S₁.toNonUnitalSubsemiring with\n    carrier := S ×ˢ S₁\n    smul_mem' := fun r _x hx => ⟨SMulMemClass.smul_mem r hx.1, SMulMemClass.smul_mem r hx.2⟩ }\n\n@[simp]\ntheorem coe_prod : (prod S S₁ : Set (A × B)) = (S : Set A) ×ˢ S₁ :=\n  rfl\n\ntheorem prod_toSubmodule : (S.prod S₁).toSubmodule = S.toSubmodule.prod S₁.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem mem_prod {S : NonUnitalSubalgebra R A} {S₁ : NonUnitalSubalgebra R B} {x : A × B} :\n    x ∈ prod S S₁ ↔ x.1 ∈ S ∧ x.2 ∈ S₁ :=\n  Set.mem_prod\n\n@[simp]\ntheorem prod_top : (prod ⊤ ⊤ : NonUnitalSubalgebra R (A × B)) = ⊤ := by ext; simp\n\ntheorem prod_mono {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S ≤ T → S₁ ≤ T₁ → prod S S₁ ≤ prod T T₁ :=\n  Set.prod_mono\n\n@[simp]\ntheorem prod_inf_prod {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S.prod S₁ ⊓ T.prod T₁ = (S ⊓ T).prod (S₁ ⊓ T₁) :=\n  SetLike.coe_injective Set.prod_inter_prod\n\nend Prod\n\nsection SuprLift\n\nvariable {ι : Type*}\n\ntheorem coe_iSup_of_directed [Nonempty ι] {S : ι → NonUnitalSubalgebra R A}\n    (dir : Directed (· ≤ ·) S) : ↑(iSup S) = ⋃ i, (S i : Set A) :=\n  let K : NonUnitalSubalgebra R A :=\n    { __ := NonUnitalSubsemiring.copy _ _ (NonUnitalSubsemiring.coe_iSup_of_directed dir).symm\n      smul_mem' := fun r _x hx ↦\n        let ⟨i, hi⟩ := Set.mem_iUnion.1 hx\n        Set.mem_iUnion.2 ⟨i, (S i).smul_mem' r hi⟩ }\n  have : iSup S = K := le_antisymm\n    (iSup_le fun i ↦ le_iSup (fun i ↦ (S i : Set A)) i) (Set.iUnion_subset fun _ ↦ le_iSup S _)\n  this.symm ▸ rfl\n\n/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              rw [hf j k hjk]\n              rfl)\n            (↑(iSup K)) (by rw [coe_iSup_of_directed dir])\n        map_zero' := by\n          dsimp\n          exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl) _ (by simp)\n        map_mul' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· * ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_add' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· + ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_smul' := fun r => by\n          dsimp\n          apply Set.iUnionLift_unary (coe_iSup_of_directed dir) _ (fun _ x => r • x)\n            (fun _ _ => rfl)\n          on_goal 2 => rw [coe_iSup_of_directed dir]\n          all_goals simp }\n\nvariable [Nonempty ι] {K : ι → NonUnitalSubalgebra R A} {dir : Directed (· ≤ ·) K}\n  {f : ∀ i, K i →ₙₐ[R] B} {hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h)}\n  {T : NonUnitalSubalgebra R A} {hT : T = iSup K}\n\n@[simp]\ntheorem iSupLift_inclusion {i : ι} (x : K i) (h : K i ≤ T) :\n    iSupLift K dir f hf T hT (inclusion h x) = f i x := by\n  subst T\n  dsimp [iSupLift]\n  ","nextTactic":"apply Set.iUnionLift_inclusion","declUpToTactic":"@[simp]\ntheorem iSupLift_inclusion {i : ι} (x : K i) (h : K i ≤ T) :\n    iSupLift K dir f hf T hT (inclusion h x) = f i x := by\n  subst T\n  dsimp [iSupLift]\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.933_0.jQvdAcOcyVirh23","decl":"@[simp]\ntheorem iSupLift_inclusion {i : ι} (x : K i) (h : K i ≤ T) :\n    iSupLift K dir f hf T hT (inclusion h x) = f i x "}
-{"state":"case h\nR : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\nS : NonUnitalSubalgebra R A\nι : Type u_1\ninst✝ : Nonempty ι\nK : ι → NonUnitalSubalgebra R A\ndir : Directed (fun x x_1 => x ≤ x_1) K\nf : (i : ι) → ↥(K i) →ₙₐ[R] B\nhf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalAlgHom.comp (f j) (inclusion h)\ni : ι\nx : ↥(K i)\nh : K i ≤ iSup K\n⊢ ↑(K i) ⊆ ↑(iSup K)","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact NonUnitalAlgebra.gc\n\n@[simp]\ntheorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=\n  eq_top_iff.2 fun _x hx => subset_adjoin R hx\n\nvariable {R A}\n\n@[simp]\ntheorem coe_top : (↑(⊤ : NonUnitalSubalgebra R A) : Set A) = Set.univ :=\n  rfl\n\n@[simp]\ntheorem mem_top {x : A} : x ∈ (⊤ : NonUnitalSubalgebra R A) :=\n  Set.mem_univ x\n\n@[simp]\ntheorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toNonUnitalSubsemiring : (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubsemiring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toSubring {R A : Type*} [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    [IsScalarTower R A A] [SMulCommClass R A A] :\n    (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem toSubmodule_eq_top {S : NonUnitalSubalgebra R A} : S.toSubmodule = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toSubmodule'.injective.eq_iff' top_toSubmodule\n\n@[simp]\ntheorem toNonUnitalSubsemiring_eq_top {S : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubsemiring_injective.eq_iff' top_toNonUnitalSubsemiring\n\n@[simp]\ntheorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubring_injective.eq_iff' top_toSubring\n\ntheorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_left\n\ntheorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_right\n\ntheorem mul_mem_sup {S T : NonUnitalSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) :\n    x * y ∈ S ⊔ T :=\n  mul_mem (mem_sup_left hx) (mem_sup_right hy)\n\ntheorem map_sup (f : F) (S T : NonUnitalSubalgebra R A) :\n    ((S ⊔ T).map f : NonUnitalSubalgebra R B) = S.map f ⊔ T.map f :=\n  (@NonUnitalSubalgebra.gc_map_comap F R A B _ _ _ _ _ _ f).l_sup\n\n@[simp, norm_cast]\ntheorem coe_inf (S T : NonUnitalSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T :=\n  rfl\n\n@[simp]\ntheorem mem_inf {S T : NonUnitalSubalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T :=\n  Iff.rfl\n\n@[simp]\ntheorem inf_toSubmodule (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toSubmodule = S.toSubmodule ⊓ T.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem inf_toNonUnitalSubsemiring (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring ⊓ T.toNonUnitalSubsemiring :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sInf (S : Set (NonUnitalSubalgebra R A)) : (↑(sInf S) : Set A) = ⋂ s ∈ S, ↑s :=\n  sInf_image\n\ntheorem mem_sInf {S : Set (NonUnitalSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p := by\n  simp only [← SetLike.mem_coe, coe_sInf, Set.mem_iInter₂]\n\n@[simp]\ntheorem sInf_toSubmodule (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toSubmodule = sInf (NonUnitalSubalgebra.toSubmodule '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp]\ntheorem sInf_toNonUnitalSubsemiring (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toNonUnitalSubsemiring = sInf (NonUnitalSubalgebra.toNonUnitalSubsemiring '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp, norm_cast]\ntheorem coe_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} :\n    (↑(⨅ i, S i) : Set A) = ⋂ i, S i := by simp [iInf]\n\ntheorem mem_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} {x : A} :\n    (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by simp only [iInf, mem_sInf, Set.forall_range_iff]\n\n@[simp]\ntheorem iInf_toSubmodule {ι : Sort*} (S : ι → NonUnitalSubalgebra R A) :\n    (⨅ i, S i).toSubmodule = ⨅ i, (S i).toSubmodule :=\n  SetLike.coe_injective <| by simp\n\ninstance : Inhabited (NonUnitalSubalgebra R A) :=\n  ⟨⊥⟩\n\ntheorem mem_bot {x : A} : x ∈ (⊥ : NonUnitalSubalgebra R A) ↔ x = 0 :=\n  show x ∈ Submodule.span R (NonUnitalSubsemiring.closure (∅ : Set A) : Set A) ↔ x = 0 by\n    rw [NonUnitalSubsemiring.closure_empty, NonUnitalSubsemiring.coe_bot,\n      Submodule.span_zero_singleton, Submodule.mem_bot]\n\ntheorem toSubmodule_bot : (⊥ : NonUnitalSubalgebra R A).toSubmodule = ⊥ := by\n  ext\n  simp only [mem_bot, NonUnitalSubalgebra.mem_toSubmodule, Submodule.mem_bot]\n\n@[simp]\ntheorem coe_bot : ((⊥ : NonUnitalSubalgebra R A) : Set A) = {0} := by\n  simp [Set.ext_iff, NonUnitalAlgebra.mem_bot]\n\ntheorem eq_top_iff {S : NonUnitalSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S :=\n  ⟨fun h x => by rw [h]; exact mem_top, fun h => by ext x; exact ⟨fun _ => mem_top, fun _ => h x⟩⟩\n\ntheorem range_top_iff_surjective (f : A →ₙₐ[R] B) :\n    NonUnitalAlgHom.range f = (⊤ : NonUnitalSubalgebra R B) ↔ Function.Surjective f :=\n  NonUnitalAlgebra.eq_top_iff\n\n@[simp]\ntheorem range_id : NonUnitalAlgHom.range (NonUnitalAlgHom.id R A) = ⊤ :=\n  SetLike.coe_injective Set.range_id\n\n@[simp]\ntheorem map_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R A).map f = NonUnitalAlgHom.range f :=\n  SetLike.coe_injective Set.image_univ\n\n@[simp]\ntheorem map_bot (f : A →ₙₐ[R] B) : (⊥ : NonUnitalSubalgebra R A).map f = ⊥ :=\n  SetLike.coe_injective <| by simp [NonUnitalAlgebra.coe_bot, NonUnitalSubalgebra.coe_map]\n\n@[simp]\ntheorem comap_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R B).comap f = ⊤ :=\n  eq_top_iff.2 fun _ => mem_top\n\n/-- `NonUnitalAlgHom` to `⊤ : NonUnitalSubalgebra R A`. -/\ndef toTop : A →ₙₐ[R] (⊤ : NonUnitalSubalgebra R A) :=\n  NonUnitalAlgHom.codRestrict (NonUnitalAlgHom.id R A) ⊤ fun _ => mem_top\n\nend NonUnitalAlgebra\n\nnamespace NonUnitalSubalgebra\n\nopen NonUnitalAlgebra\n\nsection NonAssoc\n\nvariable {R : Type u} {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable (S : NonUnitalSubalgebra R A)\n\ninstance subsingleton_of_subsingleton [Subsingleton A] : Subsingleton (NonUnitalSubalgebra R A) :=\n  ⟨fun B C => ext fun x => by simp only [Subsingleton.elim x 0, zero_mem B, zero_mem C]⟩\n\ninstance _root_.NonUnitalAlgHom.subsingleton [Subsingleton (NonUnitalSubalgebra R A)] :\n    Subsingleton (A →ₙₐ[R] B) :=\n  ⟨fun f g =>\n    NonUnitalAlgHom.ext fun a =>\n      have : a ∈ (⊥ : NonUnitalSubalgebra R A) :=\n        Subsingleton.elim (⊤ : NonUnitalSubalgebra R A) ⊥ ▸ mem_top\n      (mem_bot.mp this).symm ▸ (map_zero f).trans (map_zero g).symm⟩\n\ntheorem range_val : NonUnitalAlgHom.range (NonUnitalSubalgebraClass.subtype S) = S :=\n  ext <| Set.ext_iff.1 <| (NonUnitalSubalgebraClass.subtype S).coe_range.trans Subtype.range_val\n\n/-- The map `S → T` when `S` is a non-unital subalgebra contained in the non-unital subalgebra `T`.\n\nThis is the non-unital subalgebra version of `Submodule.inclusion`, or `Subring.inclusion`  -/\ndef inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) : S →ₙₐ[R] T\n    where\n  toFun := Set.inclusion h\n  map_add' _ _ := rfl\n  map_mul' _ _ := rfl\n  map_zero' := rfl\n  map_smul' _ _ := rfl\n\ntheorem inclusion_injective {S T : NonUnitalSubalgebra R A} (h : S ≤ T) :\n    Function.Injective (inclusion h) := fun _ _ => Subtype.ext ∘ Subtype.mk.inj\n\n@[simp]\ntheorem inclusion_self {S : NonUnitalSubalgebra R A} :\n    inclusion (le_refl S) = NonUnitalAlgHom.id R S :=\n  NonUnitalAlgHom.ext fun _ => Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_mk {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : A) (hx : x ∈ S) :\n    inclusion h ⟨x, hx⟩ = ⟨x, h hx⟩ :=\n  rfl\n\ntheorem inclusion_right {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : T) (m : (x : A) ∈ S) :\n    inclusion h ⟨x, m⟩ = x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_inclusion {S T U : NonUnitalSubalgebra R A} (hst : S ≤ T) (htu : T ≤ U) (x : S) :\n    inclusion htu (inclusion hst x) = inclusion (le_trans hst htu) x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem coe_inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (s : S) :\n    (inclusion h s : A) = s :=\n  rfl\n\nsection Prod\n\nvariable (S₁ : NonUnitalSubalgebra R B)\n\n/-- The product of two non-unital subalgebras is a non-unital subalgebra. -/\ndef prod : NonUnitalSubalgebra R (A × B) :=\n  { S.toNonUnitalSubsemiring.prod S₁.toNonUnitalSubsemiring with\n    carrier := S ×ˢ S₁\n    smul_mem' := fun r _x hx => ⟨SMulMemClass.smul_mem r hx.1, SMulMemClass.smul_mem r hx.2⟩ }\n\n@[simp]\ntheorem coe_prod : (prod S S₁ : Set (A × B)) = (S : Set A) ×ˢ S₁ :=\n  rfl\n\ntheorem prod_toSubmodule : (S.prod S₁).toSubmodule = S.toSubmodule.prod S₁.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem mem_prod {S : NonUnitalSubalgebra R A} {S₁ : NonUnitalSubalgebra R B} {x : A × B} :\n    x ∈ prod S S₁ ↔ x.1 ∈ S ∧ x.2 ∈ S₁ :=\n  Set.mem_prod\n\n@[simp]\ntheorem prod_top : (prod ⊤ ⊤ : NonUnitalSubalgebra R (A × B)) = ⊤ := by ext; simp\n\ntheorem prod_mono {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S ≤ T → S₁ ≤ T₁ → prod S S₁ ≤ prod T T₁ :=\n  Set.prod_mono\n\n@[simp]\ntheorem prod_inf_prod {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S.prod S₁ ⊓ T.prod T₁ = (S ⊓ T).prod (S₁ ⊓ T₁) :=\n  SetLike.coe_injective Set.prod_inter_prod\n\nend Prod\n\nsection SuprLift\n\nvariable {ι : Type*}\n\ntheorem coe_iSup_of_directed [Nonempty ι] {S : ι → NonUnitalSubalgebra R A}\n    (dir : Directed (· ≤ ·) S) : ↑(iSup S) = ⋃ i, (S i : Set A) :=\n  let K : NonUnitalSubalgebra R A :=\n    { __ := NonUnitalSubsemiring.copy _ _ (NonUnitalSubsemiring.coe_iSup_of_directed dir).symm\n      smul_mem' := fun r _x hx ↦\n        let ⟨i, hi⟩ := Set.mem_iUnion.1 hx\n        Set.mem_iUnion.2 ⟨i, (S i).smul_mem' r hi⟩ }\n  have : iSup S = K := le_antisymm\n    (iSup_le fun i ↦ le_iSup (fun i ↦ (S i : Set A)) i) (Set.iUnion_subset fun _ ↦ le_iSup S _)\n  this.symm ▸ rfl\n\n/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              rw [hf j k hjk]\n              rfl)\n            (↑(iSup K)) (by rw [coe_iSup_of_directed dir])\n        map_zero' := by\n          dsimp\n          exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl) _ (by simp)\n        map_mul' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· * ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_add' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· + ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_smul' := fun r => by\n          dsimp\n          apply Set.iUnionLift_unary (coe_iSup_of_directed dir) _ (fun _ x => r • x)\n            (fun _ _ => rfl)\n          on_goal 2 => rw [coe_iSup_of_directed dir]\n          all_goals simp }\n\nvariable [Nonempty ι] {K : ι → NonUnitalSubalgebra R A} {dir : Directed (· ≤ ·) K}\n  {f : ∀ i, K i →ₙₐ[R] B} {hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h)}\n  {T : NonUnitalSubalgebra R A} {hT : T = iSup K}\n\n@[simp]\ntheorem iSupLift_inclusion {i : ι} (x : K i) (h : K i ≤ T) :\n    iSupLift K dir f hf T hT (inclusion h x) = f i x := by\n  subst T\n  dsimp [iSupLift]\n  apply Set.iUnionLift_inclusion\n  ","nextTactic":"exact h","declUpToTactic":"@[simp]\ntheorem iSupLift_inclusion {i : ι} (x : K i) (h : K i ≤ T) :\n    iSupLift K dir f hf T hT (inclusion h x) = f i x := by\n  subst T\n  dsimp [iSupLift]\n  apply Set.iUnionLift_inclusion\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.933_0.jQvdAcOcyVirh23","decl":"@[simp]\ntheorem iSupLift_inclusion {i : ι} (x : K i) (h : K i ≤ T) :\n    iSupLift K dir f hf T hT (inclusion h x) = f i x "}
-{"state":"R : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\nS : NonUnitalSubalgebra R A\nι : Type u_1\ninst✝ : Nonempty ι\nK : ι → NonUnitalSubalgebra R A\ndir : Directed (fun x x_1 => x ≤ x_1) K\nf : (i : ι) → ↥(K i) →ₙₐ[R] B\nhf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalAlgHom.comp (f j) (inclusion h)\nT : NonUnitalSubalgebra R A\nhT : T = iSup K\ni : ι\nh : K i ≤ T\n⊢ NonUnitalAlgHom.comp (iSupLift K dir f hf T hT) (inclusion h) = f i","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact NonUnitalAlgebra.gc\n\n@[simp]\ntheorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=\n  eq_top_iff.2 fun _x hx => subset_adjoin R hx\n\nvariable {R A}\n\n@[simp]\ntheorem coe_top : (↑(⊤ : NonUnitalSubalgebra R A) : Set A) = Set.univ :=\n  rfl\n\n@[simp]\ntheorem mem_top {x : A} : x ∈ (⊤ : NonUnitalSubalgebra R A) :=\n  Set.mem_univ x\n\n@[simp]\ntheorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toNonUnitalSubsemiring : (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubsemiring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toSubring {R A : Type*} [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    [IsScalarTower R A A] [SMulCommClass R A A] :\n    (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem toSubmodule_eq_top {S : NonUnitalSubalgebra R A} : S.toSubmodule = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toSubmodule'.injective.eq_iff' top_toSubmodule\n\n@[simp]\ntheorem toNonUnitalSubsemiring_eq_top {S : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubsemiring_injective.eq_iff' top_toNonUnitalSubsemiring\n\n@[simp]\ntheorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubring_injective.eq_iff' top_toSubring\n\ntheorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_left\n\ntheorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_right\n\ntheorem mul_mem_sup {S T : NonUnitalSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) :\n    x * y ∈ S ⊔ T :=\n  mul_mem (mem_sup_left hx) (mem_sup_right hy)\n\ntheorem map_sup (f : F) (S T : NonUnitalSubalgebra R A) :\n    ((S ⊔ T).map f : NonUnitalSubalgebra R B) = S.map f ⊔ T.map f :=\n  (@NonUnitalSubalgebra.gc_map_comap F R A B _ _ _ _ _ _ f).l_sup\n\n@[simp, norm_cast]\ntheorem coe_inf (S T : NonUnitalSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T :=\n  rfl\n\n@[simp]\ntheorem mem_inf {S T : NonUnitalSubalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T :=\n  Iff.rfl\n\n@[simp]\ntheorem inf_toSubmodule (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toSubmodule = S.toSubmodule ⊓ T.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem inf_toNonUnitalSubsemiring (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring ⊓ T.toNonUnitalSubsemiring :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sInf (S : Set (NonUnitalSubalgebra R A)) : (↑(sInf S) : Set A) = ⋂ s ∈ S, ↑s :=\n  sInf_image\n\ntheorem mem_sInf {S : Set (NonUnitalSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p := by\n  simp only [← SetLike.mem_coe, coe_sInf, Set.mem_iInter₂]\n\n@[simp]\ntheorem sInf_toSubmodule (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toSubmodule = sInf (NonUnitalSubalgebra.toSubmodule '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp]\ntheorem sInf_toNonUnitalSubsemiring (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toNonUnitalSubsemiring = sInf (NonUnitalSubalgebra.toNonUnitalSubsemiring '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp, norm_cast]\ntheorem coe_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} :\n    (↑(⨅ i, S i) : Set A) = ⋂ i, S i := by simp [iInf]\n\ntheorem mem_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} {x : A} :\n    (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by simp only [iInf, mem_sInf, Set.forall_range_iff]\n\n@[simp]\ntheorem iInf_toSubmodule {ι : Sort*} (S : ι → NonUnitalSubalgebra R A) :\n    (⨅ i, S i).toSubmodule = ⨅ i, (S i).toSubmodule :=\n  SetLike.coe_injective <| by simp\n\ninstance : Inhabited (NonUnitalSubalgebra R A) :=\n  ⟨⊥⟩\n\ntheorem mem_bot {x : A} : x ∈ (⊥ : NonUnitalSubalgebra R A) ↔ x = 0 :=\n  show x ∈ Submodule.span R (NonUnitalSubsemiring.closure (∅ : Set A) : Set A) ↔ x = 0 by\n    rw [NonUnitalSubsemiring.closure_empty, NonUnitalSubsemiring.coe_bot,\n      Submodule.span_zero_singleton, Submodule.mem_bot]\n\ntheorem toSubmodule_bot : (⊥ : NonUnitalSubalgebra R A).toSubmodule = ⊥ := by\n  ext\n  simp only [mem_bot, NonUnitalSubalgebra.mem_toSubmodule, Submodule.mem_bot]\n\n@[simp]\ntheorem coe_bot : ((⊥ : NonUnitalSubalgebra R A) : Set A) = {0} := by\n  simp [Set.ext_iff, NonUnitalAlgebra.mem_bot]\n\ntheorem eq_top_iff {S : NonUnitalSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S :=\n  ⟨fun h x => by rw [h]; exact mem_top, fun h => by ext x; exact ⟨fun _ => mem_top, fun _ => h x⟩⟩\n\ntheorem range_top_iff_surjective (f : A →ₙₐ[R] B) :\n    NonUnitalAlgHom.range f = (⊤ : NonUnitalSubalgebra R B) ↔ Function.Surjective f :=\n  NonUnitalAlgebra.eq_top_iff\n\n@[simp]\ntheorem range_id : NonUnitalAlgHom.range (NonUnitalAlgHom.id R A) = ⊤ :=\n  SetLike.coe_injective Set.range_id\n\n@[simp]\ntheorem map_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R A).map f = NonUnitalAlgHom.range f :=\n  SetLike.coe_injective Set.image_univ\n\n@[simp]\ntheorem map_bot (f : A →ₙₐ[R] B) : (⊥ : NonUnitalSubalgebra R A).map f = ⊥ :=\n  SetLike.coe_injective <| by simp [NonUnitalAlgebra.coe_bot, NonUnitalSubalgebra.coe_map]\n\n@[simp]\ntheorem comap_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R B).comap f = ⊤ :=\n  eq_top_iff.2 fun _ => mem_top\n\n/-- `NonUnitalAlgHom` to `⊤ : NonUnitalSubalgebra R A`. -/\ndef toTop : A →ₙₐ[R] (⊤ : NonUnitalSubalgebra R A) :=\n  NonUnitalAlgHom.codRestrict (NonUnitalAlgHom.id R A) ⊤ fun _ => mem_top\n\nend NonUnitalAlgebra\n\nnamespace NonUnitalSubalgebra\n\nopen NonUnitalAlgebra\n\nsection NonAssoc\n\nvariable {R : Type u} {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable (S : NonUnitalSubalgebra R A)\n\ninstance subsingleton_of_subsingleton [Subsingleton A] : Subsingleton (NonUnitalSubalgebra R A) :=\n  ⟨fun B C => ext fun x => by simp only [Subsingleton.elim x 0, zero_mem B, zero_mem C]⟩\n\ninstance _root_.NonUnitalAlgHom.subsingleton [Subsingleton (NonUnitalSubalgebra R A)] :\n    Subsingleton (A →ₙₐ[R] B) :=\n  ⟨fun f g =>\n    NonUnitalAlgHom.ext fun a =>\n      have : a ∈ (⊥ : NonUnitalSubalgebra R A) :=\n        Subsingleton.elim (⊤ : NonUnitalSubalgebra R A) ⊥ ▸ mem_top\n      (mem_bot.mp this).symm ▸ (map_zero f).trans (map_zero g).symm⟩\n\ntheorem range_val : NonUnitalAlgHom.range (NonUnitalSubalgebraClass.subtype S) = S :=\n  ext <| Set.ext_iff.1 <| (NonUnitalSubalgebraClass.subtype S).coe_range.trans Subtype.range_val\n\n/-- The map `S → T` when `S` is a non-unital subalgebra contained in the non-unital subalgebra `T`.\n\nThis is the non-unital subalgebra version of `Submodule.inclusion`, or `Subring.inclusion`  -/\ndef inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) : S →ₙₐ[R] T\n    where\n  toFun := Set.inclusion h\n  map_add' _ _ := rfl\n  map_mul' _ _ := rfl\n  map_zero' := rfl\n  map_smul' _ _ := rfl\n\ntheorem inclusion_injective {S T : NonUnitalSubalgebra R A} (h : S ≤ T) :\n    Function.Injective (inclusion h) := fun _ _ => Subtype.ext ∘ Subtype.mk.inj\n\n@[simp]\ntheorem inclusion_self {S : NonUnitalSubalgebra R A} :\n    inclusion (le_refl S) = NonUnitalAlgHom.id R S :=\n  NonUnitalAlgHom.ext fun _ => Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_mk {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : A) (hx : x ∈ S) :\n    inclusion h ⟨x, hx⟩ = ⟨x, h hx⟩ :=\n  rfl\n\ntheorem inclusion_right {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : T) (m : (x : A) ∈ S) :\n    inclusion h ⟨x, m⟩ = x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_inclusion {S T U : NonUnitalSubalgebra R A} (hst : S ≤ T) (htu : T ≤ U) (x : S) :\n    inclusion htu (inclusion hst x) = inclusion (le_trans hst htu) x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem coe_inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (s : S) :\n    (inclusion h s : A) = s :=\n  rfl\n\nsection Prod\n\nvariable (S₁ : NonUnitalSubalgebra R B)\n\n/-- The product of two non-unital subalgebras is a non-unital subalgebra. -/\ndef prod : NonUnitalSubalgebra R (A × B) :=\n  { S.toNonUnitalSubsemiring.prod S₁.toNonUnitalSubsemiring with\n    carrier := S ×ˢ S₁\n    smul_mem' := fun r _x hx => ⟨SMulMemClass.smul_mem r hx.1, SMulMemClass.smul_mem r hx.2⟩ }\n\n@[simp]\ntheorem coe_prod : (prod S S₁ : Set (A × B)) = (S : Set A) ×ˢ S₁ :=\n  rfl\n\ntheorem prod_toSubmodule : (S.prod S₁).toSubmodule = S.toSubmodule.prod S₁.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem mem_prod {S : NonUnitalSubalgebra R A} {S₁ : NonUnitalSubalgebra R B} {x : A × B} :\n    x ∈ prod S S₁ ↔ x.1 ∈ S ∧ x.2 ∈ S₁ :=\n  Set.mem_prod\n\n@[simp]\ntheorem prod_top : (prod ⊤ ⊤ : NonUnitalSubalgebra R (A × B)) = ⊤ := by ext; simp\n\ntheorem prod_mono {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S ≤ T → S₁ ≤ T₁ → prod S S₁ ≤ prod T T₁ :=\n  Set.prod_mono\n\n@[simp]\ntheorem prod_inf_prod {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S.prod S₁ ⊓ T.prod T₁ = (S ⊓ T).prod (S₁ ⊓ T₁) :=\n  SetLike.coe_injective Set.prod_inter_prod\n\nend Prod\n\nsection SuprLift\n\nvariable {ι : Type*}\n\ntheorem coe_iSup_of_directed [Nonempty ι] {S : ι → NonUnitalSubalgebra R A}\n    (dir : Directed (· ≤ ·) S) : ↑(iSup S) = ⋃ i, (S i : Set A) :=\n  let K : NonUnitalSubalgebra R A :=\n    { __ := NonUnitalSubsemiring.copy _ _ (NonUnitalSubsemiring.coe_iSup_of_directed dir).symm\n      smul_mem' := fun r _x hx ↦\n        let ⟨i, hi⟩ := Set.mem_iUnion.1 hx\n        Set.mem_iUnion.2 ⟨i, (S i).smul_mem' r hi⟩ }\n  have : iSup S = K := le_antisymm\n    (iSup_le fun i ↦ le_iSup (fun i ↦ (S i : Set A)) i) (Set.iUnion_subset fun _ ↦ le_iSup S _)\n  this.symm ▸ rfl\n\n/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              rw [hf j k hjk]\n              rfl)\n            (↑(iSup K)) (by rw [coe_iSup_of_directed dir])\n        map_zero' := by\n          dsimp\n          exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl) _ (by simp)\n        map_mul' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· * ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_add' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· + ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_smul' := fun r => by\n          dsimp\n          apply Set.iUnionLift_unary (coe_iSup_of_directed dir) _ (fun _ x => r • x)\n            (fun _ _ => rfl)\n          on_goal 2 => rw [coe_iSup_of_directed dir]\n          all_goals simp }\n\nvariable [Nonempty ι] {K : ι → NonUnitalSubalgebra R A} {dir : Directed (· ≤ ·) K}\n  {f : ∀ i, K i →ₙₐ[R] B} {hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h)}\n  {T : NonUnitalSubalgebra R A} {hT : T = iSup K}\n\n@[simp]\ntheorem iSupLift_inclusion {i : ι} (x : K i) (h : K i ≤ T) :\n    iSupLift K dir f hf T hT (inclusion h x) = f i x := by\n  subst T\n  dsimp [iSupLift]\n  apply Set.iUnionLift_inclusion\n  exact h\n\n@[simp]\ntheorem iSupLift_comp_inclusion {i : ι} (h : K i ≤ T) :\n    (iSupLift K dir f hf T hT).comp (inclusion h) = f i := by ","nextTactic":"ext","declUpToTactic":"@[simp]\ntheorem iSupLift_comp_inclusion {i : ι} (h : K i ≤ T) :\n    (iSupLift K dir f hf T hT).comp (inclusion h) = f i := by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.941_0.jQvdAcOcyVirh23","decl":"@[simp]\ntheorem iSupLift_comp_inclusion {i : ι} (h : K i ≤ T) :\n    (iSupLift K dir f hf T hT).comp (inclusion h) = f i "}
-{"state":"case h\nR : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\nS : NonUnitalSubalgebra R A\nι : Type u_1\ninst✝ : Nonempty ι\nK : ι → NonUnitalSubalgebra R A\ndir : Directed (fun x x_1 => x ≤ x_1) K\nf : (i : ι) → ↥(K i) →ₙₐ[R] B\nhf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalAlgHom.comp (f j) (inclusion h)\nT : NonUnitalSubalgebra R A\nhT : T = iSup K\ni : ι\nh : K i ≤ T\nx✝ : ↥(K i)\n⊢ (NonUnitalAlgHom.comp (iSupLift K dir f hf T hT) (inclusion h)) x✝ = (f i) x✝","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact NonUnitalAlgebra.gc\n\n@[simp]\ntheorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=\n  eq_top_iff.2 fun _x hx => subset_adjoin R hx\n\nvariable {R A}\n\n@[simp]\ntheorem coe_top : (↑(⊤ : NonUnitalSubalgebra R A) : Set A) = Set.univ :=\n  rfl\n\n@[simp]\ntheorem mem_top {x : A} : x ∈ (⊤ : NonUnitalSubalgebra R A) :=\n  Set.mem_univ x\n\n@[simp]\ntheorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toNonUnitalSubsemiring : (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubsemiring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toSubring {R A : Type*} [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    [IsScalarTower R A A] [SMulCommClass R A A] :\n    (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem toSubmodule_eq_top {S : NonUnitalSubalgebra R A} : S.toSubmodule = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toSubmodule'.injective.eq_iff' top_toSubmodule\n\n@[simp]\ntheorem toNonUnitalSubsemiring_eq_top {S : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubsemiring_injective.eq_iff' top_toNonUnitalSubsemiring\n\n@[simp]\ntheorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubring_injective.eq_iff' top_toSubring\n\ntheorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_left\n\ntheorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_right\n\ntheorem mul_mem_sup {S T : NonUnitalSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) :\n    x * y ∈ S ⊔ T :=\n  mul_mem (mem_sup_left hx) (mem_sup_right hy)\n\ntheorem map_sup (f : F) (S T : NonUnitalSubalgebra R A) :\n    ((S ⊔ T).map f : NonUnitalSubalgebra R B) = S.map f ⊔ T.map f :=\n  (@NonUnitalSubalgebra.gc_map_comap F R A B _ _ _ _ _ _ f).l_sup\n\n@[simp, norm_cast]\ntheorem coe_inf (S T : NonUnitalSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T :=\n  rfl\n\n@[simp]\ntheorem mem_inf {S T : NonUnitalSubalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T :=\n  Iff.rfl\n\n@[simp]\ntheorem inf_toSubmodule (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toSubmodule = S.toSubmodule ⊓ T.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem inf_toNonUnitalSubsemiring (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring ⊓ T.toNonUnitalSubsemiring :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sInf (S : Set (NonUnitalSubalgebra R A)) : (↑(sInf S) : Set A) = ⋂ s ∈ S, ↑s :=\n  sInf_image\n\ntheorem mem_sInf {S : Set (NonUnitalSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p := by\n  simp only [← SetLike.mem_coe, coe_sInf, Set.mem_iInter₂]\n\n@[simp]\ntheorem sInf_toSubmodule (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toSubmodule = sInf (NonUnitalSubalgebra.toSubmodule '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp]\ntheorem sInf_toNonUnitalSubsemiring (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toNonUnitalSubsemiring = sInf (NonUnitalSubalgebra.toNonUnitalSubsemiring '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp, norm_cast]\ntheorem coe_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} :\n    (↑(⨅ i, S i) : Set A) = ⋂ i, S i := by simp [iInf]\n\ntheorem mem_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} {x : A} :\n    (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by simp only [iInf, mem_sInf, Set.forall_range_iff]\n\n@[simp]\ntheorem iInf_toSubmodule {ι : Sort*} (S : ι → NonUnitalSubalgebra R A) :\n    (⨅ i, S i).toSubmodule = ⨅ i, (S i).toSubmodule :=\n  SetLike.coe_injective <| by simp\n\ninstance : Inhabited (NonUnitalSubalgebra R A) :=\n  ⟨⊥⟩\n\ntheorem mem_bot {x : A} : x ∈ (⊥ : NonUnitalSubalgebra R A) ↔ x = 0 :=\n  show x ∈ Submodule.span R (NonUnitalSubsemiring.closure (∅ : Set A) : Set A) ↔ x = 0 by\n    rw [NonUnitalSubsemiring.closure_empty, NonUnitalSubsemiring.coe_bot,\n      Submodule.span_zero_singleton, Submodule.mem_bot]\n\ntheorem toSubmodule_bot : (⊥ : NonUnitalSubalgebra R A).toSubmodule = ⊥ := by\n  ext\n  simp only [mem_bot, NonUnitalSubalgebra.mem_toSubmodule, Submodule.mem_bot]\n\n@[simp]\ntheorem coe_bot : ((⊥ : NonUnitalSubalgebra R A) : Set A) = {0} := by\n  simp [Set.ext_iff, NonUnitalAlgebra.mem_bot]\n\ntheorem eq_top_iff {S : NonUnitalSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S :=\n  ⟨fun h x => by rw [h]; exact mem_top, fun h => by ext x; exact ⟨fun _ => mem_top, fun _ => h x⟩⟩\n\ntheorem range_top_iff_surjective (f : A →ₙₐ[R] B) :\n    NonUnitalAlgHom.range f = (⊤ : NonUnitalSubalgebra R B) ↔ Function.Surjective f :=\n  NonUnitalAlgebra.eq_top_iff\n\n@[simp]\ntheorem range_id : NonUnitalAlgHom.range (NonUnitalAlgHom.id R A) = ⊤ :=\n  SetLike.coe_injective Set.range_id\n\n@[simp]\ntheorem map_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R A).map f = NonUnitalAlgHom.range f :=\n  SetLike.coe_injective Set.image_univ\n\n@[simp]\ntheorem map_bot (f : A →ₙₐ[R] B) : (⊥ : NonUnitalSubalgebra R A).map f = ⊥ :=\n  SetLike.coe_injective <| by simp [NonUnitalAlgebra.coe_bot, NonUnitalSubalgebra.coe_map]\n\n@[simp]\ntheorem comap_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R B).comap f = ⊤ :=\n  eq_top_iff.2 fun _ => mem_top\n\n/-- `NonUnitalAlgHom` to `⊤ : NonUnitalSubalgebra R A`. -/\ndef toTop : A →ₙₐ[R] (⊤ : NonUnitalSubalgebra R A) :=\n  NonUnitalAlgHom.codRestrict (NonUnitalAlgHom.id R A) ⊤ fun _ => mem_top\n\nend NonUnitalAlgebra\n\nnamespace NonUnitalSubalgebra\n\nopen NonUnitalAlgebra\n\nsection NonAssoc\n\nvariable {R : Type u} {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable (S : NonUnitalSubalgebra R A)\n\ninstance subsingleton_of_subsingleton [Subsingleton A] : Subsingleton (NonUnitalSubalgebra R A) :=\n  ⟨fun B C => ext fun x => by simp only [Subsingleton.elim x 0, zero_mem B, zero_mem C]⟩\n\ninstance _root_.NonUnitalAlgHom.subsingleton [Subsingleton (NonUnitalSubalgebra R A)] :\n    Subsingleton (A →ₙₐ[R] B) :=\n  ⟨fun f g =>\n    NonUnitalAlgHom.ext fun a =>\n      have : a ∈ (⊥ : NonUnitalSubalgebra R A) :=\n        Subsingleton.elim (⊤ : NonUnitalSubalgebra R A) ⊥ ▸ mem_top\n      (mem_bot.mp this).symm ▸ (map_zero f).trans (map_zero g).symm⟩\n\ntheorem range_val : NonUnitalAlgHom.range (NonUnitalSubalgebraClass.subtype S) = S :=\n  ext <| Set.ext_iff.1 <| (NonUnitalSubalgebraClass.subtype S).coe_range.trans Subtype.range_val\n\n/-- The map `S → T` when `S` is a non-unital subalgebra contained in the non-unital subalgebra `T`.\n\nThis is the non-unital subalgebra version of `Submodule.inclusion`, or `Subring.inclusion`  -/\ndef inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) : S →ₙₐ[R] T\n    where\n  toFun := Set.inclusion h\n  map_add' _ _ := rfl\n  map_mul' _ _ := rfl\n  map_zero' := rfl\n  map_smul' _ _ := rfl\n\ntheorem inclusion_injective {S T : NonUnitalSubalgebra R A} (h : S ≤ T) :\n    Function.Injective (inclusion h) := fun _ _ => Subtype.ext ∘ Subtype.mk.inj\n\n@[simp]\ntheorem inclusion_self {S : NonUnitalSubalgebra R A} :\n    inclusion (le_refl S) = NonUnitalAlgHom.id R S :=\n  NonUnitalAlgHom.ext fun _ => Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_mk {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : A) (hx : x ∈ S) :\n    inclusion h ⟨x, hx⟩ = ⟨x, h hx⟩ :=\n  rfl\n\ntheorem inclusion_right {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : T) (m : (x : A) ∈ S) :\n    inclusion h ⟨x, m⟩ = x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_inclusion {S T U : NonUnitalSubalgebra R A} (hst : S ≤ T) (htu : T ≤ U) (x : S) :\n    inclusion htu (inclusion hst x) = inclusion (le_trans hst htu) x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem coe_inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (s : S) :\n    (inclusion h s : A) = s :=\n  rfl\n\nsection Prod\n\nvariable (S₁ : NonUnitalSubalgebra R B)\n\n/-- The product of two non-unital subalgebras is a non-unital subalgebra. -/\ndef prod : NonUnitalSubalgebra R (A × B) :=\n  { S.toNonUnitalSubsemiring.prod S₁.toNonUnitalSubsemiring with\n    carrier := S ×ˢ S₁\n    smul_mem' := fun r _x hx => ⟨SMulMemClass.smul_mem r hx.1, SMulMemClass.smul_mem r hx.2⟩ }\n\n@[simp]\ntheorem coe_prod : (prod S S₁ : Set (A × B)) = (S : Set A) ×ˢ S₁ :=\n  rfl\n\ntheorem prod_toSubmodule : (S.prod S₁).toSubmodule = S.toSubmodule.prod S₁.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem mem_prod {S : NonUnitalSubalgebra R A} {S₁ : NonUnitalSubalgebra R B} {x : A × B} :\n    x ∈ prod S S₁ ↔ x.1 ∈ S ∧ x.2 ∈ S₁ :=\n  Set.mem_prod\n\n@[simp]\ntheorem prod_top : (prod ⊤ ⊤ : NonUnitalSubalgebra R (A × B)) = ⊤ := by ext; simp\n\ntheorem prod_mono {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S ≤ T → S₁ ≤ T₁ → prod S S₁ ≤ prod T T₁ :=\n  Set.prod_mono\n\n@[simp]\ntheorem prod_inf_prod {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S.prod S₁ ⊓ T.prod T₁ = (S ⊓ T).prod (S₁ ⊓ T₁) :=\n  SetLike.coe_injective Set.prod_inter_prod\n\nend Prod\n\nsection SuprLift\n\nvariable {ι : Type*}\n\ntheorem coe_iSup_of_directed [Nonempty ι] {S : ι → NonUnitalSubalgebra R A}\n    (dir : Directed (· ≤ ·) S) : ↑(iSup S) = ⋃ i, (S i : Set A) :=\n  let K : NonUnitalSubalgebra R A :=\n    { __ := NonUnitalSubsemiring.copy _ _ (NonUnitalSubsemiring.coe_iSup_of_directed dir).symm\n      smul_mem' := fun r _x hx ↦\n        let ⟨i, hi⟩ := Set.mem_iUnion.1 hx\n        Set.mem_iUnion.2 ⟨i, (S i).smul_mem' r hi⟩ }\n  have : iSup S = K := le_antisymm\n    (iSup_le fun i ↦ le_iSup (fun i ↦ (S i : Set A)) i) (Set.iUnion_subset fun _ ↦ le_iSup S _)\n  this.symm ▸ rfl\n\n/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              rw [hf j k hjk]\n              rfl)\n            (↑(iSup K)) (by rw [coe_iSup_of_directed dir])\n        map_zero' := by\n          dsimp\n          exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl) _ (by simp)\n        map_mul' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· * ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_add' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· + ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_smul' := fun r => by\n          dsimp\n          apply Set.iUnionLift_unary (coe_iSup_of_directed dir) _ (fun _ x => r • x)\n            (fun _ _ => rfl)\n          on_goal 2 => rw [coe_iSup_of_directed dir]\n          all_goals simp }\n\nvariable [Nonempty ι] {K : ι → NonUnitalSubalgebra R A} {dir : Directed (· ≤ ·) K}\n  {f : ∀ i, K i →ₙₐ[R] B} {hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h)}\n  {T : NonUnitalSubalgebra R A} {hT : T = iSup K}\n\n@[simp]\ntheorem iSupLift_inclusion {i : ι} (x : K i) (h : K i ≤ T) :\n    iSupLift K dir f hf T hT (inclusion h x) = f i x := by\n  subst T\n  dsimp [iSupLift]\n  apply Set.iUnionLift_inclusion\n  exact h\n\n@[simp]\ntheorem iSupLift_comp_inclusion {i : ι} (h : K i ≤ T) :\n    (iSupLift K dir f hf T hT).comp (inclusion h) = f i := by ext; ","nextTactic":"simp","declUpToTactic":"@[simp]\ntheorem iSupLift_comp_inclusion {i : ι} (h : K i ≤ T) :\n    (iSupLift K dir f hf T hT).comp (inclusion h) = f i := by ext; ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.941_0.jQvdAcOcyVirh23","decl":"@[simp]\ntheorem iSupLift_comp_inclusion {i : ι} (h : K i ≤ T) :\n    (iSupLift K dir f hf T hT).comp (inclusion h) = f i "}
-{"state":"R : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\nS : NonUnitalSubalgebra R A\nι : Type u_1\ninst✝ : Nonempty ι\nK : ι → NonUnitalSubalgebra R A\ndir : Directed (fun x x_1 => x ≤ x_1) K\nf : (i : ι) → ↥(K i) →ₙₐ[R] B\nhf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalAlgHom.comp (f j) (inclusion h)\nT : NonUnitalSubalgebra R A\nhT : T = iSup K\ni : ι\nx : ↥(K i)\nhx : ↑x ∈ T\n⊢ (iSupLift K dir f hf T hT) { val := ↑x, property := hx } = (f i) x","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact NonUnitalAlgebra.gc\n\n@[simp]\ntheorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=\n  eq_top_iff.2 fun _x hx => subset_adjoin R hx\n\nvariable {R A}\n\n@[simp]\ntheorem coe_top : (↑(⊤ : NonUnitalSubalgebra R A) : Set A) = Set.univ :=\n  rfl\n\n@[simp]\ntheorem mem_top {x : A} : x ∈ (⊤ : NonUnitalSubalgebra R A) :=\n  Set.mem_univ x\n\n@[simp]\ntheorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toNonUnitalSubsemiring : (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubsemiring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toSubring {R A : Type*} [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    [IsScalarTower R A A] [SMulCommClass R A A] :\n    (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem toSubmodule_eq_top {S : NonUnitalSubalgebra R A} : S.toSubmodule = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toSubmodule'.injective.eq_iff' top_toSubmodule\n\n@[simp]\ntheorem toNonUnitalSubsemiring_eq_top {S : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubsemiring_injective.eq_iff' top_toNonUnitalSubsemiring\n\n@[simp]\ntheorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubring_injective.eq_iff' top_toSubring\n\ntheorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_left\n\ntheorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_right\n\ntheorem mul_mem_sup {S T : NonUnitalSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) :\n    x * y ∈ S ⊔ T :=\n  mul_mem (mem_sup_left hx) (mem_sup_right hy)\n\ntheorem map_sup (f : F) (S T : NonUnitalSubalgebra R A) :\n    ((S ⊔ T).map f : NonUnitalSubalgebra R B) = S.map f ⊔ T.map f :=\n  (@NonUnitalSubalgebra.gc_map_comap F R A B _ _ _ _ _ _ f).l_sup\n\n@[simp, norm_cast]\ntheorem coe_inf (S T : NonUnitalSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T :=\n  rfl\n\n@[simp]\ntheorem mem_inf {S T : NonUnitalSubalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T :=\n  Iff.rfl\n\n@[simp]\ntheorem inf_toSubmodule (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toSubmodule = S.toSubmodule ⊓ T.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem inf_toNonUnitalSubsemiring (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring ⊓ T.toNonUnitalSubsemiring :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sInf (S : Set (NonUnitalSubalgebra R A)) : (↑(sInf S) : Set A) = ⋂ s ∈ S, ↑s :=\n  sInf_image\n\ntheorem mem_sInf {S : Set (NonUnitalSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p := by\n  simp only [← SetLike.mem_coe, coe_sInf, Set.mem_iInter₂]\n\n@[simp]\ntheorem sInf_toSubmodule (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toSubmodule = sInf (NonUnitalSubalgebra.toSubmodule '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp]\ntheorem sInf_toNonUnitalSubsemiring (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toNonUnitalSubsemiring = sInf (NonUnitalSubalgebra.toNonUnitalSubsemiring '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp, norm_cast]\ntheorem coe_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} :\n    (↑(⨅ i, S i) : Set A) = ⋂ i, S i := by simp [iInf]\n\ntheorem mem_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} {x : A} :\n    (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by simp only [iInf, mem_sInf, Set.forall_range_iff]\n\n@[simp]\ntheorem iInf_toSubmodule {ι : Sort*} (S : ι → NonUnitalSubalgebra R A) :\n    (⨅ i, S i).toSubmodule = ⨅ i, (S i).toSubmodule :=\n  SetLike.coe_injective <| by simp\n\ninstance : Inhabited (NonUnitalSubalgebra R A) :=\n  ⟨⊥⟩\n\ntheorem mem_bot {x : A} : x ∈ (⊥ : NonUnitalSubalgebra R A) ↔ x = 0 :=\n  show x ∈ Submodule.span R (NonUnitalSubsemiring.closure (∅ : Set A) : Set A) ↔ x = 0 by\n    rw [NonUnitalSubsemiring.closure_empty, NonUnitalSubsemiring.coe_bot,\n      Submodule.span_zero_singleton, Submodule.mem_bot]\n\ntheorem toSubmodule_bot : (⊥ : NonUnitalSubalgebra R A).toSubmodule = ⊥ := by\n  ext\n  simp only [mem_bot, NonUnitalSubalgebra.mem_toSubmodule, Submodule.mem_bot]\n\n@[simp]\ntheorem coe_bot : ((⊥ : NonUnitalSubalgebra R A) : Set A) = {0} := by\n  simp [Set.ext_iff, NonUnitalAlgebra.mem_bot]\n\ntheorem eq_top_iff {S : NonUnitalSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S :=\n  ⟨fun h x => by rw [h]; exact mem_top, fun h => by ext x; exact ⟨fun _ => mem_top, fun _ => h x⟩⟩\n\ntheorem range_top_iff_surjective (f : A →ₙₐ[R] B) :\n    NonUnitalAlgHom.range f = (⊤ : NonUnitalSubalgebra R B) ↔ Function.Surjective f :=\n  NonUnitalAlgebra.eq_top_iff\n\n@[simp]\ntheorem range_id : NonUnitalAlgHom.range (NonUnitalAlgHom.id R A) = ⊤ :=\n  SetLike.coe_injective Set.range_id\n\n@[simp]\ntheorem map_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R A).map f = NonUnitalAlgHom.range f :=\n  SetLike.coe_injective Set.image_univ\n\n@[simp]\ntheorem map_bot (f : A →ₙₐ[R] B) : (⊥ : NonUnitalSubalgebra R A).map f = ⊥ :=\n  SetLike.coe_injective <| by simp [NonUnitalAlgebra.coe_bot, NonUnitalSubalgebra.coe_map]\n\n@[simp]\ntheorem comap_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R B).comap f = ⊤ :=\n  eq_top_iff.2 fun _ => mem_top\n\n/-- `NonUnitalAlgHom` to `⊤ : NonUnitalSubalgebra R A`. -/\ndef toTop : A →ₙₐ[R] (⊤ : NonUnitalSubalgebra R A) :=\n  NonUnitalAlgHom.codRestrict (NonUnitalAlgHom.id R A) ⊤ fun _ => mem_top\n\nend NonUnitalAlgebra\n\nnamespace NonUnitalSubalgebra\n\nopen NonUnitalAlgebra\n\nsection NonAssoc\n\nvariable {R : Type u} {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable (S : NonUnitalSubalgebra R A)\n\ninstance subsingleton_of_subsingleton [Subsingleton A] : Subsingleton (NonUnitalSubalgebra R A) :=\n  ⟨fun B C => ext fun x => by simp only [Subsingleton.elim x 0, zero_mem B, zero_mem C]⟩\n\ninstance _root_.NonUnitalAlgHom.subsingleton [Subsingleton (NonUnitalSubalgebra R A)] :\n    Subsingleton (A →ₙₐ[R] B) :=\n  ⟨fun f g =>\n    NonUnitalAlgHom.ext fun a =>\n      have : a ∈ (⊥ : NonUnitalSubalgebra R A) :=\n        Subsingleton.elim (⊤ : NonUnitalSubalgebra R A) ⊥ ▸ mem_top\n      (mem_bot.mp this).symm ▸ (map_zero f).trans (map_zero g).symm⟩\n\ntheorem range_val : NonUnitalAlgHom.range (NonUnitalSubalgebraClass.subtype S) = S :=\n  ext <| Set.ext_iff.1 <| (NonUnitalSubalgebraClass.subtype S).coe_range.trans Subtype.range_val\n\n/-- The map `S → T` when `S` is a non-unital subalgebra contained in the non-unital subalgebra `T`.\n\nThis is the non-unital subalgebra version of `Submodule.inclusion`, or `Subring.inclusion`  -/\ndef inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) : S →ₙₐ[R] T\n    where\n  toFun := Set.inclusion h\n  map_add' _ _ := rfl\n  map_mul' _ _ := rfl\n  map_zero' := rfl\n  map_smul' _ _ := rfl\n\ntheorem inclusion_injective {S T : NonUnitalSubalgebra R A} (h : S ≤ T) :\n    Function.Injective (inclusion h) := fun _ _ => Subtype.ext ∘ Subtype.mk.inj\n\n@[simp]\ntheorem inclusion_self {S : NonUnitalSubalgebra R A} :\n    inclusion (le_refl S) = NonUnitalAlgHom.id R S :=\n  NonUnitalAlgHom.ext fun _ => Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_mk {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : A) (hx : x ∈ S) :\n    inclusion h ⟨x, hx⟩ = ⟨x, h hx⟩ :=\n  rfl\n\ntheorem inclusion_right {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : T) (m : (x : A) ∈ S) :\n    inclusion h ⟨x, m⟩ = x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_inclusion {S T U : NonUnitalSubalgebra R A} (hst : S ≤ T) (htu : T ≤ U) (x : S) :\n    inclusion htu (inclusion hst x) = inclusion (le_trans hst htu) x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem coe_inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (s : S) :\n    (inclusion h s : A) = s :=\n  rfl\n\nsection Prod\n\nvariable (S₁ : NonUnitalSubalgebra R B)\n\n/-- The product of two non-unital subalgebras is a non-unital subalgebra. -/\ndef prod : NonUnitalSubalgebra R (A × B) :=\n  { S.toNonUnitalSubsemiring.prod S₁.toNonUnitalSubsemiring with\n    carrier := S ×ˢ S₁\n    smul_mem' := fun r _x hx => ⟨SMulMemClass.smul_mem r hx.1, SMulMemClass.smul_mem r hx.2⟩ }\n\n@[simp]\ntheorem coe_prod : (prod S S₁ : Set (A × B)) = (S : Set A) ×ˢ S₁ :=\n  rfl\n\ntheorem prod_toSubmodule : (S.prod S₁).toSubmodule = S.toSubmodule.prod S₁.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem mem_prod {S : NonUnitalSubalgebra R A} {S₁ : NonUnitalSubalgebra R B} {x : A × B} :\n    x ∈ prod S S₁ ↔ x.1 ∈ S ∧ x.2 ∈ S₁ :=\n  Set.mem_prod\n\n@[simp]\ntheorem prod_top : (prod ⊤ ⊤ : NonUnitalSubalgebra R (A × B)) = ⊤ := by ext; simp\n\ntheorem prod_mono {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S ≤ T → S₁ ≤ T₁ → prod S S₁ ≤ prod T T₁ :=\n  Set.prod_mono\n\n@[simp]\ntheorem prod_inf_prod {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S.prod S₁ ⊓ T.prod T₁ = (S ⊓ T).prod (S₁ ⊓ T₁) :=\n  SetLike.coe_injective Set.prod_inter_prod\n\nend Prod\n\nsection SuprLift\n\nvariable {ι : Type*}\n\ntheorem coe_iSup_of_directed [Nonempty ι] {S : ι → NonUnitalSubalgebra R A}\n    (dir : Directed (· ≤ ·) S) : ↑(iSup S) = ⋃ i, (S i : Set A) :=\n  let K : NonUnitalSubalgebra R A :=\n    { __ := NonUnitalSubsemiring.copy _ _ (NonUnitalSubsemiring.coe_iSup_of_directed dir).symm\n      smul_mem' := fun r _x hx ↦\n        let ⟨i, hi⟩ := Set.mem_iUnion.1 hx\n        Set.mem_iUnion.2 ⟨i, (S i).smul_mem' r hi⟩ }\n  have : iSup S = K := le_antisymm\n    (iSup_le fun i ↦ le_iSup (fun i ↦ (S i : Set A)) i) (Set.iUnion_subset fun _ ↦ le_iSup S _)\n  this.symm ▸ rfl\n\n/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              rw [hf j k hjk]\n              rfl)\n            (↑(iSup K)) (by rw [coe_iSup_of_directed dir])\n        map_zero' := by\n          dsimp\n          exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl) _ (by simp)\n        map_mul' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· * ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_add' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· + ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_smul' := fun r => by\n          dsimp\n          apply Set.iUnionLift_unary (coe_iSup_of_directed dir) _ (fun _ x => r • x)\n            (fun _ _ => rfl)\n          on_goal 2 => rw [coe_iSup_of_directed dir]\n          all_goals simp }\n\nvariable [Nonempty ι] {K : ι → NonUnitalSubalgebra R A} {dir : Directed (· ≤ ·) K}\n  {f : ∀ i, K i →ₙₐ[R] B} {hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h)}\n  {T : NonUnitalSubalgebra R A} {hT : T = iSup K}\n\n@[simp]\ntheorem iSupLift_inclusion {i : ι} (x : K i) (h : K i ≤ T) :\n    iSupLift K dir f hf T hT (inclusion h x) = f i x := by\n  subst T\n  dsimp [iSupLift]\n  apply Set.iUnionLift_inclusion\n  exact h\n\n@[simp]\ntheorem iSupLift_comp_inclusion {i : ι} (h : K i ≤ T) :\n    (iSupLift K dir f hf T hT).comp (inclusion h) = f i := by ext; simp\n\n@[simp]\ntheorem iSupLift_mk {i : ι} (x : K i) (hx : (x : A) ∈ T) :\n    iSupLift K dir f hf T hT ⟨x, hx⟩ = f i x := by\n  ","nextTactic":"subst hT","declUpToTactic":"@[simp]\ntheorem iSupLift_mk {i : ι} (x : K i) (hx : (x : A) ∈ T) :\n    iSupLift K dir f hf T hT ⟨x, hx⟩ = f i x := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.945_0.jQvdAcOcyVirh23","decl":"@[simp]\ntheorem iSupLift_mk {i : ι} (x : K i) (hx : (x : A) ∈ T) :\n    iSupLift K dir f hf T hT ⟨x, hx⟩ = f i x "}
-{"state":"R : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\nS : NonUnitalSubalgebra R A\nι : Type u_1\ninst✝ : Nonempty ι\nK : ι → NonUnitalSubalgebra R A\ndir : Directed (fun x x_1 => x ≤ x_1) K\nf : (i : ι) → ↥(K i) →ₙₐ[R] B\nhf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalAlgHom.comp (f j) (inclusion h)\ni : ι\nx : ↥(K i)\nhx : ↑x ∈ iSup K\n⊢ (iSupLift K dir f hf (iSup K) (_ : iSup K = iSup K)) { val := ↑x, property := hx } = (f i) x","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact NonUnitalAlgebra.gc\n\n@[simp]\ntheorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=\n  eq_top_iff.2 fun _x hx => subset_adjoin R hx\n\nvariable {R A}\n\n@[simp]\ntheorem coe_top : (↑(⊤ : NonUnitalSubalgebra R A) : Set A) = Set.univ :=\n  rfl\n\n@[simp]\ntheorem mem_top {x : A} : x ∈ (⊤ : NonUnitalSubalgebra R A) :=\n  Set.mem_univ x\n\n@[simp]\ntheorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toNonUnitalSubsemiring : (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubsemiring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toSubring {R A : Type*} [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    [IsScalarTower R A A] [SMulCommClass R A A] :\n    (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem toSubmodule_eq_top {S : NonUnitalSubalgebra R A} : S.toSubmodule = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toSubmodule'.injective.eq_iff' top_toSubmodule\n\n@[simp]\ntheorem toNonUnitalSubsemiring_eq_top {S : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubsemiring_injective.eq_iff' top_toNonUnitalSubsemiring\n\n@[simp]\ntheorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubring_injective.eq_iff' top_toSubring\n\ntheorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_left\n\ntheorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_right\n\ntheorem mul_mem_sup {S T : NonUnitalSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) :\n    x * y ∈ S ⊔ T :=\n  mul_mem (mem_sup_left hx) (mem_sup_right hy)\n\ntheorem map_sup (f : F) (S T : NonUnitalSubalgebra R A) :\n    ((S ⊔ T).map f : NonUnitalSubalgebra R B) = S.map f ⊔ T.map f :=\n  (@NonUnitalSubalgebra.gc_map_comap F R A B _ _ _ _ _ _ f).l_sup\n\n@[simp, norm_cast]\ntheorem coe_inf (S T : NonUnitalSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T :=\n  rfl\n\n@[simp]\ntheorem mem_inf {S T : NonUnitalSubalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T :=\n  Iff.rfl\n\n@[simp]\ntheorem inf_toSubmodule (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toSubmodule = S.toSubmodule ⊓ T.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem inf_toNonUnitalSubsemiring (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring ⊓ T.toNonUnitalSubsemiring :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sInf (S : Set (NonUnitalSubalgebra R A)) : (↑(sInf S) : Set A) = ⋂ s ∈ S, ↑s :=\n  sInf_image\n\ntheorem mem_sInf {S : Set (NonUnitalSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p := by\n  simp only [← SetLike.mem_coe, coe_sInf, Set.mem_iInter₂]\n\n@[simp]\ntheorem sInf_toSubmodule (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toSubmodule = sInf (NonUnitalSubalgebra.toSubmodule '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp]\ntheorem sInf_toNonUnitalSubsemiring (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toNonUnitalSubsemiring = sInf (NonUnitalSubalgebra.toNonUnitalSubsemiring '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp, norm_cast]\ntheorem coe_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} :\n    (↑(⨅ i, S i) : Set A) = ⋂ i, S i := by simp [iInf]\n\ntheorem mem_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} {x : A} :\n    (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by simp only [iInf, mem_sInf, Set.forall_range_iff]\n\n@[simp]\ntheorem iInf_toSubmodule {ι : Sort*} (S : ι → NonUnitalSubalgebra R A) :\n    (⨅ i, S i).toSubmodule = ⨅ i, (S i).toSubmodule :=\n  SetLike.coe_injective <| by simp\n\ninstance : Inhabited (NonUnitalSubalgebra R A) :=\n  ⟨⊥⟩\n\ntheorem mem_bot {x : A} : x ∈ (⊥ : NonUnitalSubalgebra R A) ↔ x = 0 :=\n  show x ∈ Submodule.span R (NonUnitalSubsemiring.closure (∅ : Set A) : Set A) ↔ x = 0 by\n    rw [NonUnitalSubsemiring.closure_empty, NonUnitalSubsemiring.coe_bot,\n      Submodule.span_zero_singleton, Submodule.mem_bot]\n\ntheorem toSubmodule_bot : (⊥ : NonUnitalSubalgebra R A).toSubmodule = ⊥ := by\n  ext\n  simp only [mem_bot, NonUnitalSubalgebra.mem_toSubmodule, Submodule.mem_bot]\n\n@[simp]\ntheorem coe_bot : ((⊥ : NonUnitalSubalgebra R A) : Set A) = {0} := by\n  simp [Set.ext_iff, NonUnitalAlgebra.mem_bot]\n\ntheorem eq_top_iff {S : NonUnitalSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S :=\n  ⟨fun h x => by rw [h]; exact mem_top, fun h => by ext x; exact ⟨fun _ => mem_top, fun _ => h x⟩⟩\n\ntheorem range_top_iff_surjective (f : A →ₙₐ[R] B) :\n    NonUnitalAlgHom.range f = (⊤ : NonUnitalSubalgebra R B) ↔ Function.Surjective f :=\n  NonUnitalAlgebra.eq_top_iff\n\n@[simp]\ntheorem range_id : NonUnitalAlgHom.range (NonUnitalAlgHom.id R A) = ⊤ :=\n  SetLike.coe_injective Set.range_id\n\n@[simp]\ntheorem map_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R A).map f = NonUnitalAlgHom.range f :=\n  SetLike.coe_injective Set.image_univ\n\n@[simp]\ntheorem map_bot (f : A →ₙₐ[R] B) : (⊥ : NonUnitalSubalgebra R A).map f = ⊥ :=\n  SetLike.coe_injective <| by simp [NonUnitalAlgebra.coe_bot, NonUnitalSubalgebra.coe_map]\n\n@[simp]\ntheorem comap_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R B).comap f = ⊤ :=\n  eq_top_iff.2 fun _ => mem_top\n\n/-- `NonUnitalAlgHom` to `⊤ : NonUnitalSubalgebra R A`. -/\ndef toTop : A →ₙₐ[R] (⊤ : NonUnitalSubalgebra R A) :=\n  NonUnitalAlgHom.codRestrict (NonUnitalAlgHom.id R A) ⊤ fun _ => mem_top\n\nend NonUnitalAlgebra\n\nnamespace NonUnitalSubalgebra\n\nopen NonUnitalAlgebra\n\nsection NonAssoc\n\nvariable {R : Type u} {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable (S : NonUnitalSubalgebra R A)\n\ninstance subsingleton_of_subsingleton [Subsingleton A] : Subsingleton (NonUnitalSubalgebra R A) :=\n  ⟨fun B C => ext fun x => by simp only [Subsingleton.elim x 0, zero_mem B, zero_mem C]⟩\n\ninstance _root_.NonUnitalAlgHom.subsingleton [Subsingleton (NonUnitalSubalgebra R A)] :\n    Subsingleton (A →ₙₐ[R] B) :=\n  ⟨fun f g =>\n    NonUnitalAlgHom.ext fun a =>\n      have : a ∈ (⊥ : NonUnitalSubalgebra R A) :=\n        Subsingleton.elim (⊤ : NonUnitalSubalgebra R A) ⊥ ▸ mem_top\n      (mem_bot.mp this).symm ▸ (map_zero f).trans (map_zero g).symm⟩\n\ntheorem range_val : NonUnitalAlgHom.range (NonUnitalSubalgebraClass.subtype S) = S :=\n  ext <| Set.ext_iff.1 <| (NonUnitalSubalgebraClass.subtype S).coe_range.trans Subtype.range_val\n\n/-- The map `S → T` when `S` is a non-unital subalgebra contained in the non-unital subalgebra `T`.\n\nThis is the non-unital subalgebra version of `Submodule.inclusion`, or `Subring.inclusion`  -/\ndef inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) : S →ₙₐ[R] T\n    where\n  toFun := Set.inclusion h\n  map_add' _ _ := rfl\n  map_mul' _ _ := rfl\n  map_zero' := rfl\n  map_smul' _ _ := rfl\n\ntheorem inclusion_injective {S T : NonUnitalSubalgebra R A} (h : S ≤ T) :\n    Function.Injective (inclusion h) := fun _ _ => Subtype.ext ∘ Subtype.mk.inj\n\n@[simp]\ntheorem inclusion_self {S : NonUnitalSubalgebra R A} :\n    inclusion (le_refl S) = NonUnitalAlgHom.id R S :=\n  NonUnitalAlgHom.ext fun _ => Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_mk {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : A) (hx : x ∈ S) :\n    inclusion h ⟨x, hx⟩ = ⟨x, h hx⟩ :=\n  rfl\n\ntheorem inclusion_right {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : T) (m : (x : A) ∈ S) :\n    inclusion h ⟨x, m⟩ = x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_inclusion {S T U : NonUnitalSubalgebra R A} (hst : S ≤ T) (htu : T ≤ U) (x : S) :\n    inclusion htu (inclusion hst x) = inclusion (le_trans hst htu) x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem coe_inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (s : S) :\n    (inclusion h s : A) = s :=\n  rfl\n\nsection Prod\n\nvariable (S₁ : NonUnitalSubalgebra R B)\n\n/-- The product of two non-unital subalgebras is a non-unital subalgebra. -/\ndef prod : NonUnitalSubalgebra R (A × B) :=\n  { S.toNonUnitalSubsemiring.prod S₁.toNonUnitalSubsemiring with\n    carrier := S ×ˢ S₁\n    smul_mem' := fun r _x hx => ⟨SMulMemClass.smul_mem r hx.1, SMulMemClass.smul_mem r hx.2⟩ }\n\n@[simp]\ntheorem coe_prod : (prod S S₁ : Set (A × B)) = (S : Set A) ×ˢ S₁ :=\n  rfl\n\ntheorem prod_toSubmodule : (S.prod S₁).toSubmodule = S.toSubmodule.prod S₁.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem mem_prod {S : NonUnitalSubalgebra R A} {S₁ : NonUnitalSubalgebra R B} {x : A × B} :\n    x ∈ prod S S₁ ↔ x.1 ∈ S ∧ x.2 ∈ S₁ :=\n  Set.mem_prod\n\n@[simp]\ntheorem prod_top : (prod ⊤ ⊤ : NonUnitalSubalgebra R (A × B)) = ⊤ := by ext; simp\n\ntheorem prod_mono {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S ≤ T → S₁ ≤ T₁ → prod S S₁ ≤ prod T T₁ :=\n  Set.prod_mono\n\n@[simp]\ntheorem prod_inf_prod {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S.prod S₁ ⊓ T.prod T₁ = (S ⊓ T).prod (S₁ ⊓ T₁) :=\n  SetLike.coe_injective Set.prod_inter_prod\n\nend Prod\n\nsection SuprLift\n\nvariable {ι : Type*}\n\ntheorem coe_iSup_of_directed [Nonempty ι] {S : ι → NonUnitalSubalgebra R A}\n    (dir : Directed (· ≤ ·) S) : ↑(iSup S) = ⋃ i, (S i : Set A) :=\n  let K : NonUnitalSubalgebra R A :=\n    { __ := NonUnitalSubsemiring.copy _ _ (NonUnitalSubsemiring.coe_iSup_of_directed dir).symm\n      smul_mem' := fun r _x hx ↦\n        let ⟨i, hi⟩ := Set.mem_iUnion.1 hx\n        Set.mem_iUnion.2 ⟨i, (S i).smul_mem' r hi⟩ }\n  have : iSup S = K := le_antisymm\n    (iSup_le fun i ↦ le_iSup (fun i ↦ (S i : Set A)) i) (Set.iUnion_subset fun _ ↦ le_iSup S _)\n  this.symm ▸ rfl\n\n/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              rw [hf j k hjk]\n              rfl)\n            (↑(iSup K)) (by rw [coe_iSup_of_directed dir])\n        map_zero' := by\n          dsimp\n          exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl) _ (by simp)\n        map_mul' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· * ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_add' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· + ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_smul' := fun r => by\n          dsimp\n          apply Set.iUnionLift_unary (coe_iSup_of_directed dir) _ (fun _ x => r • x)\n            (fun _ _ => rfl)\n          on_goal 2 => rw [coe_iSup_of_directed dir]\n          all_goals simp }\n\nvariable [Nonempty ι] {K : ι → NonUnitalSubalgebra R A} {dir : Directed (· ≤ ·) K}\n  {f : ∀ i, K i →ₙₐ[R] B} {hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h)}\n  {T : NonUnitalSubalgebra R A} {hT : T = iSup K}\n\n@[simp]\ntheorem iSupLift_inclusion {i : ι} (x : K i) (h : K i ≤ T) :\n    iSupLift K dir f hf T hT (inclusion h x) = f i x := by\n  subst T\n  dsimp [iSupLift]\n  apply Set.iUnionLift_inclusion\n  exact h\n\n@[simp]\ntheorem iSupLift_comp_inclusion {i : ι} (h : K i ≤ T) :\n    (iSupLift K dir f hf T hT).comp (inclusion h) = f i := by ext; simp\n\n@[simp]\ntheorem iSupLift_mk {i : ι} (x : K i) (hx : (x : A) ∈ T) :\n    iSupLift K dir f hf T hT ⟨x, hx⟩ = f i x := by\n  subst hT\n  ","nextTactic":"dsimp [iSupLift]","declUpToTactic":"@[simp]\ntheorem iSupLift_mk {i : ι} (x : K i) (hx : (x : A) ∈ T) :\n    iSupLift K dir f hf T hT ⟨x, hx⟩ = f i x := by\n  subst hT\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.945_0.jQvdAcOcyVirh23","decl":"@[simp]\ntheorem iSupLift_mk {i : ι} (x : K i) (hx : (x : A) ∈ T) :\n    iSupLift K dir f hf T hT ⟨x, hx⟩ = f i x "}
-{"state":"R : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\nS : NonUnitalSubalgebra R A\nι : Type u_1\ninst✝ : Nonempty ι\nK : ι → NonUnitalSubalgebra R A\ndir : Directed (fun x x_1 => x ≤ x_1) K\nf : (i : ι) → ↥(K i) →ₙₐ[R] B\nhf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalAlgHom.comp (f j) (inclusion h)\ni : ι\nx : ↥(K i)\nhx : ↑x ∈ iSup K\n⊢ Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x)\n      (_ :\n        ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),\n          (fun i x => (f i) x) i { val := x, property := hxi } = (fun i x => (f i) x) j { val := x, property := hxj })\n      ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ i, ↑(K i)) { val := ↑x, property := hx } =\n    (f i) x","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact NonUnitalAlgebra.gc\n\n@[simp]\ntheorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=\n  eq_top_iff.2 fun _x hx => subset_adjoin R hx\n\nvariable {R A}\n\n@[simp]\ntheorem coe_top : (↑(⊤ : NonUnitalSubalgebra R A) : Set A) = Set.univ :=\n  rfl\n\n@[simp]\ntheorem mem_top {x : A} : x ∈ (⊤ : NonUnitalSubalgebra R A) :=\n  Set.mem_univ x\n\n@[simp]\ntheorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toNonUnitalSubsemiring : (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubsemiring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toSubring {R A : Type*} [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    [IsScalarTower R A A] [SMulCommClass R A A] :\n    (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem toSubmodule_eq_top {S : NonUnitalSubalgebra R A} : S.toSubmodule = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toSubmodule'.injective.eq_iff' top_toSubmodule\n\n@[simp]\ntheorem toNonUnitalSubsemiring_eq_top {S : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubsemiring_injective.eq_iff' top_toNonUnitalSubsemiring\n\n@[simp]\ntheorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubring_injective.eq_iff' top_toSubring\n\ntheorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_left\n\ntheorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_right\n\ntheorem mul_mem_sup {S T : NonUnitalSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) :\n    x * y ∈ S ⊔ T :=\n  mul_mem (mem_sup_left hx) (mem_sup_right hy)\n\ntheorem map_sup (f : F) (S T : NonUnitalSubalgebra R A) :\n    ((S ⊔ T).map f : NonUnitalSubalgebra R B) = S.map f ⊔ T.map f :=\n  (@NonUnitalSubalgebra.gc_map_comap F R A B _ _ _ _ _ _ f).l_sup\n\n@[simp, norm_cast]\ntheorem coe_inf (S T : NonUnitalSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T :=\n  rfl\n\n@[simp]\ntheorem mem_inf {S T : NonUnitalSubalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T :=\n  Iff.rfl\n\n@[simp]\ntheorem inf_toSubmodule (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toSubmodule = S.toSubmodule ⊓ T.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem inf_toNonUnitalSubsemiring (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring ⊓ T.toNonUnitalSubsemiring :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sInf (S : Set (NonUnitalSubalgebra R A)) : (↑(sInf S) : Set A) = ⋂ s ∈ S, ↑s :=\n  sInf_image\n\ntheorem mem_sInf {S : Set (NonUnitalSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p := by\n  simp only [← SetLike.mem_coe, coe_sInf, Set.mem_iInter₂]\n\n@[simp]\ntheorem sInf_toSubmodule (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toSubmodule = sInf (NonUnitalSubalgebra.toSubmodule '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp]\ntheorem sInf_toNonUnitalSubsemiring (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toNonUnitalSubsemiring = sInf (NonUnitalSubalgebra.toNonUnitalSubsemiring '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp, norm_cast]\ntheorem coe_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} :\n    (↑(⨅ i, S i) : Set A) = ⋂ i, S i := by simp [iInf]\n\ntheorem mem_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} {x : A} :\n    (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by simp only [iInf, mem_sInf, Set.forall_range_iff]\n\n@[simp]\ntheorem iInf_toSubmodule {ι : Sort*} (S : ι → NonUnitalSubalgebra R A) :\n    (⨅ i, S i).toSubmodule = ⨅ i, (S i).toSubmodule :=\n  SetLike.coe_injective <| by simp\n\ninstance : Inhabited (NonUnitalSubalgebra R A) :=\n  ⟨⊥⟩\n\ntheorem mem_bot {x : A} : x ∈ (⊥ : NonUnitalSubalgebra R A) ↔ x = 0 :=\n  show x ∈ Submodule.span R (NonUnitalSubsemiring.closure (∅ : Set A) : Set A) ↔ x = 0 by\n    rw [NonUnitalSubsemiring.closure_empty, NonUnitalSubsemiring.coe_bot,\n      Submodule.span_zero_singleton, Submodule.mem_bot]\n\ntheorem toSubmodule_bot : (⊥ : NonUnitalSubalgebra R A).toSubmodule = ⊥ := by\n  ext\n  simp only [mem_bot, NonUnitalSubalgebra.mem_toSubmodule, Submodule.mem_bot]\n\n@[simp]\ntheorem coe_bot : ((⊥ : NonUnitalSubalgebra R A) : Set A) = {0} := by\n  simp [Set.ext_iff, NonUnitalAlgebra.mem_bot]\n\ntheorem eq_top_iff {S : NonUnitalSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S :=\n  ⟨fun h x => by rw [h]; exact mem_top, fun h => by ext x; exact ⟨fun _ => mem_top, fun _ => h x⟩⟩\n\ntheorem range_top_iff_surjective (f : A →ₙₐ[R] B) :\n    NonUnitalAlgHom.range f = (⊤ : NonUnitalSubalgebra R B) ↔ Function.Surjective f :=\n  NonUnitalAlgebra.eq_top_iff\n\n@[simp]\ntheorem range_id : NonUnitalAlgHom.range (NonUnitalAlgHom.id R A) = ⊤ :=\n  SetLike.coe_injective Set.range_id\n\n@[simp]\ntheorem map_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R A).map f = NonUnitalAlgHom.range f :=\n  SetLike.coe_injective Set.image_univ\n\n@[simp]\ntheorem map_bot (f : A →ₙₐ[R] B) : (⊥ : NonUnitalSubalgebra R A).map f = ⊥ :=\n  SetLike.coe_injective <| by simp [NonUnitalAlgebra.coe_bot, NonUnitalSubalgebra.coe_map]\n\n@[simp]\ntheorem comap_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R B).comap f = ⊤ :=\n  eq_top_iff.2 fun _ => mem_top\n\n/-- `NonUnitalAlgHom` to `⊤ : NonUnitalSubalgebra R A`. -/\ndef toTop : A →��ₐ[R] (⊤ : NonUnitalSubalgebra R A) :=\n  NonUnitalAlgHom.codRestrict (NonUnitalAlgHom.id R A) ⊤ fun _ => mem_top\n\nend NonUnitalAlgebra\n\nnamespace NonUnitalSubalgebra\n\nopen NonUnitalAlgebra\n\nsection NonAssoc\n\nvariable {R : Type u} {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable (S : NonUnitalSubalgebra R A)\n\ninstance subsingleton_of_subsingleton [Subsingleton A] : Subsingleton (NonUnitalSubalgebra R A) :=\n  ⟨fun B C => ext fun x => by simp only [Subsingleton.elim x 0, zero_mem B, zero_mem C]⟩\n\ninstance _root_.NonUnitalAlgHom.subsingleton [Subsingleton (NonUnitalSubalgebra R A)] :\n    Subsingleton (A →ₙₐ[R] B) :=\n  ⟨fun f g =>\n    NonUnitalAlgHom.ext fun a =>\n      have : a ∈ (⊥ : NonUnitalSubalgebra R A) :=\n        Subsingleton.elim (⊤ : NonUnitalSubalgebra R A) ⊥ ▸ mem_top\n      (mem_bot.mp this).symm ▸ (map_zero f).trans (map_zero g).symm⟩\n\ntheorem range_val : NonUnitalAlgHom.range (NonUnitalSubalgebraClass.subtype S) = S :=\n  ext <| Set.ext_iff.1 <| (NonUnitalSubalgebraClass.subtype S).coe_range.trans Subtype.range_val\n\n/-- The map `S → T` when `S` is a non-unital subalgebra contained in the non-unital subalgebra `T`.\n\nThis is the non-unital subalgebra version of `Submodule.inclusion`, or `Subring.inclusion`  -/\ndef inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) : S →ₙₐ[R] T\n    where\n  toFun := Set.inclusion h\n  map_add' _ _ := rfl\n  map_mul' _ _ := rfl\n  map_zero' := rfl\n  map_smul' _ _ := rfl\n\ntheorem inclusion_injective {S T : NonUnitalSubalgebra R A} (h : S ≤ T) :\n    Function.Injective (inclusion h) := fun _ _ => Subtype.ext ∘ Subtype.mk.inj\n\n@[simp]\ntheorem inclusion_self {S : NonUnitalSubalgebra R A} :\n    inclusion (le_refl S) = NonUnitalAlgHom.id R S :=\n  NonUnitalAlgHom.ext fun _ => Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_mk {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : A) (hx : x ∈ S) :\n    inclusion h ⟨x, hx⟩ = ⟨x, h hx⟩ :=\n  rfl\n\ntheorem inclusion_right {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : T) (m : (x : A) ∈ S) :\n    inclusion h ⟨x, m⟩ = x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_inclusion {S T U : NonUnitalSubalgebra R A} (hst : S ≤ T) (htu : T ≤ U) (x : S) :\n    inclusion htu (inclusion hst x) = inclusion (le_trans hst htu) x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem coe_inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (s : S) :\n    (inclusion h s : A) = s :=\n  rfl\n\nsection Prod\n\nvariable (S₁ : NonUnitalSubalgebra R B)\n\n/-- The product of two non-unital subalgebras is a non-unital subalgebra. -/\ndef prod : NonUnitalSubalgebra R (A × B) :=\n  { S.toNonUnitalSubsemiring.prod S₁.toNonUnitalSubsemiring with\n    carrier := S ×ˢ S₁\n    smul_mem' := fun r _x hx => ⟨SMulMemClass.smul_mem r hx.1, SMulMemClass.smul_mem r hx.2⟩ }\n\n@[simp]\ntheorem coe_prod : (prod S S₁ : Set (A × B)) = (S : Set A) ×ˢ S₁ :=\n  rfl\n\ntheorem prod_toSubmodule : (S.prod S₁).toSubmodule = S.toSubmodule.prod S₁.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem mem_prod {S : NonUnitalSubalgebra R A} {S₁ : NonUnitalSubalgebra R B} {x : A × B} :\n    x ∈ prod S S₁ ↔ x.1 ∈ S ∧ x.2 ∈ S₁ :=\n  Set.mem_prod\n\n@[simp]\ntheorem prod_top : (prod ⊤ ⊤ : NonUnitalSubalgebra R (A × B)) = ⊤ := by ext; simp\n\ntheorem prod_mono {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S ≤ T → S₁ ≤ T₁ → prod S S₁ ≤ prod T T₁ :=\n  Set.prod_mono\n\n@[simp]\ntheorem prod_inf_prod {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S.prod S₁ ⊓ T.prod T₁ = (S ⊓ T).prod (S₁ ⊓ T₁) :=\n  SetLike.coe_injective Set.prod_inter_prod\n\nend Prod\n\nsection SuprLift\n\nvariable {ι : Type*}\n\ntheorem coe_iSup_of_directed [Nonempty ι] {S : ι → NonUnitalSubalgebra R A}\n    (dir : Directed (· ≤ ·) S) : ↑(iSup S) = ⋃ i, (S i : Set A) :=\n  let K : NonUnitalSubalgebra R A :=\n    { __ := NonUnitalSubsemiring.copy _ _ (NonUnitalSubsemiring.coe_iSup_of_directed dir).symm\n      smul_mem' := fun r _x hx ↦\n        let ⟨i, hi⟩ := Set.mem_iUnion.1 hx\n        Set.mem_iUnion.2 ⟨i, (S i).smul_mem' r hi⟩ }\n  have : iSup S = K := le_antisymm\n    (iSup_le fun i ↦ le_iSup (fun i ↦ (S i : Set A)) i) (Set.iUnion_subset fun _ ↦ le_iSup S _)\n  this.symm ▸ rfl\n\n/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              rw [hf j k hjk]\n              rfl)\n            (↑(iSup K)) (by rw [coe_iSup_of_directed dir])\n        map_zero' := by\n          dsimp\n          exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl) _ (by simp)\n        map_mul' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· * ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_add' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· + ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_smul' := fun r => by\n          dsimp\n          apply Set.iUnionLift_unary (coe_iSup_of_directed dir) _ (fun _ x => r • x)\n            (fun _ _ => rfl)\n          on_goal 2 => rw [coe_iSup_of_directed dir]\n          all_goals simp }\n\nvariable [Nonempty ι] {K : ι → NonUnitalSubalgebra R A} {dir : Directed (· ≤ ·) K}\n  {f : ∀ i, K i →ₙₐ[R] B} {hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h)}\n  {T : NonUnitalSubalgebra R A} {hT : T = iSup K}\n\n@[simp]\ntheorem iSupLift_inclusion {i : ι} (x : K i) (h : K i ≤ T) :\n    iSupLift K dir f hf T hT (inclusion h x) = f i x := by\n  subst T\n  dsimp [iSupLift]\n  apply Set.iUnionLift_inclusion\n  exact h\n\n@[simp]\ntheorem iSupLift_comp_inclusion {i : ι} (h : K i ≤ T) :\n    (iSupLift K dir f hf T hT).comp (inclusion h) = f i := by ext; simp\n\n@[simp]\ntheorem iSupLift_mk {i : ι} (x : K i) (hx : (x : A) ∈ T) :\n    iSupLift K dir f hf T hT ⟨x, hx⟩ = f i x := by\n  subst hT\n  dsimp [iSupLift]\n  ","nextTactic":"apply Set.iUnionLift_mk","declUpToTactic":"@[simp]\ntheorem iSupLift_mk {i : ι} (x : K i) (hx : (x : A) ∈ T) :\n    iSupLift K dir f hf T hT ⟨x, hx⟩ = f i x := by\n  subst hT\n  dsimp [iSupLift]\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.945_0.jQvdAcOcyVirh23","decl":"@[simp]\ntheorem iSupLift_mk {i : ι} (x : K i) (hx : (x : A) ∈ T) :\n    iSupLift K dir f hf T hT ⟨x, hx⟩ = f i x "}
-{"state":"R : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\nS : NonUnitalSubalgebra R A\nι : Type u_1\ninst✝ : Nonempty ι\nK : ι → NonUnitalSubalgebra R A\ndir : Directed (fun x x_1 => x ≤ x_1) K\nf : (i : ι) → ↥(K i) →ₙₐ[R] B\nhf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalAlgHom.comp (f j) (inclusion h)\nT : NonUnitalSubalgebra R A\nhT : T = iSup K\ni : ι\nx : ↥T\nhx : ↑x ∈ K i\n⊢ (iSupLift K dir f hf T hT) x = (f i) { val := ↑x, property := hx }","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact NonUnitalAlgebra.gc\n\n@[simp]\ntheorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=\n  eq_top_iff.2 fun _x hx => subset_adjoin R hx\n\nvariable {R A}\n\n@[simp]\ntheorem coe_top : (↑(⊤ : NonUnitalSubalgebra R A) : Set A) = Set.univ :=\n  rfl\n\n@[simp]\ntheorem mem_top {x : A} : x ∈ (⊤ : NonUnitalSubalgebra R A) :=\n  Set.mem_univ x\n\n@[simp]\ntheorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toNonUnitalSubsemiring : (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubsemiring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toSubring {R A : Type*} [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    [IsScalarTower R A A] [SMulCommClass R A A] :\n    (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem toSubmodule_eq_top {S : NonUnitalSubalgebra R A} : S.toSubmodule = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toSubmodule'.injective.eq_iff' top_toSubmodule\n\n@[simp]\ntheorem toNonUnitalSubsemiring_eq_top {S : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubsemiring_injective.eq_iff' top_toNonUnitalSubsemiring\n\n@[simp]\ntheorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubring_injective.eq_iff' top_toSubring\n\ntheorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_left\n\ntheorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_right\n\ntheorem mul_mem_sup {S T : NonUnitalSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) :\n    x * y ∈ S ⊔ T :=\n  mul_mem (mem_sup_left hx) (mem_sup_right hy)\n\ntheorem map_sup (f : F) (S T : NonUnitalSubalgebra R A) :\n    ((S ⊔ T).map f : NonUnitalSubalgebra R B) = S.map f ⊔ T.map f :=\n  (@NonUnitalSubalgebra.gc_map_comap F R A B _ _ _ _ _ _ f).l_sup\n\n@[simp, norm_cast]\ntheorem coe_inf (S T : NonUnitalSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T :=\n  rfl\n\n@[simp]\ntheorem mem_inf {S T : NonUnitalSubalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T :=\n  Iff.rfl\n\n@[simp]\ntheorem inf_toSubmodule (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toSubmodule = S.toSubmodule ⊓ T.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem inf_toNonUnitalSubsemiring (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring ⊓ T.toNonUnitalSubsemiring :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sInf (S : Set (NonUnitalSubalgebra R A)) : (↑(sInf S) : Set A) = ⋂ s ∈ S, ↑s :=\n  sInf_image\n\ntheorem mem_sInf {S : Set (NonUnitalSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p := by\n  simp only [← SetLike.mem_coe, coe_sInf, Set.mem_iInter₂]\n\n@[simp]\ntheorem sInf_toSubmodule (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toSubmodule = sInf (NonUnitalSubalgebra.toSubmodule '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp]\ntheorem sInf_toNonUnitalSubsemiring (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toNonUnitalSubsemiring = sInf (NonUnitalSubalgebra.toNonUnitalSubsemiring '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp, norm_cast]\ntheorem coe_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} :\n    (↑(⨅ i, S i) : Set A) = ⋂ i, S i := by simp [iInf]\n\ntheorem mem_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} {x : A} :\n    (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by simp only [iInf, mem_sInf, Set.forall_range_iff]\n\n@[simp]\ntheorem iInf_toSubmodule {ι : Sort*} (S : ι → NonUnitalSubalgebra R A) :\n    (⨅ i, S i).toSubmodule = ⨅ i, (S i).toSubmodule :=\n  SetLike.coe_injective <| by simp\n\ninstance : Inhabited (NonUnitalSubalgebra R A) :=\n  ⟨⊥⟩\n\ntheorem mem_bot {x : A} : x ∈ (⊥ : NonUnitalSubalgebra R A) ↔ x = 0 :=\n  show x ∈ Submodule.span R (NonUnitalSubsemiring.closure (∅ : Set A) : Set A) ↔ x = 0 by\n    rw [NonUnitalSubsemiring.closure_empty, NonUnitalSubsemiring.coe_bot,\n      Submodule.span_zero_singleton, Submodule.mem_bot]\n\ntheorem toSubmodule_bot : (⊥ : NonUnitalSubalgebra R A).toSubmodule = ⊥ := by\n  ext\n  simp only [mem_bot, NonUnitalSubalgebra.mem_toSubmodule, Submodule.mem_bot]\n\n@[simp]\ntheorem coe_bot : ((⊥ : NonUnitalSubalgebra R A) : Set A) = {0} := by\n  simp [Set.ext_iff, NonUnitalAlgebra.mem_bot]\n\ntheorem eq_top_iff {S : NonUnitalSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S :=\n  ⟨fun h x => by rw [h]; exact mem_top, fun h => by ext x; exact ⟨fun _ => mem_top, fun _ => h x⟩⟩\n\ntheorem range_top_iff_surjective (f : A →ₙₐ[R] B) :\n    NonUnitalAlgHom.range f = (⊤ : NonUnitalSubalgebra R B) ↔ Function.Surjective f :=\n  NonUnitalAlgebra.eq_top_iff\n\n@[simp]\ntheorem range_id : NonUnitalAlgHom.range (NonUnitalAlgHom.id R A) = ⊤ :=\n  SetLike.coe_injective Set.range_id\n\n@[simp]\ntheorem map_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R A).map f = NonUnitalAlgHom.range f :=\n  SetLike.coe_injective Set.image_univ\n\n@[simp]\ntheorem map_bot (f : A →ₙₐ[R] B) : (⊥ : NonUnitalSubalgebra R A).map f = ⊥ :=\n  SetLike.coe_injective <| by simp [NonUnitalAlgebra.coe_bot, NonUnitalSubalgebra.coe_map]\n\n@[simp]\ntheorem comap_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R B).comap f = ⊤ :=\n  eq_top_iff.2 fun _ => mem_top\n\n/-- `NonUnitalAlgHom` to `⊤ : NonUnitalSubalgebra R A`. -/\ndef toTop : A →ₙₐ[R] (⊤ : NonUnitalSubalgebra R A) :=\n  NonUnitalAlgHom.codRestrict (NonUnitalAlgHom.id R A) ⊤ fun _ => mem_top\n\nend NonUnitalAlgebra\n\nnamespace NonUnitalSubalgebra\n\nopen NonUnitalAlgebra\n\nsection NonAssoc\n\nvariable {R : Type u} {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable (S : NonUnitalSubalgebra R A)\n\ninstance subsingleton_of_subsingleton [Subsingleton A] : Subsingleton (NonUnitalSubalgebra R A) :=\n  ⟨fun B C => ext fun x => by simp only [Subsingleton.elim x 0, zero_mem B, zero_mem C]⟩\n\ninstance _root_.NonUnitalAlgHom.subsingleton [Subsingleton (NonUnitalSubalgebra R A)] :\n    Subsingleton (A →ₙₐ[R] B) :=\n  ⟨fun f g =>\n    NonUnitalAlgHom.ext fun a =>\n      have : a ∈ (⊥ : NonUnitalSubalgebra R A) :=\n        Subsingleton.elim (⊤ : NonUnitalSubalgebra R A) ⊥ ▸ mem_top\n      (mem_bot.mp this).symm ▸ (map_zero f).trans (map_zero g).symm⟩\n\ntheorem range_val : NonUnitalAlgHom.range (NonUnitalSubalgebraClass.subtype S) = S :=\n  ext <| Set.ext_iff.1 <| (NonUnitalSubalgebraClass.subtype S).coe_range.trans Subtype.range_val\n\n/-- The map `S → T` when `S` is a non-unital subalgebra contained in the non-unital subalgebra `T`.\n\nThis is the non-unital subalgebra version of `Submodule.inclusion`, or `Subring.inclusion`  -/\ndef inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) : S →ₙₐ[R] T\n    where\n  toFun := Set.inclusion h\n  map_add' _ _ := rfl\n  map_mul' _ _ := rfl\n  map_zero' := rfl\n  map_smul' _ _ := rfl\n\ntheorem inclusion_injective {S T : NonUnitalSubalgebra R A} (h : S ≤ T) :\n    Function.Injective (inclusion h) := fun _ _ => Subtype.ext ∘ Subtype.mk.inj\n\n@[simp]\ntheorem inclusion_self {S : NonUnitalSubalgebra R A} :\n    inclusion (le_refl S) = NonUnitalAlgHom.id R S :=\n  NonUnitalAlgHom.ext fun _ => Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_mk {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : A) (hx : x ∈ S) :\n    inclusion h ⟨x, hx⟩ = ⟨x, h hx⟩ :=\n  rfl\n\ntheorem inclusion_right {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : T) (m : (x : A) ∈ S) :\n    inclusion h ⟨x, m⟩ = x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_inclusion {S T U : NonUnitalSubalgebra R A} (hst : S ≤ T) (htu : T ≤ U) (x : S) :\n    inclusion htu (inclusion hst x) = inclusion (le_trans hst htu) x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem coe_inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (s : S) :\n    (inclusion h s : A) = s :=\n  rfl\n\nsection Prod\n\nvariable (S₁ : NonUnitalSubalgebra R B)\n\n/-- The product of two non-unital subalgebras is a non-unital subalgebra. -/\ndef prod : NonUnitalSubalgebra R (A × B) :=\n  { S.toNonUnitalSubsemiring.prod S₁.toNonUnitalSubsemiring with\n    carrier := S ×ˢ S₁\n    smul_mem' := fun r _x hx => ⟨SMulMemClass.smul_mem r hx.1, SMulMemClass.smul_mem r hx.2⟩ }\n\n@[simp]\ntheorem coe_prod : (prod S S₁ : Set (A × B)) = (S : Set A) ×ˢ S₁ :=\n  rfl\n\ntheorem prod_toSubmodule : (S.prod S₁).toSubmodule = S.toSubmodule.prod S₁.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem mem_prod {S : NonUnitalSubalgebra R A} {S₁ : NonUnitalSubalgebra R B} {x : A × B} :\n    x ∈ prod S S₁ ↔ x.1 ∈ S ∧ x.2 ∈ S₁ :=\n  Set.mem_prod\n\n@[simp]\ntheorem prod_top : (prod ⊤ ⊤ : NonUnitalSubalgebra R (A × B)) = ⊤ := by ext; simp\n\ntheorem prod_mono {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S ≤ T → S₁ ≤ T₁ → prod S S₁ ≤ prod T T₁ :=\n  Set.prod_mono\n\n@[simp]\ntheorem prod_inf_prod {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S.prod S₁ ⊓ T.prod T₁ = (S ⊓ T).prod (S₁ ⊓ T₁) :=\n  SetLike.coe_injective Set.prod_inter_prod\n\nend Prod\n\nsection SuprLift\n\nvariable {ι : Type*}\n\ntheorem coe_iSup_of_directed [Nonempty ι] {S : ι → NonUnitalSubalgebra R A}\n    (dir : Directed (· ≤ ·) S) : ↑(iSup S) = ⋃ i, (S i : Set A) :=\n  let K : NonUnitalSubalgebra R A :=\n    { __ := NonUnitalSubsemiring.copy _ _ (NonUnitalSubsemiring.coe_iSup_of_directed dir).symm\n      smul_mem' := fun r _x hx ↦\n        let ⟨i, hi⟩ := Set.mem_iUnion.1 hx\n        Set.mem_iUnion.2 ⟨i, (S i).smul_mem' r hi⟩ }\n  have : iSup S = K := le_antisymm\n    (iSup_le fun i ↦ le_iSup (fun i ↦ (S i : Set A)) i) (Set.iUnion_subset fun _ ↦ le_iSup S _)\n  this.symm ▸ rfl\n\n/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              rw [hf j k hjk]\n              rfl)\n            (↑(iSup K)) (by rw [coe_iSup_of_directed dir])\n        map_zero' := by\n          dsimp\n          exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl) _ (by simp)\n        map_mul' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· * ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_add' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· + ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_smul' := fun r => by\n          dsimp\n          apply Set.iUnionLift_unary (coe_iSup_of_directed dir) _ (fun _ x => r • x)\n            (fun _ _ => rfl)\n          on_goal 2 => rw [coe_iSup_of_directed dir]\n          all_goals simp }\n\nvariable [Nonempty ι] {K : ι → NonUnitalSubalgebra R A} {dir : Directed (· ≤ ·) K}\n  {f : ∀ i, K i →ₙₐ[R] B} {hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h)}\n  {T : NonUnitalSubalgebra R A} {hT : T = iSup K}\n\n@[simp]\ntheorem iSupLift_inclusion {i : ι} (x : K i) (h : K i ≤ T) :\n    iSupLift K dir f hf T hT (inclusion h x) = f i x := by\n  subst T\n  dsimp [iSupLift]\n  apply Set.iUnionLift_inclusion\n  exact h\n\n@[simp]\ntheorem iSupLift_comp_inclusion {i : ι} (h : K i ≤ T) :\n    (iSupLift K dir f hf T hT).comp (inclusion h) = f i := by ext; simp\n\n@[simp]\ntheorem iSupLift_mk {i : ι} (x : K i) (hx : (x : A) ∈ T) :\n    iSupLift K dir f hf T hT ⟨x, hx⟩ = f i x := by\n  subst hT\n  dsimp [iSupLift]\n  apply Set.iUnionLift_mk\n\ntheorem iSupLift_of_mem {i : ι} (x : T) (hx : (x : A) ∈ K i) :\n    iSupLift K dir f hf T hT x = f i ⟨x, hx⟩ := by\n  ","nextTactic":"subst hT","declUpToTactic":"theorem iSupLift_of_mem {i : ι} (x : T) (hx : (x : A) ∈ K i) :\n    iSupLift K dir f hf T hT x = f i ⟨x, hx⟩ := by\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.952_0.jQvdAcOcyVirh23","decl":"theorem iSupLift_of_mem {i : ι} (x : T) (hx : (x : A) ∈ K i) :\n    iSupLift K dir f hf T hT x = f i ⟨x, hx⟩ "}
-{"state":"R : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\nS : NonUnitalSubalgebra R A\nι : Type u_1\ninst✝ : Nonempty ι\nK : ι → NonUnitalSubalgebra R A\ndir : Directed (fun x x_1 => x ≤ x_1) K\nf : (i : ι) → ↥(K i) →ₙₐ[R] B\nhf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalAlgHom.comp (f j) (inclusion h)\ni : ι\nx : ↥(iSup K)\nhx : ↑x ∈ K i\n⊢ (iSupLift K dir f hf (iSup K) (_ : iSup K = iSup K)) x = (f i) { val := ↑x, property := hx }","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact NonUnitalAlgebra.gc\n\n@[simp]\ntheorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=\n  eq_top_iff.2 fun _x hx => subset_adjoin R hx\n\nvariable {R A}\n\n@[simp]\ntheorem coe_top : (↑(⊤ : NonUnitalSubalgebra R A) : Set A) = Set.univ :=\n  rfl\n\n@[simp]\ntheorem mem_top {x : A} : x ∈ (⊤ : NonUnitalSubalgebra R A) :=\n  Set.mem_univ x\n\n@[simp]\ntheorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toNonUnitalSubsemiring : (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubsemiring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toSubring {R A : Type*} [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    [IsScalarTower R A A] [SMulCommClass R A A] :\n    (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem toSubmodule_eq_top {S : NonUnitalSubalgebra R A} : S.toSubmodule = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toSubmodule'.injective.eq_iff' top_toSubmodule\n\n@[simp]\ntheorem toNonUnitalSubsemiring_eq_top {S : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubsemiring_injective.eq_iff' top_toNonUnitalSubsemiring\n\n@[simp]\ntheorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubring_injective.eq_iff' top_toSubring\n\ntheorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_left\n\ntheorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_right\n\ntheorem mul_mem_sup {S T : NonUnitalSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) :\n    x * y ∈ S ⊔ T :=\n  mul_mem (mem_sup_left hx) (mem_sup_right hy)\n\ntheorem map_sup (f : F) (S T : NonUnitalSubalgebra R A) :\n    ((S ⊔ T).map f : NonUnitalSubalgebra R B) = S.map f ⊔ T.map f :=\n  (@NonUnitalSubalgebra.gc_map_comap F R A B _ _ _ _ _ _ f).l_sup\n\n@[simp, norm_cast]\ntheorem coe_inf (S T : NonUnitalSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T :=\n  rfl\n\n@[simp]\ntheorem mem_inf {S T : NonUnitalSubalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T :=\n  Iff.rfl\n\n@[simp]\ntheorem inf_toSubmodule (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toSubmodule = S.toSubmodule ⊓ T.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem inf_toNonUnitalSubsemiring (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring ⊓ T.toNonUnitalSubsemiring :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sInf (S : Set (NonUnitalSubalgebra R A)) : (↑(sInf S) : Set A) = ⋂ s ∈ S, ↑s :=\n  sInf_image\n\ntheorem mem_sInf {S : Set (NonUnitalSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p := by\n  simp only [← SetLike.mem_coe, coe_sInf, Set.mem_iInter₂]\n\n@[simp]\ntheorem sInf_toSubmodule (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toSubmodule = sInf (NonUnitalSubalgebra.toSubmodule '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp]\ntheorem sInf_toNonUnitalSubsemiring (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toNonUnitalSubsemiring = sInf (NonUnitalSubalgebra.toNonUnitalSubsemiring '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp, norm_cast]\ntheorem coe_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} :\n    (↑(⨅ i, S i) : Set A) = ⋂ i, S i := by simp [iInf]\n\ntheorem mem_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} {x : A} :\n    (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by simp only [iInf, mem_sInf, Set.forall_range_iff]\n\n@[simp]\ntheorem iInf_toSubmodule {ι : Sort*} (S : ι → NonUnitalSubalgebra R A) :\n    (⨅ i, S i).toSubmodule = ⨅ i, (S i).toSubmodule :=\n  SetLike.coe_injective <| by simp\n\ninstance : Inhabited (NonUnitalSubalgebra R A) :=\n  ⟨⊥⟩\n\ntheorem mem_bot {x : A} : x ∈ (⊥ : NonUnitalSubalgebra R A) ↔ x = 0 :=\n  show x ∈ Submodule.span R (NonUnitalSubsemiring.closure (∅ : Set A) : Set A) ↔ x = 0 by\n    rw [NonUnitalSubsemiring.closure_empty, NonUnitalSubsemiring.coe_bot,\n      Submodule.span_zero_singleton, Submodule.mem_bot]\n\ntheorem toSubmodule_bot : (⊥ : NonUnitalSubalgebra R A).toSubmodule = ⊥ := by\n  ext\n  simp only [mem_bot, NonUnitalSubalgebra.mem_toSubmodule, Submodule.mem_bot]\n\n@[simp]\ntheorem coe_bot : ((⊥ : NonUnitalSubalgebra R A) : Set A) = {0} := by\n  simp [Set.ext_iff, NonUnitalAlgebra.mem_bot]\n\ntheorem eq_top_iff {S : NonUnitalSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S :=\n  ⟨fun h x => by rw [h]; exact mem_top, fun h => by ext x; exact ⟨fun _ => mem_top, fun _ => h x⟩⟩\n\ntheorem range_top_iff_surjective (f : A →ₙₐ[R] B) :\n    NonUnitalAlgHom.range f = (⊤ : NonUnitalSubalgebra R B) ↔ Function.Surjective f :=\n  NonUnitalAlgebra.eq_top_iff\n\n@[simp]\ntheorem range_id : NonUnitalAlgHom.range (NonUnitalAlgHom.id R A) = ⊤ :=\n  SetLike.coe_injective Set.range_id\n\n@[simp]\ntheorem map_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R A).map f = NonUnitalAlgHom.range f :=\n  SetLike.coe_injective Set.image_univ\n\n@[simp]\ntheorem map_bot (f : A →ₙₐ[R] B) : (⊥ : NonUnitalSubalgebra R A).map f = ⊥ :=\n  SetLike.coe_injective <| by simp [NonUnitalAlgebra.coe_bot, NonUnitalSubalgebra.coe_map]\n\n@[simp]\ntheorem comap_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R B).comap f = ⊤ :=\n  eq_top_iff.2 fun _ => mem_top\n\n/-- `NonUnitalAlgHom` to `⊤ : NonUnitalSubalgebra R A`. -/\ndef toTop : A →ₙₐ[R] (⊤ : NonUnitalSubalgebra R A) :=\n  NonUnitalAlgHom.codRestrict (NonUnitalAlgHom.id R A) ⊤ fun _ => mem_top\n\nend NonUnitalAlgebra\n\nnamespace NonUnitalSubalgebra\n\nopen NonUnitalAlgebra\n\nsection NonAssoc\n\nvariable {R : Type u} {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable (S : NonUnitalSubalgebra R A)\n\ninstance subsingleton_of_subsingleton [Subsingleton A] : Subsingleton (NonUnitalSubalgebra R A) :=\n  ⟨fun B C => ext fun x => by simp only [Subsingleton.elim x 0, zero_mem B, zero_mem C]⟩\n\ninstance _root_.NonUnitalAlgHom.subsingleton [Subsingleton (NonUnitalSubalgebra R A)] :\n    Subsingleton (A →ₙₐ[R] B) :=\n  ⟨fun f g =>\n    NonUnitalAlgHom.ext fun a =>\n      have : a ∈ (⊥ : NonUnitalSubalgebra R A) :=\n        Subsingleton.elim (⊤ : NonUnitalSubalgebra R A) ⊥ ▸ mem_top\n      (mem_bot.mp this).symm ▸ (map_zero f).trans (map_zero g).symm⟩\n\ntheorem range_val : NonUnitalAlgHom.range (NonUnitalSubalgebraClass.subtype S) = S :=\n  ext <| Set.ext_iff.1 <| (NonUnitalSubalgebraClass.subtype S).coe_range.trans Subtype.range_val\n\n/-- The map `S → T` when `S` is a non-unital subalgebra contained in the non-unital subalgebra `T`.\n\nThis is the non-unital subalgebra version of `Submodule.inclusion`, or `Subring.inclusion`  -/\ndef inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) : S →ₙₐ[R] T\n    where\n  toFun := Set.inclusion h\n  map_add' _ _ := rfl\n  map_mul' _ _ := rfl\n  map_zero' := rfl\n  map_smul' _ _ := rfl\n\ntheorem inclusion_injective {S T : NonUnitalSubalgebra R A} (h : S ≤ T) :\n    Function.Injective (inclusion h) := fun _ _ => Subtype.ext ∘ Subtype.mk.inj\n\n@[simp]\ntheorem inclusion_self {S : NonUnitalSubalgebra R A} :\n    inclusion (le_refl S) = NonUnitalAlgHom.id R S :=\n  NonUnitalAlgHom.ext fun _ => Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_mk {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : A) (hx : x ∈ S) :\n    inclusion h ⟨x, hx⟩ = ⟨x, h hx⟩ :=\n  rfl\n\ntheorem inclusion_right {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : T) (m : (x : A) ∈ S) :\n    inclusion h ⟨x, m⟩ = x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_inclusion {S T U : NonUnitalSubalgebra R A} (hst : S ≤ T) (htu : T ≤ U) (x : S) :\n    inclusion htu (inclusion hst x) = inclusion (le_trans hst htu) x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem coe_inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (s : S) :\n    (inclusion h s : A) = s :=\n  rfl\n\nsection Prod\n\nvariable (S₁ : NonUnitalSubalgebra R B)\n\n/-- The product of two non-unital subalgebras is a non-unital subalgebra. -/\ndef prod : NonUnitalSubalgebra R (A × B) :=\n  { S.toNonUnitalSubsemiring.prod S₁.toNonUnitalSubsemiring with\n    carrier := S ×ˢ S₁\n    smul_mem' := fun r _x hx => ⟨SMulMemClass.smul_mem r hx.1, SMulMemClass.smul_mem r hx.2⟩ }\n\n@[simp]\ntheorem coe_prod : (prod S S₁ : Set (A × B)) = (S : Set A) ×ˢ S₁ :=\n  rfl\n\ntheorem prod_toSubmodule : (S.prod S₁).toSubmodule = S.toSubmodule.prod S₁.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem mem_prod {S : NonUnitalSubalgebra R A} {S₁ : NonUnitalSubalgebra R B} {x : A × B} :\n    x ∈ prod S S₁ ↔ x.1 ∈ S ∧ x.2 ∈ S₁ :=\n  Set.mem_prod\n\n@[simp]\ntheorem prod_top : (prod ⊤ ⊤ : NonUnitalSubalgebra R (A × B)) = ⊤ := by ext; simp\n\ntheorem prod_mono {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S ≤ T → S₁ ≤ T₁ → prod S S₁ ≤ prod T T₁ :=\n  Set.prod_mono\n\n@[simp]\ntheorem prod_inf_prod {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S.prod S₁ ⊓ T.prod T₁ = (S ⊓ T).prod (S₁ ⊓ T₁) :=\n  SetLike.coe_injective Set.prod_inter_prod\n\nend Prod\n\nsection SuprLift\n\nvariable {ι : Type*}\n\ntheorem coe_iSup_of_directed [Nonempty ι] {S : ι → NonUnitalSubalgebra R A}\n    (dir : Directed (· ≤ ·) S) : ↑(iSup S) = ⋃ i, (S i : Set A) :=\n  let K : NonUnitalSubalgebra R A :=\n    { __ := NonUnitalSubsemiring.copy _ _ (NonUnitalSubsemiring.coe_iSup_of_directed dir).symm\n      smul_mem' := fun r _x hx ↦\n        let ⟨i, hi⟩ := Set.mem_iUnion.1 hx\n        Set.mem_iUnion.2 ⟨i, (S i).smul_mem' r hi⟩ }\n  have : iSup S = K := le_antisymm\n    (iSup_le fun i ↦ le_iSup (fun i ↦ (S i : Set A)) i) (Set.iUnion_subset fun _ ↦ le_iSup S _)\n  this.symm ▸ rfl\n\n/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              rw [hf j k hjk]\n              rfl)\n            (↑(iSup K)) (by rw [coe_iSup_of_directed dir])\n        map_zero' := by\n          dsimp\n          exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl) _ (by simp)\n        map_mul' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· * ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_add' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· + ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_smul' := fun r => by\n          dsimp\n          apply Set.iUnionLift_unary (coe_iSup_of_directed dir) _ (fun _ x => r • x)\n            (fun _ _ => rfl)\n          on_goal 2 => rw [coe_iSup_of_directed dir]\n          all_goals simp }\n\nvariable [Nonempty ι] {K : ι → NonUnitalSubalgebra R A} {dir : Directed (· ≤ ·) K}\n  {f : ∀ i, K i →ₙₐ[R] B} {hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h)}\n  {T : NonUnitalSubalgebra R A} {hT : T = iSup K}\n\n@[simp]\ntheorem iSupLift_inclusion {i : ι} (x : K i) (h : K i ≤ T) :\n    iSupLift K dir f hf T hT (inclusion h x) = f i x := by\n  subst T\n  dsimp [iSupLift]\n  apply Set.iUnionLift_inclusion\n  exact h\n\n@[simp]\ntheorem iSupLift_comp_inclusion {i : ι} (h : K i ≤ T) :\n    (iSupLift K dir f hf T hT).comp (inclusion h) = f i := by ext; simp\n\n@[simp]\ntheorem iSupLift_mk {i : ι} (x : K i) (hx : (x : A) ∈ T) :\n    iSupLift K dir f hf T hT ⟨x, hx⟩ = f i x := by\n  subst hT\n  dsimp [iSupLift]\n  apply Set.iUnionLift_mk\n\ntheorem iSupLift_of_mem {i : ι} (x : T) (hx : (x : A) ∈ K i) :\n    iSupLift K dir f hf T hT x = f i ⟨x, hx⟩ := by\n  subst hT\n  ","nextTactic":"dsimp [iSupLift]","declUpToTactic":"theorem iSupLift_of_mem {i : ι} (x : T) (hx : (x : A) ∈ K i) :\n    iSupLift K dir f hf T hT x = f i ⟨x, hx⟩ := by\n  subst hT\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.952_0.jQvdAcOcyVirh23","decl":"theorem iSupLift_of_mem {i : ι} (x : T) (hx : (x : A) ∈ K i) :\n    iSupLift K dir f hf T hT x = f i ⟨x, hx⟩ "}
-{"state":"R : Type u\nA : Type v\nB : Type w\ninst✝⁹ : CommSemiring R\ninst✝⁸ : NonUnitalNonAssocSemiring A\ninst✝⁷ : Module R A\ninst✝⁶ : IsScalarTower R A A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : NonUnitalNonAssocSemiring B\ninst✝³ : Module R B\ninst✝² : IsScalarTower R B B\ninst✝¹ : SMulCommClass R B B\nS : NonUnitalSubalgebra R A\nι : Type u_1\ninst✝ : Nonempty ι\nK : ι → NonUnitalSubalgebra R A\ndir : Directed (fun x x_1 => x ≤ x_1) K\nf : (i : ι) → ↥(K i) →ₙₐ[R] B\nhf : ∀ (i j : ι) (h : K i ≤ K j), f i = NonUnitalAlgHom.comp (f j) (inclusion h)\ni : ι\nx : ↥(iSup K)\nhx : ↑x ∈ K i\n⊢ Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x)\n      (_ :\n        ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),\n          (fun i x => (f i) x) i { val := x, property := hxi } = (fun i x => (f i) x) j { val := x, property := hxj })\n      ↑(iSup K) (_ : ↑(iSup K) ⊆ ⋃ i, ↑(K i)) x =\n    (f i) { val := ↑x, property := hx }","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact NonUnitalAlgebra.gc\n\n@[simp]\ntheorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=\n  eq_top_iff.2 fun _x hx => subset_adjoin R hx\n\nvariable {R A}\n\n@[simp]\ntheorem coe_top : (↑(⊤ : NonUnitalSubalgebra R A) : Set A) = Set.univ :=\n  rfl\n\n@[simp]\ntheorem mem_top {x : A} : x ∈ (⊤ : NonUnitalSubalgebra R A) :=\n  Set.mem_univ x\n\n@[simp]\ntheorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toNonUnitalSubsemiring : (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubsemiring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toSubring {R A : Type*} [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    [IsScalarTower R A A] [SMulCommClass R A A] :\n    (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem toSubmodule_eq_top {S : NonUnitalSubalgebra R A} : S.toSubmodule = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toSubmodule'.injective.eq_iff' top_toSubmodule\n\n@[simp]\ntheorem toNonUnitalSubsemiring_eq_top {S : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubsemiring_injective.eq_iff' top_toNonUnitalSubsemiring\n\n@[simp]\ntheorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubring_injective.eq_iff' top_toSubring\n\ntheorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_left\n\ntheorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_right\n\ntheorem mul_mem_sup {S T : NonUnitalSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) :\n    x * y ∈ S ⊔ T :=\n  mul_mem (mem_sup_left hx) (mem_sup_right hy)\n\ntheorem map_sup (f : F) (S T : NonUnitalSubalgebra R A) :\n    ((S ⊔ T).map f : NonUnitalSubalgebra R B) = S.map f ⊔ T.map f :=\n  (@NonUnitalSubalgebra.gc_map_comap F R A B _ _ _ _ _ _ f).l_sup\n\n@[simp, norm_cast]\ntheorem coe_inf (S T : NonUnitalSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T :=\n  rfl\n\n@[simp]\ntheorem mem_inf {S T : NonUnitalSubalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T :=\n  Iff.rfl\n\n@[simp]\ntheorem inf_toSubmodule (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toSubmodule = S.toSubmodule ⊓ T.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem inf_toNonUnitalSubsemiring (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring ⊓ T.toNonUnitalSubsemiring :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sInf (S : Set (NonUnitalSubalgebra R A)) : (↑(sInf S) : Set A) = ⋂ s ∈ S, ↑s :=\n  sInf_image\n\ntheorem mem_sInf {S : Set (NonUnitalSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p := by\n  simp only [← SetLike.mem_coe, coe_sInf, Set.mem_iInter₂]\n\n@[simp]\ntheorem sInf_toSubmodule (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toSubmodule = sInf (NonUnitalSubalgebra.toSubmodule '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp]\ntheorem sInf_toNonUnitalSubsemiring (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toNonUnitalSubsemiring = sInf (NonUnitalSubalgebra.toNonUnitalSubsemiring '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp, norm_cast]\ntheorem coe_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} :\n    (↑(⨅ i, S i) : Set A) = ⋂ i, S i := by simp [iInf]\n\ntheorem mem_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} {x : A} :\n    (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by simp only [iInf, mem_sInf, Set.forall_range_iff]\n\n@[simp]\ntheorem iInf_toSubmodule {ι : Sort*} (S : ι → NonUnitalSubalgebra R A) :\n    (⨅ i, S i).toSubmodule = ⨅ i, (S i).toSubmodule :=\n  SetLike.coe_injective <| by simp\n\ninstance : Inhabited (NonUnitalSubalgebra R A) :=\n  ⟨⊥⟩\n\ntheorem mem_bot {x : A} : x ∈ (⊥ : NonUnitalSubalgebra R A) ↔ x = 0 :=\n  show x ∈ Submodule.span R (NonUnitalSubsemiring.closure (∅ : Set A) : Set A) ↔ x = 0 by\n    rw [NonUnitalSubsemiring.closure_empty, NonUnitalSubsemiring.coe_bot,\n      Submodule.span_zero_singleton, Submodule.mem_bot]\n\ntheorem toSubmodule_bot : (⊥ : NonUnitalSubalgebra R A).toSubmodule = ⊥ := by\n  ext\n  simp only [mem_bot, NonUnitalSubalgebra.mem_toSubmodule, Submodule.mem_bot]\n\n@[simp]\ntheorem coe_bot : ((⊥ : NonUnitalSubalgebra R A) : Set A) = {0} := by\n  simp [Set.ext_iff, NonUnitalAlgebra.mem_bot]\n\ntheorem eq_top_iff {S : NonUnitalSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S :=\n  ⟨fun h x => by rw [h]; exact mem_top, fun h => by ext x; exact ⟨fun _ => mem_top, fun _ => h x⟩⟩\n\ntheorem range_top_iff_surjective (f : A →ₙₐ[R] B) :\n    NonUnitalAlgHom.range f = (⊤ : NonUnitalSubalgebra R B) ↔ Function.Surjective f :=\n  NonUnitalAlgebra.eq_top_iff\n\n@[simp]\ntheorem range_id : NonUnitalAlgHom.range (NonUnitalAlgHom.id R A) = ⊤ :=\n  SetLike.coe_injective Set.range_id\n\n@[simp]\ntheorem map_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R A).map f = NonUnitalAlgHom.range f :=\n  SetLike.coe_injective Set.image_univ\n\n@[simp]\ntheorem map_bot (f : A →ₙₐ[R] B) : (⊥ : NonUnitalSubalgebra R A).map f = ⊥ :=\n  SetLike.coe_injective <| by simp [NonUnitalAlgebra.coe_bot, NonUnitalSubalgebra.coe_map]\n\n@[simp]\ntheorem comap_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R B).comap f = ⊤ :=\n  eq_top_iff.2 fun _ => mem_top\n\n/-- `NonUnitalAlgHom` to `⊤ : NonUnitalSubalgebra R A`. -/\ndef toTop : A →ₙₐ[R] (⊤ : NonUnitalSubalgebra R A) :=\n  NonUnitalAlgHom.codRestrict (NonUnitalAlgHom.id R A) ⊤ fun _ => mem_top\n\nend NonUnitalAlgebra\n\nnamespace NonUnitalSubalgebra\n\nopen NonUnitalAlgebra\n\nsection NonAssoc\n\nvariable {R : Type u} {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable (S : NonUnitalSubalgebra R A)\n\ninstance subsingleton_of_subsingleton [Subsingleton A] : Subsingleton (NonUnitalSubalgebra R A) :=\n  ⟨fun B C => ext fun x => by simp only [Subsingleton.elim x 0, zero_mem B, zero_mem C]⟩\n\ninstance _root_.NonUnitalAlgHom.subsingleton [Subsingleton (NonUnitalSubalgebra R A)] :\n    Subsingleton (A →ₙₐ[R] B) :=\n  ⟨fun f g =>\n    NonUnitalAlgHom.ext fun a =>\n      have : a ∈ (⊥ : NonUnitalSubalgebra R A) :=\n        Subsingleton.elim (⊤ : NonUnitalSubalgebra R A) ⊥ ▸ mem_top\n      (mem_bot.mp this).symm ▸ (map_zero f).trans (map_zero g).symm⟩\n\ntheorem range_val : NonUnitalAlgHom.range (NonUnitalSubalgebraClass.subtype S) = S :=\n  ext <| Set.ext_iff.1 <| (NonUnitalSubalgebraClass.subtype S).coe_range.trans Subtype.range_val\n\n/-- The map `S → T` when `S` is a non-unital subalgebra contained in the non-unital subalgebra `T`.\n\nThis is the non-unital subalgebra version of `Submodule.inclusion`, or `Subring.inclusion`  -/\ndef inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) : S →ₙₐ[R] T\n    where\n  toFun := Set.inclusion h\n  map_add' _ _ := rfl\n  map_mul' _ _ := rfl\n  map_zero' := rfl\n  map_smul' _ _ := rfl\n\ntheorem inclusion_injective {S T : NonUnitalSubalgebra R A} (h : S ≤ T) :\n    Function.Injective (inclusion h) := fun _ _ => Subtype.ext ∘ Subtype.mk.inj\n\n@[simp]\ntheorem inclusion_self {S : NonUnitalSubalgebra R A} :\n    inclusion (le_refl S) = NonUnitalAlgHom.id R S :=\n  NonUnitalAlgHom.ext fun _ => Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_mk {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : A) (hx : x ∈ S) :\n    inclusion h ⟨x, hx⟩ = ⟨x, h hx⟩ :=\n  rfl\n\ntheorem inclusion_right {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : T) (m : (x : A) ∈ S) :\n    inclusion h ⟨x, m⟩ = x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_inclusion {S T U : NonUnitalSubalgebra R A} (hst : S ≤ T) (htu : T ≤ U) (x : S) :\n    inclusion htu (inclusion hst x) = inclusion (le_trans hst htu) x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem coe_inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (s : S) :\n    (inclusion h s : A) = s :=\n  rfl\n\nsection Prod\n\nvariable (S₁ : NonUnitalSubalgebra R B)\n\n/-- The product of two non-unital subalgebras is a non-unital subalgebra. -/\ndef prod : NonUnitalSubalgebra R (A × B) :=\n  { S.toNonUnitalSubsemiring.prod S₁.toNonUnitalSubsemiring with\n    carrier := S ×ˢ S₁\n    smul_mem' := fun r _x hx => ⟨SMulMemClass.smul_mem r hx.1, SMulMemClass.smul_mem r hx.2⟩ }\n\n@[simp]\ntheorem coe_prod : (prod S S₁ : Set (A × B)) = (S : Set A) ×ˢ S₁ :=\n  rfl\n\ntheorem prod_toSubmodule : (S.prod S₁).toSubmodule = S.toSubmodule.prod S₁.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem mem_prod {S : NonUnitalSubalgebra R A} {S₁ : NonUnitalSubalgebra R B} {x : A × B} :\n    x ∈ prod S S₁ ↔ x.1 ∈ S ∧ x.2 ∈ S₁ :=\n  Set.mem_prod\n\n@[simp]\ntheorem prod_top : (prod ⊤ ⊤ : NonUnitalSubalgebra R (A × B)) = ⊤ := by ext; simp\n\ntheorem prod_mono {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S ≤ T → S₁ ≤ T₁ → prod S S₁ ≤ prod T T₁ :=\n  Set.prod_mono\n\n@[simp]\ntheorem prod_inf_prod {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S.prod S₁ ⊓ T.prod T₁ = (S ⊓ T).prod (S₁ ⊓ T₁) :=\n  SetLike.coe_injective Set.prod_inter_prod\n\nend Prod\n\nsection SuprLift\n\nvariable {ι : Type*}\n\ntheorem coe_iSup_of_directed [Nonempty ι] {S : ι → NonUnitalSubalgebra R A}\n    (dir : Directed (· ≤ ·) S) : ↑(iSup S) = ⋃ i, (S i : Set A) :=\n  let K : NonUnitalSubalgebra R A :=\n    { __ := NonUnitalSubsemiring.copy _ _ (NonUnitalSubsemiring.coe_iSup_of_directed dir).symm\n      smul_mem' := fun r _x hx ↦\n        let ⟨i, hi⟩ := Set.mem_iUnion.1 hx\n        Set.mem_iUnion.2 ⟨i, (S i).smul_mem' r hi⟩ }\n  have : iSup S = K := le_antisymm\n    (iSup_le fun i ↦ le_iSup (fun i ↦ (S i : Set A)) i) (Set.iUnion_subset fun _ ↦ le_iSup S _)\n  this.symm ▸ rfl\n\n/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              rw [hf j k hjk]\n              rfl)\n            (↑(iSup K)) (by rw [coe_iSup_of_directed dir])\n        map_zero' := by\n          dsimp\n          exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl) _ (by simp)\n        map_mul' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· * ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_add' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· + ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_smul' := fun r => by\n          dsimp\n          apply Set.iUnionLift_unary (coe_iSup_of_directed dir) _ (fun _ x => r • x)\n            (fun _ _ => rfl)\n          on_goal 2 => rw [coe_iSup_of_directed dir]\n          all_goals simp }\n\nvariable [Nonempty ι] {K : ι → NonUnitalSubalgebra R A} {dir : Directed (· ≤ ·) K}\n  {f : ∀ i, K i →ₙₐ[R] B} {hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h)}\n  {T : NonUnitalSubalgebra R A} {hT : T = iSup K}\n\n@[simp]\ntheorem iSupLift_inclusion {i : ι} (x : K i) (h : K i ≤ T) :\n    iSupLift K dir f hf T hT (inclusion h x) = f i x := by\n  subst T\n  dsimp [iSupLift]\n  apply Set.iUnionLift_inclusion\n  exact h\n\n@[simp]\ntheorem iSupLift_comp_inclusion {i : ι} (h : K i ≤ T) :\n    (iSupLift K dir f hf T hT).comp (inclusion h) = f i := by ext; simp\n\n@[simp]\ntheorem iSupLift_mk {i : ι} (x : K i) (hx : (x : A) ∈ T) :\n    iSupLift K dir f hf T hT ⟨x, hx⟩ = f i x := by\n  subst hT\n  dsimp [iSupLift]\n  apply Set.iUnionLift_mk\n\ntheorem iSupLift_of_mem {i : ι} (x : T) (hx : (x : A) ∈ K i) :\n    iSupLift K dir f hf T hT x = f i ⟨x, hx⟩ := by\n  subst hT\n  dsimp [iSupLift]\n  ","nextTactic":"apply Set.iUnionLift_of_mem","declUpToTactic":"theorem iSupLift_of_mem {i : ι} (x : T) (hx : (x : A) ∈ K i) :\n    iSupLift K dir f hf T hT x = f i ⟨x, hx⟩ := by\n  subst hT\n  dsimp [iSupLift]\n  ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.952_0.jQvdAcOcyVirh23","decl":"theorem iSupLift_of_mem {i : ι} (x : T) (hx : (x : A) ∈ K i) :\n    iSupLift K dir f hf T hT x = f i ⟨x, hx⟩ "}
-{"state":"R : Type u_1\nA : Type u_2\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nr : R\na : A\nha : a ∈ Set.center A\nb : A\n⊢ r • a * b = b * r • a","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact NonUnitalAlgebra.gc\n\n@[simp]\ntheorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=\n  eq_top_iff.2 fun _x hx => subset_adjoin R hx\n\nvariable {R A}\n\n@[simp]\ntheorem coe_top : (↑(⊤ : NonUnitalSubalgebra R A) : Set A) = Set.univ :=\n  rfl\n\n@[simp]\ntheorem mem_top {x : A} : x ∈ (⊤ : NonUnitalSubalgebra R A) :=\n  Set.mem_univ x\n\n@[simp]\ntheorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toNonUnitalSubsemiring : (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubsemiring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toSubring {R A : Type*} [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    [IsScalarTower R A A] [SMulCommClass R A A] :\n    (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem toSubmodule_eq_top {S : NonUnitalSubalgebra R A} : S.toSubmodule = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toSubmodule'.injective.eq_iff' top_toSubmodule\n\n@[simp]\ntheorem toNonUnitalSubsemiring_eq_top {S : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubsemiring_injective.eq_iff' top_toNonUnitalSubsemiring\n\n@[simp]\ntheorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubring_injective.eq_iff' top_toSubring\n\ntheorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_left\n\ntheorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_right\n\ntheorem mul_mem_sup {S T : NonUnitalSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) :\n    x * y ∈ S ⊔ T :=\n  mul_mem (mem_sup_left hx) (mem_sup_right hy)\n\ntheorem map_sup (f : F) (S T : NonUnitalSubalgebra R A) :\n    ((S ⊔ T).map f : NonUnitalSubalgebra R B) = S.map f ⊔ T.map f :=\n  (@NonUnitalSubalgebra.gc_map_comap F R A B _ _ _ _ _ _ f).l_sup\n\n@[simp, norm_cast]\ntheorem coe_inf (S T : NonUnitalSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T :=\n  rfl\n\n@[simp]\ntheorem mem_inf {S T : NonUnitalSubalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T :=\n  Iff.rfl\n\n@[simp]\ntheorem inf_toSubmodule (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toSubmodule = S.toSubmodule ⊓ T.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem inf_toNonUnitalSubsemiring (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring ⊓ T.toNonUnitalSubsemiring :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sInf (S : Set (NonUnitalSubalgebra R A)) : (↑(sInf S) : Set A) = ⋂ s ∈ S, ↑s :=\n  sInf_image\n\ntheorem mem_sInf {S : Set (NonUnitalSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p := by\n  simp only [← SetLike.mem_coe, coe_sInf, Set.mem_iInter₂]\n\n@[simp]\ntheorem sInf_toSubmodule (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toSubmodule = sInf (NonUnitalSubalgebra.toSubmodule '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp]\ntheorem sInf_toNonUnitalSubsemiring (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toNonUnitalSubsemiring = sInf (NonUnitalSubalgebra.toNonUnitalSubsemiring '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp, norm_cast]\ntheorem coe_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} :\n    (↑(⨅ i, S i) : Set A) = ⋂ i, S i := by simp [iInf]\n\ntheorem mem_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} {x : A} :\n    (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by simp only [iInf, mem_sInf, Set.forall_range_iff]\n\n@[simp]\ntheorem iInf_toSubmodule {ι : Sort*} (S : ι → NonUnitalSubalgebra R A) :\n    (⨅ i, S i).toSubmodule = ⨅ i, (S i).toSubmodule :=\n  SetLike.coe_injective <| by simp\n\ninstance : Inhabited (NonUnitalSubalgebra R A) :=\n  ⟨⊥⟩\n\ntheorem mem_bot {x : A} : x ∈ (⊥ : NonUnitalSubalgebra R A) ↔ x = 0 :=\n  show x ∈ Submodule.span R (NonUnitalSubsemiring.closure (∅ : Set A) : Set A) ↔ x = 0 by\n    rw [NonUnitalSubsemiring.closure_empty, NonUnitalSubsemiring.coe_bot,\n      Submodule.span_zero_singleton, Submodule.mem_bot]\n\ntheorem toSubmodule_bot : (⊥ : NonUnitalSubalgebra R A).toSubmodule = ⊥ := by\n  ext\n  simp only [mem_bot, NonUnitalSubalgebra.mem_toSubmodule, Submodule.mem_bot]\n\n@[simp]\ntheorem coe_bot : ((⊥ : NonUnitalSubalgebra R A) : Set A) = {0} := by\n  simp [Set.ext_iff, NonUnitalAlgebra.mem_bot]\n\ntheorem eq_top_iff {S : NonUnitalSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S :=\n  ⟨fun h x => by rw [h]; exact mem_top, fun h => by ext x; exact ⟨fun _ => mem_top, fun _ => h x⟩⟩\n\ntheorem range_top_iff_surjective (f : A →ₙₐ[R] B) :\n    NonUnitalAlgHom.range f = (⊤ : NonUnitalSubalgebra R B) ↔ Function.Surjective f :=\n  NonUnitalAlgebra.eq_top_iff\n\n@[simp]\ntheorem range_id : NonUnitalAlgHom.range (NonUnitalAlgHom.id R A) = ⊤ :=\n  SetLike.coe_injective Set.range_id\n\n@[simp]\ntheorem map_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R A).map f = NonUnitalAlgHom.range f :=\n  SetLike.coe_injective Set.image_univ\n\n@[simp]\ntheorem map_bot (f : A →ₙₐ[R] B) : (⊥ : NonUnitalSubalgebra R A).map f = ⊥ :=\n  SetLike.coe_injective <| by simp [NonUnitalAlgebra.coe_bot, NonUnitalSubalgebra.coe_map]\n\n@[simp]\ntheorem comap_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R B).comap f = ⊤ :=\n  eq_top_iff.2 fun _ => mem_top\n\n/-- `NonUnitalAlgHom` to `⊤ : NonUnitalSubalgebra R A`. -/\ndef toTop : A →ₙₐ[R] (⊤ : NonUnitalSubalgebra R A) :=\n  NonUnitalAlgHom.codRestrict (NonUnitalAlgHom.id R A) ⊤ fun _ => mem_top\n\nend NonUnitalAlgebra\n\nnamespace NonUnitalSubalgebra\n\nopen NonUnitalAlgebra\n\nsection NonAssoc\n\nvariable {R : Type u} {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable (S : NonUnitalSubalgebra R A)\n\ninstance subsingleton_of_subsingleton [Subsingleton A] : Subsingleton (NonUnitalSubalgebra R A) :=\n  ⟨fun B C => ext fun x => by simp only [Subsingleton.elim x 0, zero_mem B, zero_mem C]⟩\n\ninstance _root_.NonUnitalAlgHom.subsingleton [Subsingleton (NonUnitalSubalgebra R A)] :\n    Subsingleton (A →ₙₐ[R] B) :=\n  ⟨fun f g =>\n    NonUnitalAlgHom.ext fun a =>\n      have : a ∈ (⊥ : NonUnitalSubalgebra R A) :=\n        Subsingleton.elim (⊤ : NonUnitalSubalgebra R A) ⊥ ▸ mem_top\n      (mem_bot.mp this).symm ▸ (map_zero f).trans (map_zero g).symm⟩\n\ntheorem range_val : NonUnitalAlgHom.range (NonUnitalSubalgebraClass.subtype S) = S :=\n  ext <| Set.ext_iff.1 <| (NonUnitalSubalgebraClass.subtype S).coe_range.trans Subtype.range_val\n\n/-- The map `S → T` when `S` is a non-unital subalgebra contained in the non-unital subalgebra `T`.\n\nThis is the non-unital subalgebra version of `Submodule.inclusion`, or `Subring.inclusion`  -/\ndef inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) : S →ₙₐ[R] T\n    where\n  toFun := Set.inclusion h\n  map_add' _ _ := rfl\n  map_mul' _ _ := rfl\n  map_zero' := rfl\n  map_smul' _ _ := rfl\n\ntheorem inclusion_injective {S T : NonUnitalSubalgebra R A} (h : S ≤ T) :\n    Function.Injective (inclusion h) := fun _ _ => Subtype.ext ∘ Subtype.mk.inj\n\n@[simp]\ntheorem inclusion_self {S : NonUnitalSubalgebra R A} :\n    inclusion (le_refl S) = NonUnitalAlgHom.id R S :=\n  NonUnitalAlgHom.ext fun _ => Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_mk {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : A) (hx : x ∈ S) :\n    inclusion h ⟨x, hx⟩ = ⟨x, h hx⟩ :=\n  rfl\n\ntheorem inclusion_right {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : T) (m : (x : A) ∈ S) :\n    inclusion h ⟨x, m⟩ = x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_inclusion {S T U : NonUnitalSubalgebra R A} (hst : S ≤ T) (htu : T ≤ U) (x : S) :\n    inclusion htu (inclusion hst x) = inclusion (le_trans hst htu) x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem coe_inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (s : S) :\n    (inclusion h s : A) = s :=\n  rfl\n\nsection Prod\n\nvariable (S₁ : NonUnitalSubalgebra R B)\n\n/-- The product of two non-unital subalgebras is a non-unital subalgebra. -/\ndef prod : NonUnitalSubalgebra R (A × B) :=\n  { S.toNonUnitalSubsemiring.prod S₁.toNonUnitalSubsemiring with\n    carrier := S ×ˢ S₁\n    smul_mem' := fun r _x hx => ⟨SMulMemClass.smul_mem r hx.1, SMulMemClass.smul_mem r hx.2⟩ }\n\n@[simp]\ntheorem coe_prod : (prod S S₁ : Set (A × B)) = (S : Set A) ×ˢ S₁ :=\n  rfl\n\ntheorem prod_toSubmodule : (S.prod S₁).toSubmodule = S.toSubmodule.prod S₁.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem mem_prod {S : NonUnitalSubalgebra R A} {S₁ : NonUnitalSubalgebra R B} {x : A × B} :\n    x ∈ prod S S₁ ↔ x.1 ∈ S ∧ x.2 ∈ S₁ :=\n  Set.mem_prod\n\n@[simp]\ntheorem prod_top : (prod ⊤ ⊤ : NonUnitalSubalgebra R (A × B)) = ⊤ := by ext; simp\n\ntheorem prod_mono {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S ≤ T → S₁ ≤ T₁ → prod S S₁ ≤ prod T T₁ :=\n  Set.prod_mono\n\n@[simp]\ntheorem prod_inf_prod {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S.prod S₁ ⊓ T.prod T₁ = (S ⊓ T).prod (S₁ ⊓ T₁) :=\n  SetLike.coe_injective Set.prod_inter_prod\n\nend Prod\n\nsection SuprLift\n\nvariable {ι : Type*}\n\ntheorem coe_iSup_of_directed [Nonempty ι] {S : ι → NonUnitalSubalgebra R A}\n    (dir : Directed (· ≤ ·) S) : ↑(iSup S) = ⋃ i, (S i : Set A) :=\n  let K : NonUnitalSubalgebra R A :=\n    { __ := NonUnitalSubsemiring.copy _ _ (NonUnitalSubsemiring.coe_iSup_of_directed dir).symm\n      smul_mem' := fun r _x hx ↦\n        let ⟨i, hi⟩ := Set.mem_iUnion.1 hx\n        Set.mem_iUnion.2 ⟨i, (S i).smul_mem' r hi⟩ }\n  have : iSup S = K := le_antisymm\n    (iSup_le fun i ↦ le_iSup (fun i ↦ (S i : Set A)) i) (Set.iUnion_subset fun _ ↦ le_iSup S _)\n  this.symm ▸ rfl\n\n/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              rw [hf j k hjk]\n              rfl)\n            (↑(iSup K)) (by rw [coe_iSup_of_directed dir])\n        map_zero' := by\n          dsimp\n          exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl) _ (by simp)\n        map_mul' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· * ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_add' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· + ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_smul' := fun r => by\n          dsimp\n          apply Set.iUnionLift_unary (coe_iSup_of_directed dir) _ (fun _ x => r • x)\n            (fun _ _ => rfl)\n          on_goal 2 => rw [coe_iSup_of_directed dir]\n          all_goals simp }\n\nvariable [Nonempty ι] {K : ι → NonUnitalSubalgebra R A} {dir : Directed (· ≤ ·) K}\n  {f : ∀ i, K i →ₙₐ[R] B} {hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h)}\n  {T : NonUnitalSubalgebra R A} {hT : T = iSup K}\n\n@[simp]\ntheorem iSupLift_inclusion {i : ι} (x : K i) (h : K i ≤ T) :\n    iSupLift K dir f hf T hT (inclusion h x) = f i x := by\n  subst T\n  dsimp [iSupLift]\n  apply Set.iUnionLift_inclusion\n  exact h\n\n@[simp]\ntheorem iSupLift_comp_inclusion {i : ι} (h : K i ≤ T) :\n    (iSupLift K dir f hf T hT).comp (inclusion h) = f i := by ext; simp\n\n@[simp]\ntheorem iSupLift_mk {i : ι} (x : K i) (hx : (x : A) ∈ T) :\n    iSupLift K dir f hf T hT ⟨x, hx⟩ = f i x := by\n  subst hT\n  dsimp [iSupLift]\n  apply Set.iUnionLift_mk\n\ntheorem iSupLift_of_mem {i : ι} (x : T) (hx : (x : A) ∈ K i) :\n    iSupLift K dir f hf T hT x = f i ⟨x, hx⟩ := by\n  subst hT\n  dsimp [iSupLift]\n  apply Set.iUnionLift_of_mem\n\nend SuprLift\n\nend NonAssoc\n\nsection Center\n\nsection NonUnitalNonAssocSemiring\nvariable {R A : Type*}\nvariable [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [IsScalarTower R A A] [SMulCommClass R A A]\n\ntheorem _root_.Set.smul_mem_center (r : R) {a : A} (ha : a ∈ Set.center A) :\n    r • a ∈ Set.center A where\n  comm b := by ","nextTactic":"rw [mul_smul_comm, smul_mul_assoc, ha.comm]","declUpToTactic":"theorem _root_.Set.smul_mem_center (r : R) {a : A} (ha : a ∈ Set.center A) :\n    r • a ∈ Set.center A where\n  comm b := by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.969_0.jQvdAcOcyVirh23","decl":"theorem _root_.Set.smul_mem_center (r : R) {a : A} (ha : a ∈ Set.center A) :\n    r • a ∈ Set.center A where\n  comm b "}
-{"state":"R : Type u_1\nA : Type u_2\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nr : R\na : A\nha : a ∈ Set.center A\nb c : A\n⊢ r • a * (b * c) = r • a * b * c","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact NonUnitalAlgebra.gc\n\n@[simp]\ntheorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=\n  eq_top_iff.2 fun _x hx => subset_adjoin R hx\n\nvariable {R A}\n\n@[simp]\ntheorem coe_top : (↑(⊤ : NonUnitalSubalgebra R A) : Set A) = Set.univ :=\n  rfl\n\n@[simp]\ntheorem mem_top {x : A} : x ∈ (⊤ : NonUnitalSubalgebra R A) :=\n  Set.mem_univ x\n\n@[simp]\ntheorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toNonUnitalSubsemiring : (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubsemiring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toSubring {R A : Type*} [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    [IsScalarTower R A A] [SMulCommClass R A A] :\n    (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem toSubmodule_eq_top {S : NonUnitalSubalgebra R A} : S.toSubmodule = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toSubmodule'.injective.eq_iff' top_toSubmodule\n\n@[simp]\ntheorem toNonUnitalSubsemiring_eq_top {S : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubsemiring_injective.eq_iff' top_toNonUnitalSubsemiring\n\n@[simp]\ntheorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubring_injective.eq_iff' top_toSubring\n\ntheorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_left\n\ntheorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_right\n\ntheorem mul_mem_sup {S T : NonUnitalSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) :\n    x * y ∈ S ⊔ T :=\n  mul_mem (mem_sup_left hx) (mem_sup_right hy)\n\ntheorem map_sup (f : F) (S T : NonUnitalSubalgebra R A) :\n    ((S ⊔ T).map f : NonUnitalSubalgebra R B) = S.map f ⊔ T.map f :=\n  (@NonUnitalSubalgebra.gc_map_comap F R A B _ _ _ _ _ _ f).l_sup\n\n@[simp, norm_cast]\ntheorem coe_inf (S T : NonUnitalSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T :=\n  rfl\n\n@[simp]\ntheorem mem_inf {S T : NonUnitalSubalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T :=\n  Iff.rfl\n\n@[simp]\ntheorem inf_toSubmodule (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toSubmodule = S.toSubmodule ⊓ T.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem inf_toNonUnitalSubsemiring (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring ⊓ T.toNonUnitalSubsemiring :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sInf (S : Set (NonUnitalSubalgebra R A)) : (↑(sInf S) : Set A) = ⋂ s ∈ S, ↑s :=\n  sInf_image\n\ntheorem mem_sInf {S : Set (NonUnitalSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p := by\n  simp only [← SetLike.mem_coe, coe_sInf, Set.mem_iInter₂]\n\n@[simp]\ntheorem sInf_toSubmodule (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toSubmodule = sInf (NonUnitalSubalgebra.toSubmodule '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp]\ntheorem sInf_toNonUnitalSubsemiring (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toNonUnitalSubsemiring = sInf (NonUnitalSubalgebra.toNonUnitalSubsemiring '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp, norm_cast]\ntheorem coe_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} :\n    (↑(⨅ i, S i) : Set A) = ⋂ i, S i := by simp [iInf]\n\ntheorem mem_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} {x : A} :\n    (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by simp only [iInf, mem_sInf, Set.forall_range_iff]\n\n@[simp]\ntheorem iInf_toSubmodule {ι : Sort*} (S : ι → NonUnitalSubalgebra R A) :\n    (⨅ i, S i).toSubmodule = ⨅ i, (S i).toSubmodule :=\n  SetLike.coe_injective <| by simp\n\ninstance : Inhabited (NonUnitalSubalgebra R A) :=\n  ⟨⊥⟩\n\ntheorem mem_bot {x : A} : x ∈ (⊥ : NonUnitalSubalgebra R A) ↔ x = 0 :=\n  show x ∈ Submodule.span R (NonUnitalSubsemiring.closure (∅ : Set A) : Set A) ↔ x = 0 by\n    rw [NonUnitalSubsemiring.closure_empty, NonUnitalSubsemiring.coe_bot,\n      Submodule.span_zero_singleton, Submodule.mem_bot]\n\ntheorem toSubmodule_bot : (��� : NonUnitalSubalgebra R A).toSubmodule = ⊥ := by\n  ext\n  simp only [mem_bot, NonUnitalSubalgebra.mem_toSubmodule, Submodule.mem_bot]\n\n@[simp]\ntheorem coe_bot : ((⊥ : NonUnitalSubalgebra R A) : Set A) = {0} := by\n  simp [Set.ext_iff, NonUnitalAlgebra.mem_bot]\n\ntheorem eq_top_iff {S : NonUnitalSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S :=\n  ⟨fun h x => by rw [h]; exact mem_top, fun h => by ext x; exact ⟨fun _ => mem_top, fun _ => h x⟩⟩\n\ntheorem range_top_iff_surjective (f : A →ₙₐ[R] B) :\n    NonUnitalAlgHom.range f = (⊤ : NonUnitalSubalgebra R B) ↔ Function.Surjective f :=\n  NonUnitalAlgebra.eq_top_iff\n\n@[simp]\ntheorem range_id : NonUnitalAlgHom.range (NonUnitalAlgHom.id R A) = ⊤ :=\n  SetLike.coe_injective Set.range_id\n\n@[simp]\ntheorem map_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R A).map f = NonUnitalAlgHom.range f :=\n  SetLike.coe_injective Set.image_univ\n\n@[simp]\ntheorem map_bot (f : A →ₙₐ[R] B) : (⊥ : NonUnitalSubalgebra R A).map f = ⊥ :=\n  SetLike.coe_injective <| by simp [NonUnitalAlgebra.coe_bot, NonUnitalSubalgebra.coe_map]\n\n@[simp]\ntheorem comap_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R B).comap f = ⊤ :=\n  eq_top_iff.2 fun _ => mem_top\n\n/-- `NonUnitalAlgHom` to `⊤ : NonUnitalSubalgebra R A`. -/\ndef toTop : A →ₙₐ[R] (⊤ : NonUnitalSubalgebra R A) :=\n  NonUnitalAlgHom.codRestrict (NonUnitalAlgHom.id R A) ⊤ fun _ => mem_top\n\nend NonUnitalAlgebra\n\nnamespace NonUnitalSubalgebra\n\nopen NonUnitalAlgebra\n\nsection NonAssoc\n\nvariable {R : Type u} {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable (S : NonUnitalSubalgebra R A)\n\ninstance subsingleton_of_subsingleton [Subsingleton A] : Subsingleton (NonUnitalSubalgebra R A) :=\n  ⟨fun B C => ext fun x => by simp only [Subsingleton.elim x 0, zero_mem B, zero_mem C]⟩\n\ninstance _root_.NonUnitalAlgHom.subsingleton [Subsingleton (NonUnitalSubalgebra R A)] :\n    Subsingleton (A →ₙₐ[R] B) :=\n  ⟨fun f g =>\n    NonUnitalAlgHom.ext fun a =>\n      have : a ∈ (⊥ : NonUnitalSubalgebra R A) :=\n        Subsingleton.elim (⊤ : NonUnitalSubalgebra R A) ⊥ ▸ mem_top\n      (mem_bot.mp this).symm ▸ (map_zero f).trans (map_zero g).symm⟩\n\ntheorem range_val : NonUnitalAlgHom.range (NonUnitalSubalgebraClass.subtype S) = S :=\n  ext <| Set.ext_iff.1 <| (NonUnitalSubalgebraClass.subtype S).coe_range.trans Subtype.range_val\n\n/-- The map `S → T` when `S` is a non-unital subalgebra contained in the non-unital subalgebra `T`.\n\nThis is the non-unital subalgebra version of `Submodule.inclusion`, or `Subring.inclusion`  -/\ndef inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) : S →ₙₐ[R] T\n    where\n  toFun := Set.inclusion h\n  map_add' _ _ := rfl\n  map_mul' _ _ := rfl\n  map_zero' := rfl\n  map_smul' _ _ := rfl\n\ntheorem inclusion_injective {S T : NonUnitalSubalgebra R A} (h : S ≤ T) :\n    Function.Injective (inclusion h) := fun _ _ => Subtype.ext ∘ Subtype.mk.inj\n\n@[simp]\ntheorem inclusion_self {S : NonUnitalSubalgebra R A} :\n    inclusion (le_refl S) = NonUnitalAlgHom.id R S :=\n  NonUnitalAlgHom.ext fun _ => Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_mk {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : A) (hx : x ∈ S) :\n    inclusion h ⟨x, hx⟩ = ⟨x, h hx⟩ :=\n  rfl\n\ntheorem inclusion_right {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : T) (m : (x : A) ∈ S) :\n    inclusion h ⟨x, m⟩ = x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_inclusion {S T U : NonUnitalSubalgebra R A} (hst : S ≤ T) (htu : T ≤ U) (x : S) :\n    inclusion htu (inclusion hst x) = inclusion (le_trans hst htu) x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem coe_inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (s : S) :\n    (inclusion h s : A) = s :=\n  rfl\n\nsection Prod\n\nvariable (S₁ : NonUnitalSubalgebra R B)\n\n/-- The product of two non-unital subalgebras is a non-unital subalgebra. -/\ndef prod : NonUnitalSubalgebra R (A × B) :=\n  { S.toNonUnitalSubsemiring.prod S₁.toNonUnitalSubsemiring with\n    carrier := S ×ˢ S₁\n    smul_mem' := fun r _x hx => ⟨SMulMemClass.smul_mem r hx.1, SMulMemClass.smul_mem r hx.2⟩ }\n\n@[simp]\ntheorem coe_prod : (prod S S₁ : Set (A × B)) = (S : Set A) ×ˢ S₁ :=\n  rfl\n\ntheorem prod_toSubmodule : (S.prod S₁).toSubmodule = S.toSubmodule.prod S₁.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem mem_prod {S : NonUnitalSubalgebra R A} {S₁ : NonUnitalSubalgebra R B} {x : A × B} :\n    x ∈ prod S S₁ ↔ x.1 ∈ S ∧ x.2 ∈ S₁ :=\n  Set.mem_prod\n\n@[simp]\ntheorem prod_top : (prod ⊤ ⊤ : NonUnitalSubalgebra R (A × B)) = ⊤ := by ext; simp\n\ntheorem prod_mono {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S ≤ T → S₁ ≤ T₁ → prod S S₁ ≤ prod T T₁ :=\n  Set.prod_mono\n\n@[simp]\ntheorem prod_inf_prod {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S.prod S₁ ⊓ T.prod T₁ = (S ⊓ T).prod (S₁ ⊓ T₁) :=\n  SetLike.coe_injective Set.prod_inter_prod\n\nend Prod\n\nsection SuprLift\n\nvariable {ι : Type*}\n\ntheorem coe_iSup_of_directed [Nonempty ι] {S : ι → NonUnitalSubalgebra R A}\n    (dir : Directed (· ≤ ·) S) : ↑(iSup S) = ⋃ i, (S i : Set A) :=\n  let K : NonUnitalSubalgebra R A :=\n    { __ := NonUnitalSubsemiring.copy _ _ (NonUnitalSubsemiring.coe_iSup_of_directed dir).symm\n      smul_mem' := fun r _x hx ↦\n        let ⟨i, hi⟩ := Set.mem_iUnion.1 hx\n        Set.mem_iUnion.2 ⟨i, (S i).smul_mem' r hi⟩ }\n  have : iSup S = K := le_antisymm\n    (iSup_le fun i ↦ le_iSup (fun i ↦ (S i : Set A)) i) (Set.iUnion_subset fun _ ↦ le_iSup S _)\n  this.symm ▸ rfl\n\n/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              rw [hf j k hjk]\n              rfl)\n            (↑(iSup K)) (by rw [coe_iSup_of_directed dir])\n        map_zero' := by\n          dsimp\n          exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl) _ (by simp)\n        map_mul' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· * ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_add' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· + ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_smul' := fun r => by\n          dsimp\n          apply Set.iUnionLift_unary (coe_iSup_of_directed dir) _ (fun _ x => r • x)\n            (fun _ _ => rfl)\n          on_goal 2 => rw [coe_iSup_of_directed dir]\n          all_goals simp }\n\nvariable [Nonempty ι] {K : ι → NonUnitalSubalgebra R A} {dir : Directed (· ≤ ·) K}\n  {f : ∀ i, K i →ₙₐ[R] B} {hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h)}\n  {T : NonUnitalSubalgebra R A} {hT : T = iSup K}\n\n@[simp]\ntheorem iSupLift_inclusion {i : ι} (x : K i) (h : K i ≤ T) :\n    iSupLift K dir f hf T hT (inclusion h x) = f i x := by\n  subst T\n  dsimp [iSupLift]\n  apply Set.iUnionLift_inclusion\n  exact h\n\n@[simp]\ntheorem iSupLift_comp_inclusion {i : ι} (h : K i ≤ T) :\n    (iSupLift K dir f hf T hT).comp (inclusion h) = f i := by ext; simp\n\n@[simp]\ntheorem iSupLift_mk {i : ι} (x : K i) (hx : (x : A) ∈ T) :\n    iSupLift K dir f hf T hT ⟨x, hx⟩ = f i x := by\n  subst hT\n  dsimp [iSupLift]\n  apply Set.iUnionLift_mk\n\ntheorem iSupLift_of_mem {i : ι} (x : T) (hx : (x : A) ∈ K i) :\n    iSupLift K dir f hf T hT x = f i ⟨x, hx⟩ := by\n  subst hT\n  dsimp [iSupLift]\n  apply Set.iUnionLift_of_mem\n\nend SuprLift\n\nend NonAssoc\n\nsection Center\n\nsection NonUnitalNonAssocSemiring\nvariable {R A : Type*}\nvariable [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [IsScalarTower R A A] [SMulCommClass R A A]\n\ntheorem _root_.Set.smul_mem_center (r : R) {a : A} (ha : a ∈ Set.center A) :\n    r • a ∈ Set.center A where\n  comm b := by rw [mul_smul_comm, smul_mul_assoc, ha.comm]\n  left_assoc b c := by ","nextTactic":"rw [smul_mul_assoc, smul_mul_assoc, smul_mul_assoc, ha.left_assoc]","declUpToTactic":"theorem _root_.Set.smul_mem_center (r : R) {a : A} (ha : a ∈ Set.center A) :\n    r • a ∈ Set.center A where\n  comm b := by rw [mul_smul_comm, smul_mul_assoc, ha.comm]\n  left_assoc b c := by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.969_0.jQvdAcOcyVirh23","decl":"theorem _root_.Set.smul_mem_center (r : R) {a : A} (ha : a ∈ Set.center A) :\n    r • a ∈ Set.center A where\n  comm b "}
-{"state":"R : Type u_1\nA : Type u_2\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nr : R\na : A\nha : a ∈ Set.center A\nb c : A\n⊢ b * r • a * c = b * (r • a * c)","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact NonUnitalAlgebra.gc\n\n@[simp]\ntheorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=\n  eq_top_iff.2 fun _x hx => subset_adjoin R hx\n\nvariable {R A}\n\n@[simp]\ntheorem coe_top : (↑(⊤ : NonUnitalSubalgebra R A) : Set A) = Set.univ :=\n  rfl\n\n@[simp]\ntheorem mem_top {x : A} : x ∈ (⊤ : NonUnitalSubalgebra R A) :=\n  Set.mem_univ x\n\n@[simp]\ntheorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toNonUnitalSubsemiring : (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubsemiring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toSubring {R A : Type*} [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    [IsScalarTower R A A] [SMulCommClass R A A] :\n    (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem toSubmodule_eq_top {S : NonUnitalSubalgebra R A} : S.toSubmodule = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toSubmodule'.injective.eq_iff' top_toSubmodule\n\n@[simp]\ntheorem toNonUnitalSubsemiring_eq_top {S : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubsemiring_injective.eq_iff' top_toNonUnitalSubsemiring\n\n@[simp]\ntheorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubring_injective.eq_iff' top_toSubring\n\ntheorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_left\n\ntheorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_right\n\ntheorem mul_mem_sup {S T : NonUnitalSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) :\n    x * y ∈ S ⊔ T :=\n  mul_mem (mem_sup_left hx) (mem_sup_right hy)\n\ntheorem map_sup (f : F) (S T : NonUnitalSubalgebra R A) :\n    ((S ⊔ T).map f : NonUnitalSubalgebra R B) = S.map f ⊔ T.map f :=\n  (@NonUnitalSubalgebra.gc_map_comap F R A B _ _ _ _ _ _ f).l_sup\n\n@[simp, norm_cast]\ntheorem coe_inf (S T : NonUnitalSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T :=\n  rfl\n\n@[simp]\ntheorem mem_inf {S T : NonUnitalSubalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T :=\n  Iff.rfl\n\n@[simp]\ntheorem inf_toSubmodule (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toSubmodule = S.toSubmodule ⊓ T.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem inf_toNonUnitalSubsemiring (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring ⊓ T.toNonUnitalSubsemiring :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sInf (S : Set (NonUnitalSubalgebra R A)) : (↑(sInf S) : Set A) = ⋂ s ∈ S, ↑s :=\n  sInf_image\n\ntheorem mem_sInf {S : Set (NonUnitalSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p := by\n  simp only [← SetLike.mem_coe, coe_sInf, Set.mem_iInter₂]\n\n@[simp]\ntheorem sInf_toSubmodule (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toSubmodule = sInf (NonUnitalSubalgebra.toSubmodule '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp]\ntheorem sInf_toNonUnitalSubsemiring (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toNonUnitalSubsemiring = sInf (NonUnitalSubalgebra.toNonUnitalSubsemiring '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp, norm_cast]\ntheorem coe_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} :\n    (↑(⨅ i, S i) : Set A) = ⋂ i, S i := by simp [iInf]\n\ntheorem mem_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} {x : A} :\n    (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by simp only [iInf, mem_sInf, Set.forall_range_iff]\n\n@[simp]\ntheorem iInf_toSubmodule {ι : Sort*} (S : ι → NonUnitalSubalgebra R A) :\n    (⨅ i, S i).toSubmodule = ⨅ i, (S i).toSubmodule :=\n  SetLike.coe_injective <| by simp\n\ninstance : Inhabited (NonUnitalSubalgebra R A) :=\n  ⟨⊥⟩\n\ntheorem mem_bot {x : A} : x ∈ (⊥ : NonUnitalSubalgebra R A) ↔ x = 0 :=\n  show x ∈ Submodule.span R (NonUnitalSubsemiring.closure (∅ : Set A) : Set A) ↔ x = 0 by\n    rw [NonUnitalSubsemiring.closure_empty, NonUnitalSubsemiring.coe_bot,\n      Submodule.span_zero_singleton, Submodule.mem_bot]\n\ntheorem toSubmodule_bot : (⊥ : NonUnitalSubalgebra R A).toSubmodule = ⊥ := by\n  ext\n  simp only [mem_bot, NonUnitalSubalgebra.mem_toSubmodule, Submodule.mem_bot]\n\n@[simp]\ntheorem coe_bot : ((⊥ : NonUnitalSubalgebra R A) : Set A) = {0} := by\n  simp [Set.ext_iff, NonUnitalAlgebra.mem_bot]\n\ntheorem eq_top_iff {S : NonUnitalSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S :=\n  ⟨fun h x => by rw [h]; exact mem_top, fun h => by ext x; exact ⟨fun _ => mem_top, fun _ => h x⟩⟩\n\ntheorem range_top_iff_surjective (f : A →ₙₐ[R] B) :\n    NonUnitalAlgHom.range f = (��� : NonUnitalSubalgebra R B) ↔ Function.Surjective f :=\n  NonUnitalAlgebra.eq_top_iff\n\n@[simp]\ntheorem range_id : NonUnitalAlgHom.range (NonUnitalAlgHom.id R A) = ⊤ :=\n  SetLike.coe_injective Set.range_id\n\n@[simp]\ntheorem map_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R A).map f = NonUnitalAlgHom.range f :=\n  SetLike.coe_injective Set.image_univ\n\n@[simp]\ntheorem map_bot (f : A →ₙₐ[R] B) : (⊥ : NonUnitalSubalgebra R A).map f = ⊥ :=\n  SetLike.coe_injective <| by simp [NonUnitalAlgebra.coe_bot, NonUnitalSubalgebra.coe_map]\n\n@[simp]\ntheorem comap_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R B).comap f = ⊤ :=\n  eq_top_iff.2 fun _ => mem_top\n\n/-- `NonUnitalAlgHom` to `⊤ : NonUnitalSubalgebra R A`. -/\ndef toTop : A →ₙₐ[R] (⊤ : NonUnitalSubalgebra R A) :=\n  NonUnitalAlgHom.codRestrict (NonUnitalAlgHom.id R A) ⊤ fun _ => mem_top\n\nend NonUnitalAlgebra\n\nnamespace NonUnitalSubalgebra\n\nopen NonUnitalAlgebra\n\nsection NonAssoc\n\nvariable {R : Type u} {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable (S : NonUnitalSubalgebra R A)\n\ninstance subsingleton_of_subsingleton [Subsingleton A] : Subsingleton (NonUnitalSubalgebra R A) :=\n  ⟨fun B C => ext fun x => by simp only [Subsingleton.elim x 0, zero_mem B, zero_mem C]⟩\n\ninstance _root_.NonUnitalAlgHom.subsingleton [Subsingleton (NonUnitalSubalgebra R A)] :\n    Subsingleton (A →ₙₐ[R] B) :=\n  ⟨fun f g =>\n    NonUnitalAlgHom.ext fun a =>\n      have : a ∈ (⊥ : NonUnitalSubalgebra R A) :=\n        Subsingleton.elim (⊤ : NonUnitalSubalgebra R A) ⊥ ▸ mem_top\n      (mem_bot.mp this).symm ▸ (map_zero f).trans (map_zero g).symm⟩\n\ntheorem range_val : NonUnitalAlgHom.range (NonUnitalSubalgebraClass.subtype S) = S :=\n  ext <| Set.ext_iff.1 <| (NonUnitalSubalgebraClass.subtype S).coe_range.trans Subtype.range_val\n\n/-- The map `S → T` when `S` is a non-unital subalgebra contained in the non-unital subalgebra `T`.\n\nThis is the non-unital subalgebra version of `Submodule.inclusion`, or `Subring.inclusion`  -/\ndef inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) : S →ₙₐ[R] T\n    where\n  toFun := Set.inclusion h\n  map_add' _ _ := rfl\n  map_mul' _ _ := rfl\n  map_zero' := rfl\n  map_smul' _ _ := rfl\n\ntheorem inclusion_injective {S T : NonUnitalSubalgebra R A} (h : S ≤ T) :\n    Function.Injective (inclusion h) := fun _ _ => Subtype.ext ∘ Subtype.mk.inj\n\n@[simp]\ntheorem inclusion_self {S : NonUnitalSubalgebra R A} :\n    inclusion (le_refl S) = NonUnitalAlgHom.id R S :=\n  NonUnitalAlgHom.ext fun _ => Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_mk {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : A) (hx : x ∈ S) :\n    inclusion h ⟨x, hx⟩ = ⟨x, h hx⟩ :=\n  rfl\n\ntheorem inclusion_right {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : T) (m : (x : A) ∈ S) :\n    inclusion h ⟨x, m⟩ = x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_inclusion {S T U : NonUnitalSubalgebra R A} (hst : S ≤ T) (htu : T ≤ U) (x : S) :\n    inclusion htu (inclusion hst x) = inclusion (le_trans hst htu) x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem coe_inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (s : S) :\n    (inclusion h s : A) = s :=\n  rfl\n\nsection Prod\n\nvariable (S₁ : NonUnitalSubalgebra R B)\n\n/-- The product of two non-unital subalgebras is a non-unital subalgebra. -/\ndef prod : NonUnitalSubalgebra R (A × B) :=\n  { S.toNonUnitalSubsemiring.prod S₁.toNonUnitalSubsemiring with\n    carrier := S ×ˢ S₁\n    smul_mem' := fun r _x hx => ⟨SMulMemClass.smul_mem r hx.1, SMulMemClass.smul_mem r hx.2⟩ }\n\n@[simp]\ntheorem coe_prod : (prod S S₁ : Set (A × B)) = (S : Set A) ×ˢ S₁ :=\n  rfl\n\ntheorem prod_toSubmodule : (S.prod S₁).toSubmodule = S.toSubmodule.prod S₁.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem mem_prod {S : NonUnitalSubalgebra R A} {S₁ : NonUnitalSubalgebra R B} {x : A × B} :\n    x ∈ prod S S₁ ↔ x.1 ∈ S ∧ x.2 ∈ S₁ :=\n  Set.mem_prod\n\n@[simp]\ntheorem prod_top : (prod ⊤ ⊤ : NonUnitalSubalgebra R (A × B)) = ⊤ := by ext; simp\n\ntheorem prod_mono {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S ≤ T → S₁ ≤ T₁ → prod S S₁ ≤ prod T T₁ :=\n  Set.prod_mono\n\n@[simp]\ntheorem prod_inf_prod {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S.prod S₁ ⊓ T.prod T₁ = (S ⊓ T).prod (S₁ ⊓ T₁) :=\n  SetLike.coe_injective Set.prod_inter_prod\n\nend Prod\n\nsection SuprLift\n\nvariable {ι : Type*}\n\ntheorem coe_iSup_of_directed [Nonempty ι] {S : ι → NonUnitalSubalgebra R A}\n    (dir : Directed (· ≤ ·) S) : ↑(iSup S) = ⋃ i, (S i : Set A) :=\n  let K : NonUnitalSubalgebra R A :=\n    { __ := NonUnitalSubsemiring.copy _ _ (NonUnitalSubsemiring.coe_iSup_of_directed dir).symm\n      smul_mem' := fun r _x hx ↦\n        let ⟨i, hi⟩ := Set.mem_iUnion.1 hx\n        Set.mem_iUnion.2 ⟨i, (S i).smul_mem' r hi⟩ }\n  have : iSup S = K := le_antisymm\n    (iSup_le fun i ↦ le_iSup (fun i ↦ (S i : Set A)) i) (Set.iUnion_subset fun _ ↦ le_iSup S _)\n  this.symm ▸ rfl\n\n/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              rw [hf j k hjk]\n              rfl)\n            (↑(iSup K)) (by rw [coe_iSup_of_directed dir])\n        map_zero' := by\n          dsimp\n          exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl) _ (by simp)\n        map_mul' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· * ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_add' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· + ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_smul' := fun r => by\n          dsimp\n          apply Set.iUnionLift_unary (coe_iSup_of_directed dir) _ (fun _ x => r • x)\n            (fun _ _ => rfl)\n          on_goal 2 => rw [coe_iSup_of_directed dir]\n          all_goals simp }\n\nvariable [Nonempty ι] {K : ι → NonUnitalSubalgebra R A} {dir : Directed (· ≤ ·) K}\n  {f : ∀ i, K i →ₙₐ[R] B} {hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h)}\n  {T : NonUnitalSubalgebra R A} {hT : T = iSup K}\n\n@[simp]\ntheorem iSupLift_inclusion {i : ι} (x : K i) (h : K i ≤ T) :\n    iSupLift K dir f hf T hT (inclusion h x) = f i x := by\n  subst T\n  dsimp [iSupLift]\n  apply Set.iUnionLift_inclusion\n  exact h\n\n@[simp]\ntheorem iSupLift_comp_inclusion {i : ι} (h : K i ≤ T) :\n    (iSupLift K dir f hf T hT).comp (inclusion h) = f i := by ext; simp\n\n@[simp]\ntheorem iSupLift_mk {i : ι} (x : K i) (hx : (x : A) ∈ T) :\n    iSupLift K dir f hf T hT ⟨x, hx⟩ = f i x := by\n  subst hT\n  dsimp [iSupLift]\n  apply Set.iUnionLift_mk\n\ntheorem iSupLift_of_mem {i : ι} (x : T) (hx : (x : A) ∈ K i) :\n    iSupLift K dir f hf T hT x = f i ⟨x, hx⟩ := by\n  subst hT\n  dsimp [iSupLift]\n  apply Set.iUnionLift_of_mem\n\nend SuprLift\n\nend NonAssoc\n\nsection Center\n\nsection NonUnitalNonAssocSemiring\nvariable {R A : Type*}\nvariable [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [IsScalarTower R A A] [SMulCommClass R A A]\n\ntheorem _root_.Set.smul_mem_center (r : R) {a : A} (ha : a ∈ Set.center A) :\n    r • a ∈ Set.center A where\n  comm b := by rw [mul_smul_comm, smul_mul_assoc, ha.comm]\n  left_assoc b c := by rw [smul_mul_assoc, smul_mul_assoc, smul_mul_assoc, ha.left_assoc]\n  mid_assoc b c := by\n    ","nextTactic":"rw [mul_smul_comm]","declUpToTactic":"theorem _root_.Set.smul_mem_center (r : R) {a : A} (ha : a ∈ Set.center A) :\n    r • a ∈ Set.center A where\n  comm b := by rw [mul_smul_comm, smul_mul_assoc, ha.comm]\n  left_assoc b c := by rw [smul_mul_assoc, smul_mul_assoc, smul_mul_assoc, ha.left_assoc]\n  mid_assoc b c := by\n    ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.969_0.jQvdAcOcyVirh23","decl":"theorem _root_.Set.smul_mem_center (r : R) {a : A} (ha : a ∈ Set.center A) :\n    r • a ∈ Set.center A where\n  comm b "}
-{"state":"R : Type u_1\nA : Type u_2\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nr : R\na : A\nha : a ∈ Set.center A\nb c : A\n⊢ r • (b * a) * c = b * (r • a * c)","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact NonUnitalAlgebra.gc\n\n@[simp]\ntheorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=\n  eq_top_iff.2 fun _x hx => subset_adjoin R hx\n\nvariable {R A}\n\n@[simp]\ntheorem coe_top : (↑(⊤ : NonUnitalSubalgebra R A) : Set A) = Set.univ :=\n  rfl\n\n@[simp]\ntheorem mem_top {x : A} : x ∈ (⊤ : NonUnitalSubalgebra R A) :=\n  Set.mem_univ x\n\n@[simp]\ntheorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toNonUnitalSubsemiring : (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubsemiring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toSubring {R A : Type*} [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    [IsScalarTower R A A] [SMulCommClass R A A] :\n    (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem toSubmodule_eq_top {S : NonUnitalSubalgebra R A} : S.toSubmodule = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toSubmodule'.injective.eq_iff' top_toSubmodule\n\n@[simp]\ntheorem toNonUnitalSubsemiring_eq_top {S : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubsemiring_injective.eq_iff' top_toNonUnitalSubsemiring\n\n@[simp]\ntheorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubring_injective.eq_iff' top_toSubring\n\ntheorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_left\n\ntheorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_right\n\ntheorem mul_mem_sup {S T : NonUnitalSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) :\n    x * y ∈ S ⊔ T :=\n  mul_mem (mem_sup_left hx) (mem_sup_right hy)\n\ntheorem map_sup (f : F) (S T : NonUnitalSubalgebra R A) :\n    ((S ⊔ T).map f : NonUnitalSubalgebra R B) = S.map f ⊔ T.map f :=\n  (@NonUnitalSubalgebra.gc_map_comap F R A B _ _ _ _ _ _ f).l_sup\n\n@[simp, norm_cast]\ntheorem coe_inf (S T : NonUnitalSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T :=\n  rfl\n\n@[simp]\ntheorem mem_inf {S T : NonUnitalSubalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T :=\n  Iff.rfl\n\n@[simp]\ntheorem inf_toSubmodule (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toSubmodule = S.toSubmodule ⊓ T.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem inf_toNonUnitalSubsemiring (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring ⊓ T.toNonUnitalSubsemiring :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sInf (S : Set (NonUnitalSubalgebra R A)) : (↑(sInf S) : Set A) = ⋂ s ∈ S, ↑s :=\n  sInf_image\n\ntheorem mem_sInf {S : Set (NonUnitalSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p := by\n  simp only [← SetLike.mem_coe, coe_sInf, Set.mem_iInter₂]\n\n@[simp]\ntheorem sInf_toSubmodule (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toSubmodule = sInf (NonUnitalSubalgebra.toSubmodule '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp]\ntheorem sInf_toNonUnitalSubsemiring (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toNonUnitalSubsemiring = sInf (NonUnitalSubalgebra.toNonUnitalSubsemiring '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp, norm_cast]\ntheorem coe_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} :\n    (↑(⨅ i, S i) : Set A) = ⋂ i, S i := by simp [iInf]\n\ntheorem mem_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} {x : A} :\n    (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by simp only [iInf, mem_sInf, Set.forall_range_iff]\n\n@[simp]\ntheorem iInf_toSubmodule {ι : Sort*} (S : ι → NonUnitalSubalgebra R A) :\n    (⨅ i, S i).toSubmodule = ⨅ i, (S i).toSubmodule :=\n  SetLike.coe_injective <| by simp\n\ninstance : Inhabited (NonUnitalSubalgebra R A) :=\n  ⟨⊥⟩\n\ntheorem mem_bot {x : A} : x ∈ (⊥ : NonUnitalSubalgebra R A) ↔ x = 0 :=\n  show x ∈ Submodule.span R (NonUnitalSubsemiring.closure (∅ : Set A) : Set A) ↔ x = 0 by\n    rw [NonUnitalSubsemiring.closure_empty, NonUnitalSubsemiring.coe_bot,\n      Submodule.span_zero_singleton, Submodule.mem_bot]\n\ntheorem toSubmodule_bot : (⊥ : NonUnitalSubalgebra R A).toSubmodule = ⊥ := by\n  ext\n  simp only [mem_bot, NonUnitalSubalgebra.mem_toSubmodule, Submodule.mem_bot]\n\n@[simp]\ntheorem coe_bot : ((⊥ : NonUnitalSubalgebra R A) : Set A) = {0} := by\n  simp [Set.ext_iff, NonUnitalAlgebra.mem_bot]\n\ntheorem eq_top_iff {S : NonUnitalSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S :=\n  ⟨fun h x => by rw [h]; exact mem_top, fun h => by ext x; exact ⟨fun _ => mem_top, fun _ => h x⟩⟩\n\ntheorem range_top_iff_surjective (f : A →ₙₐ[R] B) :\n    NonUnitalAlgHom.range f = (⊤ : NonUnitalSubalgebra R B) ↔ Function.Surjective f :=\n  NonUnitalAlgebra.eq_top_iff\n\n@[simp]\ntheorem range_id : NonUnitalAlgHom.range (NonUnitalAlgHom.id R A) = ⊤ :=\n  SetLike.coe_injective Set.range_id\n\n@[simp]\ntheorem map_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R A).map f = NonUnitalAlgHom.range f :=\n  SetLike.coe_injective Set.image_univ\n\n@[simp]\ntheorem map_bot (f : A →ₙ��[R] B) : (⊥ : NonUnitalSubalgebra R A).map f = ⊥ :=\n  SetLike.coe_injective <| by simp [NonUnitalAlgebra.coe_bot, NonUnitalSubalgebra.coe_map]\n\n@[simp]\ntheorem comap_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R B).comap f = ⊤ :=\n  eq_top_iff.2 fun _ => mem_top\n\n/-- `NonUnitalAlgHom` to `⊤ : NonUnitalSubalgebra R A`. -/\ndef toTop : A →ₙₐ[R] (⊤ : NonUnitalSubalgebra R A) :=\n  NonUnitalAlgHom.codRestrict (NonUnitalAlgHom.id R A) ⊤ fun _ => mem_top\n\nend NonUnitalAlgebra\n\nnamespace NonUnitalSubalgebra\n\nopen NonUnitalAlgebra\n\nsection NonAssoc\n\nvariable {R : Type u} {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable (S : NonUnitalSubalgebra R A)\n\ninstance subsingleton_of_subsingleton [Subsingleton A] : Subsingleton (NonUnitalSubalgebra R A) :=\n  ⟨fun B C => ext fun x => by simp only [Subsingleton.elim x 0, zero_mem B, zero_mem C]⟩\n\ninstance _root_.NonUnitalAlgHom.subsingleton [Subsingleton (NonUnitalSubalgebra R A)] :\n    Subsingleton (A →ₙₐ[R] B) :=\n  ⟨fun f g =>\n    NonUnitalAlgHom.ext fun a =>\n      have : a ∈ (⊥ : NonUnitalSubalgebra R A) :=\n        Subsingleton.elim (⊤ : NonUnitalSubalgebra R A) ⊥ ▸ mem_top\n      (mem_bot.mp this).symm ▸ (map_zero f).trans (map_zero g).symm⟩\n\ntheorem range_val : NonUnitalAlgHom.range (NonUnitalSubalgebraClass.subtype S) = S :=\n  ext <| Set.ext_iff.1 <| (NonUnitalSubalgebraClass.subtype S).coe_range.trans Subtype.range_val\n\n/-- The map `S → T` when `S` is a non-unital subalgebra contained in the non-unital subalgebra `T`.\n\nThis is the non-unital subalgebra version of `Submodule.inclusion`, or `Subring.inclusion`  -/\ndef inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) : S →ₙₐ[R] T\n    where\n  toFun := Set.inclusion h\n  map_add' _ _ := rfl\n  map_mul' _ _ := rfl\n  map_zero' := rfl\n  map_smul' _ _ := rfl\n\ntheorem inclusion_injective {S T : NonUnitalSubalgebra R A} (h : S ≤ T) :\n    Function.Injective (inclusion h) := fun _ _ => Subtype.ext ∘ Subtype.mk.inj\n\n@[simp]\ntheorem inclusion_self {S : NonUnitalSubalgebra R A} :\n    inclusion (le_refl S) = NonUnitalAlgHom.id R S :=\n  NonUnitalAlgHom.ext fun _ => Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_mk {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : A) (hx : x ∈ S) :\n    inclusion h ⟨x, hx⟩ = ⟨x, h hx⟩ :=\n  rfl\n\ntheorem inclusion_right {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : T) (m : (x : A) ∈ S) :\n    inclusion h ⟨x, m⟩ = x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_inclusion {S T U : NonUnitalSubalgebra R A} (hst : S ≤ T) (htu : T ≤ U) (x : S) :\n    inclusion htu (inclusion hst x) = inclusion (le_trans hst htu) x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem coe_inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (s : S) :\n    (inclusion h s : A) = s :=\n  rfl\n\nsection Prod\n\nvariable (S₁ : NonUnitalSubalgebra R B)\n\n/-- The product of two non-unital subalgebras is a non-unital subalgebra. -/\ndef prod : NonUnitalSubalgebra R (A × B) :=\n  { S.toNonUnitalSubsemiring.prod S₁.toNonUnitalSubsemiring with\n    carrier := S ×ˢ S₁\n    smul_mem' := fun r _x hx => ⟨SMulMemClass.smul_mem r hx.1, SMulMemClass.smul_mem r hx.2⟩ }\n\n@[simp]\ntheorem coe_prod : (prod S S₁ : Set (A × B)) = (S : Set A) ×ˢ S₁ :=\n  rfl\n\ntheorem prod_toSubmodule : (S.prod S₁).toSubmodule = S.toSubmodule.prod S₁.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem mem_prod {S : NonUnitalSubalgebra R A} {S₁ : NonUnitalSubalgebra R B} {x : A × B} :\n    x ∈ prod S S₁ ↔ x.1 ∈ S ∧ x.2 ∈ S₁ :=\n  Set.mem_prod\n\n@[simp]\ntheorem prod_top : (prod ⊤ ⊤ : NonUnitalSubalgebra R (A × B)) = ⊤ := by ext; simp\n\ntheorem prod_mono {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S ≤ T → S₁ ≤ T₁ → prod S S₁ ≤ prod T T₁ :=\n  Set.prod_mono\n\n@[simp]\ntheorem prod_inf_prod {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S.prod S₁ ⊓ T.prod T₁ = (S ⊓ T).prod (S₁ ⊓ T₁) :=\n  SetLike.coe_injective Set.prod_inter_prod\n\nend Prod\n\nsection SuprLift\n\nvariable {ι : Type*}\n\ntheorem coe_iSup_of_directed [Nonempty ι] {S : ι → NonUnitalSubalgebra R A}\n    (dir : Directed (· ≤ ·) S) : ↑(iSup S) = ⋃ i, (S i : Set A) :=\n  let K : NonUnitalSubalgebra R A :=\n    { __ := NonUnitalSubsemiring.copy _ _ (NonUnitalSubsemiring.coe_iSup_of_directed dir).symm\n      smul_mem' := fun r _x hx ↦\n        let ⟨i, hi⟩ := Set.mem_iUnion.1 hx\n        Set.mem_iUnion.2 ⟨i, (S i).smul_mem' r hi⟩ }\n  have : iSup S = K := le_antisymm\n    (iSup_le fun i ↦ le_iSup (fun i ↦ (S i : Set A)) i) (Set.iUnion_subset fun _ ↦ le_iSup S _)\n  this.symm ▸ rfl\n\n/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              rw [hf j k hjk]\n              rfl)\n            (↑(iSup K)) (by rw [coe_iSup_of_directed dir])\n        map_zero' := by\n          dsimp\n          exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl) _ (by simp)\n        map_mul' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· * ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_add' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· + ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_smul' := fun r => by\n          dsimp\n          apply Set.iUnionLift_unary (coe_iSup_of_directed dir) _ (fun _ x => r • x)\n            (fun _ _ => rfl)\n          on_goal 2 => rw [coe_iSup_of_directed dir]\n          all_goals simp }\n\nvariable [Nonempty ι] {K : ι → NonUnitalSubalgebra R A} {dir : Directed (· ≤ ·) K}\n  {f : ∀ i, K i →ₙₐ[R] B} {hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h)}\n  {T : NonUnitalSubalgebra R A} {hT : T = iSup K}\n\n@[simp]\ntheorem iSupLift_inclusion {i : ι} (x : K i) (h : K i ≤ T) :\n    iSupLift K dir f hf T hT (inclusion h x) = f i x := by\n  subst T\n  dsimp [iSupLift]\n  apply Set.iUnionLift_inclusion\n  exact h\n\n@[simp]\ntheorem iSupLift_comp_inclusion {i : ι} (h : K i ≤ T) :\n    (iSupLift K dir f hf T hT).comp (inclusion h) = f i := by ext; simp\n\n@[simp]\ntheorem iSupLift_mk {i : ι} (x : K i) (hx : (x : A) ∈ T) :\n    iSupLift K dir f hf T hT ⟨x, hx⟩ = f i x := by\n  subst hT\n  dsimp [iSupLift]\n  apply Set.iUnionLift_mk\n\ntheorem iSupLift_of_mem {i : ι} (x : T) (hx : (x : A) ∈ K i) :\n    iSupLift K dir f hf T hT x = f i ⟨x, hx⟩ := by\n  subst hT\n  dsimp [iSupLift]\n  apply Set.iUnionLift_of_mem\n\nend SuprLift\n\nend NonAssoc\n\nsection Center\n\nsection NonUnitalNonAssocSemiring\nvariable {R A : Type*}\nvariable [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [IsScalarTower R A A] [SMulCommClass R A A]\n\ntheorem _root_.Set.smul_mem_center (r : R) {a : A} (ha : a ∈ Set.center A) :\n    r • a ∈ Set.center A where\n  comm b := by rw [mul_smul_comm, smul_mul_assoc, ha.comm]\n  left_assoc b c := by rw [smul_mul_assoc, smul_mul_assoc, smul_mul_assoc, ha.left_assoc]\n  mid_assoc b c := by\n    rw [mul_smul_comm]\n    ","nextTactic":"rw [smul_mul_assoc]","declUpToTactic":"theorem _root_.Set.smul_mem_center (r : R) {a : A} (ha : a ∈ Set.center A) :\n    r • a ∈ Set.center A where\n  comm b := by rw [mul_smul_comm, smul_mul_assoc, ha.comm]\n  left_assoc b c := by rw [smul_mul_assoc, smul_mul_assoc, smul_mul_assoc, ha.left_assoc]\n  mid_assoc b c := by\n    rw [mul_smul_comm]\n    ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.969_0.jQvdAcOcyVirh23","decl":"theorem _root_.Set.smul_mem_center (r : R) {a : A} (ha : a ∈ Set.center A) :\n    r • a ∈ Set.center A where\n  comm b "}
-{"state":"R : Type u_1\nA : Type u_2\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nr : R\na : A\nha : a ∈ Set.center A\nb c : A\n⊢ r • (b * a * c) = b * (r • a * c)","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact NonUnitalAlgebra.gc\n\n@[simp]\ntheorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=\n  eq_top_iff.2 fun _x hx => subset_adjoin R hx\n\nvariable {R A}\n\n@[simp]\ntheorem coe_top : (↑(⊤ : NonUnitalSubalgebra R A) : Set A) = Set.univ :=\n  rfl\n\n@[simp]\ntheorem mem_top {x : A} : x ∈ (⊤ : NonUnitalSubalgebra R A) :=\n  Set.mem_univ x\n\n@[simp]\ntheorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toNonUnitalSubsemiring : (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubsemiring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toSubring {R A : Type*} [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    [IsScalarTower R A A] [SMulCommClass R A A] :\n    (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem toSubmodule_eq_top {S : NonUnitalSubalgebra R A} : S.toSubmodule = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toSubmodule'.injective.eq_iff' top_toSubmodule\n\n@[simp]\ntheorem toNonUnitalSubsemiring_eq_top {S : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubsemiring_injective.eq_iff' top_toNonUnitalSubsemiring\n\n@[simp]\ntheorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubring_injective.eq_iff' top_toSubring\n\ntheorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_left\n\ntheorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_right\n\ntheorem mul_mem_sup {S T : NonUnitalSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) :\n    x * y ∈ S ⊔ T :=\n  mul_mem (mem_sup_left hx) (mem_sup_right hy)\n\ntheorem map_sup (f : F) (S T : NonUnitalSubalgebra R A) :\n    ((S ⊔ T).map f : NonUnitalSubalgebra R B) = S.map f ⊔ T.map f :=\n  (@NonUnitalSubalgebra.gc_map_comap F R A B _ _ _ _ _ _ f).l_sup\n\n@[simp, norm_cast]\ntheorem coe_inf (S T : NonUnitalSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T :=\n  rfl\n\n@[simp]\ntheorem mem_inf {S T : NonUnitalSubalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T :=\n  Iff.rfl\n\n@[simp]\ntheorem inf_toSubmodule (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toSubmodule = S.toSubmodule ⊓ T.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem inf_toNonUnitalSubsemiring (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring ⊓ T.toNonUnitalSubsemiring :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sInf (S : Set (NonUnitalSubalgebra R A)) : (↑(sInf S) : Set A) = ⋂ s ∈ S, ↑s :=\n  sInf_image\n\ntheorem mem_sInf {S : Set (NonUnitalSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p := by\n  simp only [← SetLike.mem_coe, coe_sInf, Set.mem_iInter₂]\n\n@[simp]\ntheorem sInf_toSubmodule (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toSubmodule = sInf (NonUnitalSubalgebra.toSubmodule '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp]\ntheorem sInf_toNonUnitalSubsemiring (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toNonUnitalSubsemiring = sInf (NonUnitalSubalgebra.toNonUnitalSubsemiring '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp, norm_cast]\ntheorem coe_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} :\n    (↑(⨅ i, S i) : Set A) = ⋂ i, S i := by simp [iInf]\n\ntheorem mem_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} {x : A} :\n    (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by simp only [iInf, mem_sInf, Set.forall_range_iff]\n\n@[simp]\ntheorem iInf_toSubmodule {ι : Sort*} (S : ι → NonUnitalSubalgebra R A) :\n    (⨅ i, S i).toSubmodule = ⨅ i, (S i).toSubmodule :=\n  SetLike.coe_injective <| by simp\n\ninstance : Inhabited (NonUnitalSubalgebra R A) :=\n  ⟨⊥⟩\n\ntheorem mem_bot {x : A} : x ∈ (⊥ : NonUnitalSubalgebra R A) ↔ x = 0 :=\n  show x ∈ Submodule.span R (NonUnitalSubsemiring.closure (∅ : Set A) : Set A) ↔ x = 0 by\n    rw [NonUnitalSubsemiring.closure_empty, NonUnitalSubsemiring.coe_bot,\n      Submodule.span_zero_singleton, Submodule.mem_bot]\n\ntheorem toSubmodule_bot : (⊥ : NonUnitalSubalgebra R A).toSubmodule = ⊥ := by\n  ext\n  simp only [mem_bot, NonUnitalSubalgebra.mem_toSubmodule, Submodule.mem_bot]\n\n@[simp]\ntheorem coe_bot : ((⊥ : NonUnitalSubalgebra R A) : Set A) = {0} := by\n  simp [Set.ext_iff, NonUnitalAlgebra.mem_bot]\n\ntheorem eq_top_iff {S : NonUnitalSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S :=\n  ⟨fun h x => by rw [h]; exact mem_top, fun h => by ext x; exact ⟨fun _ => mem_top, fun _ => h x⟩⟩\n\ntheorem range_top_iff_surjective (f : A →ₙₐ[R] B) :\n    NonUnitalAlgHom.range f = (⊤ : NonUnitalSubalgebra R B) ↔ Function.Surjective f :=\n  NonUnitalAlgebra.eq_top_iff\n\n@[simp]\ntheorem range_id : NonUnitalAlgHom.range (NonUnitalAlgHom.id R A) = ⊤ :=\n  SetLike.coe_injective Set.range_id\n\n@[simp]\ntheorem map_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R A).map f = NonUnitalAlgHom.range f :=\n  SetLike.coe_injective Set.image_univ\n\n@[simp]\ntheorem map_bot (f : A →ₙₐ[R] B) : (⊥ : NonUnitalSubalgebra R A).map f = ⊥ :=\n  SetLike.coe_injective <| by simp [NonUnitalAlgebra.coe_bot, NonUnitalSubalgebra.coe_map]\n\n@[simp]\ntheorem comap_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R B).comap f = ⊤ :=\n  eq_top_iff.2 fun _ => mem_top\n\n/-- `NonUnitalAlgHom` to `⊤ : NonUnitalSubalgebra R A`. -/\ndef toTop : A →���ₐ[R] (⊤ : NonUnitalSubalgebra R A) :=\n  NonUnitalAlgHom.codRestrict (NonUnitalAlgHom.id R A) ⊤ fun _ => mem_top\n\nend NonUnitalAlgebra\n\nnamespace NonUnitalSubalgebra\n\nopen NonUnitalAlgebra\n\nsection NonAssoc\n\nvariable {R : Type u} {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable (S : NonUnitalSubalgebra R A)\n\ninstance subsingleton_of_subsingleton [Subsingleton A] : Subsingleton (NonUnitalSubalgebra R A) :=\n  ⟨fun B C => ext fun x => by simp only [Subsingleton.elim x 0, zero_mem B, zero_mem C]⟩\n\ninstance _root_.NonUnitalAlgHom.subsingleton [Subsingleton (NonUnitalSubalgebra R A)] :\n    Subsingleton (A →ₙₐ[R] B) :=\n  ⟨fun f g =>\n    NonUnitalAlgHom.ext fun a =>\n      have : a ∈ (⊥ : NonUnitalSubalgebra R A) :=\n        Subsingleton.elim (⊤ : NonUnitalSubalgebra R A) ⊥ ▸ mem_top\n      (mem_bot.mp this).symm ▸ (map_zero f).trans (map_zero g).symm⟩\n\ntheorem range_val : NonUnitalAlgHom.range (NonUnitalSubalgebraClass.subtype S) = S :=\n  ext <| Set.ext_iff.1 <| (NonUnitalSubalgebraClass.subtype S).coe_range.trans Subtype.range_val\n\n/-- The map `S → T` when `S` is a non-unital subalgebra contained in the non-unital subalgebra `T`.\n\nThis is the non-unital subalgebra version of `Submodule.inclusion`, or `Subring.inclusion`  -/\ndef inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) : S →ₙₐ[R] T\n    where\n  toFun := Set.inclusion h\n  map_add' _ _ := rfl\n  map_mul' _ _ := rfl\n  map_zero' := rfl\n  map_smul' _ _ := rfl\n\ntheorem inclusion_injective {S T : NonUnitalSubalgebra R A} (h : S ≤ T) :\n    Function.Injective (inclusion h) := fun _ _ => Subtype.ext ∘ Subtype.mk.inj\n\n@[simp]\ntheorem inclusion_self {S : NonUnitalSubalgebra R A} :\n    inclusion (le_refl S) = NonUnitalAlgHom.id R S :=\n  NonUnitalAlgHom.ext fun _ => Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_mk {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : A) (hx : x ∈ S) :\n    inclusion h ⟨x, hx⟩ = ⟨x, h hx⟩ :=\n  rfl\n\ntheorem inclusion_right {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : T) (m : (x : A) ∈ S) :\n    inclusion h ⟨x, m⟩ = x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_inclusion {S T U : NonUnitalSubalgebra R A} (hst : S ≤ T) (htu : T ≤ U) (x : S) :\n    inclusion htu (inclusion hst x) = inclusion (le_trans hst htu) x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem coe_inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (s : S) :\n    (inclusion h s : A) = s :=\n  rfl\n\nsection Prod\n\nvariable (S₁ : NonUnitalSubalgebra R B)\n\n/-- The product of two non-unital subalgebras is a non-unital subalgebra. -/\ndef prod : NonUnitalSubalgebra R (A × B) :=\n  { S.toNonUnitalSubsemiring.prod S₁.toNonUnitalSubsemiring with\n    carrier := S ×ˢ S₁\n    smul_mem' := fun r _x hx => ⟨SMulMemClass.smul_mem r hx.1, SMulMemClass.smul_mem r hx.2⟩ }\n\n@[simp]\ntheorem coe_prod : (prod S S₁ : Set (A × B)) = (S : Set A) ×ˢ S₁ :=\n  rfl\n\ntheorem prod_toSubmodule : (S.prod S₁).toSubmodule = S.toSubmodule.prod S₁.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem mem_prod {S : NonUnitalSubalgebra R A} {S₁ : NonUnitalSubalgebra R B} {x : A × B} :\n    x ∈ prod S S₁ ↔ x.1 ∈ S ∧ x.2 ∈ S₁ :=\n  Set.mem_prod\n\n@[simp]\ntheorem prod_top : (prod ⊤ ⊤ : NonUnitalSubalgebra R (A × B)) = ⊤ := by ext; simp\n\ntheorem prod_mono {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S ≤ T → S₁ ≤ T₁ → prod S S₁ ≤ prod T T₁ :=\n  Set.prod_mono\n\n@[simp]\ntheorem prod_inf_prod {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S.prod S₁ ⊓ T.prod T₁ = (S ⊓ T).prod (S₁ ⊓ T₁) :=\n  SetLike.coe_injective Set.prod_inter_prod\n\nend Prod\n\nsection SuprLift\n\nvariable {ι : Type*}\n\ntheorem coe_iSup_of_directed [Nonempty ι] {S : ι → NonUnitalSubalgebra R A}\n    (dir : Directed (· ≤ ·) S) : ↑(iSup S) = ⋃ i, (S i : Set A) :=\n  let K : NonUnitalSubalgebra R A :=\n    { __ := NonUnitalSubsemiring.copy _ _ (NonUnitalSubsemiring.coe_iSup_of_directed dir).symm\n      smul_mem' := fun r _x hx ↦\n        let ⟨i, hi⟩ := Set.mem_iUnion.1 hx\n        Set.mem_iUnion.2 ⟨i, (S i).smul_mem' r hi⟩ }\n  have : iSup S = K := le_antisymm\n    (iSup_le fun i ↦ le_iSup (fun i ↦ (S i : Set A)) i) (Set.iUnion_subset fun _ ↦ le_iSup S _)\n  this.symm ▸ rfl\n\n/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              rw [hf j k hjk]\n              rfl)\n            (↑(iSup K)) (by rw [coe_iSup_of_directed dir])\n        map_zero' := by\n          dsimp\n          exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl) _ (by simp)\n        map_mul' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· * ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_add' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· + ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_smul' := fun r => by\n          dsimp\n          apply Set.iUnionLift_unary (coe_iSup_of_directed dir) _ (fun _ x => r • x)\n            (fun _ _ => rfl)\n          on_goal 2 => rw [coe_iSup_of_directed dir]\n          all_goals simp }\n\nvariable [Nonempty ι] {K : ι → NonUnitalSubalgebra R A} {dir : Directed (· ≤ ·) K}\n  {f : ∀ i, K i →ₙₐ[R] B} {hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h)}\n  {T : NonUnitalSubalgebra R A} {hT : T = iSup K}\n\n@[simp]\ntheorem iSupLift_inclusion {i : ι} (x : K i) (h : K i ≤ T) :\n    iSupLift K dir f hf T hT (inclusion h x) = f i x := by\n  subst T\n  dsimp [iSupLift]\n  apply Set.iUnionLift_inclusion\n  exact h\n\n@[simp]\ntheorem iSupLift_comp_inclusion {i : ι} (h : K i ≤ T) :\n    (iSupLift K dir f hf T hT).comp (inclusion h) = f i := by ext; simp\n\n@[simp]\ntheorem iSupLift_mk {i : ι} (x : K i) (hx : (x : A) ∈ T) :\n    iSupLift K dir f hf T hT ⟨x, hx⟩ = f i x := by\n  subst hT\n  dsimp [iSupLift]\n  apply Set.iUnionLift_mk\n\ntheorem iSupLift_of_mem {i : ι} (x : T) (hx : (x : A) ∈ K i) :\n    iSupLift K dir f hf T hT x = f i ⟨x, hx⟩ := by\n  subst hT\n  dsimp [iSupLift]\n  apply Set.iUnionLift_of_mem\n\nend SuprLift\n\nend NonAssoc\n\nsection Center\n\nsection NonUnitalNonAssocSemiring\nvariable {R A : Type*}\nvariable [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [IsScalarTower R A A] [SMulCommClass R A A]\n\ntheorem _root_.Set.smul_mem_center (r : R) {a : A} (ha : a ∈ Set.center A) :\n    r • a ∈ Set.center A where\n  comm b := by rw [mul_smul_comm, smul_mul_assoc, ha.comm]\n  left_assoc b c := by rw [smul_mul_assoc, smul_mul_assoc, smul_mul_assoc, ha.left_assoc]\n  mid_assoc b c := by\n    rw [mul_smul_comm]\n    rw [smul_mul_assoc]\n    ","nextTactic":"rw [smul_mul_assoc]","declUpToTactic":"theorem _root_.Set.smul_mem_center (r : R) {a : A} (ha : a ∈ Set.center A) :\n    r • a ∈ Set.center A where\n  comm b := by rw [mul_smul_comm, smul_mul_assoc, ha.comm]\n  left_assoc b c := by rw [smul_mul_assoc, smul_mul_assoc, smul_mul_assoc, ha.left_assoc]\n  mid_assoc b c := by\n    rw [mul_smul_comm]\n    rw [smul_mul_assoc]\n    ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.969_0.jQvdAcOcyVirh23","decl":"theorem _root_.Set.smul_mem_center (r : R) {a : A} (ha : a ∈ Set.center A) :\n    r • a ∈ Set.center A where\n  comm b "}
-{"state":"R : Type u_1\nA : Type u_2\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nr : R\na : A\nha : a ∈ Set.center A\nb c : A\n⊢ r • (b * a * c) = b * r • (a * c)","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact NonUnitalAlgebra.gc\n\n@[simp]\ntheorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=\n  eq_top_iff.2 fun _x hx => subset_adjoin R hx\n\nvariable {R A}\n\n@[simp]\ntheorem coe_top : (↑(⊤ : NonUnitalSubalgebra R A) : Set A) = Set.univ :=\n  rfl\n\n@[simp]\ntheorem mem_top {x : A} : x ∈ (⊤ : NonUnitalSubalgebra R A) :=\n  Set.mem_univ x\n\n@[simp]\ntheorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toNonUnitalSubsemiring : (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubsemiring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toSubring {R A : Type*} [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    [IsScalarTower R A A] [SMulCommClass R A A] :\n    (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem toSubmodule_eq_top {S : NonUnitalSubalgebra R A} : S.toSubmodule = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toSubmodule'.injective.eq_iff' top_toSubmodule\n\n@[simp]\ntheorem toNonUnitalSubsemiring_eq_top {S : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubsemiring_injective.eq_iff' top_toNonUnitalSubsemiring\n\n@[simp]\ntheorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubring_injective.eq_iff' top_toSubring\n\ntheorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_left\n\ntheorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_right\n\ntheorem mul_mem_sup {S T : NonUnitalSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) :\n    x * y ∈ S ⊔ T :=\n  mul_mem (mem_sup_left hx) (mem_sup_right hy)\n\ntheorem map_sup (f : F) (S T : NonUnitalSubalgebra R A) :\n    ((S ⊔ T).map f : NonUnitalSubalgebra R B) = S.map f ⊔ T.map f :=\n  (@NonUnitalSubalgebra.gc_map_comap F R A B _ _ _ _ _ _ f).l_sup\n\n@[simp, norm_cast]\ntheorem coe_inf (S T : NonUnitalSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T :=\n  rfl\n\n@[simp]\ntheorem mem_inf {S T : NonUnitalSubalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T :=\n  Iff.rfl\n\n@[simp]\ntheorem inf_toSubmodule (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toSubmodule = S.toSubmodule ⊓ T.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem inf_toNonUnitalSubsemiring (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring ⊓ T.toNonUnitalSubsemiring :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sInf (S : Set (NonUnitalSubalgebra R A)) : (↑(sInf S) : Set A) = ⋂ s ∈ S, ↑s :=\n  sInf_image\n\ntheorem mem_sInf {S : Set (NonUnitalSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p := by\n  simp only [← SetLike.mem_coe, coe_sInf, Set.mem_iInter₂]\n\n@[simp]\ntheorem sInf_toSubmodule (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toSubmodule = sInf (NonUnitalSubalgebra.toSubmodule '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp]\ntheorem sInf_toNonUnitalSubsemiring (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toNonUnitalSubsemiring = sInf (NonUnitalSubalgebra.toNonUnitalSubsemiring '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp, norm_cast]\ntheorem coe_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} :\n    (↑(⨅ i, S i) : Set A) = ⋂ i, S i := by simp [iInf]\n\ntheorem mem_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} {x : A} :\n    (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by simp only [iInf, mem_sInf, Set.forall_range_iff]\n\n@[simp]\ntheorem iInf_toSubmodule {ι : Sort*} (S : ι → NonUnitalSubalgebra R A) :\n    (⨅ i, S i).toSubmodule = ⨅ i, (S i).toSubmodule :=\n  SetLike.coe_injective <| by simp\n\ninstance : Inhabited (NonUnitalSubalgebra R A) :=\n  ⟨⊥⟩\n\ntheorem mem_bot {x : A} : x ∈ (⊥ : NonUnitalSubalgebra R A) ↔ x = 0 :=\n  show x ∈ Submodule.span R (NonUnitalSubsemiring.closure (∅ : Set A) : Set A) ↔ x = 0 by\n    rw [NonUnitalSubsemiring.closure_empty, NonUnitalSubsemiring.coe_bot,\n      Submodule.span_zero_singleton, Submodule.mem_bot]\n\ntheorem toSubmodule_bot : (⊥ : NonUnitalSubalgebra R A).toSubmodule = ⊥ := by\n  ext\n  simp only [mem_bot, NonUnitalSubalgebra.mem_toSubmodule, Submodule.mem_bot]\n\n@[simp]\ntheorem coe_bot : ((⊥ : NonUnitalSubalgebra R A) : Set A) = {0} := by\n  simp [Set.ext_iff, NonUnitalAlgebra.mem_bot]\n\ntheorem eq_top_iff {S : NonUnitalSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S :=\n  ⟨fun h x => by rw [h]; exact mem_top, fun h => by ext x; exact ⟨fun _ => mem_top, fun _ => h x⟩⟩\n\ntheorem range_top_iff_surjective (f : A →ₙₐ[R] B) :\n    NonUnitalAlgHom.range f = (⊤ : NonUnitalSubalgebra R B) ↔ Function.Surjective f :=\n  NonUnitalAlgebra.eq_top_iff\n\n@[simp]\ntheorem range_id : NonUnitalAlgHom.range (NonUnitalAlgHom.id R A) = ⊤ :=\n  SetLike.coe_injective Set.range_id\n\n@[simp]\ntheorem map_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R A).map f = NonUnitalAlgHom.range f :=\n  SetLike.coe_injective Set.image_univ\n\n@[simp]\ntheorem map_bot (f : A →ₙₐ[R] B) : (⊥ : NonUnitalSubalgebra R A).map f = ⊥ :=\n  SetLike.coe_injective <| by simp [NonUnitalAlgebra.coe_bot, NonUnitalSubalgebra.coe_map]\n\n@[simp]\ntheorem comap_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R B).comap f = ⊤ :=\n  eq_top_iff.2 fun _ => mem_top\n\n/-- `NonUnitalAlgHom` to `⊤ : NonUnitalSubalgebra R A`. -/\ndef toTop : A →ₙₐ[R] (⊤ : NonUnitalSubalgebra R A) :=\n  NonUnitalAlgHom.codRestrict (NonUnitalAlgHom.id R A) ⊤ fun _ => mem_top\n\nend NonUnitalAlgebra\n\nnamespace NonUnitalSubalgebra\n\nopen NonUnitalAlgebra\n\nsection NonAssoc\n\nvariable {R : Type u} {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable (S : NonUnitalSubalgebra R A)\n\ninstance subsingleton_of_subsingleton [Subsingleton A] : Subsingleton (NonUnitalSubalgebra R A) :=\n  ⟨fun B C => ext fun x => by simp only [Subsingleton.elim x 0, zero_mem B, zero_mem C]⟩\n\ninstance _root_.NonUnitalAlgHom.subsingleton [Subsingleton (NonUnitalSubalgebra R A)] :\n    Subsingleton (A →ₙₐ[R] B) :=\n  ⟨fun f g =>\n    NonUnitalAlgHom.ext fun a =>\n      have : a ∈ (⊥ : NonUnitalSubalgebra R A) :=\n        Subsingleton.elim (⊤ : NonUnitalSubalgebra R A) ⊥ ▸ mem_top\n      (mem_bot.mp this).symm ▸ (map_zero f).trans (map_zero g).symm⟩\n\ntheorem range_val : NonUnitalAlgHom.range (NonUnitalSubalgebraClass.subtype S) = S :=\n  ext <| Set.ext_iff.1 <| (NonUnitalSubalgebraClass.subtype S).coe_range.trans Subtype.range_val\n\n/-- The map `S → T` when `S` is a non-unital subalgebra contained in the non-unital subalgebra `T`.\n\nThis is the non-unital subalgebra version of `Submodule.inclusion`, or `Subring.inclusion`  -/\ndef inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) : S →ₙₐ[R] T\n    where\n  toFun := Set.inclusion h\n  map_add' _ _ := rfl\n  map_mul' _ _ := rfl\n  map_zero' := rfl\n  map_smul' _ _ := rfl\n\ntheorem inclusion_injective {S T : NonUnitalSubalgebra R A} (h : S ≤ T) :\n    Function.Injective (inclusion h) := fun _ _ => Subtype.ext ∘ Subtype.mk.inj\n\n@[simp]\ntheorem inclusion_self {S : NonUnitalSubalgebra R A} :\n    inclusion (le_refl S) = NonUnitalAlgHom.id R S :=\n  NonUnitalAlgHom.ext fun _ => Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_mk {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : A) (hx : x ∈ S) :\n    inclusion h ⟨x, hx⟩ = ⟨x, h hx⟩ :=\n  rfl\n\ntheorem inclusion_right {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : T) (m : (x : A) ∈ S) :\n    inclusion h ⟨x, m⟩ = x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_inclusion {S T U : NonUnitalSubalgebra R A} (hst : S ≤ T) (htu : T ≤ U) (x : S) :\n    inclusion htu (inclusion hst x) = inclusion (le_trans hst htu) x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem coe_inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (s : S) :\n    (inclusion h s : A) = s :=\n  rfl\n\nsection Prod\n\nvariable (S₁ : NonUnitalSubalgebra R B)\n\n/-- The product of two non-unital subalgebras is a non-unital subalgebra. -/\ndef prod : NonUnitalSubalgebra R (A × B) :=\n  { S.toNonUnitalSubsemiring.prod S₁.toNonUnitalSubsemiring with\n    carrier := S ×ˢ S₁\n    smul_mem' := fun r _x hx => ⟨SMulMemClass.smul_mem r hx.1, SMulMemClass.smul_mem r hx.2⟩ }\n\n@[simp]\ntheorem coe_prod : (prod S S₁ : Set (A × B)) = (S : Set A) ×ˢ S₁ :=\n  rfl\n\ntheorem prod_toSubmodule : (S.prod S₁).toSubmodule = S.toSubmodule.prod S₁.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem mem_prod {S : NonUnitalSubalgebra R A} {S₁ : NonUnitalSubalgebra R B} {x : A × B} :\n    x ∈ prod S S₁ ↔ x.1 ∈ S ∧ x.2 ∈ S₁ :=\n  Set.mem_prod\n\n@[simp]\ntheorem prod_top : (prod ⊤ ⊤ : NonUnitalSubalgebra R (A × B)) = ⊤ := by ext; simp\n\ntheorem prod_mono {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S ≤ T → S₁ ≤ T₁ → prod S S₁ ≤ prod T T₁ :=\n  Set.prod_mono\n\n@[simp]\ntheorem prod_inf_prod {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S.prod S₁ ⊓ T.prod T₁ = (S ⊓ T).prod (S₁ ⊓ T₁) :=\n  SetLike.coe_injective Set.prod_inter_prod\n\nend Prod\n\nsection SuprLift\n\nvariable {ι : Type*}\n\ntheorem coe_iSup_of_directed [Nonempty ι] {S : ι → NonUnitalSubalgebra R A}\n    (dir : Directed (· ≤ ·) S) : ↑(iSup S) = ⋃ i, (S i : Set A) :=\n  let K : NonUnitalSubalgebra R A :=\n    { __ := NonUnitalSubsemiring.copy _ _ (NonUnitalSubsemiring.coe_iSup_of_directed dir).symm\n      smul_mem' := fun r _x hx ↦\n        let ⟨i, hi⟩ := Set.mem_iUnion.1 hx\n        Set.mem_iUnion.2 ⟨i, (S i).smul_mem' r hi⟩ }\n  have : iSup S = K := le_antisymm\n    (iSup_le fun i ↦ le_iSup (fun i ↦ (S i : Set A)) i) (Set.iUnion_subset fun _ ↦ le_iSup S _)\n  this.symm ▸ rfl\n\n/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              rw [hf j k hjk]\n              rfl)\n            (↑(iSup K)) (by rw [coe_iSup_of_directed dir])\n        map_zero' := by\n          dsimp\n          exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl) _ (by simp)\n        map_mul' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· * ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_add' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· + ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_smul' := fun r => by\n          dsimp\n          apply Set.iUnionLift_unary (coe_iSup_of_directed dir) _ (fun _ x => r • x)\n            (fun _ _ => rfl)\n          on_goal 2 => rw [coe_iSup_of_directed dir]\n          all_goals simp }\n\nvariable [Nonempty ι] {K : ι → NonUnitalSubalgebra R A} {dir : Directed (· ≤ ·) K}\n  {f : ∀ i, K i →ₙₐ[R] B} {hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h)}\n  {T : NonUnitalSubalgebra R A} {hT : T = iSup K}\n\n@[simp]\ntheorem iSupLift_inclusion {i : ι} (x : K i) (h : K i ≤ T) :\n    iSupLift K dir f hf T hT (inclusion h x) = f i x := by\n  subst T\n  dsimp [iSupLift]\n  apply Set.iUnionLift_inclusion\n  exact h\n\n@[simp]\ntheorem iSupLift_comp_inclusion {i : ι} (h : K i ≤ T) :\n    (iSupLift K dir f hf T hT).comp (inclusion h) = f i := by ext; simp\n\n@[simp]\ntheorem iSupLift_mk {i : ι} (x : K i) (hx : (x : A) ∈ T) :\n    iSupLift K dir f hf T hT ⟨x, hx⟩ = f i x := by\n  subst hT\n  dsimp [iSupLift]\n  apply Set.iUnionLift_mk\n\ntheorem iSupLift_of_mem {i : ι} (x : T) (hx : (x : A) ∈ K i) :\n    iSupLift K dir f hf T hT x = f i ⟨x, hx⟩ := by\n  subst hT\n  dsimp [iSupLift]\n  apply Set.iUnionLift_of_mem\n\nend SuprLift\n\nend NonAssoc\n\nsection Center\n\nsection NonUnitalNonAssocSemiring\nvariable {R A : Type*}\nvariable [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [IsScalarTower R A A] [SMulCommClass R A A]\n\ntheorem _root_.Set.smul_mem_center (r : R) {a : A} (ha : a ∈ Set.center A) :\n    r • a ∈ Set.center A where\n  comm b := by rw [mul_smul_comm, smul_mul_assoc, ha.comm]\n  left_assoc b c := by rw [smul_mul_assoc, smul_mul_assoc, smul_mul_assoc, ha.left_assoc]\n  mid_assoc b c := by\n    rw [mul_smul_comm]\n    rw [smul_mul_assoc]\n    rw [smul_mul_assoc]\n    ","nextTactic":"rw [mul_smul_comm]","declUpToTactic":"theorem _root_.Set.smul_mem_center (r : R) {a : A} (ha : a ∈ Set.center A) :\n    r • a ∈ Set.center A where\n  comm b := by rw [mul_smul_comm, smul_mul_assoc, ha.comm]\n  left_assoc b c := by rw [smul_mul_assoc, smul_mul_assoc, smul_mul_assoc, ha.left_assoc]\n  mid_assoc b c := by\n    rw [mul_smul_comm]\n    rw [smul_mul_assoc]\n    rw [smul_mul_assoc]\n    ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.969_0.jQvdAcOcyVirh23","decl":"theorem _root_.Set.smul_mem_center (r : R) {a : A} (ha : a ∈ Set.center A) :\n    r • a ∈ Set.center A where\n  comm b "}
-{"state":"R : Type u_1\nA : Type u_2\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nr : R\na : A\nha : a ∈ Set.center A\nb c : A\n⊢ r • (b * a * c) = r • (b * (a * c))","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact NonUnitalAlgebra.gc\n\n@[simp]\ntheorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=\n  eq_top_iff.2 fun _x hx => subset_adjoin R hx\n\nvariable {R A}\n\n@[simp]\ntheorem coe_top : (↑(⊤ : NonUnitalSubalgebra R A) : Set A) = Set.univ :=\n  rfl\n\n@[simp]\ntheorem mem_top {x : A} : x ∈ (⊤ : NonUnitalSubalgebra R A) :=\n  Set.mem_univ x\n\n@[simp]\ntheorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toNonUnitalSubsemiring : (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubsemiring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toSubring {R A : Type*} [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    [IsScalarTower R A A] [SMulCommClass R A A] :\n    (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem toSubmodule_eq_top {S : NonUnitalSubalgebra R A} : S.toSubmodule = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toSubmodule'.injective.eq_iff' top_toSubmodule\n\n@[simp]\ntheorem toNonUnitalSubsemiring_eq_top {S : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubsemiring_injective.eq_iff' top_toNonUnitalSubsemiring\n\n@[simp]\ntheorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubring_injective.eq_iff' top_toSubring\n\ntheorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_left\n\ntheorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_right\n\ntheorem mul_mem_sup {S T : NonUnitalSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) :\n    x * y ∈ S ⊔ T :=\n  mul_mem (mem_sup_left hx) (mem_sup_right hy)\n\ntheorem map_sup (f : F) (S T : NonUnitalSubalgebra R A) :\n    ((S ⊔ T).map f : NonUnitalSubalgebra R B) = S.map f ⊔ T.map f :=\n  (@NonUnitalSubalgebra.gc_map_comap F R A B _ _ _ _ _ _ f).l_sup\n\n@[simp, norm_cast]\ntheorem coe_inf (S T : NonUnitalSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T :=\n  rfl\n\n@[simp]\ntheorem mem_inf {S T : NonUnitalSubalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T :=\n  Iff.rfl\n\n@[simp]\ntheorem inf_toSubmodule (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toSubmodule = S.toSubmodule ⊓ T.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem inf_toNonUnitalSubsemiring (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring ⊓ T.toNonUnitalSubsemiring :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sInf (S : Set (NonUnitalSubalgebra R A)) : (↑(sInf S) : Set A) = ⋂ s ∈ S, ↑s :=\n  sInf_image\n\ntheorem mem_sInf {S : Set (NonUnitalSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p := by\n  simp only [← SetLike.mem_coe, coe_sInf, Set.mem_iInter₂]\n\n@[simp]\ntheorem sInf_toSubmodule (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toSubmodule = sInf (NonUnitalSubalgebra.toSubmodule '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp]\ntheorem sInf_toNonUnitalSubsemiring (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toNonUnitalSubsemiring = sInf (NonUnitalSubalgebra.toNonUnitalSubsemiring '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp, norm_cast]\ntheorem coe_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} :\n    (↑(⨅ i, S i) : Set A) = ⋂ i, S i := by simp [iInf]\n\ntheorem mem_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} {x : A} :\n    (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by simp only [iInf, mem_sInf, Set.forall_range_iff]\n\n@[simp]\ntheorem iInf_toSubmodule {ι : Sort*} (S : ι → NonUnitalSubalgebra R A) :\n    (⨅ i, S i).toSubmodule = ⨅ i, (S i).toSubmodule :=\n  SetLike.coe_injective <| by simp\n\ninstance : Inhabited (NonUnitalSubalgebra R A) :=\n  ⟨⊥⟩\n\ntheorem mem_bot {x : A} : x ∈ (⊥ : NonUnitalSubalgebra R A) ↔ x = 0 :=\n  show x ∈ Submodule.span R (NonUnitalSubsemiring.closure (∅ : Set A) : Set A) ↔ x = 0 by\n    rw [NonUnitalSubsemiring.closure_empty, NonUnitalSubsemiring.coe_bot,\n      Submodule.span_zero_singleton, Submodule.mem_bot]\n\ntheorem toSubmodule_bot : (⊥ : NonUnitalSubalgebra R A).toSubmodule = ⊥ := by\n  ext\n  simp only [mem_bot, NonUnitalSubalgebra.mem_toSubmodule, Submodule.mem_bot]\n\n@[simp]\ntheorem coe_bot : ((⊥ : NonUnitalSubalgebra R A) : Set A) = {0} := by\n  simp [Set.ext_iff, NonUnitalAlgebra.mem_bot]\n\ntheorem eq_top_iff {S : NonUnitalSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S :=\n  ⟨fun h x => by rw [h]; exact mem_top, fun h => by ext x; exact ⟨fun _ => mem_top, fun _ => h x⟩⟩\n\ntheorem range_top_iff_surjective (f : A →ₙₐ[R] B) :\n    NonUnitalAlgHom.range f = (⊤ : NonUnitalSubalgebra R B) ↔ Function.Surjective f :=\n  NonUnitalAlgebra.eq_top_iff\n\n@[simp]\ntheorem range_id : NonUnitalAlgHom.range (NonUnitalAlgHom.id R A) = ⊤ :=\n  SetLike.coe_injective Set.range_id\n\n@[simp]\ntheorem map_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R A).map f = NonUnitalAlgHom.range f :=\n  SetLike.coe_injective Set.image_univ\n\n@[simp]\ntheorem map_bot (f : A →ₙₐ[R] B) : (⊥ : NonUnitalSubalgebra R A).map f = ⊥ :=\n  SetLike.coe_injective <| by simp [NonUnitalAlgebra.coe_bot, NonUnitalSubalgebra.coe_map]\n\n@[simp]\ntheorem comap_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R B).comap f = ⊤ :=\n  eq_top_iff.2 fun _ => mem_top\n\n/-- `NonUnitalAlgHom` to `⊤ : NonUnitalSubalgebra R A`. -/\ndef toTop : A →ₙₐ[R] (⊤ : NonUnitalSubalgebra R A) :=\n  NonUnitalAlgHom.codRestrict (NonUnitalAlgHom.id R A) ⊤ fun _ => mem_top\n\nend NonUnitalAlgebra\n\nnamespace NonUnitalSubalgebra\n\nopen NonUnitalAlgebra\n\nsection NonAssoc\n\nvariable {R : Type u} {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable (S : NonUnitalSubalgebra R A)\n\ninstance subsingleton_of_subsingleton [Subsingleton A] : Subsingleton (NonUnitalSubalgebra R A) :=\n  ⟨fun B C => ext fun x => by simp only [Subsingleton.elim x 0, zero_mem B, zero_mem C]⟩\n\ninstance _root_.NonUnitalAlgHom.subsingleton [Subsingleton (NonUnitalSubalgebra R A)] :\n    Subsingleton (A →ₙₐ[R] B) :=\n  ⟨fun f g =>\n    NonUnitalAlgHom.ext fun a =>\n      have : a ∈ (⊥ : NonUnitalSubalgebra R A) :=\n        Subsingleton.elim (⊤ : NonUnitalSubalgebra R A) ⊥ ▸ mem_top\n      (mem_bot.mp this).symm ▸ (map_zero f).trans (map_zero g).symm⟩\n\ntheorem range_val : NonUnitalAlgHom.range (NonUnitalSubalgebraClass.subtype S) = S :=\n  ext <| Set.ext_iff.1 <| (NonUnitalSubalgebraClass.subtype S).coe_range.trans Subtype.range_val\n\n/-- The map `S → T` when `S` is a non-unital subalgebra contained in the non-unital subalgebra `T`.\n\nThis is the non-unital subalgebra version of `Submodule.inclusion`, or `Subring.inclusion`  -/\ndef inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) : S →ₙₐ[R] T\n    where\n  toFun := Set.inclusion h\n  map_add' _ _ := rfl\n  map_mul' _ _ := rfl\n  map_zero' := rfl\n  map_smul' _ _ := rfl\n\ntheorem inclusion_injective {S T : NonUnitalSubalgebra R A} (h : S ≤ T) :\n    Function.Injective (inclusion h) := fun _ _ => Subtype.ext ∘ Subtype.mk.inj\n\n@[simp]\ntheorem inclusion_self {S : NonUnitalSubalgebra R A} :\n    inclusion (le_refl S) = NonUnitalAlgHom.id R S :=\n  NonUnitalAlgHom.ext fun _ => Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_mk {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : A) (hx : x ∈ S) :\n    inclusion h ⟨x, hx⟩ = ⟨x, h hx⟩ :=\n  rfl\n\ntheorem inclusion_right {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : T) (m : (x : A) ∈ S) :\n    inclusion h ⟨x, m⟩ = x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_inclusion {S T U : NonUnitalSubalgebra R A} (hst : S ≤ T) (htu : T ≤ U) (x : S) :\n    inclusion htu (inclusion hst x) = inclusion (le_trans hst htu) x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem coe_inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (s : S) :\n    (inclusion h s : A) = s :=\n  rfl\n\nsection Prod\n\nvariable (S₁ : NonUnitalSubalgebra R B)\n\n/-- The product of two non-unital subalgebras is a non-unital subalgebra. -/\ndef prod : NonUnitalSubalgebra R (A × B) :=\n  { S.toNonUnitalSubsemiring.prod S₁.toNonUnitalSubsemiring with\n    carrier := S ×ˢ S₁\n    smul_mem' := fun r _x hx => ⟨SMulMemClass.smul_mem r hx.1, SMulMemClass.smul_mem r hx.2⟩ }\n\n@[simp]\ntheorem coe_prod : (prod S S₁ : Set (A × B)) = (S : Set A) ×ˢ S₁ :=\n  rfl\n\ntheorem prod_toSubmodule : (S.prod S₁).toSubmodule = S.toSubmodule.prod S₁.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem mem_prod {S : NonUnitalSubalgebra R A} {S₁ : NonUnitalSubalgebra R B} {x : A × B} :\n    x ∈ prod S S₁ ↔ x.1 ∈ S ∧ x.2 ∈ S₁ :=\n  Set.mem_prod\n\n@[simp]\ntheorem prod_top : (prod ⊤ ⊤ : NonUnitalSubalgebra R (A × B)) = ⊤ := by ext; simp\n\ntheorem prod_mono {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S ≤ T → S₁ ≤ T₁ → prod S S₁ ≤ prod T T₁ :=\n  Set.prod_mono\n\n@[simp]\ntheorem prod_inf_prod {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S.prod S₁ ⊓ T.prod T₁ = (S ⊓ T).prod (S₁ ⊓ T₁) :=\n  SetLike.coe_injective Set.prod_inter_prod\n\nend Prod\n\nsection SuprLift\n\nvariable {ι : Type*}\n\ntheorem coe_iSup_of_directed [Nonempty ι] {S : ι → NonUnitalSubalgebra R A}\n    (dir : Directed (· ≤ ·) S) : ↑(iSup S) = ⋃ i, (S i : Set A) :=\n  let K : NonUnitalSubalgebra R A :=\n    { __ := NonUnitalSubsemiring.copy _ _ (NonUnitalSubsemiring.coe_iSup_of_directed dir).symm\n      smul_mem' := fun r _x hx ↦\n        let ⟨i, hi⟩ := Set.mem_iUnion.1 hx\n        Set.mem_iUnion.2 ⟨i, (S i).smul_mem' r hi⟩ }\n  have : iSup S = K := le_antisymm\n    (iSup_le fun i ↦ le_iSup (fun i ↦ (S i : Set A)) i) (Set.iUnion_subset fun _ ↦ le_iSup S _)\n  this.symm ▸ rfl\n\n/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              rw [hf j k hjk]\n              rfl)\n            (↑(iSup K)) (by rw [coe_iSup_of_directed dir])\n        map_zero' := by\n          dsimp\n          exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl) _ (by simp)\n        map_mul' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· * ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_add' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· + ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_smul' := fun r => by\n          dsimp\n          apply Set.iUnionLift_unary (coe_iSup_of_directed dir) _ (fun _ x => r • x)\n            (fun _ _ => rfl)\n          on_goal 2 => rw [coe_iSup_of_directed dir]\n          all_goals simp }\n\nvariable [Nonempty ι] {K : ι → NonUnitalSubalgebra R A} {dir : Directed (· ≤ ·) K}\n  {f : ∀ i, K i →ₙₐ[R] B} {hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h)}\n  {T : NonUnitalSubalgebra R A} {hT : T = iSup K}\n\n@[simp]\ntheorem iSupLift_inclusion {i : ι} (x : K i) (h : K i ≤ T) :\n    iSupLift K dir f hf T hT (inclusion h x) = f i x := by\n  subst T\n  dsimp [iSupLift]\n  apply Set.iUnionLift_inclusion\n  exact h\n\n@[simp]\ntheorem iSupLift_comp_inclusion {i : ι} (h : K i ≤ T) :\n    (iSupLift K dir f hf T hT).comp (inclusion h) = f i := by ext; simp\n\n@[simp]\ntheorem iSupLift_mk {i : ι} (x : K i) (hx : (x : A) ∈ T) :\n    iSupLift K dir f hf T hT ⟨x, hx⟩ = f i x := by\n  subst hT\n  dsimp [iSupLift]\n  apply Set.iUnionLift_mk\n\ntheorem iSupLift_of_mem {i : ι} (x : T) (hx : (x : A) ∈ K i) :\n    iSupLift K dir f hf T hT x = f i ⟨x, hx⟩ := by\n  subst hT\n  dsimp [iSupLift]\n  apply Set.iUnionLift_of_mem\n\nend SuprLift\n\nend NonAssoc\n\nsection Center\n\nsection NonUnitalNonAssocSemiring\nvariable {R A : Type*}\nvariable [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [IsScalarTower R A A] [SMulCommClass R A A]\n\ntheorem _root_.Set.smul_mem_center (r : R) {a : A} (ha : a ∈ Set.center A) :\n    r • a ∈ Set.center A where\n  comm b := by rw [mul_smul_comm, smul_mul_assoc, ha.comm]\n  left_assoc b c := by rw [smul_mul_assoc, smul_mul_assoc, smul_mul_assoc, ha.left_assoc]\n  mid_assoc b c := by\n    rw [mul_smul_comm]\n    rw [smul_mul_assoc]\n    rw [smul_mul_assoc]\n    rw [mul_smul_comm]\n    ","nextTactic":"rw [ha.mid_assoc]","declUpToTactic":"theorem _root_.Set.smul_mem_center (r : R) {a : A} (ha : a ∈ Set.center A) :\n    r • a ∈ Set.center A where\n  comm b := by rw [mul_smul_comm, smul_mul_assoc, ha.comm]\n  left_assoc b c := by rw [smul_mul_assoc, smul_mul_assoc, smul_mul_assoc, ha.left_assoc]\n  mid_assoc b c := by\n    rw [mul_smul_comm]\n    rw [smul_mul_assoc]\n    rw [smul_mul_assoc]\n    rw [mul_smul_comm]\n    ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.969_0.jQvdAcOcyVirh23","decl":"theorem _root_.Set.smul_mem_center (r : R) {a : A} (ha : a ∈ Set.center A) :\n    r • a ∈ Set.center A where\n  comm b "}
-{"state":"R : Type u_1\nA : Type u_2\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nr : R\na : A\nha : a ∈ Set.center A\nb c : A\n⊢ b * c * r • a = b * (c * r • a)","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact NonUnitalAlgebra.gc\n\n@[simp]\ntheorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=\n  eq_top_iff.2 fun _x hx => subset_adjoin R hx\n\nvariable {R A}\n\n@[simp]\ntheorem coe_top : (↑(⊤ : NonUnitalSubalgebra R A) : Set A) = Set.univ :=\n  rfl\n\n@[simp]\ntheorem mem_top {x : A} : x ∈ (⊤ : NonUnitalSubalgebra R A) :=\n  Set.mem_univ x\n\n@[simp]\ntheorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toNonUnitalSubsemiring : (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubsemiring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toSubring {R A : Type*} [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    [IsScalarTower R A A] [SMulCommClass R A A] :\n    (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem toSubmodule_eq_top {S : NonUnitalSubalgebra R A} : S.toSubmodule = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toSubmodule'.injective.eq_iff' top_toSubmodule\n\n@[simp]\ntheorem toNonUnitalSubsemiring_eq_top {S : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubsemiring_injective.eq_iff' top_toNonUnitalSubsemiring\n\n@[simp]\ntheorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubring_injective.eq_iff' top_toSubring\n\ntheorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_left\n\ntheorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_right\n\ntheorem mul_mem_sup {S T : NonUnitalSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) :\n    x * y ∈ S ⊔ T :=\n  mul_mem (mem_sup_left hx) (mem_sup_right hy)\n\ntheorem map_sup (f : F) (S T : NonUnitalSubalgebra R A) :\n    ((S ⊔ T).map f : NonUnitalSubalgebra R B) = S.map f ⊔ T.map f :=\n  (@NonUnitalSubalgebra.gc_map_comap F R A B _ _ _ _ _ _ f).l_sup\n\n@[simp, norm_cast]\ntheorem coe_inf (S T : NonUnitalSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T :=\n  rfl\n\n@[simp]\ntheorem mem_inf {S T : NonUnitalSubalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T :=\n  Iff.rfl\n\n@[simp]\ntheorem inf_toSubmodule (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toSubmodule = S.toSubmodule ⊓ T.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem inf_toNonUnitalSubsemiring (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring ⊓ T.toNonUnitalSubsemiring :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sInf (S : Set (NonUnitalSubalgebra R A)) : (↑(sInf S) : Set A) = ⋂ s ∈ S, ↑s :=\n  sInf_image\n\ntheorem mem_sInf {S : Set (NonUnitalSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p := by\n  simp only [← SetLike.mem_coe, coe_sInf, Set.mem_iInter₂]\n\n@[simp]\ntheorem sInf_toSubmodule (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toSubmodule = sInf (NonUnitalSubalgebra.toSubmodule '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp]\ntheorem sInf_toNonUnitalSubsemiring (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toNonUnitalSubsemiring = sInf (NonUnitalSubalgebra.toNonUnitalSubsemiring '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp, norm_cast]\ntheorem coe_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} :\n    (↑(⨅ i, S i) : Set A) = ⋂ i, S i := by simp [iInf]\n\ntheorem mem_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} {x : A} :\n    (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by simp only [iInf, mem_sInf, Set.forall_range_iff]\n\n@[simp]\ntheorem iInf_toSubmodule {ι : Sort*} (S : ι → NonUnitalSubalgebra R A) :\n    (⨅ i, S i).toSubmodule = ⨅ i, (S i).toSubmodule :=\n  SetLike.coe_injective <| by simp\n\ninstance : Inhabited (NonUnitalSubalgebra R A) :=\n  ⟨⊥⟩\n\ntheorem mem_bot {x : A} : x ∈ (⊥ : NonUnitalSubalgebra R A) ↔ x = 0 :=\n  show x ∈ Submodule.span R (NonUnitalSubsemiring.closure (∅ : Set A) : Set A) ↔ x = 0 by\n    rw [NonUnitalSubsemiring.closure_empty, NonUnitalSubsemiring.coe_bot,\n      Submodule.span_zero_singleton, Submodule.mem_bot]\n\ntheorem toSubmodule_bot : (⊥ : NonUnitalSubalgebra R A).toSubmodule = ⊥ := by\n  ext\n  simp only [mem_bot, NonUnitalSubalgebra.mem_toSubmodule, Submodule.mem_bot]\n\n@[simp]\ntheorem coe_bot : ((⊥ : NonUnitalSubalgebra R A) : Set A) = {0} := by\n  simp [Set.ext_iff, NonUnitalAlgebra.mem_bot]\n\ntheorem eq_top_iff {S : NonUnitalSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S :=\n  ⟨fun h x => by rw [h]; exact mem_top, fun h => by ext x; exact ⟨fun _ => mem_top, fun _ => h x⟩⟩\n\ntheorem range_top_iff_surjective (f : A →ₙₐ[R] B) :\n    NonUnitalAlgHom.range f = (⊤ : NonUnitalSubalgebra R B) ↔ Function.Surjective f :=\n  NonUnitalAlgebra.eq_top_iff\n\n@[simp]\ntheorem range_id : NonUnitalAlgHom.range (NonUnitalAlgHom.id R A) = ⊤ :=\n  SetLike.coe_injective Set.range_id\n\n@[simp]\ntheorem map_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R A).map f = NonUnitalAlgHom.range f :=\n  SetLike.coe_injective Set.image_univ\n\n@[simp]\ntheorem map_bot (f : A →ₙₐ[R] B) : (⊥ : NonUnitalSubalgebra R A).map f = ⊥ :=\n  SetLike.coe_injective <| by simp [NonUnitalAlgebra.coe_bot, NonUnitalSubalgebra.coe_map]\n\n@[simp]\ntheorem comap_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R B).comap f = ⊤ :=\n  eq_top_iff.2 fun _ => mem_top\n\n/-- `NonUnitalAlgHom` to `⊤ : NonUnitalSubalgebra R A`. -/\ndef toTop : A →ₙₐ[R] (⊤ : NonUnitalSubalgebra R A) :=\n  NonUnitalAlgHom.codRestrict (NonUnitalAlgHom.id R A) ⊤ fun _ => mem_top\n\nend NonUnitalAlgebra\n\nnamespace NonUnitalSubalgebra\n\nopen NonUnitalAlgebra\n\nsection NonAssoc\n\nvariable {R : Type u} {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable (S : NonUnitalSubalgebra R A)\n\ninstance subsingleton_of_subsingleton [Subsingleton A] : Subsingleton (NonUnitalSubalgebra R A) :=\n  ⟨fun B C => ext fun x => by simp only [Subsingleton.elim x 0, zero_mem B, zero_mem C]⟩\n\ninstance _root_.NonUnitalAlgHom.subsingleton [Subsingleton (NonUnitalSubalgebra R A)] :\n    Subsingleton (A →ₙₐ[R] B) :=\n  ⟨fun f g =>\n    NonUnitalAlgHom.ext fun a =>\n      have : a ∈ (⊥ : NonUnitalSubalgebra R A) :=\n        Subsingleton.elim (⊤ : NonUnitalSubalgebra R A) ⊥ ▸ mem_top\n      (mem_bot.mp this).symm ▸ (map_zero f).trans (map_zero g).symm⟩\n\ntheorem range_val : NonUnitalAlgHom.range (NonUnitalSubalgebraClass.subtype S) = S :=\n  ext <| Set.ext_iff.1 <| (NonUnitalSubalgebraClass.subtype S).coe_range.trans Subtype.range_val\n\n/-- The map `S → T` when `S` is a non-unital subalgebra contained in the non-unital subalgebra `T`.\n\nThis is the non-unital subalgebra version of `Submodule.inclusion`, or `Subring.inclusion`  -/\ndef inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) : S →ₙₐ[R] T\n    where\n  toFun := Set.inclusion h\n  map_add' _ _ := rfl\n  map_mul' _ _ := rfl\n  map_zero' := rfl\n  map_smul' _ _ := rfl\n\ntheorem inclusion_injective {S T : NonUnitalSubalgebra R A} (h : S ≤ T) :\n    Function.Injective (inclusion h) := fun _ _ => Subtype.ext ∘ Subtype.mk.inj\n\n@[simp]\ntheorem inclusion_self {S : NonUnitalSubalgebra R A} :\n    inclusion (le_refl S) = NonUnitalAlgHom.id R S :=\n  NonUnitalAlgHom.ext fun _ => Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_mk {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : A) (hx : x ∈ S) :\n    inclusion h ⟨x, hx⟩ = ⟨x, h hx⟩ :=\n  rfl\n\ntheorem inclusion_right {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : T) (m : (x : A) ∈ S) :\n    inclusion h ⟨x, m⟩ = x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_inclusion {S T U : NonUnitalSubalgebra R A} (hst : S ≤ T) (htu : T ≤ U) (x : S) :\n    inclusion htu (inclusion hst x) = inclusion (le_trans hst htu) x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem coe_inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (s : S) :\n    (inclusion h s : A) = s :=\n  rfl\n\nsection Prod\n\nvariable (S₁ : NonUnitalSubalgebra R B)\n\n/-- The product of two non-unital subalgebras is a non-unital subalgebra. -/\ndef prod : NonUnitalSubalgebra R (A × B) :=\n  { S.toNonUnitalSubsemiring.prod S₁.toNonUnitalSubsemiring with\n    carrier := S ×ˢ S₁\n    smul_mem' := fun r _x hx => ⟨SMulMemClass.smul_mem r hx.1, SMulMemClass.smul_mem r hx.2⟩ }\n\n@[simp]\ntheorem coe_prod : (prod S S₁ : Set (A × B)) = (S : Set A) ×ˢ S₁ :=\n  rfl\n\ntheorem prod_toSubmodule : (S.prod S₁).toSubmodule = S.toSubmodule.prod S₁.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem mem_prod {S : NonUnitalSubalgebra R A} {S₁ : NonUnitalSubalgebra R B} {x : A × B} :\n    x ∈ prod S S₁ ↔ x.1 ∈ S ∧ x.2 ∈ S₁ :=\n  Set.mem_prod\n\n@[simp]\ntheorem prod_top : (prod ⊤ ⊤ : NonUnitalSubalgebra R (A × B)) = ⊤ := by ext; simp\n\ntheorem prod_mono {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S ≤ T → S₁ ≤ T₁ → prod S S₁ ≤ prod T T₁ :=\n  Set.prod_mono\n\n@[simp]\ntheorem prod_inf_prod {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S.prod S₁ ⊓ T.prod T₁ = (S ⊓ T).prod (S₁ ⊓ T₁) :=\n  SetLike.coe_injective Set.prod_inter_prod\n\nend Prod\n\nsection SuprLift\n\nvariable {ι : Type*}\n\ntheorem coe_iSup_of_directed [Nonempty ι] {S : ι → NonUnitalSubalgebra R A}\n    (dir : Directed (· ≤ ·) S) : ↑(iSup S) = ⋃ i, (S i : Set A) :=\n  let K : NonUnitalSubalgebra R A :=\n    { __ := NonUnitalSubsemiring.copy _ _ (NonUnitalSubsemiring.coe_iSup_of_directed dir).symm\n      smul_mem' := fun r _x hx ↦\n        let ⟨i, hi⟩ := Set.mem_iUnion.1 hx\n        Set.mem_iUnion.2 ⟨i, (S i).smul_mem' r hi⟩ }\n  have : iSup S = K := le_antisymm\n    (iSup_le fun i ↦ le_iSup (fun i ↦ (S i : Set A)) i) (Set.iUnion_subset fun _ ↦ le_iSup S _)\n  this.symm ▸ rfl\n\n/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              rw [hf j k hjk]\n              rfl)\n            (↑(iSup K)) (by rw [coe_iSup_of_directed dir])\n        map_zero' := by\n          dsimp\n          exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl) _ (by simp)\n        map_mul' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· * ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_add' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· + ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_smul' := fun r => by\n          dsimp\n          apply Set.iUnionLift_unary (coe_iSup_of_directed dir) _ (fun _ x => r • x)\n            (fun _ _ => rfl)\n          on_goal 2 => rw [coe_iSup_of_directed dir]\n          all_goals simp }\n\nvariable [Nonempty ι] {K : ι → NonUnitalSubalgebra R A} {dir : Directed (· ≤ ·) K}\n  {f : ∀ i, K i →ₙₐ[R] B} {hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h)}\n  {T : NonUnitalSubalgebra R A} {hT : T = iSup K}\n\n@[simp]\ntheorem iSupLift_inclusion {i : ι} (x : K i) (h : K i ≤ T) :\n    iSupLift K dir f hf T hT (inclusion h x) = f i x := by\n  subst T\n  dsimp [iSupLift]\n  apply Set.iUnionLift_inclusion\n  exact h\n\n@[simp]\ntheorem iSupLift_comp_inclusion {i : ι} (h : K i ≤ T) :\n    (iSupLift K dir f hf T hT).comp (inclusion h) = f i := by ext; simp\n\n@[simp]\ntheorem iSupLift_mk {i : ι} (x : K i) (hx : (x : A) ∈ T) :\n    iSupLift K dir f hf T hT ⟨x, hx⟩ = f i x := by\n  subst hT\n  dsimp [iSupLift]\n  apply Set.iUnionLift_mk\n\ntheorem iSupLift_of_mem {i : ι} (x : T) (hx : (x : A) ∈ K i) :\n    iSupLift K dir f hf T hT x = f i ⟨x, hx⟩ := by\n  subst hT\n  dsimp [iSupLift]\n  apply Set.iUnionLift_of_mem\n\nend SuprLift\n\nend NonAssoc\n\nsection Center\n\nsection NonUnitalNonAssocSemiring\nvariable {R A : Type*}\nvariable [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [IsScalarTower R A A] [SMulCommClass R A A]\n\ntheorem _root_.Set.smul_mem_center (r : R) {a : A} (ha : a ∈ Set.center A) :\n    r • a ∈ Set.center A where\n  comm b := by rw [mul_smul_comm, smul_mul_assoc, ha.comm]\n  left_assoc b c := by rw [smul_mul_assoc, smul_mul_assoc, smul_mul_assoc, ha.left_assoc]\n  mid_assoc b c := by\n    rw [mul_smul_comm]\n    rw [smul_mul_assoc]\n    rw [smul_mul_assoc]\n    rw [mul_smul_comm]\n    rw [ha.mid_assoc]\n  right_assoc b c := by\n    ","nextTactic":"rw [mul_smul_comm]","declUpToTactic":"theorem _root_.Set.smul_mem_center (r : R) {a : A} (ha : a ∈ Set.center A) :\n    r • a ∈ Set.center A where\n  comm b := by rw [mul_smul_comm, smul_mul_assoc, ha.comm]\n  left_assoc b c := by rw [smul_mul_assoc, smul_mul_assoc, smul_mul_assoc, ha.left_assoc]\n  mid_assoc b c := by\n    rw [mul_smul_comm]\n    rw [smul_mul_assoc]\n    rw [smul_mul_assoc]\n    rw [mul_smul_comm]\n    rw [ha.mid_assoc]\n  right_assoc b c := by\n    ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.969_0.jQvdAcOcyVirh23","decl":"theorem _root_.Set.smul_mem_center (r : R) {a : A} (ha : a ∈ Set.center A) :\n    r • a ∈ Set.center A where\n  comm b "}
-{"state":"R : Type u_1\nA : Type u_2\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nr : R\na : A\nha : a ∈ Set.center A\nb c : A\n⊢ r • (b * c * a) = b * (c * r • a)","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact NonUnitalAlgebra.gc\n\n@[simp]\ntheorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=\n  eq_top_iff.2 fun _x hx => subset_adjoin R hx\n\nvariable {R A}\n\n@[simp]\ntheorem coe_top : (↑(⊤ : NonUnitalSubalgebra R A) : Set A) = Set.univ :=\n  rfl\n\n@[simp]\ntheorem mem_top {x : A} : x ∈ (⊤ : NonUnitalSubalgebra R A) :=\n  Set.mem_univ x\n\n@[simp]\ntheorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toNonUnitalSubsemiring : (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubsemiring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toSubring {R A : Type*} [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    [IsScalarTower R A A] [SMulCommClass R A A] :\n    (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem toSubmodule_eq_top {S : NonUnitalSubalgebra R A} : S.toSubmodule = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toSubmodule'.injective.eq_iff' top_toSubmodule\n\n@[simp]\ntheorem toNonUnitalSubsemiring_eq_top {S : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubsemiring_injective.eq_iff' top_toNonUnitalSubsemiring\n\n@[simp]\ntheorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubring_injective.eq_iff' top_toSubring\n\ntheorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ��� {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_left\n\ntheorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_right\n\ntheorem mul_mem_sup {S T : NonUnitalSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) :\n    x * y ∈ S ⊔ T :=\n  mul_mem (mem_sup_left hx) (mem_sup_right hy)\n\ntheorem map_sup (f : F) (S T : NonUnitalSubalgebra R A) :\n    ((S ⊔ T).map f : NonUnitalSubalgebra R B) = S.map f ⊔ T.map f :=\n  (@NonUnitalSubalgebra.gc_map_comap F R A B _ _ _ _ _ _ f).l_sup\n\n@[simp, norm_cast]\ntheorem coe_inf (S T : NonUnitalSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T :=\n  rfl\n\n@[simp]\ntheorem mem_inf {S T : NonUnitalSubalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T :=\n  Iff.rfl\n\n@[simp]\ntheorem inf_toSubmodule (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toSubmodule = S.toSubmodule ⊓ T.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem inf_toNonUnitalSubsemiring (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring ⊓ T.toNonUnitalSubsemiring :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sInf (S : Set (NonUnitalSubalgebra R A)) : (↑(sInf S) : Set A) = ⋂ s ∈ S, ↑s :=\n  sInf_image\n\ntheorem mem_sInf {S : Set (NonUnitalSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p := by\n  simp only [← SetLike.mem_coe, coe_sInf, Set.mem_iInter₂]\n\n@[simp]\ntheorem sInf_toSubmodule (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toSubmodule = sInf (NonUnitalSubalgebra.toSubmodule '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp]\ntheorem sInf_toNonUnitalSubsemiring (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toNonUnitalSubsemiring = sInf (NonUnitalSubalgebra.toNonUnitalSubsemiring '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp, norm_cast]\ntheorem coe_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} :\n    (↑(⨅ i, S i) : Set A) = ⋂ i, S i := by simp [iInf]\n\ntheorem mem_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} {x : A} :\n    (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by simp only [iInf, mem_sInf, Set.forall_range_iff]\n\n@[simp]\ntheorem iInf_toSubmodule {ι : Sort*} (S : ι → NonUnitalSubalgebra R A) :\n    (⨅ i, S i).toSubmodule = ⨅ i, (S i).toSubmodule :=\n  SetLike.coe_injective <| by simp\n\ninstance : Inhabited (NonUnitalSubalgebra R A) :=\n  ⟨⊥⟩\n\ntheorem mem_bot {x : A} : x ∈ (⊥ : NonUnitalSubalgebra R A) ↔ x = 0 :=\n  show x ∈ Submodule.span R (NonUnitalSubsemiring.closure (∅ : Set A) : Set A) ↔ x = 0 by\n    rw [NonUnitalSubsemiring.closure_empty, NonUnitalSubsemiring.coe_bot,\n      Submodule.span_zero_singleton, Submodule.mem_bot]\n\ntheorem toSubmodule_bot : (⊥ : NonUnitalSubalgebra R A).toSubmodule = ⊥ := by\n  ext\n  simp only [mem_bot, NonUnitalSubalgebra.mem_toSubmodule, Submodule.mem_bot]\n\n@[simp]\ntheorem coe_bot : ((⊥ : NonUnitalSubalgebra R A) : Set A) = {0} := by\n  simp [Set.ext_iff, NonUnitalAlgebra.mem_bot]\n\ntheorem eq_top_iff {S : NonUnitalSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S :=\n  ⟨fun h x => by rw [h]; exact mem_top, fun h => by ext x; exact ⟨fun _ => mem_top, fun _ => h x⟩⟩\n\ntheorem range_top_iff_surjective (f : A →ₙₐ[R] B) :\n    NonUnitalAlgHom.range f = (⊤ : NonUnitalSubalgebra R B) ↔ Function.Surjective f :=\n  NonUnitalAlgebra.eq_top_iff\n\n@[simp]\ntheorem range_id : NonUnitalAlgHom.range (NonUnitalAlgHom.id R A) = ⊤ :=\n  SetLike.coe_injective Set.range_id\n\n@[simp]\ntheorem map_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R A).map f = NonUnitalAlgHom.range f :=\n  SetLike.coe_injective Set.image_univ\n\n@[simp]\ntheorem map_bot (f : A →ₙₐ[R] B) : (⊥ : NonUnitalSubalgebra R A).map f = ⊥ :=\n  SetLike.coe_injective <| by simp [NonUnitalAlgebra.coe_bot, NonUnitalSubalgebra.coe_map]\n\n@[simp]\ntheorem comap_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R B).comap f = ⊤ :=\n  eq_top_iff.2 fun _ => mem_top\n\n/-- `NonUnitalAlgHom` to `⊤ : NonUnitalSubalgebra R A`. -/\ndef toTop : A →ₙₐ[R] (⊤ : NonUnitalSubalgebra R A) :=\n  NonUnitalAlgHom.codRestrict (NonUnitalAlgHom.id R A) ⊤ fun _ => mem_top\n\nend NonUnitalAlgebra\n\nnamespace NonUnitalSubalgebra\n\nopen NonUnitalAlgebra\n\nsection NonAssoc\n\nvariable {R : Type u} {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable (S : NonUnitalSubalgebra R A)\n\ninstance subsingleton_of_subsingleton [Subsingleton A] : Subsingleton (NonUnitalSubalgebra R A) :=\n  ⟨fun B C => ext fun x => by simp only [Subsingleton.elim x 0, zero_mem B, zero_mem C]⟩\n\ninstance _root_.NonUnitalAlgHom.subsingleton [Subsingleton (NonUnitalSubalgebra R A)] :\n    Subsingleton (A →ₙₐ[R] B) :=\n  ⟨fun f g =>\n    NonUnitalAlgHom.ext fun a =>\n      have : a ∈ (⊥ : NonUnitalSubalgebra R A) :=\n        Subsingleton.elim (⊤ : NonUnitalSubalgebra R A) ⊥ ▸ mem_top\n      (mem_bot.mp this).symm ▸ (map_zero f).trans (map_zero g).symm⟩\n\ntheorem range_val : NonUnitalAlgHom.range (NonUnitalSubalgebraClass.subtype S) = S :=\n  ext <| Set.ext_iff.1 <| (NonUnitalSubalgebraClass.subtype S).coe_range.trans Subtype.range_val\n\n/-- The map `S → T` when `S` is a non-unital subalgebra contained in the non-unital subalgebra `T`.\n\nThis is the non-unital subalgebra version of `Submodule.inclusion`, or `Subring.inclusion`  -/\ndef inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) : S →ₙₐ[R] T\n    where\n  toFun := Set.inclusion h\n  map_add' _ _ := rfl\n  map_mul' _ _ := rfl\n  map_zero' := rfl\n  map_smul' _ _ := rfl\n\ntheorem inclusion_injective {S T : NonUnitalSubalgebra R A} (h : S ≤ T) :\n    Function.Injective (inclusion h) := fun _ _ => Subtype.ext ∘ Subtype.mk.inj\n\n@[simp]\ntheorem inclusion_self {S : NonUnitalSubalgebra R A} :\n    inclusion (le_refl S) = NonUnitalAlgHom.id R S :=\n  NonUnitalAlgHom.ext fun _ => Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_mk {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : A) (hx : x ∈ S) :\n    inclusion h ⟨x, hx⟩ = ⟨x, h hx⟩ :=\n  rfl\n\ntheorem inclusion_right {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : T) (m : (x : A) ∈ S) :\n    inclusion h ⟨x, m⟩ = x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_inclusion {S T U : NonUnitalSubalgebra R A} (hst : S ≤ T) (htu : T ≤ U) (x : S) :\n    inclusion htu (inclusion hst x) = inclusion (le_trans hst htu) x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem coe_inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (s : S) :\n    (inclusion h s : A) = s :=\n  rfl\n\nsection Prod\n\nvariable (S₁ : NonUnitalSubalgebra R B)\n\n/-- The product of two non-unital subalgebras is a non-unital subalgebra. -/\ndef prod : NonUnitalSubalgebra R (A × B) :=\n  { S.toNonUnitalSubsemiring.prod S₁.toNonUnitalSubsemiring with\n    carrier := S ×ˢ S₁\n    smul_mem' := fun r _x hx => ⟨SMulMemClass.smul_mem r hx.1, SMulMemClass.smul_mem r hx.2⟩ }\n\n@[simp]\ntheorem coe_prod : (prod S S₁ : Set (A × B)) = (S : Set A) ×ˢ S₁ :=\n  rfl\n\ntheorem prod_toSubmodule : (S.prod S₁).toSubmodule = S.toSubmodule.prod S₁.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem mem_prod {S : NonUnitalSubalgebra R A} {S₁ : NonUnitalSubalgebra R B} {x : A × B} :\n    x ∈ prod S S₁ ↔ x.1 ∈ S ∧ x.2 ∈ S₁ :=\n  Set.mem_prod\n\n@[simp]\ntheorem prod_top : (prod ⊤ ⊤ : NonUnitalSubalgebra R (A × B)) = ⊤ := by ext; simp\n\ntheorem prod_mono {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S ≤ T → S₁ ≤ T₁ → prod S S₁ ≤ prod T T₁ :=\n  Set.prod_mono\n\n@[simp]\ntheorem prod_inf_prod {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S.prod S₁ ⊓ T.prod T₁ = (S ⊓ T).prod (S₁ ⊓ T₁) :=\n  SetLike.coe_injective Set.prod_inter_prod\n\nend Prod\n\nsection SuprLift\n\nvariable {ι : Type*}\n\ntheorem coe_iSup_of_directed [Nonempty ι] {S : ι → NonUnitalSubalgebra R A}\n    (dir : Directed (· ≤ ·) S) : ↑(iSup S) = ⋃ i, (S i : Set A) :=\n  let K : NonUnitalSubalgebra R A :=\n    { __ := NonUnitalSubsemiring.copy _ _ (NonUnitalSubsemiring.coe_iSup_of_directed dir).symm\n      smul_mem' := fun r _x hx ↦\n        let ⟨i, hi⟩ := Set.mem_iUnion.1 hx\n        Set.mem_iUnion.2 ⟨i, (S i).smul_mem' r hi⟩ }\n  have : iSup S = K := le_antisymm\n    (iSup_le fun i ↦ le_iSup (fun i ↦ (S i : Set A)) i) (Set.iUnion_subset fun _ ↦ le_iSup S _)\n  this.symm ▸ rfl\n\n/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              rw [hf j k hjk]\n              rfl)\n            (↑(iSup K)) (by rw [coe_iSup_of_directed dir])\n        map_zero' := by\n          dsimp\n          exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl) _ (by simp)\n        map_mul' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· * ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_add' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· + ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_smul' := fun r => by\n          dsimp\n          apply Set.iUnionLift_unary (coe_iSup_of_directed dir) _ (fun _ x => r • x)\n            (fun _ _ => rfl)\n          on_goal 2 => rw [coe_iSup_of_directed dir]\n          all_goals simp }\n\nvariable [Nonempty ι] {K : ι → NonUnitalSubalgebra R A} {dir : Directed (· ≤ ·) K}\n  {f : ∀ i, K i →ₙₐ[R] B} {hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h)}\n  {T : NonUnitalSubalgebra R A} {hT : T = iSup K}\n\n@[simp]\ntheorem iSupLift_inclusion {i : ι} (x : K i) (h : K i ≤ T) :\n    iSupLift K dir f hf T hT (inclusion h x) = f i x := by\n  subst T\n  dsimp [iSupLift]\n  apply Set.iUnionLift_inclusion\n  exact h\n\n@[simp]\ntheorem iSupLift_comp_inclusion {i : ι} (h : K i ≤ T) :\n    (iSupLift K dir f hf T hT).comp (inclusion h) = f i := by ext; simp\n\n@[simp]\ntheorem iSupLift_mk {i : ι} (x : K i) (hx : (x : A) ∈ T) :\n    iSupLift K dir f hf T hT ⟨x, hx⟩ = f i x := by\n  subst hT\n  dsimp [iSupLift]\n  apply Set.iUnionLift_mk\n\ntheorem iSupLift_of_mem {i : ι} (x : T) (hx : (x : A) ∈ K i) :\n    iSupLift K dir f hf T hT x = f i ⟨x, hx⟩ := by\n  subst hT\n  dsimp [iSupLift]\n  apply Set.iUnionLift_of_mem\n\nend SuprLift\n\nend NonAssoc\n\nsection Center\n\nsection NonUnitalNonAssocSemiring\nvariable {R A : Type*}\nvariable [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [IsScalarTower R A A] [SMulCommClass R A A]\n\ntheorem _root_.Set.smul_mem_center (r : R) {a : A} (ha : a ∈ Set.center A) :\n    r • a ∈ Set.center A where\n  comm b := by rw [mul_smul_comm, smul_mul_assoc, ha.comm]\n  left_assoc b c := by rw [smul_mul_assoc, smul_mul_assoc, smul_mul_assoc, ha.left_assoc]\n  mid_assoc b c := by\n    rw [mul_smul_comm]\n    rw [smul_mul_assoc]\n    rw [smul_mul_assoc]\n    rw [mul_smul_comm]\n    rw [ha.mid_assoc]\n  right_assoc b c := by\n    rw [mul_smul_comm]\n    ","nextTactic":"rw [mul_smul_comm]","declUpToTactic":"theorem _root_.Set.smul_mem_center (r : R) {a : A} (ha : a ∈ Set.center A) :\n    r • a ∈ Set.center A where\n  comm b := by rw [mul_smul_comm, smul_mul_assoc, ha.comm]\n  left_assoc b c := by rw [smul_mul_assoc, smul_mul_assoc, smul_mul_assoc, ha.left_assoc]\n  mid_assoc b c := by\n    rw [mul_smul_comm]\n    rw [smul_mul_assoc]\n    rw [smul_mul_assoc]\n    rw [mul_smul_comm]\n    rw [ha.mid_assoc]\n  right_assoc b c := by\n    rw [mul_smul_comm]\n    ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.969_0.jQvdAcOcyVirh23","decl":"theorem _root_.Set.smul_mem_center (r : R) {a : A} (ha : a ∈ Set.center A) :\n    r • a ∈ Set.center A where\n  comm b "}
-{"state":"R : Type u_1\nA : Type u_2\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nr : R\na : A\nha : a ∈ Set.center A\nb c : A\n⊢ r • (b * c * a) = b * r • (c * a)","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact NonUnitalAlgebra.gc\n\n@[simp]\ntheorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=\n  eq_top_iff.2 fun _x hx => subset_adjoin R hx\n\nvariable {R A}\n\n@[simp]\ntheorem coe_top : (↑(⊤ : NonUnitalSubalgebra R A) : Set A) = Set.univ :=\n  rfl\n\n@[simp]\ntheorem mem_top {x : A} : x ∈ (⊤ : NonUnitalSubalgebra R A) :=\n  Set.mem_univ x\n\n@[simp]\ntheorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toNonUnitalSubsemiring : (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubsemiring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toSubring {R A : Type*} [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    [IsScalarTower R A A] [SMulCommClass R A A] :\n    (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem toSubmodule_eq_top {S : NonUnitalSubalgebra R A} : S.toSubmodule = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toSubmodule'.injective.eq_iff' top_toSubmodule\n\n@[simp]\ntheorem toNonUnitalSubsemiring_eq_top {S : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubsemiring_injective.eq_iff' top_toNonUnitalSubsemiring\n\n@[simp]\ntheorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubring_injective.eq_iff' top_toSubring\n\ntheorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_left\n\ntheorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_right\n\ntheorem mul_mem_sup {S T : NonUnitalSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) :\n    x * y ∈ S ⊔ T :=\n  mul_mem (mem_sup_left hx) (mem_sup_right hy)\n\ntheorem map_sup (f : F) (S T : NonUnitalSubalgebra R A) :\n    ((S ⊔ T).map f : NonUnitalSubalgebra R B) = S.map f ⊔ T.map f :=\n  (@NonUnitalSubalgebra.gc_map_comap F R A B _ _ _ _ _ _ f).l_sup\n\n@[simp, norm_cast]\ntheorem coe_inf (S T : NonUnitalSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T :=\n  rfl\n\n@[simp]\ntheorem mem_inf {S T : NonUnitalSubalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T :=\n  Iff.rfl\n\n@[simp]\ntheorem inf_toSubmodule (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toSubmodule = S.toSubmodule ⊓ T.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem inf_toNonUnitalSubsemiring (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring ⊓ T.toNonUnitalSubsemiring :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sInf (S : Set (NonUnitalSubalgebra R A)) : (↑(sInf S) : Set A) = ⋂ s ∈ S, ↑s :=\n  sInf_image\n\ntheorem mem_sInf {S : Set (NonUnitalSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p := by\n  simp only [← SetLike.mem_coe, coe_sInf, Set.mem_iInter₂]\n\n@[simp]\ntheorem sInf_toSubmodule (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toSubmodule = sInf (NonUnitalSubalgebra.toSubmodule '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp]\ntheorem sInf_toNonUnitalSubsemiring (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toNonUnitalSubsemiring = sInf (NonUnitalSubalgebra.toNonUnitalSubsemiring '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp, norm_cast]\ntheorem coe_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} :\n    (↑(⨅ i, S i) : Set A) = ⋂ i, S i := by simp [iInf]\n\ntheorem mem_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} {x : A} :\n    (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by simp only [iInf, mem_sInf, Set.forall_range_iff]\n\n@[simp]\ntheorem iInf_toSubmodule {ι : Sort*} (S : ι → NonUnitalSubalgebra R A) :\n    (⨅ i, S i).toSubmodule = ⨅ i, (S i).toSubmodule :=\n  SetLike.coe_injective <| by simp\n\ninstance : Inhabited (NonUnitalSubalgebra R A) :=\n  ⟨⊥⟩\n\ntheorem mem_bot {x : A} : x ∈ (⊥ : NonUnitalSubalgebra R A) ↔ x = 0 :=\n  show x ∈ Submodule.span R (NonUnitalSubsemiring.closure (∅ : Set A) : Set A) ↔ x = 0 by\n    rw [NonUnitalSubsemiring.closure_empty, NonUnitalSubsemiring.coe_bot,\n      Submodule.span_zero_singleton, Submodule.mem_bot]\n\ntheorem toSubmodule_bot : (⊥ : NonUnitalSubalgebra R A).toSubmodule = ⊥ := by\n  ext\n  simp only [mem_bot, NonUnitalSubalgebra.mem_toSubmodule, Submodule.mem_bot]\n\n@[simp]\ntheorem coe_bot : ((⊥ : NonUnitalSubalgebra R A) : Set A) = {0} := by\n  simp [Set.ext_iff, NonUnitalAlgebra.mem_bot]\n\ntheorem eq_top_iff {S : NonUnitalSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S :=\n  ⟨fun h x => by rw [h]; exact mem_top, fun h => by ext x; exact ⟨fun _ => mem_top, fun _ => h x⟩⟩\n\ntheorem range_top_iff_surjective (f : A →ₙₐ[R] B) :\n    NonUnitalAlgHom.range f = (⊤ : NonUnitalSubalgebra R B) ↔ Function.Surjective f :=\n  NonUnitalAlgebra.eq_top_iff\n\n@[simp]\ntheorem range_id : NonUnitalAlgHom.range (NonUnitalAlgHom.id R A) = ⊤ :=\n  SetLike.coe_injective Set.range_id\n\n@[simp]\ntheorem map_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R A).map f = NonUnitalAlgHom.range f :=\n  SetLike.coe_injective Set.image_univ\n\n@[simp]\ntheorem map_bot (f : A →ₙₐ[R] B) : (⊥ : NonUnitalSubalgebra R A).map f = ⊥ :=\n  SetLike.coe_injective <| by simp [NonUnitalAlgebra.coe_bot, NonUnitalSubalgebra.coe_map]\n\n@[simp]\ntheorem comap_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R B).comap f = ⊤ :=\n  eq_top_iff.2 fun _ => mem_top\n\n/-- `NonUnitalAlgHom` to `⊤ : NonUnitalSubalgebra R A`. -/\ndef toTop : A →ₙₐ[R] (⊤ : NonUnitalSubalgebra R A) :=\n  NonUnitalAlgHom.codRestrict (NonUnitalAlgHom.id R A) ⊤ fun _ => mem_top\n\nend NonUnitalAlgebra\n\nnamespace NonUnitalSubalgebra\n\nopen NonUnitalAlgebra\n\nsection NonAssoc\n\nvariable {R : Type u} {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable (S : NonUnitalSubalgebra R A)\n\ninstance subsingleton_of_subsingleton [Subsingleton A] : Subsingleton (NonUnitalSubalgebra R A) :=\n  ⟨fun B C => ext fun x => by simp only [Subsingleton.elim x 0, zero_mem B, zero_mem C]⟩\n\ninstance _root_.NonUnitalAlgHom.subsingleton [Subsingleton (NonUnitalSubalgebra R A)] :\n    Subsingleton (A →ₙₐ[R] B) :=\n  ⟨fun f g =>\n    NonUnitalAlgHom.ext fun a =>\n      have : a ∈ (⊥ : NonUnitalSubalgebra R A) :=\n        Subsingleton.elim (⊤ : NonUnitalSubalgebra R A) ⊥ ▸ mem_top\n      (mem_bot.mp this).symm ▸ (map_zero f).trans (map_zero g).symm⟩\n\ntheorem range_val : NonUnitalAlgHom.range (NonUnitalSubalgebraClass.subtype S) = S :=\n  ext <| Set.ext_iff.1 <| (NonUnitalSubalgebraClass.subtype S).coe_range.trans Subtype.range_val\n\n/-- The map `S → T` when `S` is a non-unital subalgebra contained in the non-unital subalgebra `T`.\n\nThis is the non-unital subalgebra version of `Submodule.inclusion`, or `Subring.inclusion`  -/\ndef inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) : S →ₙₐ[R] T\n    where\n  toFun := Set.inclusion h\n  map_add' _ _ := rfl\n  map_mul' _ _ := rfl\n  map_zero' := rfl\n  map_smul' _ _ := rfl\n\ntheorem inclusion_injective {S T : NonUnitalSubalgebra R A} (h : S ≤ T) :\n    Function.Injective (inclusion h) := fun _ _ => Subtype.ext ∘ Subtype.mk.inj\n\n@[simp]\ntheorem inclusion_self {S : NonUnitalSubalgebra R A} :\n    inclusion (le_refl S) = NonUnitalAlgHom.id R S :=\n  NonUnitalAlgHom.ext fun _ => Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_mk {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : A) (hx : x ∈ S) :\n    inclusion h ⟨x, hx⟩ = ⟨x, h hx⟩ :=\n  rfl\n\ntheorem inclusion_right {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : T) (m : (x : A) ∈ S) :\n    inclusion h ⟨x, m⟩ = x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_inclusion {S T U : NonUnitalSubalgebra R A} (hst : S ≤ T) (htu : T ≤ U) (x : S) :\n    inclusion htu (inclusion hst x) = inclusion (le_trans hst htu) x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem coe_inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (s : S) :\n    (inclusion h s : A) = s :=\n  rfl\n\nsection Prod\n\nvariable (S₁ : NonUnitalSubalgebra R B)\n\n/-- The product of two non-unital subalgebras is a non-unital subalgebra. -/\ndef prod : NonUnitalSubalgebra R (A × B) :=\n  { S.toNonUnitalSubsemiring.prod S₁.toNonUnitalSubsemiring with\n    carrier := S ×ˢ S₁\n    smul_mem' := fun r _x hx => ⟨SMulMemClass.smul_mem r hx.1, SMulMemClass.smul_mem r hx.2⟩ }\n\n@[simp]\ntheorem coe_prod : (prod S S₁ : Set (A × B)) = (S : Set A) ×ˢ S₁ :=\n  rfl\n\ntheorem prod_toSubmodule : (S.prod S₁).toSubmodule = S.toSubmodule.prod S₁.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem mem_prod {S : NonUnitalSubalgebra R A} {S₁ : NonUnitalSubalgebra R B} {x : A × B} :\n    x ∈ prod S S₁ ↔ x.1 ∈ S ∧ x.2 ∈ S₁ :=\n  Set.mem_prod\n\n@[simp]\ntheorem prod_top : (prod ⊤ ⊤ : NonUnitalSubalgebra R (A × B)) = ⊤ := by ext; simp\n\ntheorem prod_mono {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S ≤ T → S₁ ≤ T₁ → prod S S₁ ≤ prod T T₁ :=\n  Set.prod_mono\n\n@[simp]\ntheorem prod_inf_prod {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S.prod S₁ ⊓ T.prod T₁ = (S ⊓ T).prod (S₁ ⊓ T₁) :=\n  SetLike.coe_injective Set.prod_inter_prod\n\nend Prod\n\nsection SuprLift\n\nvariable {ι : Type*}\n\ntheorem coe_iSup_of_directed [Nonempty ι] {S : ι → NonUnitalSubalgebra R A}\n    (dir : Directed (· ≤ ·) S) : ↑(iSup S) = ⋃ i, (S i : Set A) :=\n  let K : NonUnitalSubalgebra R A :=\n    { __ := NonUnitalSubsemiring.copy _ _ (NonUnitalSubsemiring.coe_iSup_of_directed dir).symm\n      smul_mem' := fun r _x hx ↦\n        let ⟨i, hi⟩ := Set.mem_iUnion.1 hx\n        Set.mem_iUnion.2 ⟨i, (S i).smul_mem' r hi⟩ }\n  have : iSup S = K := le_antisymm\n    (iSup_le fun i ↦ le_iSup (fun i ↦ (S i : Set A)) i) (Set.iUnion_subset fun _ ↦ le_iSup S _)\n  this.symm ▸ rfl\n\n/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              rw [hf j k hjk]\n              rfl)\n            (↑(iSup K)) (by rw [coe_iSup_of_directed dir])\n        map_zero' := by\n          dsimp\n          exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl) _ (by simp)\n        map_mul' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· * ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_add' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· + ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_smul' := fun r => by\n          dsimp\n          apply Set.iUnionLift_unary (coe_iSup_of_directed dir) _ (fun _ x => r • x)\n            (fun _ _ => rfl)\n          on_goal 2 => rw [coe_iSup_of_directed dir]\n          all_goals simp }\n\nvariable [Nonempty ι] {K : ι → NonUnitalSubalgebra R A} {dir : Directed (· ≤ ·) K}\n  {f : ∀ i, K i →ₙₐ[R] B} {hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h)}\n  {T : NonUnitalSubalgebra R A} {hT : T = iSup K}\n\n@[simp]\ntheorem iSupLift_inclusion {i : ι} (x : K i) (h : K i ≤ T) :\n    iSupLift K dir f hf T hT (inclusion h x) = f i x := by\n  subst T\n  dsimp [iSupLift]\n  apply Set.iUnionLift_inclusion\n  exact h\n\n@[simp]\ntheorem iSupLift_comp_inclusion {i : ι} (h : K i ≤ T) :\n    (iSupLift K dir f hf T hT).comp (inclusion h) = f i := by ext; simp\n\n@[simp]\ntheorem iSupLift_mk {i : ι} (x : K i) (hx : (x : A) ∈ T) :\n    iSupLift K dir f hf T hT ⟨x, hx⟩ = f i x := by\n  subst hT\n  dsimp [iSupLift]\n  apply Set.iUnionLift_mk\n\ntheorem iSupLift_of_mem {i : ι} (x : T) (hx : (x : A) ∈ K i) :\n    iSupLift K dir f hf T hT x = f i ⟨x, hx⟩ := by\n  subst hT\n  dsimp [iSupLift]\n  apply Set.iUnionLift_of_mem\n\nend SuprLift\n\nend NonAssoc\n\nsection Center\n\nsection NonUnitalNonAssocSemiring\nvariable {R A : Type*}\nvariable [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [IsScalarTower R A A] [SMulCommClass R A A]\n\ntheorem _root_.Set.smul_mem_center (r : R) {a : A} (ha : a ∈ Set.center A) :\n    r • a ∈ Set.center A where\n  comm b := by rw [mul_smul_comm, smul_mul_assoc, ha.comm]\n  left_assoc b c := by rw [smul_mul_assoc, smul_mul_assoc, smul_mul_assoc, ha.left_assoc]\n  mid_assoc b c := by\n    rw [mul_smul_comm]\n    rw [smul_mul_assoc]\n    rw [smul_mul_assoc]\n    rw [mul_smul_comm]\n    rw [ha.mid_assoc]\n  right_assoc b c := by\n    rw [mul_smul_comm]\n    rw [mul_smul_comm]\n    ","nextTactic":"rw [mul_smul_comm]","declUpToTactic":"theorem _root_.Set.smul_mem_center (r : R) {a : A} (ha : a ∈ Set.center A) :\n    r • a ∈ Set.center A where\n  comm b := by rw [mul_smul_comm, smul_mul_assoc, ha.comm]\n  left_assoc b c := by rw [smul_mul_assoc, smul_mul_assoc, smul_mul_assoc, ha.left_assoc]\n  mid_assoc b c := by\n    rw [mul_smul_comm]\n    rw [smul_mul_assoc]\n    rw [smul_mul_assoc]\n    rw [mul_smul_comm]\n    rw [ha.mid_assoc]\n  right_assoc b c := by\n    rw [mul_smul_comm]\n    rw [mul_smul_comm]\n    ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.969_0.jQvdAcOcyVirh23","decl":"theorem _root_.Set.smul_mem_center (r : R) {a : A} (ha : a ∈ Set.center A) :\n    r • a ∈ Set.center A where\n  comm b "}
-{"state":"R : Type u_1\nA : Type u_2\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nr : R\na : A\nha : a ∈ Set.center A\nb c : A\n⊢ r • (b * c * a) = r • (b * (c * a))","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact NonUnitalAlgebra.gc\n\n@[simp]\ntheorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=\n  eq_top_iff.2 fun _x hx => subset_adjoin R hx\n\nvariable {R A}\n\n@[simp]\ntheorem coe_top : (↑(⊤ : NonUnitalSubalgebra R A) : Set A) = Set.univ :=\n  rfl\n\n@[simp]\ntheorem mem_top {x : A} : x ∈ (⊤ : NonUnitalSubalgebra R A) :=\n  Set.mem_univ x\n\n@[simp]\ntheorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toNonUnitalSubsemiring : (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubsemiring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toSubring {R A : Type*} [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    [IsScalarTower R A A] [SMulCommClass R A A] :\n    (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem toSubmodule_eq_top {S : NonUnitalSubalgebra R A} : S.toSubmodule = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toSubmodule'.injective.eq_iff' top_toSubmodule\n\n@[simp]\ntheorem toNonUnitalSubsemiring_eq_top {S : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubsemiring_injective.eq_iff' top_toNonUnitalSubsemiring\n\n@[simp]\ntheorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubring_injective.eq_iff' top_toSubring\n\ntheorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_left\n\ntheorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_right\n\ntheorem mul_mem_sup {S T : NonUnitalSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) :\n    x * y ∈ S ⊔ T :=\n  mul_mem (mem_sup_left hx) (mem_sup_right hy)\n\ntheorem map_sup (f : F) (S T : NonUnitalSubalgebra R A) :\n    ((S ⊔ T).map f : NonUnitalSubalgebra R B) = S.map f ⊔ T.map f :=\n  (@NonUnitalSubalgebra.gc_map_comap F R A B _ _ _ _ _ _ f).l_sup\n\n@[simp, norm_cast]\ntheorem coe_inf (S T : NonUnitalSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T :=\n  rfl\n\n@[simp]\ntheorem mem_inf {S T : NonUnitalSubalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T :=\n  Iff.rfl\n\n@[simp]\ntheorem inf_toSubmodule (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toSubmodule = S.toSubmodule ⊓ T.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem inf_toNonUnitalSubsemiring (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring ⊓ T.toNonUnitalSubsemiring :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sInf (S : Set (NonUnitalSubalgebra R A)) : (↑(sInf S) : Set A) = ⋂ s ∈ S, ↑s :=\n  sInf_image\n\ntheorem mem_sInf {S : Set (NonUnitalSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p := by\n  simp only [← SetLike.mem_coe, coe_sInf, Set.mem_iInter₂]\n\n@[simp]\ntheorem sInf_toSubmodule (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toSubmodule = sInf (NonUnitalSubalgebra.toSubmodule '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp]\ntheorem sInf_toNonUnitalSubsemiring (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toNonUnitalSubsemiring = sInf (NonUnitalSubalgebra.toNonUnitalSubsemiring '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp, norm_cast]\ntheorem coe_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} :\n    (↑(⨅ i, S i) : Set A) = ⋂ i, S i := by simp [iInf]\n\ntheorem mem_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} {x : A} :\n    (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by simp only [iInf, mem_sInf, Set.forall_range_iff]\n\n@[simp]\ntheorem iInf_toSubmodule {ι : Sort*} (S : ι → NonUnitalSubalgebra R A) :\n    (⨅ i, S i).toSubmodule = ⨅ i, (S i).toSubmodule :=\n  SetLike.coe_injective <| by simp\n\ninstance : Inhabited (NonUnitalSubalgebra R A) :=\n  ⟨⊥⟩\n\ntheorem mem_bot {x : A} : x ∈ (⊥ : NonUnitalSubalgebra R A) ↔ x = 0 :=\n  show x ∈ Submodule.span R (NonUnitalSubsemiring.closure (∅ : Set A) : Set A) ↔ x = 0 by\n    rw [NonUnitalSubsemiring.closure_empty, NonUnitalSubsemiring.coe_bot,\n      Submodule.span_zero_singleton, Submodule.mem_bot]\n\ntheorem toSubmodule_bot : (⊥ : NonUnitalSubalgebra R A).toSubmodule = ⊥ := by\n  ext\n  simp only [mem_bot, NonUnitalSubalgebra.mem_toSubmodule, Submodule.mem_bot]\n\n@[simp]\ntheorem coe_bot : ((⊥ : NonUnitalSubalgebra R A) : Set A) = {0} := by\n  simp [Set.ext_iff, NonUnitalAlgebra.mem_bot]\n\ntheorem eq_top_iff {S : NonUnitalSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S :=\n  ⟨fun h x => by rw [h]; exact mem_top, fun h => by ext x; exact ⟨fun _ => mem_top, fun _ => h x⟩⟩\n\ntheorem range_top_iff_surjective (f : A →ₙₐ[R] B) :\n    NonUnitalAlgHom.range f = (⊤ : NonUnitalSubalgebra R B) ↔ Function.Surjective f :=\n  NonUnitalAlgebra.eq_top_iff\n\n@[simp]\ntheorem range_id : NonUnitalAlgHom.range (NonUnitalAlgHom.id R A) = ⊤ :=\n  SetLike.coe_injective Set.range_id\n\n@[simp]\ntheorem map_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R A).map f = NonUnitalAlgHom.range f :=\n  SetLike.coe_injective Set.image_univ\n\n@[simp]\ntheorem map_bot (f : A →ₙₐ[R] B) : (⊥ : NonUnitalSubalgebra R A).map f = ⊥ :=\n  SetLike.coe_injective <| by simp [NonUnitalAlgebra.coe_bot, NonUnitalSubalgebra.coe_map]\n\n@[simp]\ntheorem comap_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R B).comap f = ⊤ :=\n  eq_top_iff.2 fun _ => mem_top\n\n/-- `NonUnitalAlgHom` to `⊤ : NonUnitalSubalgebra R A`. -/\ndef toTop : A →ₙₐ[R] (⊤ : NonUnitalSubalgebra R A) :=\n  NonUnitalAlgHom.codRestrict (NonUnitalAlgHom.id R A) ⊤ fun _ => mem_top\n\nend NonUnitalAlgebra\n\nnamespace NonUnitalSubalgebra\n\nopen NonUnitalAlgebra\n\nsection NonAssoc\n\nvariable {R : Type u} {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable (S : NonUnitalSubalgebra R A)\n\ninstance subsingleton_of_subsingleton [Subsingleton A] : Subsingleton (NonUnitalSubalgebra R A) :=\n  ⟨fun B C => ext fun x => by simp only [Subsingleton.elim x 0, zero_mem B, zero_mem C]⟩\n\ninstance _root_.NonUnitalAlgHom.subsingleton [Subsingleton (NonUnitalSubalgebra R A)] :\n    Subsingleton (A →ₙₐ[R] B) :=\n  ⟨fun f g =>\n    NonUnitalAlgHom.ext fun a =>\n      have : a ∈ (⊥ : NonUnitalSubalgebra R A) :=\n        Subsingleton.elim (⊤ : NonUnitalSubalgebra R A) ⊥ ▸ mem_top\n      (mem_bot.mp this).symm ▸ (map_zero f).trans (map_zero g).symm⟩\n\ntheorem range_val : NonUnitalAlgHom.range (NonUnitalSubalgebraClass.subtype S) = S :=\n  ext <| Set.ext_iff.1 <| (NonUnitalSubalgebraClass.subtype S).coe_range.trans Subtype.range_val\n\n/-- The map `S → T` when `S` is a non-unital subalgebra contained in the non-unital subalgebra `T`.\n\nThis is the non-unital subalgebra version of `Submodule.inclusion`, or `Subring.inclusion`  -/\ndef inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) : S →ₙₐ[R] T\n    where\n  toFun := Set.inclusion h\n  map_add' _ _ := rfl\n  map_mul' _ _ := rfl\n  map_zero' := rfl\n  map_smul' _ _ := rfl\n\ntheorem inclusion_injective {S T : NonUnitalSubalgebra R A} (h : S ≤ T) :\n    Function.Injective (inclusion h) := fun _ _ => Subtype.ext ∘ Subtype.mk.inj\n\n@[simp]\ntheorem inclusion_self {S : NonUnitalSubalgebra R A} :\n    inclusion (le_refl S) = NonUnitalAlgHom.id R S :=\n  NonUnitalAlgHom.ext fun _ => Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_mk {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : A) (hx : x ∈ S) :\n    inclusion h ⟨x, hx⟩ = ⟨x, h hx⟩ :=\n  rfl\n\ntheorem inclusion_right {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : T) (m : (x : A) ∈ S) :\n    inclusion h ⟨x, m⟩ = x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_inclusion {S T U : NonUnitalSubalgebra R A} (hst : S ≤ T) (htu : T ≤ U) (x : S) :\n    inclusion htu (inclusion hst x) = inclusion (le_trans hst htu) x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem coe_inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (s : S) :\n    (inclusion h s : A) = s :=\n  rfl\n\nsection Prod\n\nvariable (S₁ : NonUnitalSubalgebra R B)\n\n/-- The product of two non-unital subalgebras is a non-unital subalgebra. -/\ndef prod : NonUnitalSubalgebra R (A × B) :=\n  { S.toNonUnitalSubsemiring.prod S₁.toNonUnitalSubsemiring with\n    carrier := S ×ˢ S₁\n    smul_mem' := fun r _x hx => ⟨SMulMemClass.smul_mem r hx.1, SMulMemClass.smul_mem r hx.2⟩ }\n\n@[simp]\ntheorem coe_prod : (prod S S₁ : Set (A × B)) = (S : Set A) ×ˢ S₁ :=\n  rfl\n\ntheorem prod_toSubmodule : (S.prod S₁).toSubmodule = S.toSubmodule.prod S₁.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem mem_prod {S : NonUnitalSubalgebra R A} {S₁ : NonUnitalSubalgebra R B} {x : A × B} :\n    x ∈ prod S S₁ ↔ x.1 ∈ S ∧ x.2 ∈ S₁ :=\n  Set.mem_prod\n\n@[simp]\ntheorem prod_top : (prod ⊤ ⊤ : NonUnitalSubalgebra R (A × B)) = ⊤ := by ext; simp\n\ntheorem prod_mono {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S ≤ T → S₁ ≤ T₁ → prod S S₁ ≤ prod T T₁ :=\n  Set.prod_mono\n\n@[simp]\ntheorem prod_inf_prod {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S.prod S₁ ⊓ T.prod T₁ = (S ⊓ T).prod (S₁ ⊓ T₁) :=\n  SetLike.coe_injective Set.prod_inter_prod\n\nend Prod\n\nsection SuprLift\n\nvariable {ι : Type*}\n\ntheorem coe_iSup_of_directed [Nonempty ι] {S : ι → NonUnitalSubalgebra R A}\n    (dir : Directed (· ≤ ·) S) : ↑(iSup S) = ⋃ i, (S i : Set A) :=\n  let K : NonUnitalSubalgebra R A :=\n    { __ := NonUnitalSubsemiring.copy _ _ (NonUnitalSubsemiring.coe_iSup_of_directed dir).symm\n      smul_mem' := fun r _x hx ↦\n        let ⟨i, hi⟩ := Set.mem_iUnion.1 hx\n        Set.mem_iUnion.2 ⟨i, (S i).smul_mem' r hi⟩ }\n  have : iSup S = K := le_antisymm\n    (iSup_le fun i ↦ le_iSup (fun i ↦ (S i : Set A)) i) (Set.iUnion_subset fun _ ↦ le_iSup S _)\n  this.symm ▸ rfl\n\n/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              rw [hf j k hjk]\n              rfl)\n            (↑(iSup K)) (by rw [coe_iSup_of_directed dir])\n        map_zero' := by\n          dsimp\n          exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl) _ (by simp)\n        map_mul' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· * ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_add' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· + ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_smul' := fun r => by\n          dsimp\n          apply Set.iUnionLift_unary (coe_iSup_of_directed dir) _ (fun _ x => r • x)\n            (fun _ _ => rfl)\n          on_goal 2 => rw [coe_iSup_of_directed dir]\n          all_goals simp }\n\nvariable [Nonempty ι] {K : ι → NonUnitalSubalgebra R A} {dir : Directed (· ≤ ·) K}\n  {f : ∀ i, K i →ₙₐ[R] B} {hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h)}\n  {T : NonUnitalSubalgebra R A} {hT : T = iSup K}\n\n@[simp]\ntheorem iSupLift_inclusion {i : ι} (x : K i) (h : K i ≤ T) :\n    iSupLift K dir f hf T hT (inclusion h x) = f i x := by\n  subst T\n  dsimp [iSupLift]\n  apply Set.iUnionLift_inclusion\n  exact h\n\n@[simp]\ntheorem iSupLift_comp_inclusion {i : ι} (h : K i ≤ T) :\n    (iSupLift K dir f hf T hT).comp (inclusion h) = f i := by ext; simp\n\n@[simp]\ntheorem iSupLift_mk {i : ι} (x : K i) (hx : (x : A) ∈ T) :\n    iSupLift K dir f hf T hT ⟨x, hx⟩ = f i x := by\n  subst hT\n  dsimp [iSupLift]\n  apply Set.iUnionLift_mk\n\ntheorem iSupLift_of_mem {i : ι} (x : T) (hx : (x : A) ∈ K i) :\n    iSupLift K dir f hf T hT x = f i ⟨x, hx⟩ := by\n  subst hT\n  dsimp [iSupLift]\n  apply Set.iUnionLift_of_mem\n\nend SuprLift\n\nend NonAssoc\n\nsection Center\n\nsection NonUnitalNonAssocSemiring\nvariable {R A : Type*}\nvariable [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [IsScalarTower R A A] [SMulCommClass R A A]\n\ntheorem _root_.Set.smul_mem_center (r : R) {a : A} (ha : a ∈ Set.center A) :\n    r • a ∈ Set.center A where\n  comm b := by rw [mul_smul_comm, smul_mul_assoc, ha.comm]\n  left_assoc b c := by rw [smul_mul_assoc, smul_mul_assoc, smul_mul_assoc, ha.left_assoc]\n  mid_assoc b c := by\n    rw [mul_smul_comm]\n    rw [smul_mul_assoc]\n    rw [smul_mul_assoc]\n    rw [mul_smul_comm]\n    rw [ha.mid_assoc]\n  right_assoc b c := by\n    rw [mul_smul_comm]\n    rw [mul_smul_comm]\n    rw [mul_smul_comm]\n    ","nextTactic":"rw [ha.right_assoc]","declUpToTactic":"theorem _root_.Set.smul_mem_center (r : R) {a : A} (ha : a ∈ Set.center A) :\n    r • a ∈ Set.center A where\n  comm b := by rw [mul_smul_comm, smul_mul_assoc, ha.comm]\n  left_assoc b c := by rw [smul_mul_assoc, smul_mul_assoc, smul_mul_assoc, ha.left_assoc]\n  mid_assoc b c := by\n    rw [mul_smul_comm]\n    rw [smul_mul_assoc]\n    rw [smul_mul_assoc]\n    rw [mul_smul_comm]\n    rw [ha.mid_assoc]\n  right_assoc b c := by\n    rw [mul_smul_comm]\n    rw [mul_smul_comm]\n    rw [mul_smul_comm]\n    ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.969_0.jQvdAcOcyVirh23","decl":"theorem _root_.Set.smul_mem_center (r : R) {a : A} (ha : a ∈ Set.center A) :\n    r • a ∈ Set.center A where\n  comm b "}
-{"state":"R : Type u_1\nA : Type u_2\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\ns : Set A\nr : R\na : A\nha : a ∈ Set.centralizer s\nx : A\nhx : x ∈ s\n⊢ x * r • a = r • a * x","srcUpToTactic":"/-\nCopyright (c) 2023 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport Mathlib.Algebra.Algebra.NonUnitalHom\nimport Mathlib.Data.Set.UnionLift\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.RingTheory.NonUnitalSubring.Basic\n\n/-!\n# Non-unital Subalgebras over Commutative Semirings\n\nIn this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).\n\n## TODO\n\n* once we have scalar actions by semigroups (as opposed to monoids), implement the action of a\n  non-unital subalgebra on the larger algebra.\n-/\n\nuniverse u u' v v' w w'\n\nopen scoped BigOperators\n\nsection NonUnitalSubalgebraClass\n\nvariable {S R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [SetLike S A] [NonUnitalSubsemiringClass S A] [hSR : SMulMemClass S R A] (s : S)\n\nnamespace NonUnitalSubalgebraClass\n\n/-- Embedding of a non-unital subalgebra into the non-unital algebra. -/\ndef subtype (s : S) : s →ₙₐ[R] A :=\n  { NonUnitalSubsemiringClass.subtype s, SMulMemClass.subtype s with toFun := (↑) }\n\n@[simp]\ntheorem coeSubtype : (subtype s : s → A) = ((↑) : s → A) :=\n  rfl\n\nend NonUnitalSubalgebraClass\n\nend NonUnitalSubalgebraClass\n\n/-- A non-unital subalgebra is a sub(semi)ring that is also a submodule. -/\nstructure NonUnitalSubalgebra (R : Type u) (A : Type v) [CommSemiring R]\n    [NonUnitalNonAssocSemiring A] [Module R A]\n    extends NonUnitalSubsemiring A, Submodule R A : Type v\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `NonUnitalSubsemiring`. -/\nadd_decl_doc NonUnitalSubalgebra.toNonUnitalSubsemiring\n\n/-- Reinterpret a `NonUnitalSubalgebra` as a `Submodule`. -/\nadd_decl_doc NonUnitalSubalgebra.toSubmodule\n\nnamespace NonUnitalSubalgebra\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\n\nsection NonUnitalNonAssocSemiring\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance : SetLike (NonUnitalSubalgebra R A) A\n    where\n  coe s := s.carrier\n  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.coe_injective h\n\ninstance instNonUnitalSubsemiringClass : NonUnitalSubsemiringClass (NonUnitalSubalgebra R A) A\n    where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  zero_mem {s} := s.zero_mem'\n\ninstance instSMulMemClass : SMulMemClass (NonUnitalSubalgebra R A) R A where\n  smul_mem := @fun s => s.smul_mem'\n\ntheorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n\n@[ext]\ntheorem ext {S T : NonUnitalSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem mem_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubsemiring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubsemiring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubsemiring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A → NonUnitalSubsemiring A) :=\n  fun S T h =>\n  ext fun x => by rw [← mem_toNonUnitalSubsemiring, ← mem_toNonUnitalSubsemiring, h]\n\ntheorem toNonUnitalSubsemiring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = U.toNonUnitalSubsemiring ↔ S = U :=\n  toNonUnitalSubsemiring_injective.eq_iff\n\ntheorem mem_toSubmodule (S : NonUnitalSubalgebra R A) {x} : x ∈ S.toSubmodule ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toSubmodule (S : NonUnitalSubalgebra R A) : (↑S.toSubmodule : Set A) = S :=\n  rfl\n\ntheorem toSubmodule_injective :\n    Function.Injective (toSubmodule : NonUnitalSubalgebra R A → Submodule R A) := fun S T h =>\n  ext fun x => by rw [← mem_toSubmodule, ← mem_toSubmodule, h]\n\ntheorem toSubmodule_inj {S U : NonUnitalSubalgebra R A} : S.toSubmodule = U.toSubmodule ↔ S = U :=\n  toSubmodule_injective.eq_iff\n\n/-- Copy of a non-unital subalgebra with a new `carrier` equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.copy s hs with\n    smul_mem' := fun r a (ha : a ∈ s) => by\n      show r • a ∈ s\n      rw [hs] at ha ⊢\n      exact S.smul_mem' r ha }\n\n@[simp]\ntheorem coe_copy (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) :\n    (S.copy s hs : Set A) = s :=\n  rfl\n\ntheorem copy_eq (S : NonUnitalSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n\ninstance (S : NonUnitalSubalgebra R A) : Inhabited S :=\n  ⟨(0 : S.toNonUnitalSubsemiring)⟩\n\nend NonUnitalNonAssocSemiring\n\nsection NonUnitalNonAssocRing\nvariable [CommRing R]\nvariable [NonUnitalNonAssocRing A] [NonUnitalNonAssocRing B] [NonUnitalNonAssocRing C]\nvariable [Module R A] [Module R B] [Module R C]\n\ninstance instNonUnitalSubringClass : NonUnitalSubringClass (NonUnitalSubalgebra R A) A :=\n  { NonUnitalSubalgebra.instNonUnitalSubsemiringClass with\n    neg_mem := @fun _ x hx => neg_one_smul R x ▸ SMulMemClass.smul_mem _ hx }\n\n/-- A non-unital subalgebra over a ring is also a `Subring`. -/\ndef toNonUnitalSubring (S : NonUnitalSubalgebra R A) : NonUnitalSubring A where\n  toNonUnitalSubsemiring := S.toNonUnitalSubsemiring\n  neg_mem' := neg_mem (s := S)\n\n@[simp]\ntheorem mem_toNonUnitalSubring {S : NonUnitalSubalgebra R A} {x} :\n    x ∈ S.toNonUnitalSubring ↔ x ∈ S :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubring (S : NonUnitalSubalgebra R A) :\n    (↑S.toNonUnitalSubring : Set A) = S :=\n  rfl\n\ntheorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A → NonUnitalSubring A) :=\n  fun S T h => ext fun x => by rw [← mem_toNonUnitalSubring, ← mem_toNonUnitalSubring, h]\n\ntheorem toNonUnitalSubring_inj {S U : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubring = U.toNonUnitalSubring ↔ S = U :=\n  toNonUnitalSubring_injective.eq_iff\n\nend NonUnitalNonAssocRing\n\nsection\n\n/-! `NonUnitalSubalgebra`s inherit structure from their `NonUnitalSubsemiring` / `Semiring`\ncoercions. -/\n\n\ninstance toNonUnitalNonAssocSemiring [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocSemiring S :=\n  inferInstance\n\ninstance toNonUnitalSemiring [CommSemiring R] [NonUnitalSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalSemiring S :=\n  inferInstance\n\ninstance toNonUnitalCommSemiring [CommSemiring R] [NonUnitalCommSemiring A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommSemiring S :=\n  inferInstance\n\ninstance toNonUnitalNonAssocRing [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalNonAssocRing S :=\n  inferInstance\n\ninstance toNonUnitalRing [CommRing R] [NonUnitalRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalRing S :=\n  inferInstance\n\ninstance toNonUnitalCommRing [CommRing R] [NonUnitalCommRing A] [Module R A]\n    (S : NonUnitalSubalgebra R A) : NonUnitalCommRing S :=\n  inferInstance\n\nend\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `Submodule` as an `OrderEmbedding` -/\ndef toSubmodule' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o Submodule R A where\n  toEmbedding :=\n    { toFun := fun S => S.toSubmodule\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubsemiring' [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubsemiring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubsemiring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\n/-- The forgetful map from `NonUnitalSubalgebra` to `NonUnitalSubsemiring` as an\n`OrderEmbedding` -/\ndef toNonUnitalSubring' [CommRing R] [NonUnitalNonAssocRing A] [Module R A] :\n    NonUnitalSubalgebra R A ↪o NonUnitalSubring A where\n  toEmbedding :=\n    { toFun := fun S => S.toNonUnitalSubring\n      inj' := fun S T h => ext <| by apply SetLike.ext_iff.1 h }\n  map_rel_iff' := SetLike.coe_subset_coe.symm.trans SetLike.coe_subset_coe\n\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [NonUnitalNonAssocSemiring C]\nvariable [Module R A] [Module R B] [Module R C]\nvariable {S : NonUnitalSubalgebra R A}\n\nsection\n\n/-! ### `NonUnitalSubalgebra`s inherit structure from their `Submodule` coercions. -/\n\ninstance instModule' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] : Module R' S :=\n  SMulMemClass.toModule' _ R' R A S\n\ninstance instModule : Module R S :=\n  S.instModule'\n\ninstance instIsScalarTower' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] :\n    IsScalarTower R' R S :=\n  S.toSubmodule.isScalarTower\n\ninstance [IsScalarTower R A A] : IsScalarTower R S S where\n  smul_assoc r x y := Subtype.ext <| smul_assoc r (x : A) (y : A)\n\ninstance instSMulCommClass' [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A]\n    [SMulCommClass R' R A] : SMulCommClass R' R S where\n  smul_comm r' r s := Subtype.ext <| smul_comm r' r (s : A)\n\ninstance instSMulCommClass [SMulCommClass R A A] : SMulCommClass R S S where\n  smul_comm r x y := Subtype.ext <| smul_comm r (x : A) (y : A)\n\ninstance noZeroSMulDivisors_bot [NoZeroSMulDivisors R A] : NoZeroSMulDivisors R S :=\n  ⟨fun {c x} h =>\n    have : c = 0 ∨ (x : A) = 0 := eq_zero_or_eq_zero_of_smul_eq_zero (congr_arg ((↑) : S → A) h)\n    this.imp_right (@Subtype.ext_iff _ _ x 0).mpr⟩\n\nend\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : A) = ↑x + ↑y :=\n  rfl\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : A) = ↑x * ↑y :=\n  rfl\n\nprotected theorem coe_zero : ((0 : S) : A) = 0 :=\n  rfl\n\nprotected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (↑(-x) : A) = -↑x :=\n  rfl\n\nprotected theorem coe_sub {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x y : S) : (↑(x - y) : A) = ↑x - ↑y :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    ↑(r • x) = r • (x : A) :=\n  rfl\n\nprotected theorem coe_eq_zero {x : S} : (x : A) = 0 ↔ x = 0 :=\n  ZeroMemClass.coe_eq_zero\n\n@[simp]\ntheorem toNonUnitalSubsemiring_subtype :\n    NonUnitalSubsemiringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n@[simp]\ntheorem toSubring_subtype {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    (S : NonUnitalSubalgebra R A) :\n    NonUnitalSubringClass.subtype S = NonUnitalSubalgebraClass.subtype (R := R) S :=\n  rfl\n\n/-- Linear equivalence between `S : Submodule R A` and `S`. Though these types are equal,\nwe define it as a `LinearEquiv` to avoid type equalities. -/\ndef toSubmoduleEquiv (S : NonUnitalSubalgebra R A) : S.toSubmodule ≃ₗ[R] S :=\n  LinearEquiv.ofEq _ _ rfl\n\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- Transport a non-unital subalgebra via an algebra homomorphism. -/\ndef map (f : F) (S : NonUnitalSubalgebra R A) : NonUnitalSubalgebra R B :=\n  { S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) with\n    smul_mem' := fun r b hb => by\n      rcases hb with ⟨a, ha, rfl⟩\n      exact map_smul f r a ▸ Set.mem_image_of_mem f (S.smul_mem' r ha) }\n\ntheorem map_mono {S₁ S₂ : NonUnitalSubalgebra R A} {f : F} :\n    S₁ ≤ S₂ → (map f S₁ : NonUnitalSubalgebra R B) ≤ map f S₂ :=\n  Set.image_subset f\n\ntheorem map_injective {f : F} (hf : Function.Injective f) :\n    Function.Injective (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) :=\n  fun _S₁ _S₂ ih =>\n  ext <| Set.ext_iff.1 <| Set.image_injective.2 hf <| Set.ext <| SetLike.ext_iff.mp ih\n\n@[simp]\ntheorem map_id (S : NonUnitalSubalgebra R A) : map (NonUnitalAlgHom.id R A) S = S :=\n  SetLike.coe_injective <| Set.image_id _\n\ntheorem map_map (S : NonUnitalSubalgebra R A) (g : B →ₙₐ[R] C) (f : A →ₙₐ[R] B) :\n    (S.map f).map g = S.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n\n@[simp]\ntheorem mem_map {S : NonUnitalSubalgebra R A} {f : F} {y : B} : y ∈ map f S ↔ ∃ x ∈ S, f x = y :=\n  NonUnitalSubsemiring.mem_map\n\ntheorem map_toSubmodule {S : NonUnitalSubalgebra R A} {f : F} :\n    -- TODO: introduce a better coercion from `NonUnitalAlgHomClass` to `LinearMap`\n    (map f S).toSubmodule = Submodule.map ((↑f : A →+[R] B) : A →ₗ[R] B) S.toSubmodule :=\n  SetLike.coe_injective rfl\n\ntheorem map_toNonUnitalSubsemiring {S : NonUnitalSubalgebra R A} {f : F} :\n    (map f S).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring.map (f : A →ₙ+* B) :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem coe_map (S : NonUnitalSubalgebra R A) (f : F) : (map f S : Set B) = f '' S :=\n  rfl\n\n/-- Preimage of a non-unital subalgebra under an algebra homomorphism. -/\ndef comap (f : F) (S : NonUnitalSubalgebra R B) : NonUnitalSubalgebra R A :=\n  { S.toNonUnitalSubsemiring.comap (f : A →ₙ+* B) with\n    smul_mem' := fun r a (ha : f a ∈ S) =>\n      show f (r • a) ∈ S from (map_smul f r a).symm ▸ SMulMemClass.smul_mem r ha }\n\ntheorem map_le {S : NonUnitalSubalgebra R A} {f : F} {U : NonUnitalSubalgebra R B} :\n    map f S ≤ U ↔ S ≤ comap f U :=\n  Set.image_subset_iff\n\ntheorem gc_map_comap (f : F) :\n    GaloisConnection (map f : NonUnitalSubalgebra R A → NonUnitalSubalgebra R B) (comap f) :=\n  fun _ _ => map_le\n\n@[simp]\ntheorem mem_comap (S : NonUnitalSubalgebra R B) (f : F) (x : A) : x ∈ comap f S ↔ f x ∈ S :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f ⁻¹' (S : Set B) :=\n  rfl\n\ninstance noZeroDivisors {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [NoZeroDivisors A]\n    [Module R A] (S : NonUnitalSubalgebra R A) : NoZeroDivisors S :=\n  NonUnitalSubsemiringClass.noZeroDivisors S\n\nend NonUnitalSubalgebra\n\nnamespace Submodule\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\n\n/-- A submodule closed under multiplication is a non-unital subalgebra. -/\ndef toNonUnitalSubalgebra (p : Submodule R A) (h_mul : ∀ x y, x ∈ p → y ∈ p → x * y ∈ p) :\n    NonUnitalSubalgebra R A :=\n  { p with\n    mul_mem' := h_mul _ _ }\n\n@[simp]\ntheorem mem_toNonUnitalSubalgebra {p : Submodule R A} {h_mul} {x} :\n    x ∈ p.toNonUnitalSubalgebra h_mul ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toNonUnitalSubalgebra (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul : Set A) = p :=\n  rfl\n\ntheorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk ⟨⟨⟨p, p.add_mem⟩, p.zero_mem⟩, hmul _ _⟩ p.smul_mem' :=\n  rfl\n\n@[simp]\ntheorem toNonUnitalSubalgebra_toSubmodule (p : Submodule R A) (h_mul) :\n    (p.toNonUnitalSubalgebra h_mul).toSubmodule = p :=\n  SetLike.coe_injective rfl\n\n@[simp]\ntheorem _root_.NonUnitalSubalgebra.toSubmodule_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A) :\n    (S.toSubmodule.toNonUnitalSubalgebra fun _ _ => mul_mem (s := S)) = S :=\n  SetLike.coe_injective rfl\n\nend Submodule\n\nnamespace NonUnitalAlgHom\n\nvariable {F : Type v'} {R' : Type u'} {R : Type u} {A : Type v} {B : Type w} {C : Type w'}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [NonUnitalNonAssocSemiring B] [Module R B]\nvariable [NonUnitalNonAssocSemiring C] [Module R C] [NonUnitalAlgHomClass F R A B]\n\n/-- Range of an `NonUnitalAlgHom` as a non-unital subalgebra. -/\nprotected def range (φ : F) : NonUnitalSubalgebra R B where\n  toNonUnitalSubsemiring := NonUnitalRingHom.srange (φ : A →ₙ+* B)\n  smul_mem' := fun r a => by rintro ⟨a, rfl⟩; exact ⟨r • a, map_smul φ r a⟩\n\n@[simp]\ntheorem mem_range (φ : F) {y : B} :\n    y ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) ↔ ∃ x : A, φ x = y :=\n  NonUnitalRingHom.mem_srange\n\ntheorem mem_range_self (φ : F) (x : A) :\n    φ x ∈ (NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) :=\n  (NonUnitalAlgHom.mem_range φ).2 ⟨x, rfl⟩\n\n@[simp]\ntheorem coe_range (φ : F) :\n    ((NonUnitalAlgHom.range φ : NonUnitalSubalgebra R B) : Set B) = Set.range (φ : A → B) := by\n  ext\n  rw [SetLike.mem_coe]\n  rw [mem_range]\n  rfl\n\ntheorem range_comp (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) = (NonUnitalAlgHom.range f).map g :=\n  SetLike.coe_injective (Set.range_comp g f)\n\ntheorem range_comp_le_range (f : A →ₙₐ[R] B) (g : B →ₙₐ[R] C) :\n    NonUnitalAlgHom.range (g.comp f) ≤ NonUnitalAlgHom.range g :=\n  SetLike.coe_mono (Set.range_comp_subset_range f g)\n\n/-- Restrict the codomain of a non-unital algebra homomorphism. -/\ndef codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) : A →ₙₐ[R] S :=\n  { NonUnitalRingHom.codRestrict (f : A →ₙ+* B) S.toNonUnitalSubsemiring hf with\n    map_smul' := fun r a => Subtype.ext <| map_smul f r a }\n\n@[simp]\ntheorem subtype_comp_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    (NonUnitalSubalgebraClass.subtype S).comp (NonUnitalAlgHom.codRestrict f S hf) = f :=\n  NonUnitalAlgHom.ext fun _ => rfl\n\n@[simp]\ntheorem coe_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x, f x ∈ S) (x : A) :\n    ↑(NonUnitalAlgHom.codRestrict f S hf x) = f x :=\n  rfl\n\ntheorem injective_codRestrict (f : F) (S : NonUnitalSubalgebra R B) (hf : ∀ x : A, f x ∈ S) :\n    Function.Injective (NonUnitalAlgHom.codRestrict f S hf) ↔ Function.Injective f :=\n  ⟨fun H _x _y hxy => H <| Subtype.eq hxy, fun H _x _y hxy => H (congr_arg Subtype.val hxy : _)⟩\n\n/-- Restrict the codomain of an `NonUnitalAlgHom` `f` to `f.range`.\n\nThis is the bundled version of `Set.rangeFactorization`. -/\n@[reducible]\ndef rangeRestrict (f : F) : A →ₙₐ[R] (NonUnitalAlgHom.range f : NonUnitalSubalgebra R B) :=\n  NonUnitalAlgHom.codRestrict f (NonUnitalAlgHom.range f) (NonUnitalAlgHom.mem_range_self f)\n\n/-- The equalizer of two non-unital `R`-algebra homomorphisms -/\ndef equalizer (ϕ ψ : F) : NonUnitalSubalgebra R A\n    where\n  carrier := {a | (ϕ a : B) = ψ a}\n  zero_mem' := by rw [Set.mem_setOf_eq, map_zero, map_zero]\n  add_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_add]\n    rw [map_add]\n    rw [hx]\n    rw [hy]\n  mul_mem' {x y} (hx : ϕ x = ψ x) (hy : ϕ y = ψ y) := by\n    rw [Set.mem_setOf_eq]\n    rw [map_mul]\n    rw [map_mul]\n    rw [hx]\n    rw [hy]\n  smul_mem' r x (hx : ϕ x = ψ x) := by rw [Set.mem_setOf_eq, map_smul, map_smul, hx]\n\n@[simp]\ntheorem mem_equalizer (φ ψ : F) (x : A) :\n    x ∈ @NonUnitalAlgHom.equalizer F R A B _ _ _ _ _ _ φ ψ ↔ φ x = ψ x :=\n  Iff.rfl\n\n/-- The range of a morphism of algebras is a fintype, if the domain is a fintype.\n\nNote that this instance can cause a diamond with `Subtype.fintype` if `B` is also a fintype. -/\ninstance fintypeRange [Fintype A] [DecidableEq B] (φ : F) :\n    Fintype (@NonUnitalAlgHom.range F R A B _ _ _ _ _ _ φ) :=\n  Set.fintypeRange φ\n\nend NonUnitalAlgHom\n\nnamespace NonUnitalAlgebra\n\nvariable {F : Type*} (R : Type u) {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable [NonUnitalAlgHomClass F R A B]\n\n/-- The minimal non-unital subalgebra that includes `s`. -/\ndef adjoin (s : Set A) : NonUnitalSubalgebra R A :=\n  { Submodule.span R (NonUnitalSubsemiring.closure s : Set A) with\n    mul_mem' :=\n      @fun a b (ha : a ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A))\n        (hb : b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A)) =>\n      show a * b ∈ Submodule.span R (NonUnitalSubsemiring.closure s : Set A) by\n        refine' Submodule.span_induction ha _ _ _ _\n        · refine' Submodule.span_induction hb _ _ _ _\n          · exact fun x (hx : x ∈ NonUnitalSubsemiring.closure s) y\n              (hy : y ∈ NonUnitalSubsemiring.closure s) => Submodule.subset_span (mul_mem hy hx)\n          · exact fun x _hx => (mul_zero x).symm ▸ Submodule.zero_mem _\n          · exact fun x y hx hy z hz => (mul_add z x y).symm ▸ add_mem (hx z hz) (hy z hz)\n          · exact fun r x hx y hy => (mul_smul_comm r y x).symm ▸ SMulMemClass.smul_mem r (hx y hy)\n        · exact (zero_mul b).symm ▸ Submodule.zero_mem _\n        · exact fun x y => (add_mul x y b).symm ▸ add_mem\n        · exact fun r x hx => (smul_mul_assoc r x b).symm ▸ SMulMemClass.smul_mem r hx }\n\ntheorem adjoin_toSubmodule (s : Set A) :\n    (adjoin R s).toSubmodule = Submodule.span R (NonUnitalSubsemiring.closure s : Set A) :=\n  rfl\n\n@[aesop safe 20 apply (rule_sets [SetLike])]\ntheorem subset_adjoin {s : Set A} : s ⊆ adjoin R s :=\n  NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span\n\ntheorem self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : Set A) :=\n  NonUnitalAlgebra.subset_adjoin R (Set.mem_singleton x)\n\nvariable {R}\n\n/-- If some predicate holds for all `x ∈ (s : Set A)` and this predicate is closed under the\n`algebraMap`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. -/\ntheorem adjoin_induction {s : Set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)\n    (Hs : ∀ x ∈ s, p x) (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Submodule.span_induction h\n    (fun _a ha => NonUnitalSubsemiring.closure_induction ha Hs H0 Hadd Hmul) H0 Hadd Hsmul\n\ntheorem adjoin_induction₂ {s : Set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)\n    (hb : b ∈ adjoin R s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ y, p 0 y)\n    (H0_right : ∀ x, p x 0) (Hadd_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ + x₂) y)\n    (Hadd_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ + y₂))\n    (Hmul_left : ∀ x₁ x₂ y, p x₁ y → p x₂ y → p (x₁ * x₂) y)\n    (Hmul_right : ∀ x y₁ y₂, p x y₁ → p x y₂ → p x (y₁ * y₂))\n    (Hsmul_left : ∀ (r : R) x y, p x y → p (r • x) y)\n    (Hsmul_right : ∀ (r : R) x y, p x y → p x (r • y)) : p a b :=\n  Submodule.span_induction₂ ha hb\n    (fun _x hx _y hy =>\n      NonUnitalSubsemiring.closure_induction₂ hx hy Hs H0_left H0_right Hadd_left Hadd_right\n        Hmul_left Hmul_right)\n    H0_left H0_right Hadd_left Hadd_right Hsmul_left Hsmul_right\n\n/-- The difference with `NonUnitalAlgebra.adjoin_induction` is that this acts on the subtype. -/\nlemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)\n    (Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R h⟩)\n    (Hadd : ∀ x y, p x → p y → p (x + y)) (H0 : p 0)\n    (Hmul : ∀ x y, p x → p y → p (x * y)) (Hsmul : ∀ (r : R) x, p x → p (r • x)) : p a :=\n  Subtype.recOn a <| fun b hb => by\n    refine Exists.elim ?_ (fun (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩) => hc)\n    apply adjoin_induction hb\n    · exact fun x hx => ⟨subset_adjoin R hx, Hs x hx⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨add_mem hx' hy', Hadd _ _ hx hy⟩\n    · exact ⟨_, H0⟩\n    · exact fun x y hx hy => Exists.elim hx <| fun hx' hx => Exists.elim hy <| fun hy' hy =>\n        ⟨mul_mem hx' hy', Hmul _ _ hx hy⟩\n    · exact fun r x hx => Exists.elim hx <| fun hx' hx =>\n        ⟨SMulMemClass.smul_mem r hx', Hsmul r _ hx⟩\n\nprotected theorem gc : GaloisConnection (adjoin R : Set A → NonUnitalSubalgebra R A) (↑) :=\n  fun s S =>\n  ⟨fun H => (NonUnitalSubsemiring.subset_closure.trans Submodule.subset_span).trans H,\n    fun H => show Submodule.span R _ ≤ S.toSubmodule from Submodule.span_le.mpr <|\n      show NonUnitalSubsemiring.closure s ≤ S.toNonUnitalSubsemiring from\n        NonUnitalSubsemiring.closure_le.2 H⟩\n\n/-- Galois insertion between `adjoin` and `Subtype.val`. -/\nprotected def gi : GaloisInsertion (adjoin R : Set A → NonUnitalSubalgebra R A) (↑)\n    where\n  choice s hs := (adjoin R s).copy s <| le_antisymm (NonUnitalAlgebra.gc.le_u_l s) hs\n  gc := NonUnitalAlgebra.gc\n  le_l_u S := (NonUnitalAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl\n  choice_eq _ _ := NonUnitalSubalgebra.copy_eq _ _ _\n\ninstance : CompleteLattice (NonUnitalSubalgebra R A) :=\n  GaloisInsertion.liftCompleteLattice NonUnitalAlgebra.gi\n\ntheorem adjoin_le {S : NonUnitalSubalgebra R A} {s : Set A} (hs : s ⊆ S) : adjoin R s ≤ S :=\n  NonUnitalAlgebra.gc.l_le hs\n\ntheorem adjoin_le_iff {S : NonUnitalSubalgebra R A} {s : Set A} : adjoin R s ≤ S ↔ s ⊆ S :=\n  NonUnitalAlgebra.gc _ _\n\ntheorem adjoin_union (s t : Set A) : adjoin R (s ∪ t) = adjoin R s ⊔ adjoin R t :=\n  (NonUnitalAlgebra.gc : GaloisConnection _ ((↑) : NonUnitalSubalgebra R A → Set A)).l_sup\n\nvariable (R A)\n\n@[simp]\ntheorem adjoin_empty : adjoin R (∅ : Set A) = ⊥ :=\n  show adjoin R ⊥ = ⊥ by apply GaloisConnection.l_bot; exact NonUnitalAlgebra.gc\n\n@[simp]\ntheorem adjoin_univ : adjoin R (Set.univ : Set A) = ⊤ :=\n  eq_top_iff.2 fun _x hx => subset_adjoin R hx\n\nvariable {R A}\n\n@[simp]\ntheorem coe_top : (↑(⊤ : NonUnitalSubalgebra R A) : Set A) = Set.univ :=\n  rfl\n\n@[simp]\ntheorem mem_top {x : A} : x ∈ (⊤ : NonUnitalSubalgebra R A) :=\n  Set.mem_univ x\n\n@[simp]\ntheorem top_toSubmodule : (⊤ : NonUnitalSubalgebra R A).toSubmodule = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toNonUnitalSubsemiring : (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubsemiring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem top_toSubring {R A : Type*} [CommRing R] [NonUnitalNonAssocRing A] [Module R A]\n    [IsScalarTower R A A] [SMulCommClass R A A] :\n    (⊤ : NonUnitalSubalgebra R A).toNonUnitalSubring = ⊤ :=\n  rfl\n\n@[simp]\ntheorem toSubmodule_eq_top {S : NonUnitalSubalgebra R A} : S.toSubmodule = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toSubmodule'.injective.eq_iff' top_toSubmodule\n\n@[simp]\ntheorem toNonUnitalSubsemiring_eq_top {S : NonUnitalSubalgebra R A} :\n    S.toNonUnitalSubsemiring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubsemiring_injective.eq_iff' top_toNonUnitalSubsemiring\n\n@[simp]\ntheorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = ⊤ ↔ S = ⊤ :=\n  NonUnitalSubalgebra.toNonUnitalSubring_injective.eq_iff' top_toSubring\n\ntheorem mem_sup_left {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_left\n\ntheorem mem_sup_right {S T : NonUnitalSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by\n  rw [← SetLike.le_def]\n  exact le_sup_right\n\ntheorem mul_mem_sup {S T : NonUnitalSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) :\n    x * y ∈ S ⊔ T :=\n  mul_mem (mem_sup_left hx) (mem_sup_right hy)\n\ntheorem map_sup (f : F) (S T : NonUnitalSubalgebra R A) :\n    ((S ⊔ T).map f : NonUnitalSubalgebra R B) = S.map f ⊔ T.map f :=\n  (@NonUnitalSubalgebra.gc_map_comap F R A B _ _ _ _ _ _ f).l_sup\n\n@[simp, norm_cast]\ntheorem coe_inf (S T : NonUnitalSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T :=\n  rfl\n\n@[simp]\ntheorem mem_inf {S T : NonUnitalSubalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T :=\n  Iff.rfl\n\n@[simp]\ntheorem inf_toSubmodule (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toSubmodule = S.toSubmodule ⊓ T.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem inf_toNonUnitalSubsemiring (S T : NonUnitalSubalgebra R A) :\n    (S ⊓ T).toNonUnitalSubsemiring = S.toNonUnitalSubsemiring ⊓ T.toNonUnitalSubsemiring :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sInf (S : Set (NonUnitalSubalgebra R A)) : (↑(sInf S) : Set A) = ⋂ s ∈ S, ↑s :=\n  sInf_image\n\ntheorem mem_sInf {S : Set (NonUnitalSubalgebra R A)} {x : A} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p := by\n  simp only [← SetLike.mem_coe, coe_sInf, Set.mem_iInter₂]\n\n@[simp]\ntheorem sInf_toSubmodule (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toSubmodule = sInf (NonUnitalSubalgebra.toSubmodule '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp]\ntheorem sInf_toNonUnitalSubsemiring (S : Set (NonUnitalSubalgebra R A)) :\n    (sInf S).toNonUnitalSubsemiring = sInf (NonUnitalSubalgebra.toNonUnitalSubsemiring '' S) :=\n  SetLike.coe_injective <| by simp\n\n@[simp, norm_cast]\ntheorem coe_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} :\n    (↑(⨅ i, S i) : Set A) = ⋂ i, S i := by simp [iInf]\n\ntheorem mem_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} {x : A} :\n    (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by simp only [iInf, mem_sInf, Set.forall_range_iff]\n\n@[simp]\ntheorem iInf_toSubmodule {ι : Sort*} (S : ι → NonUnitalSubalgebra R A) :\n    (⨅ i, S i).toSubmodule = ⨅ i, (S i).toSubmodule :=\n  SetLike.coe_injective <| by simp\n\ninstance : Inhabited (NonUnitalSubalgebra R A) :=\n  ⟨⊥⟩\n\ntheorem mem_bot {x : A} : x ∈ (⊥ : NonUnitalSubalgebra R A) ↔ x = 0 :=\n  show x ∈ Submodule.span R (NonUnitalSubsemiring.closure (∅ : Set A) : Set A) ↔ x = 0 by\n    rw [NonUnitalSubsemiring.closure_empty, NonUnitalSubsemiring.coe_bot,\n      Submodule.span_zero_singleton, Submodule.mem_bot]\n\ntheorem toSubmodule_bot : (⊥ : NonUnitalSubalgebra R A).toSubmodule = ⊥ := by\n  ext\n  simp only [mem_bot, NonUnitalSubalgebra.mem_toSubmodule, Submodule.mem_bot]\n\n@[simp]\ntheorem coe_bot : ((⊥ : NonUnitalSubalgebra R A) : Set A) = {0} := by\n  simp [Set.ext_iff, NonUnitalAlgebra.mem_bot]\n\ntheorem eq_top_iff {S : NonUnitalSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S :=\n  ⟨fun h x => by rw [h]; exact mem_top, fun h => by ext x; exact ⟨fun _ => mem_top, fun _ => h x⟩⟩\n\ntheorem range_top_iff_surjective (f : A →ₙₐ[R] B) :\n    NonUnitalAlgHom.range f = (⊤ : NonUnitalSubalgebra R B) ↔ Function.Surjective f :=\n  NonUnitalAlgebra.eq_top_iff\n\n@[simp]\ntheorem range_id : NonUnitalAlgHom.range (NonUnitalAlgHom.id R A) = ⊤ :=\n  SetLike.coe_injective Set.range_id\n\n@[simp]\ntheorem map_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R A).map f = NonUnitalAlgHom.range f :=\n  SetLike.coe_injective Set.image_univ\n\n@[simp]\ntheorem map_bot (f : A →ₙₐ[R] B) : (⊥ : NonUnitalSubalgebra R A).map f = ⊥ :=\n  SetLike.coe_injective <| by simp [NonUnitalAlgebra.coe_bot, NonUnitalSubalgebra.coe_map]\n\n@[simp]\ntheorem comap_top (f : A →ₙₐ[R] B) : (⊤ : NonUnitalSubalgebra R B).comap f = ⊤ :=\n  eq_top_iff.2 fun _ => mem_top\n\n/-- `NonUnitalAlgHom` to `⊤ : NonUnitalSubalgebra R A`. -/\ndef toTop : A →ₙₐ[R] (⊤ : NonUnitalSubalgebra R A) :=\n  NonUnitalAlgHom.codRestrict (NonUnitalAlgHom.id R A) ⊤ fun _ => mem_top\n\nend NonUnitalAlgebra\n\nnamespace NonUnitalSubalgebra\n\nopen NonUnitalAlgebra\n\nsection NonAssoc\n\nvariable {R : Type u} {A : Type v} {B : Type w}\nvariable [CommSemiring R]\nvariable [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]\nvariable [NonUnitalNonAssocSemiring B] [Module R B] [IsScalarTower R B B] [SMulCommClass R B B]\nvariable (S : NonUnitalSubalgebra R A)\n\ninstance subsingleton_of_subsingleton [Subsingleton A] : Subsingleton (NonUnitalSubalgebra R A) :=\n  ⟨fun B C => ext fun x => by simp only [Subsingleton.elim x 0, zero_mem B, zero_mem C]⟩\n\ninstance _root_.NonUnitalAlgHom.subsingleton [Subsingleton (NonUnitalSubalgebra R A)] :\n    Subsingleton (A →ₙₐ[R] B) :=\n  ⟨fun f g =>\n    NonUnitalAlgHom.ext fun a =>\n      have : a ∈ (⊥ : NonUnitalSubalgebra R A) :=\n        Subsingleton.elim (⊤ : NonUnitalSubalgebra R A) ⊥ ▸ mem_top\n      (mem_bot.mp this).symm ▸ (map_zero f).trans (map_zero g).symm⟩\n\ntheorem range_val : NonUnitalAlgHom.range (NonUnitalSubalgebraClass.subtype S) = S :=\n  ext <| Set.ext_iff.1 <| (NonUnitalSubalgebraClass.subtype S).coe_range.trans Subtype.range_val\n\n/-- The map `S → T` when `S` is a non-unital subalgebra contained in the non-unital subalgebra `T`.\n\nThis is the non-unital subalgebra version of `Submodule.inclusion`, or `Subring.inclusion`  -/\ndef inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) : S →ₙₐ[R] T\n    where\n  toFun := Set.inclusion h\n  map_add' _ _ := rfl\n  map_mul' _ _ := rfl\n  map_zero' := rfl\n  map_smul' _ _ := rfl\n\ntheorem inclusion_injective {S T : NonUnitalSubalgebra R A} (h : S ≤ T) :\n    Function.Injective (inclusion h) := fun _ _ => Subtype.ext ∘ Subtype.mk.inj\n\n@[simp]\ntheorem inclusion_self {S : NonUnitalSubalgebra R A} :\n    inclusion (le_refl S) = NonUnitalAlgHom.id R S :=\n  NonUnitalAlgHom.ext fun _ => Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_mk {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : A) (hx : x ∈ S) :\n    inclusion h ⟨x, hx⟩ = ⟨x, h hx⟩ :=\n  rfl\n\ntheorem inclusion_right {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (x : T) (m : (x : A) ∈ S) :\n    inclusion h ⟨x, m⟩ = x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem inclusion_inclusion {S T U : NonUnitalSubalgebra R A} (hst : S ≤ T) (htu : T ≤ U) (x : S) :\n    inclusion htu (inclusion hst x) = inclusion (le_trans hst htu) x :=\n  Subtype.ext rfl\n\n@[simp]\ntheorem coe_inclusion {S T : NonUnitalSubalgebra R A} (h : S ≤ T) (s : S) :\n    (inclusion h s : A) = s :=\n  rfl\n\nsection Prod\n\nvariable (S₁ : NonUnitalSubalgebra R B)\n\n/-- The product of two non-unital subalgebras is a non-unital subalgebra. -/\ndef prod : NonUnitalSubalgebra R (A × B) :=\n  { S.toNonUnitalSubsemiring.prod S₁.toNonUnitalSubsemiring with\n    carrier := S ×ˢ S₁\n    smul_mem' := fun r _x hx => ⟨SMulMemClass.smul_mem r hx.1, SMulMemClass.smul_mem r hx.2⟩ }\n\n@[simp]\ntheorem coe_prod : (prod S S₁ : Set (A × B)) = (S : Set A) ×ˢ S₁ :=\n  rfl\n\ntheorem prod_toSubmodule : (S.prod S₁).toSubmodule = S.toSubmodule.prod S₁.toSubmodule :=\n  rfl\n\n@[simp]\ntheorem mem_prod {S : NonUnitalSubalgebra R A} {S₁ : NonUnitalSubalgebra R B} {x : A × B} :\n    x ∈ prod S S₁ ↔ x.1 ∈ S ∧ x.2 ∈ S₁ :=\n  Set.mem_prod\n\n@[simp]\ntheorem prod_top : (prod ⊤ ⊤ : NonUnitalSubalgebra R (A × B)) = ⊤ := by ext; simp\n\ntheorem prod_mono {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S ≤ T → S₁ ≤ T₁ → prod S S₁ ≤ prod T T₁ :=\n  Set.prod_mono\n\n@[simp]\ntheorem prod_inf_prod {S T : NonUnitalSubalgebra R A} {S₁ T₁ : NonUnitalSubalgebra R B} :\n    S.prod S₁ ⊓ T.prod T₁ = (S ⊓ T).prod (S₁ ⊓ T₁) :=\n  SetLike.coe_injective Set.prod_inter_prod\n\nend Prod\n\nsection SuprLift\n\nvariable {ι : Type*}\n\ntheorem coe_iSup_of_directed [Nonempty ι] {S : ι → NonUnitalSubalgebra R A}\n    (dir : Directed (· ≤ ·) S) : ↑(iSup S) = ⋃ i, (S i : Set A) :=\n  let K : NonUnitalSubalgebra R A :=\n    { __ := NonUnitalSubsemiring.copy _ _ (NonUnitalSubsemiring.coe_iSup_of_directed dir).symm\n      smul_mem' := fun r _x hx ↦\n        let ⟨i, hi⟩ := Set.mem_iUnion.1 hx\n        Set.mem_iUnion.2 ⟨i, (S i).smul_mem' r hi⟩ }\n  have : iSup S = K := le_antisymm\n    (iSup_le fun i ↦ le_iSup (fun i ↦ (S i : Set A)) i) (Set.iUnion_subset fun _ ↦ le_iSup S _)\n  this.symm ▸ rfl\n\n/-- Define an algebra homomorphism on a directed supremum of non-unital subalgebras by defining\nit on each non-unital subalgebra, and proving that it agrees on the intersection of\nnon-unital subalgebras. -/\nnoncomputable def iSupLift [Nonempty ι] (K : ι → NonUnitalSubalgebra R A) (dir : Directed (· ≤ ·) K)\n    (f : ∀ i, K i →ₙₐ[R] B) (hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h))\n    (T : NonUnitalSubalgebra R A) (hT : T = iSup K) : ↥T →ₙₐ[R] B := by\n  subst hT\n  exact\n      { toFun :=\n          Set.iUnionLift (fun i => ↑(K i)) (fun i x => f i x)\n            (fun i j x hxi hxj => by\n              let ⟨k, hik, hjk⟩ := dir i j\n              simp only\n              rw [hf i k hik]\n              rw [hf j k hjk]\n              rfl)\n            (↑(iSup K)) (by rw [coe_iSup_of_directed dir])\n        map_zero' := by\n          dsimp\n          exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl) _ (by simp)\n        map_mul' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· * ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_add' := by\n          dsimp\n          apply Set.iUnionLift_binary (coe_iSup_of_directed dir) dir _ (fun _ => (· + ·))\n          on_goal 3 => rw [coe_iSup_of_directed dir]\n          all_goals simp\n        map_smul' := fun r => by\n          dsimp\n          apply Set.iUnionLift_unary (coe_iSup_of_directed dir) _ (fun _ x => r • x)\n            (fun _ _ => rfl)\n          on_goal 2 => rw [coe_iSup_of_directed dir]\n          all_goals simp }\n\nvariable [Nonempty ι] {K : ι → NonUnitalSubalgebra R A} {dir : Directed (· ≤ ·) K}\n  {f : ∀ i, K i →ₙₐ[R] B} {hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h)}\n  {T : NonUnitalSubalgebra R A} {hT : T = iSup K}\n\n@[simp]\ntheorem iSupLift_inclusion {i : ι} (x : K i) (h : K i ≤ T) :\n    iSupLift K dir f hf T hT (inclusion h x) = f i x := by\n  subst T\n  dsimp [iSupLift]\n  apply Set.iUnionLift_inclusion\n  exact h\n\n@[simp]\ntheorem iSupLift_comp_inclusion {i : ι} (h : K i ≤ T) :\n    (iSupLift K dir f hf T hT).comp (inclusion h) = f i := by ext; simp\n\n@[simp]\ntheorem iSupLift_mk {i : ι} (x : K i) (hx : (x : A) ∈ T) :\n    iSupLift K dir f hf T hT ⟨x, hx⟩ = f i x := by\n  subst hT\n  dsimp [iSupLift]\n  apply Set.iUnionLift_mk\n\ntheorem iSupLift_of_mem {i : ι} (x : T) (hx : (x : A) ∈ K i) :\n    iSupLift K dir f hf T hT x = f i ⟨x, hx⟩ := by\n  subst hT\n  dsimp [iSupLift]\n  apply Set.iUnionLift_of_mem\n\nend SuprLift\n\nend NonAssoc\n\nsection Center\n\nsection NonUnitalNonAssocSemiring\nvariable {R A : Type*}\nvariable [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A]\nvariable [IsScalarTower R A A] [SMulCommClass R A A]\n\ntheorem _root_.Set.smul_mem_center (r : R) {a : A} (ha : a ∈ Set.center A) :\n    r • a ∈ Set.center A where\n  comm b := by rw [mul_smul_comm, smul_mul_assoc, ha.comm]\n  left_assoc b c := by rw [smul_mul_assoc, smul_mul_assoc, smul_mul_assoc, ha.left_assoc]\n  mid_assoc b c := by\n    rw [mul_smul_comm]\n    rw [smul_mul_assoc]\n    rw [smul_mul_assoc]\n    rw [mul_smul_comm]\n    rw [ha.mid_assoc]\n  right_assoc b c := by\n    rw [mul_smul_comm]\n    rw [mul_smul_comm]\n    rw [mul_smul_comm]\n    rw [ha.right_assoc]\n\nvariable (R A) in\n/-- The center of a non-unital algebra is the set of elements which commute with every element.\nThey form a non-unital subalgebra. -/\ndef center : NonUnitalSubalgebra R A :=\n  { NonUnitalSubsemiring.center A with smul_mem' := Set.smul_mem_center }\n\ntheorem coe_center : (center R A : Set A) = Set.center A :=\n  rfl\n\n/-- The center of a non-unital algebra is a commutative and associative -/\ninstance center.instNonUnitalCommSemiring : NonUnitalCommSemiring (center R A) :=\n  NonUnitalSubsemiring.center.instNonUnitalCommSemiring _\n\ninstance center.instNonUnitalCommRing {A : Type*} [NonUnitalNonAssocRing A] [Module R A]\n    [IsScalarTower R A A] [SMulCommClass R A A] : NonUnitalCommRing (center R A) :=\n  NonUnitalSubring.center.instNonUnitalCommRing _\n\n@[simp]\ntheorem center_toNonUnitalSubsemiring :\n    (center R A).toNonUnitalSubsemiring = NonUnitalSubsemiring.center A :=\n  rfl\n\n@[simp] lemma center_toNonUnitalSubring (R A : Type*) [CommRing R] [NonUnitalRing A]\n    [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] :\n    (center R A).toNonUnitalSubring = NonUnitalSubring.center A :=\n  rfl\n\nend NonUnitalNonAssocSemiring\n\nvariable (R A : Type*) [CommSemiring R] [NonUnitalSemiring A] [Module R A] [IsScalarTower R A A]\n  [SMulCommClass R A A]\n\n-- no instance diamond, as the `npow` field isn't present in the non-unital case.\nexample :\n    center.instNonUnitalCommSemiring.toNonUnitalSemiring =\n      NonUnitalSubsemiringClass.toNonUnitalSemiring (center R A) :=\n  rfl\n\n@[simp]\ntheorem center_eq_top (A : Type*) [NonUnitalCommSemiring A] [Module R A] [IsScalarTower R A A]\n    [SMulCommClass R A A] : center R A = ⊤ :=\n  SetLike.coe_injective (Set.center_eq_univ A)\n\nvariable {R A}\n\ntheorem mem_center_iff {a : A} : a ∈ center R A ↔ ∀ b : A, b * a = a * b :=\n  Subsemigroup.mem_center_iff\n\nend Center\n\nsection Centralizer\n\nvariable {R A : Type*} [CommSemiring R] [NonUnitalSemiring A] [Module R A] [IsScalarTower R A A]\n  [SMulCommClass R A A]\n\n@[simp]\ntheorem _root_.Set.smul_mem_centralizer {s : Set A} (r : R) {a : A} (ha : a ∈ s.centralizer) :\n    r • a ∈ s.centralizer :=\n  fun x hx => by ","nextTactic":"rw [mul_smul_comm, smul_mul_assoc, ha x hx]","declUpToTactic":"@[simp]\ntheorem _root_.Set.smul_mem_centralizer {s : Set A} (r : R) {a : A} (ha : a ∈ s.centralizer) :\n    r • a ∈ s.centralizer :=\n  fun x hx => by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_NonUnitalSubalgebra.1040_0.jQvdAcOcyVirh23","decl":"@[simp]\ntheorem _root_.Set.smul_mem_centralizer {s : Set A} (r : R) {a : A} (ha : a ∈ s.centralizer) :\n    r • a ∈ s.centralizer "}