diff --git "a/Extracted/Mathlib/TacticPrediction/Mathlib_Algebra_Algebra_Hom.jsonl" "b/Extracted/Mathlib/TacticPrediction/Mathlib_Algebra_Algebra_Hom.jsonl" new file mode 100644--- /dev/null +++ "b/Extracted/Mathlib/TacticPrediction/Mathlib_Algebra_Algebra_Hom.jsonl" @@ -0,0 +1,22 @@ +{"state":"R : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝⁵ : CommSemiring R\ninst✝⁴ : Semiring A\ninst✝³ : Semiring B\ninst✝² : Algebra R A\ninst✝¹ : Algebra R B\nF : Type ?u.55413\ninst✝ : AlgHomClass F R A B\nsrc✝ : AlgHomClass F R A B := inst✝\nf : F\nr : R\nx : A\n⊢ f (r • x) = (RingHom.id R) r • f x","srcUpToTactic":"/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Yury Kudryashov\n-/\nimport Mathlib.Algebra.Algebra.Basic\nimport Mathlib.Algebra.BigOperators.Finsupp\n\n#align_import algebra.algebra.hom from \"leanprover-community/mathlib\"@\"e97cf15cd1aec9bd5c193b2ffac5a6dc9118912b\"\n\n/-!\n# Homomorphisms of `R`-algebras\n\nThis file defines bundled homomorphisms of `R`-algebras.\n\n## Main definitions\n\n* `AlgHom R A B`: the type of `R`-algebra morphisms from `A` to `B`.\n* `Algebra.ofId R A : R →ₐ[R] A`: the canonical map from `R` to `A`, as an `AlgHom`.\n\n## Notations\n\n* `A →ₐ[R] B` : `R`-algebra homomorphism from `A` to `B`.\n-/\n\nset_option autoImplicit true\n\n\nopen BigOperators\n\nuniverse u v w u₁ v₁\n\n/-- Defining the homomorphism in the category R-Alg. -/\n-- @[nolint has_nonempty_instance] -- Porting note: This linter does not exist yet.\nstructure AlgHom (R : Type u) (A : Type v) (B : Type w) [CommSemiring R] [Semiring A] [Semiring B]\n [Algebra R A] [Algebra R B] extends RingHom A B where\n commutes' : ∀ r : R, toFun (algebraMap R A r) = algebraMap R B r\n#align alg_hom AlgHom\n\n/-- Reinterpret an `AlgHom` as a `RingHom` -/\nadd_decl_doc AlgHom.toRingHom\n\n@[inherit_doc AlgHom]\ninfixr:25 \" →ₐ \" => AlgHom _\n\n@[inherit_doc]\nnotation:25 A \" →ₐ[\" R \"] \" B => AlgHom R A B\n\n/-- `AlgHomClass F R A B` asserts `F` is a type of bundled algebra homomorphisms\nfrom `A` to `B`. -/\nclass AlgHomClass (F : Type*) (R : outParam (Type*)) (A : outParam (Type*))\n (B : outParam (Type*)) [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A]\n [Algebra R B] extends RingHomClass F A B where\n commutes : ∀ (f : F) (r : R), f (algebraMap R A r) = algebraMap R B r\n#align alg_hom_class AlgHomClass\n\n-- Porting note: `dangerousInstance` linter has become smarter about `outParam`s\n-- attribute [nolint dangerousInstance] AlgHomClass.toRingHomClass\n\n-- Porting note: simp can prove this\n-- attribute [simp] AlgHomClass.commutes\n\nnamespace AlgHomClass\n\nvariable {R : Type*} {A : Type*} {B : Type*} [CommSemiring R] [Semiring A] [Semiring B]\n [Algebra R A] [Algebra R B]\n\n-- see Note [lower instance priority]\ninstance (priority := 100) linearMapClass [AlgHomClass F R A B] : LinearMapClass F R A B :=\n { ‹AlgHomClass F R A B› with\n map_smulₛₗ := fun f r x => by\n ","nextTactic":"simp only [Algebra.smul_def, map_mul, commutes, RingHom.id_apply]","declUpToTactic":"instance (priority := 100) linearMapClass [AlgHomClass F R A B] : LinearMapClass F R A B :=\n { ‹AlgHomClass F R A B› with\n map_smulₛₗ := fun f r x => by\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Hom.69_0.FxqJeqR3qcIVlrq","decl":"instance (priority "} +{"state":"R : Type u\nA : Type v\nB : Type w\nC : Type u₁\nD : Type v₁\ninst✝⁸ : CommSemiring R\ninst✝⁷ : Semiring A\ninst✝⁶ : Semiring B\ninst✝⁵ : Semiring C\ninst✝⁴ : Semiring D\ninst✝³ : Algebra R A\ninst✝² : Algebra R B\ninst✝¹ : Algebra R C\ninst✝ : Algebra R D\nf g : A →ₐ[R] B\nh : (fun f => f.toFun) f = (fun f => f.toFun) g\n⊢ f = g","srcUpToTactic":"/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Yury Kudryashov\n-/\nimport Mathlib.Algebra.Algebra.Basic\nimport Mathlib.Algebra.BigOperators.Finsupp\n\n#align_import algebra.algebra.hom from \"leanprover-community/mathlib\"@\"e97cf15cd1aec9bd5c193b2ffac5a6dc9118912b\"\n\n/-!\n# Homomorphisms of `R`-algebras\n\nThis file defines bundled homomorphisms of `R`-algebras.\n\n## Main definitions\n\n* `AlgHom R A B`: the type of `R`-algebra morphisms from `A` to `B`.\n* `Algebra.ofId R A : R →ₐ[R] A`: the canonical map from `R` to `A`, as an `AlgHom`.\n\n## Notations\n\n* `A →ₐ[R] B` : `R`-algebra homomorphism from `A` to `B`.\n-/\n\nset_option autoImplicit true\n\n\nopen BigOperators\n\nuniverse u v w u₁ v₁\n\n/-- Defining the homomorphism in the category R-Alg. -/\n-- @[nolint has_nonempty_instance] -- Porting note: This linter does not exist yet.\nstructure AlgHom (R : Type u) (A : Type v) (B : Type w) [CommSemiring R] [Semiring A] [Semiring B]\n [Algebra R A] [Algebra R B] extends RingHom A B where\n commutes' : ∀ r : R, toFun (algebraMap R A r) = algebraMap R B r\n#align alg_hom AlgHom\n\n/-- Reinterpret an `AlgHom` as a `RingHom` -/\nadd_decl_doc AlgHom.toRingHom\n\n@[inherit_doc AlgHom]\ninfixr:25 \" →ₐ \" => AlgHom _\n\n@[inherit_doc]\nnotation:25 A \" →ₐ[\" R \"] \" B => AlgHom R A B\n\n/-- `AlgHomClass F R A B` asserts `F` is a type of bundled algebra homomorphisms\nfrom `A` to `B`. -/\nclass AlgHomClass (F : Type*) (R : outParam (Type*)) (A : outParam (Type*))\n (B : outParam (Type*)) [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A]\n [Algebra R B] extends RingHomClass F A B where\n commutes : ∀ (f : F) (r : R), f (algebraMap R A r) = algebraMap R B r\n#align alg_hom_class AlgHomClass\n\n-- Porting note: `dangerousInstance` linter has become smarter about `outParam`s\n-- attribute [nolint dangerousInstance] AlgHomClass.toRingHomClass\n\n-- Porting note: simp can prove this\n-- attribute [simp] AlgHomClass.commutes\n\nnamespace AlgHomClass\n\nvariable {R : Type*} {A : Type*} {B : Type*} [CommSemiring R] [Semiring A] [Semiring B]\n [Algebra R A] [Algebra R B]\n\n-- see Note [lower instance priority]\ninstance (priority := 100) linearMapClass [AlgHomClass F R A B] : LinearMapClass F R A B :=\n { ‹AlgHomClass F R A B› with\n map_smulₛₗ := fun f r x => by\n simp only [Algebra.smul_def, map_mul, commutes, RingHom.id_apply] }\n#align alg_hom_class.linear_map_class AlgHomClass.linearMapClass\n\n-- Porting note: A new definition underlying a coercion `↑`.\n/-- Turn an element of a type `F` satisfying `AlgHomClass F α β` into an actual\n`AlgHom`. This is declared as the default coercion from `F` to `α →+* β`. -/\n@[coe]\ndef toAlgHom {F : Type*} [AlgHomClass F R A B] (f : F) : A →ₐ[R] B :=\n { (f : A →+* B) with\n toFun := f\n commutes' := AlgHomClass.commutes f }\n\ninstance coeTC {F : Type*} [AlgHomClass F R A B] : CoeTC F (A →ₐ[R] B) :=\n ⟨AlgHomClass.toAlgHom⟩\n#align alg_hom_class.alg_hom.has_coe_t AlgHomClass.coeTC\n\nend AlgHomClass\n\nnamespace AlgHom\n\nvariable {R : Type u} {A : Type v} {B : Type w} {C : Type u₁} {D : Type v₁}\n\nsection Semiring\n\nvariable [CommSemiring R] [Semiring A] [Semiring B] [Semiring C] [Semiring D]\n\nvariable [Algebra R A] [Algebra R B] [Algebra R C] [Algebra R D]\n\n-- Porting note: we don't port specialized `CoeFun` instances if there is `FunLike` instead\n#noalign alg_hom.has_coe_to_fun\n\n-- Porting note: This instance is moved.\ninstance algHomClass : AlgHomClass (A →ₐ[R] B) R A B where\n coe f := f.toFun\n coe_injective' f g h := by\n ","nextTactic":"rcases f with ⟨⟨⟨⟨_, _⟩, _⟩, _, _⟩, _⟩","declUpToTactic":"instance algHomClass : AlgHomClass (A →ₐ[R] B) R A B where\n coe f := f.toFun\n coe_injective' f g h := by\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Hom.104_0.FxqJeqR3qcIVlrq","decl":"instance algHomClass : AlgHomClass (A →ₐ[R] B) R A B where\n coe f "} +{"state":"case mk.mk.mk.mk\nR : Type u\nA : Type v\nB : Type w\nC : Type u₁\nD : Type v₁\ninst✝⁸ : CommSemiring R\ninst✝⁷ : Semiring A\ninst✝⁶ : Semiring B\ninst✝⁵ : Semiring C\ninst✝⁴ : Semiring D\ninst✝³ : Algebra R A\ninst✝² : Algebra R B\ninst✝¹ : Algebra R C\ninst✝ : Algebra R D\ng : A →ₐ[R] B\ntoFun✝ : A → B\nmap_one'✝ : toFun✝ 1 = 1\nmap_mul'✝ :\n ∀ (x y : A),\n OneHom.toFun { toFun := toFun✝, map_one' := map_one'✝ } (x * y) =\n OneHom.toFun { toFun := toFun✝, map_one' := map_one'✝ } x *\n OneHom.toFun { toFun := toFun✝, map_one' := map_one'✝ } y\nmap_zero'✝ : OneHom.toFun (↑{ toOneHom := { toFun := toFun✝, map_one' := map_one'✝ }, map_mul' := map_mul'✝ }) 0 = 0\nmap_add'✝ :\n ∀ (x y : A),\n OneHom.toFun (↑{ toOneHom := { toFun := toFun✝, map_one' := map_one'✝ }, map_mul' := map_mul'✝ }) (x + y) =\n OneHom.toFun (↑{ toOneHom := { toFun := toFun✝, map_one' := map_one'✝ }, map_mul' := map_mul'✝ }) x +\n OneHom.toFun (↑{ toOneHom := { toFun := toFun✝, map_one' := map_one'✝ }, map_mul' := map_mul'✝ }) y\ncommutes'✝ :\n ∀ (r : R),\n OneHom.toFun\n (↑↑{ toMonoidHom := { toOneHom := { toFun := toFun✝, map_one' := map_one'✝ }, map_mul' := map_mul'✝ },\n map_zero' := map_zero'✝, map_add' := map_add'✝ })\n ((algebraMap R A) r) =\n (algebraMap R B) r\nh :\n (fun f => f.toFun)\n {\n toRingHom :=\n { toMonoidHom := { toOneHom := { toFun := toFun✝, map_one' := map_one'✝ }, map_mul' := map_mul'✝ },\n map_zero' := map_zero'✝, map_add' := map_add'✝ },\n commutes' := commutes'✝ } =\n (fun f => f.toFun) g\n⊢ {\n toRingHom :=\n { toMonoidHom := { toOneHom := { toFun := toFun✝, map_one' := map_one'✝ }, map_mul' := map_mul'✝ },\n map_zero' := map_zero'✝, map_add' := map_add'✝ },\n commutes' := commutes'✝ } =\n g","srcUpToTactic":"/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Yury Kudryashov\n-/\nimport Mathlib.Algebra.Algebra.Basic\nimport Mathlib.Algebra.BigOperators.Finsupp\n\n#align_import algebra.algebra.hom from \"leanprover-community/mathlib\"@\"e97cf15cd1aec9bd5c193b2ffac5a6dc9118912b\"\n\n/-!\n# Homomorphisms of `R`-algebras\n\nThis file defines bundled homomorphisms of `R`-algebras.\n\n## Main definitions\n\n* `AlgHom R A B`: the type of `R`-algebra morphisms from `A` to `B`.\n* `Algebra.ofId R A : R →ₐ[R] A`: the canonical map from `R` to `A`, as an `AlgHom`.\n\n## Notations\n\n* `A →ₐ[R] B` : `R`-algebra homomorphism from `A` to `B`.\n-/\n\nset_option autoImplicit true\n\n\nopen BigOperators\n\nuniverse u v w u₁ v₁\n\n/-- Defining the homomorphism in the category R-Alg. -/\n-- @[nolint has_nonempty_instance] -- Porting note: This linter does not exist yet.\nstructure AlgHom (R : Type u) (A : Type v) (B : Type w) [CommSemiring R] [Semiring A] [Semiring B]\n [Algebra R A] [Algebra R B] extends RingHom A B where\n commutes' : ∀ r : R, toFun (algebraMap R A r) = algebraMap R B r\n#align alg_hom AlgHom\n\n/-- Reinterpret an `AlgHom` as a `RingHom` -/\nadd_decl_doc AlgHom.toRingHom\n\n@[inherit_doc AlgHom]\ninfixr:25 \" →ₐ \" => AlgHom _\n\n@[inherit_doc]\nnotation:25 A \" →ₐ[\" R \"] \" B => AlgHom R A B\n\n/-- `AlgHomClass F R A B` asserts `F` is a type of bundled algebra homomorphisms\nfrom `A` to `B`. -/\nclass AlgHomClass (F : Type*) (R : outParam (Type*)) (A : outParam (Type*))\n (B : outParam (Type*)) [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A]\n [Algebra R B] extends RingHomClass F A B where\n commutes : ∀ (f : F) (r : R), f (algebraMap R A r) = algebraMap R B r\n#align alg_hom_class AlgHomClass\n\n-- Porting note: `dangerousInstance` linter has become smarter about `outParam`s\n-- attribute [nolint dangerousInstance] AlgHomClass.toRingHomClass\n\n-- Porting note: simp can prove this\n-- attribute [simp] AlgHomClass.commutes\n\nnamespace AlgHomClass\n\nvariable {R : Type*} {A : Type*} {B : Type*} [CommSemiring R] [Semiring A] [Semiring B]\n [Algebra R A] [Algebra R B]\n\n-- see Note [lower instance priority]\ninstance (priority := 100) linearMapClass [AlgHomClass F R A B] : LinearMapClass F R A B :=\n { ‹AlgHomClass F R A B› with\n map_smulₛₗ := fun f r x => by\n simp only [Algebra.smul_def, map_mul, commutes, RingHom.id_apply] }\n#align alg_hom_class.linear_map_class AlgHomClass.linearMapClass\n\n-- Porting note: A new definition underlying a coercion `↑`.\n/-- Turn an element of a type `F` satisfying `AlgHomClass F α β` into an actual\n`AlgHom`. This is declared as the default coercion from `F` to `α →+* β`. -/\n@[coe]\ndef toAlgHom {F : Type*} [AlgHomClass F R A B] (f : F) : A →ₐ[R] B :=\n { (f : A →+* B) with\n toFun := f\n commutes' := AlgHomClass.commutes f }\n\ninstance coeTC {F : Type*} [AlgHomClass F R A B] : CoeTC F (A →ₐ[R] B) :=\n ⟨AlgHomClass.toAlgHom⟩\n#align alg_hom_class.alg_hom.has_coe_t AlgHomClass.coeTC\n\nend AlgHomClass\n\nnamespace AlgHom\n\nvariable {R : Type u} {A : Type v} {B : Type w} {C : Type u₁} {D : Type v₁}\n\nsection Semiring\n\nvariable [CommSemiring R] [Semiring A] [Semiring B] [Semiring C] [Semiring D]\n\nvariable [Algebra R A] [Algebra R B] [Algebra R C] [Algebra R D]\n\n-- Porting note: we don't port specialized `CoeFun` instances if there is `FunLike` instead\n#noalign alg_hom.has_coe_to_fun\n\n-- Porting note: This instance is moved.\ninstance algHomClass : AlgHomClass (A →ₐ[R] B) R A B where\n coe f := f.toFun\n coe_injective' f g h := by\n rcases f with ⟨⟨⟨⟨_, _⟩, _⟩, _, _⟩, _⟩\n ","nextTactic":"rcases g with ⟨⟨⟨⟨_, _⟩, _⟩, _, _⟩, _⟩","declUpToTactic":"instance algHomClass : AlgHomClass (A →ₐ[R] B) R A B where\n coe f := f.toFun\n coe_injective' f g h := by\n rcases f with ⟨⟨⟨⟨_, _⟩, _⟩, _, _⟩, _⟩\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Hom.104_0.FxqJeqR3qcIVlrq","decl":"instance algHomClass : AlgHomClass (A →ₐ[R] B) R A B where\n coe f "} +{"state":"case mk.mk.mk.mk.mk.mk.mk.mk\nR : Type u\nA : Type v\nB : Type w\nC : Type u₁\nD : Type v₁\ninst✝⁸ : CommSemiring R\ninst✝⁷ : Semiring A\ninst✝⁶ : Semiring B\ninst✝⁵ : Semiring C\ninst✝⁴ : Semiring D\ninst✝³ : Algebra R A\ninst✝² : Algebra R B\ninst✝¹ : Algebra R C\ninst✝ : Algebra R D\ntoFun✝¹ : A → B\nmap_one'✝¹ : toFun✝¹ 1 = 1\nmap_mul'✝¹ :\n ∀ (x y : A),\n OneHom.toFun { toFun := toFun✝¹, map_one' := map_one'✝¹ } (x * y) =\n OneHom.toFun { toFun := toFun✝¹, map_one' := map_one'✝¹ } x *\n OneHom.toFun { toFun := toFun✝¹, map_one' := map_one'✝¹ } y\nmap_zero'✝¹ : OneHom.toFun (↑{ toOneHom := { toFun := toFun✝¹, map_one' := map_one'✝¹ }, map_mul' := map_mul'✝¹ }) 0 = 0\nmap_add'✝¹ :\n ∀ (x y : A),\n OneHom.toFun (↑{ toOneHom := { toFun := toFun✝¹, map_one' := map_one'✝¹ }, map_mul' := map_mul'✝¹ }) (x + y) =\n OneHom.toFun (↑{ toOneHom := { toFun := toFun✝¹, map_one' := map_one'✝¹ }, map_mul' := map_mul'✝¹ }) x +\n OneHom.toFun (↑{ toOneHom := { toFun := toFun✝¹, map_one' := map_one'✝¹ }, map_mul' := map_mul'✝¹ }) y\ncommutes'✝¹ :\n ∀ (r : R),\n OneHom.toFun\n (↑↑{ toMonoidHom := { toOneHom := { toFun := toFun✝¹, map_one' := map_one'✝¹ }, map_mul' := map_mul'✝¹ },\n map_zero' := map_zero'✝¹, map_add' := map_add'✝¹ })\n ((algebraMap R A) r) =\n (algebraMap R B) r\ntoFun✝ : A → B\nmap_one'✝ : toFun✝ 1 = 1\nmap_mul'✝ :\n ∀ (x y : A),\n OneHom.toFun { toFun := toFun✝, map_one' := map_one'✝ } (x * y) =\n OneHom.toFun { toFun := toFun✝, map_one' := map_one'✝ } x *\n OneHom.toFun { toFun := toFun✝, map_one' := map_one'✝ } y\nmap_zero'✝ : OneHom.toFun (↑{ toOneHom := { toFun := toFun✝, map_one' := map_one'✝ }, map_mul' := map_mul'✝ }) 0 = 0\nmap_add'✝ :\n ∀ (x y : A),\n OneHom.toFun (↑{ toOneHom := { toFun := toFun✝, map_one' := map_one'✝ }, map_mul' := map_mul'✝ }) (x + y) =\n OneHom.toFun (↑{ toOneHom := { toFun := toFun✝, map_one' := map_one'✝ }, map_mul' := map_mul'✝ }) x +\n OneHom.toFun (↑{ toOneHom := { toFun := toFun✝, map_one' := map_one'✝ }, map_mul' := map_mul'✝ }) y\ncommutes'✝ :\n ∀ (r : R),\n OneHom.toFun\n (↑↑{ toMonoidHom := { toOneHom := { toFun := toFun✝, map_one' := map_one'✝ }, map_mul' := map_mul'✝ },\n map_zero' := map_zero'✝, map_add' := map_add'✝ })\n ((algebraMap R A) r) =\n (algebraMap R B) r\nh :\n (fun f => f.toFun)\n {\n toRingHom :=\n { toMonoidHom := { toOneHom := { toFun := toFun✝¹, map_one' := map_one'✝¹ }, map_mul' := map_mul'✝¹ },\n map_zero' := map_zero'✝¹, map_add' := map_add'✝¹ },\n commutes' := commutes'✝¹ } =\n (fun f => f.toFun)\n {\n toRingHom :=\n { toMonoidHom := { toOneHom := { toFun := toFun✝, map_one' := map_one'✝ }, map_mul' := map_mul'✝ },\n map_zero' := map_zero'✝, map_add' := map_add'✝ },\n commutes' := commutes'✝ }\n⊢ {\n toRingHom :=\n { toMonoidHom := { toOneHom := { toFun := toFun✝¹, map_one' := map_one'✝¹ }, map_mul' := map_mul'✝¹ },\n map_zero' := map_zero'✝¹, map_add' := map_add'✝¹ },\n commutes' := commutes'✝¹ } =\n {\n toRingHom :=\n { toMonoidHom := { toOneHom := { toFun := toFun✝, map_one' := map_one'✝ }, map_mul' := map_mul'✝ },\n map_zero' := map_zero'✝, map_add' := map_add'✝ },\n commutes' := commutes'✝ }","srcUpToTactic":"/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Yury Kudryashov\n-/\nimport Mathlib.Algebra.Algebra.Basic\nimport Mathlib.Algebra.BigOperators.Finsupp\n\n#align_import algebra.algebra.hom from \"leanprover-community/mathlib\"@\"e97cf15cd1aec9bd5c193b2ffac5a6dc9118912b\"\n\n/-!\n# Homomorphisms of `R`-algebras\n\nThis file defines bundled homomorphisms of `R`-algebras.\n\n## Main definitions\n\n* `AlgHom R A B`: the type of `R`-algebra morphisms from `A` to `B`.\n* `Algebra.ofId R A : R →ₐ[R] A`: the canonical map from `R` to `A`, as an `AlgHom`.\n\n## Notations\n\n* `A →ₐ[R] B` : `R`-algebra homomorphism from `A` to `B`.\n-/\n\nset_option autoImplicit true\n\n\nopen BigOperators\n\nuniverse u v w u₁ v₁\n\n/-- Defining the homomorphism in the category R-Alg. -/\n-- @[nolint has_nonempty_instance] -- Porting note: This linter does not exist yet.\nstructure AlgHom (R : Type u) (A : Type v) (B : Type w) [CommSemiring R] [Semiring A] [Semiring B]\n [Algebra R A] [Algebra R B] extends RingHom A B where\n commutes' : ∀ r : R, toFun (algebraMap R A r) = algebraMap R B r\n#align alg_hom AlgHom\n\n/-- Reinterpret an `AlgHom` as a `RingHom` -/\nadd_decl_doc AlgHom.toRingHom\n\n@[inherit_doc AlgHom]\ninfixr:25 \" →ₐ \" => AlgHom _\n\n@[inherit_doc]\nnotation:25 A \" →ₐ[\" R \"] \" B => AlgHom R A B\n\n/-- `AlgHomClass F R A B` asserts `F` is a type of bundled algebra homomorphisms\nfrom `A` to `B`. -/\nclass AlgHomClass (F : Type*) (R : outParam (Type*)) (A : outParam (Type*))\n (B : outParam (Type*)) [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A]\n [Algebra R B] extends RingHomClass F A B where\n commutes : ∀ (f : F) (r : R), f (algebraMap R A r) = algebraMap R B r\n#align alg_hom_class AlgHomClass\n\n-- Porting note: `dangerousInstance` linter has become smarter about `outParam`s\n-- attribute [nolint dangerousInstance] AlgHomClass.toRingHomClass\n\n-- Porting note: simp can prove this\n-- attribute [simp] AlgHomClass.commutes\n\nnamespace AlgHomClass\n\nvariable {R : Type*} {A : Type*} {B : Type*} [CommSemiring R] [Semiring A] [Semiring B]\n [Algebra R A] [Algebra R B]\n\n-- see Note [lower instance priority]\ninstance (priority := 100) linearMapClass [AlgHomClass F R A B] : LinearMapClass F R A B :=\n { ‹AlgHomClass F R A B› with\n map_smulₛₗ := fun f r x => by\n simp only [Algebra.smul_def, map_mul, commutes, RingHom.id_apply] }\n#align alg_hom_class.linear_map_class AlgHomClass.linearMapClass\n\n-- Porting note: A new definition underlying a coercion `↑`.\n/-- Turn an element of a type `F` satisfying `AlgHomClass F α β` into an actual\n`AlgHom`. This is declared as the default coercion from `F` to `α →+* β`. -/\n@[coe]\ndef toAlgHom {F : Type*} [AlgHomClass F R A B] (f : F) : A →ₐ[R] B :=\n { (f : A →+* B) with\n toFun := f\n commutes' := AlgHomClass.commutes f }\n\ninstance coeTC {F : Type*} [AlgHomClass F R A B] : CoeTC F (A →ₐ[R] B) :=\n ⟨AlgHomClass.toAlgHom⟩\n#align alg_hom_class.alg_hom.has_coe_t AlgHomClass.coeTC\n\nend AlgHomClass\n\nnamespace AlgHom\n\nvariable {R : Type u} {A : Type v} {B : Type w} {C : Type u₁} {D : Type v₁}\n\nsection Semiring\n\nvariable [CommSemiring R] [Semiring A] [Semiring B] [Semiring C] [Semiring D]\n\nvariable [Algebra R A] [Algebra R B] [Algebra R C] [Algebra R D]\n\n-- Porting note: we don't port specialized `CoeFun` instances if there is `FunLike` instead\n#noalign alg_hom.has_coe_to_fun\n\n-- Porting note: This instance is moved.\ninstance algHomClass : AlgHomClass (A →ₐ[R] B) R A B where\n coe f := f.toFun\n coe_injective' f g h := by\n rcases f with ⟨⟨⟨⟨_, _⟩, _⟩, _, _⟩, _⟩\n rcases g with ⟨⟨⟨⟨_, _⟩, _⟩, _, _⟩, _⟩\n ","nextTactic":"congr","declUpToTactic":"instance algHomClass : AlgHomClass (A →ₐ[R] B) R A B where\n coe f := f.toFun\n coe_injective' f g h := by\n rcases f with ⟨⟨⟨⟨_, _⟩, _⟩, _, _⟩, _⟩\n rcases g with ⟨⟨⟨⟨_, _⟩, _⟩, _, _⟩, _⟩\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Hom.104_0.FxqJeqR3qcIVlrq","decl":"instance algHomClass : AlgHomClass (A →ₐ[R] B) R A B where\n coe f "} +{"state":"R : Type u\nA : Type v\nB : Type w\nC : Type u₁\nD : Type v₁\ninst✝⁸ : CommSemiring R\ninst✝⁷ : Semiring A\ninst✝⁶ : Semiring B\ninst✝⁵ : Semiring C\ninst✝⁴ : Semiring D\ninst✝³ : Algebra R A\ninst✝² : Algebra R B\ninst✝¹ : Algebra R C\ninst✝ : Algebra R D\nφ : A →ₐ[R] B\nf : A →+* B\nh : ∀ (c : R) (x : A), f (c • x) = c • f x\nc : R\n⊢ OneHom.toFun\n (↑↑{\n toMonoidHom :=\n { toOneHom := { toFun := ⇑f, map_one' := (_ : OneHom.toFun (↑↑f) 1 = 1) },\n map_mul' :=\n (_ : ∀ (x y : A), OneHom.toFun (↑↑f) (x * y) = OneHom.toFun (↑↑f) x * OneHom.toFun (↑↑f) y) },\n map_zero' := (_ : OneHom.toFun (↑↑f) 0 = 0),\n map_add' := (_ : ∀ (x y : A), OneHom.toFun (↑↑f) (x + y) = OneHom.toFun (↑↑f) x + OneHom.toFun (↑↑f) y) })\n ((algebraMap R A) c) =\n (algebraMap R B) c","srcUpToTactic":"/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Yury Kudryashov\n-/\nimport Mathlib.Algebra.Algebra.Basic\nimport Mathlib.Algebra.BigOperators.Finsupp\n\n#align_import algebra.algebra.hom from \"leanprover-community/mathlib\"@\"e97cf15cd1aec9bd5c193b2ffac5a6dc9118912b\"\n\n/-!\n# Homomorphisms of `R`-algebras\n\nThis file defines bundled homomorphisms of `R`-algebras.\n\n## Main definitions\n\n* `AlgHom R A B`: the type of `R`-algebra morphisms from `A` to `B`.\n* `Algebra.ofId R A : R →ₐ[R] A`: the canonical map from `R` to `A`, as an `AlgHom`.\n\n## Notations\n\n* `A →ₐ[R] B` : `R`-algebra homomorphism from `A` to `B`.\n-/\n\nset_option autoImplicit true\n\n\nopen BigOperators\n\nuniverse u v w u₁ v₁\n\n/-- Defining the homomorphism in the category R-Alg. -/\n-- @[nolint has_nonempty_instance] -- Porting note: This linter does not exist yet.\nstructure AlgHom (R : Type u) (A : Type v) (B : Type w) [CommSemiring R] [Semiring A] [Semiring B]\n [Algebra R A] [Algebra R B] extends RingHom A B where\n commutes' : ∀ r : R, toFun (algebraMap R A r) = algebraMap R B r\n#align alg_hom AlgHom\n\n/-- Reinterpret an `AlgHom` as a `RingHom` -/\nadd_decl_doc AlgHom.toRingHom\n\n@[inherit_doc AlgHom]\ninfixr:25 \" →ₐ \" => AlgHom _\n\n@[inherit_doc]\nnotation:25 A \" →ₐ[\" R \"] \" B => AlgHom R A B\n\n/-- `AlgHomClass F R A B` asserts `F` is a type of bundled algebra homomorphisms\nfrom `A` to `B`. -/\nclass AlgHomClass (F : Type*) (R : outParam (Type*)) (A : outParam (Type*))\n (B : outParam (Type*)) [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A]\n [Algebra R B] extends RingHomClass F A B where\n commutes : ∀ (f : F) (r : R), f (algebraMap R A r) = algebraMap R B r\n#align alg_hom_class AlgHomClass\n\n-- Porting note: `dangerousInstance` linter has become smarter about `outParam`s\n-- attribute [nolint dangerousInstance] AlgHomClass.toRingHomClass\n\n-- Porting note: simp can prove this\n-- attribute [simp] AlgHomClass.commutes\n\nnamespace AlgHomClass\n\nvariable {R : Type*} {A : Type*} {B : Type*} [CommSemiring R] [Semiring A] [Semiring B]\n [Algebra R A] [Algebra R B]\n\n-- see Note [lower instance priority]\ninstance (priority := 100) linearMapClass [AlgHomClass F R A B] : LinearMapClass F R A B :=\n { ‹AlgHomClass F R A B› with\n map_smulₛₗ := fun f r x => by\n simp only [Algebra.smul_def, map_mul, commutes, RingHom.id_apply] }\n#align alg_hom_class.linear_map_class AlgHomClass.linearMapClass\n\n-- Porting note: A new definition underlying a coercion `↑`.\n/-- Turn an element of a type `F` satisfying `AlgHomClass F α β` into an actual\n`AlgHom`. This is declared as the default coercion from `F` to `α →+* β`. -/\n@[coe]\ndef toAlgHom {F : Type*} [AlgHomClass F R A B] (f : F) : A →ₐ[R] B :=\n { (f : A →+* B) with\n toFun := f\n commutes' := AlgHomClass.commutes f }\n\ninstance coeTC {F : Type*} [AlgHomClass F R A B] : CoeTC F (A →ₐ[R] B) :=\n ⟨AlgHomClass.toAlgHom⟩\n#align alg_hom_class.alg_hom.has_coe_t AlgHomClass.coeTC\n\nend AlgHomClass\n\nnamespace AlgHom\n\nvariable {R : Type u} {A : Type v} {B : Type w} {C : Type u₁} {D : Type v₁}\n\nsection Semiring\n\nvariable [CommSemiring R] [Semiring A] [Semiring B] [Semiring C] [Semiring D]\n\nvariable [Algebra R A] [Algebra R B] [Algebra R C] [Algebra R D]\n\n-- Porting note: we don't port specialized `CoeFun` instances if there is `FunLike` instead\n#noalign alg_hom.has_coe_to_fun\n\n-- Porting note: This instance is moved.\ninstance algHomClass : AlgHomClass (A →ₐ[R] B) R A B where\n coe f := f.toFun\n coe_injective' f g h := by\n rcases f with ⟨⟨⟨⟨_, _⟩, _⟩, _, _⟩, _⟩\n rcases g with ⟨⟨⟨⟨_, _⟩, _⟩, _, _⟩, _⟩\n congr\n map_add f := f.map_add'\n map_zero f := f.map_zero'\n map_mul f := f.map_mul'\n map_one f := f.map_one'\n commutes f := f.commutes'\n#align alg_hom.alg_hom_class AlgHom.algHomClass\n\n/-- See Note [custom simps projection] -/\ndef Simps.apply {R : Type u} {α : Type v} {β : Type w} [CommSemiring R]\n [Semiring α] [Semiring β] [Algebra R α] [Algebra R β] (f : α →ₐ[R] β) : α → β := f\n\ninitialize_simps_projections AlgHom (toFun → apply)\n\n@[simp]\nprotected theorem coe_coe {F : Type*} [AlgHomClass F R A B] (f : F) : ⇑(f : A →ₐ[R] B) = f :=\n rfl\n#align alg_hom.coe_coe AlgHom.coe_coe\n\n@[simp]\ntheorem toFun_eq_coe (f : A →ₐ[R] B) : f.toFun = f :=\n rfl\n#align alg_hom.to_fun_eq_coe AlgHom.toFun_eq_coe\n\nattribute [coe] AlgHom.toRingHom\n\ninstance coeOutRingHom : CoeOut (A →ₐ[R] B) (A →+* B) :=\n ⟨AlgHom.toRingHom⟩\n#align alg_hom.coe_ring_hom AlgHom.coeOutRingHom\n\n-- Porting note: A new definition underlying a coercion `↑`.\n@[coe]\ndef toMonoidHom' (f : A →ₐ[R] B) : A →* B := (f : A →+* B)\n\ninstance coeOutMonoidHom : CoeOut (A →ₐ[R] B) (A →* B) :=\n ⟨AlgHom.toMonoidHom'⟩\n#align alg_hom.coe_monoid_hom AlgHom.coeOutMonoidHom\n\n-- Porting note: A new definition underlying a coercion `↑`.\n@[coe]\ndef toAddMonoidHom' (f : A →ₐ[R] B) : A →+ B := (f : A →+* B)\n\ninstance coeOutAddMonoidHom : CoeOut (A →ₐ[R] B) (A →+ B) :=\n ⟨AlgHom.toAddMonoidHom'⟩\n#align alg_hom.coe_add_monoid_hom AlgHom.coeOutAddMonoidHom\n\n-- Porting note: Lean 3: `@[simp, norm_cast] coe_mk`\n-- Lean 4: `@[simp] coe_mk` & `@[norm_cast] coe_mks`\n@[simp]\ntheorem coe_mk {f : A →+* B} (h) : ((⟨f, h⟩ : A →ₐ[R] B) : A → B) = f :=\n rfl\n\n@[norm_cast]\ntheorem coe_mks {f : A → B} (h₁ h₂ h₃ h₄ h₅) : ⇑(⟨⟨⟨⟨f, h₁⟩, h₂⟩, h₃, h₄⟩, h₅⟩ : A →ₐ[R] B) = f :=\n rfl\n#align alg_hom.coe_mk AlgHom.coe_mks\n\n-- Porting note: This theorem is new.\n@[simp, norm_cast]\ntheorem coe_ringHom_mk {f : A →+* B} (h) : ((⟨f, h⟩ : A →ₐ[R] B) : A →+* B) = f :=\n rfl\n\n-- make the coercion the simp-normal form\n@[simp]\ntheorem toRingHom_eq_coe (f : A →ₐ[R] B) : f.toRingHom = f :=\n rfl\n#align alg_hom.to_ring_hom_eq_coe AlgHom.toRingHom_eq_coe\n\n@[simp, norm_cast]\ntheorem coe_toRingHom (f : A →ₐ[R] B) : ⇑(f : A →+* B) = f :=\n rfl\n#align alg_hom.coe_to_ring_hom AlgHom.coe_toRingHom\n\n@[simp, norm_cast]\ntheorem coe_toMonoidHom (f : A →ₐ[R] B) : ⇑(f : A →* B) = f :=\n rfl\n#align alg_hom.coe_to_monoid_hom AlgHom.coe_toMonoidHom\n\n@[simp, norm_cast]\ntheorem coe_toAddMonoidHom (f : A →ₐ[R] B) : ⇑(f : A →+ B) = f :=\n rfl\n#align alg_hom.coe_to_add_monoid_hom AlgHom.coe_toAddMonoidHom\n\nvariable (φ : A →ₐ[R] B)\n\ntheorem coe_fn_injective : @Function.Injective (A →ₐ[R] B) (A → B) (↑) :=\n FunLike.coe_injective\n#align alg_hom.coe_fn_injective AlgHom.coe_fn_injective\n\ntheorem coe_fn_inj {φ₁ φ₂ : A →ₐ[R] B} : (φ₁ : A → B) = φ₂ ↔ φ₁ = φ₂ :=\n FunLike.coe_fn_eq\n#align alg_hom.coe_fn_inj AlgHom.coe_fn_inj\n\ntheorem coe_ringHom_injective : Function.Injective ((↑) : (A →ₐ[R] B) → A →+* B) := fun φ₁ φ₂ H =>\n coe_fn_injective <| show ((φ₁ : A →+* B) : A → B) = ((φ₂ : A →+* B) : A → B) from congr_arg _ H\n#align alg_hom.coe_ring_hom_injective AlgHom.coe_ringHom_injective\n\ntheorem coe_monoidHom_injective : Function.Injective ((↑) : (A →ₐ[R] B) → A →* B) :=\n RingHom.coe_monoidHom_injective.comp coe_ringHom_injective\n#align alg_hom.coe_monoid_hom_injective AlgHom.coe_monoidHom_injective\n\ntheorem coe_addMonoidHom_injective : Function.Injective ((↑) : (A →ₐ[R] B) → A →+ B) :=\n RingHom.coe_addMonoidHom_injective.comp coe_ringHom_injective\n#align alg_hom.coe_add_monoid_hom_injective AlgHom.coe_addMonoidHom_injective\n\nprotected theorem congr_fun {φ₁ φ₂ : A →ₐ[R] B} (H : φ₁ = φ₂) (x : A) : φ₁ x = φ₂ x :=\n FunLike.congr_fun H x\n#align alg_hom.congr_fun AlgHom.congr_fun\n\nprotected theorem congr_arg (φ : A →ₐ[R] B) {x y : A} (h : x = y) : φ x = φ y :=\n FunLike.congr_arg φ h\n#align alg_hom.congr_arg AlgHom.congr_arg\n\n@[ext]\ntheorem ext {φ₁ φ₂ : A →ₐ[R] B} (H : ∀ x, φ₁ x = φ₂ x) : φ₁ = φ₂ :=\n FunLike.ext _ _ H\n#align alg_hom.ext AlgHom.ext\n\ntheorem ext_iff {φ₁ φ₂ : A →ₐ[R] B} : φ₁ = φ₂ ↔ ∀ x, φ₁ x = φ₂ x :=\n FunLike.ext_iff\n#align alg_hom.ext_iff AlgHom.ext_iff\n\n@[simp]\ntheorem mk_coe {f : A →ₐ[R] B} (h₁ h₂ h₃ h₄ h₅) : (⟨⟨⟨⟨f, h₁⟩, h₂⟩, h₃, h₄⟩, h₅⟩ : A →ₐ[R] B) = f :=\n ext fun _ => rfl\n#align alg_hom.mk_coe AlgHom.mk_coe\n\n@[simp]\ntheorem commutes (r : R) : φ (algebraMap R A r) = algebraMap R B r :=\n φ.commutes' r\n#align alg_hom.commutes AlgHom.commutes\n\ntheorem comp_algebraMap : (φ : A →+* B).comp (algebraMap R A) = algebraMap R B :=\n RingHom.ext <| φ.commutes\n#align alg_hom.comp_algebra_map AlgHom.comp_algebraMap\n\nprotected theorem map_add (r s : A) : φ (r + s) = φ r + φ s :=\n map_add _ _ _\n#align alg_hom.map_add AlgHom.map_add\n\nprotected theorem map_zero : φ 0 = 0 :=\n map_zero _\n#align alg_hom.map_zero AlgHom.map_zero\n\nprotected theorem map_mul (x y) : φ (x * y) = φ x * φ y :=\n map_mul _ _ _\n#align alg_hom.map_mul AlgHom.map_mul\n\nprotected theorem map_one : φ 1 = 1 :=\n map_one _\n#align alg_hom.map_one AlgHom.map_one\n\nprotected theorem map_pow (x : A) (n : ℕ) : φ (x ^ n) = φ x ^ n :=\n map_pow _ _ _\n#align alg_hom.map_pow AlgHom.map_pow\n\n-- @[simp] -- Porting note: simp can prove this\nprotected theorem map_smul (r : R) (x : A) : φ (r • x) = r • φ x :=\n map_smul _ _ _\n#align alg_hom.map_smul AlgHom.map_smul\n\nprotected theorem map_sum {ι : Type*} (f : ι → A) (s : Finset ι) :\n φ (∑ x in s, f x) = ∑ x in s, φ (f x) :=\n map_sum _ _ _\n#align alg_hom.map_sum AlgHom.map_sum\n\nprotected theorem map_finsupp_sum {α : Type*} [Zero α] {ι : Type*} (f : ι →₀ α) (g : ι → α → A) :\n φ (f.sum g) = f.sum fun i a => φ (g i a) :=\n map_finsupp_sum _ _ _\n#align alg_hom.map_finsupp_sum AlgHom.map_finsupp_sum\n\nset_option linter.deprecated false in\nprotected theorem map_bit0 (x) : φ (bit0 x) = bit0 (φ x) :=\n map_bit0 _ _\n#align alg_hom.map_bit0 AlgHom.map_bit0\n\nset_option linter.deprecated false in\nprotected theorem map_bit1 (x) : φ (bit1 x) = bit1 (φ x) :=\n map_bit1 _ _\n#align alg_hom.map_bit1 AlgHom.map_bit1\n\n/-- If a `RingHom` is `R`-linear, then it is an `AlgHom`. -/\ndef mk' (f : A →+* B) (h : ∀ (c : R) (x), f (c • x) = c • f x) : A →ₐ[R] B :=\n { f with\n toFun := f\n commutes' := fun c => by ","nextTactic":"simp only [Algebra.algebraMap_eq_smul_one, h, f.map_one]","declUpToTactic":"/-- If a `RingHom` is `R`-linear, then it is an `AlgHom`. -/\ndef mk' (f : A →+* B) (h : ∀ (c : R) (x), f (c • x) = c • f x) : A →ₐ[R] B :=\n { f with\n toFun := f\n commutes' := fun c => by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Hom.290_0.FxqJeqR3qcIVlrq","decl":"/-- If a `RingHom` is `R`-linear, then it is an `AlgHom`. -/\ndef mk' (f : A →+* B) (h : ∀ (c : R) (x), f (c • x) = c • f x) : A →ₐ[R] B "} +{"state":"R : Type u\nA : Type v\nB : Type w\nC : Type u₁\nD : Type v₁\ninst✝⁸ : CommSemiring R\ninst✝⁷ : Semiring A\ninst✝⁶ : Semiring B\ninst✝⁵ : Semiring C\ninst✝⁴ : Semiring D\ninst✝³ : Algebra R A\ninst✝² : Algebra R B\ninst✝¹ : Algebra R C\ninst✝ : Algebra R D\nφ : A →ₐ[R] B\nφ₁ : B →ₐ[R] C\nφ₂ : A →ₐ[R] B\nsrc✝ : A →+* C := RingHom.comp ↑φ₁ ↑φ₂\nr : R\n⊢ OneHom.toFun\n (↑↑{ toMonoidHom := ↑src✝, map_zero' := (_ : OneHom.toFun (↑↑src✝) 0 = 0),\n map_add' :=\n (_ : ∀ (x y : A), OneHom.toFun (↑↑src✝) (x + y) = OneHom.toFun (↑↑src✝) x + OneHom.toFun (↑↑src✝) y) })\n ((algebraMap R A) r) =\n (algebraMap R C) r","srcUpToTactic":"/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Yury Kudryashov\n-/\nimport Mathlib.Algebra.Algebra.Basic\nimport Mathlib.Algebra.BigOperators.Finsupp\n\n#align_import algebra.algebra.hom from \"leanprover-community/mathlib\"@\"e97cf15cd1aec9bd5c193b2ffac5a6dc9118912b\"\n\n/-!\n# Homomorphisms of `R`-algebras\n\nThis file defines bundled homomorphisms of `R`-algebras.\n\n## Main definitions\n\n* `AlgHom R A B`: the type of `R`-algebra morphisms from `A` to `B`.\n* `Algebra.ofId R A : R →ₐ[R] A`: the canonical map from `R` to `A`, as an `AlgHom`.\n\n## Notations\n\n* `A →ₐ[R] B` : `R`-algebra homomorphism from `A` to `B`.\n-/\n\nset_option autoImplicit true\n\n\nopen BigOperators\n\nuniverse u v w u��� v₁\n\n/-- Defining the homomorphism in the category R-Alg. -/\n-- @[nolint has_nonempty_instance] -- Porting note: This linter does not exist yet.\nstructure AlgHom (R : Type u) (A : Type v) (B : Type w) [CommSemiring R] [Semiring A] [Semiring B]\n [Algebra R A] [Algebra R B] extends RingHom A B where\n commutes' : ∀ r : R, toFun (algebraMap R A r) = algebraMap R B r\n#align alg_hom AlgHom\n\n/-- Reinterpret an `AlgHom` as a `RingHom` -/\nadd_decl_doc AlgHom.toRingHom\n\n@[inherit_doc AlgHom]\ninfixr:25 \" →ₐ \" => AlgHom _\n\n@[inherit_doc]\nnotation:25 A \" →ₐ[\" R \"] \" B => AlgHom R A B\n\n/-- `AlgHomClass F R A B` asserts `F` is a type of bundled algebra homomorphisms\nfrom `A` to `B`. -/\nclass AlgHomClass (F : Type*) (R : outParam (Type*)) (A : outParam (Type*))\n (B : outParam (Type*)) [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A]\n [Algebra R B] extends RingHomClass F A B where\n commutes : ∀ (f : F) (r : R), f (algebraMap R A r) = algebraMap R B r\n#align alg_hom_class AlgHomClass\n\n-- Porting note: `dangerousInstance` linter has become smarter about `outParam`s\n-- attribute [nolint dangerousInstance] AlgHomClass.toRingHomClass\n\n-- Porting note: simp can prove this\n-- attribute [simp] AlgHomClass.commutes\n\nnamespace AlgHomClass\n\nvariable {R : Type*} {A : Type*} {B : Type*} [CommSemiring R] [Semiring A] [Semiring B]\n [Algebra R A] [Algebra R B]\n\n-- see Note [lower instance priority]\ninstance (priority := 100) linearMapClass [AlgHomClass F R A B] : LinearMapClass F R A B :=\n { ‹AlgHomClass F R A B› with\n map_smulₛₗ := fun f r x => by\n simp only [Algebra.smul_def, map_mul, commutes, RingHom.id_apply] }\n#align alg_hom_class.linear_map_class AlgHomClass.linearMapClass\n\n-- Porting note: A new definition underlying a coercion `↑`.\n/-- Turn an element of a type `F` satisfying `AlgHomClass F α β` into an actual\n`AlgHom`. This is declared as the default coercion from `F` to `α →+* β`. -/\n@[coe]\ndef toAlgHom {F : Type*} [AlgHomClass F R A B] (f : F) : A →ₐ[R] B :=\n { (f : A →+* B) with\n toFun := f\n commutes' := AlgHomClass.commutes f }\n\ninstance coeTC {F : Type*} [AlgHomClass F R A B] : CoeTC F (A →ₐ[R] B) :=\n ⟨AlgHomClass.toAlgHom⟩\n#align alg_hom_class.alg_hom.has_coe_t AlgHomClass.coeTC\n\nend AlgHomClass\n\nnamespace AlgHom\n\nvariable {R : Type u} {A : Type v} {B : Type w} {C : Type u₁} {D : Type v₁}\n\nsection Semiring\n\nvariable [CommSemiring R] [Semiring A] [Semiring B] [Semiring C] [Semiring D]\n\nvariable [Algebra R A] [Algebra R B] [Algebra R C] [Algebra R D]\n\n-- Porting note: we don't port specialized `CoeFun` instances if there is `FunLike` instead\n#noalign alg_hom.has_coe_to_fun\n\n-- Porting note: This instance is moved.\ninstance algHomClass : AlgHomClass (A →ₐ[R] B) R A B where\n coe f := f.toFun\n coe_injective' f g h := by\n rcases f with ⟨⟨⟨⟨_, _⟩, _⟩, _, _⟩, _⟩\n rcases g with ⟨⟨⟨⟨_, _⟩, _⟩, _, _⟩, _⟩\n congr\n map_add f := f.map_add'\n map_zero f := f.map_zero'\n map_mul f := f.map_mul'\n map_one f := f.map_one'\n commutes f := f.commutes'\n#align alg_hom.alg_hom_class AlgHom.algHomClass\n\n/-- See Note [custom simps projection] -/\ndef Simps.apply {R : Type u} {α : Type v} {β : Type w} [CommSemiring R]\n [Semiring α] [Semiring β] [Algebra R α] [Algebra R β] (f : α →ₐ[R] β) : α → β := f\n\ninitialize_simps_projections AlgHom (toFun → apply)\n\n@[simp]\nprotected theorem coe_coe {F : Type*} [AlgHomClass F R A B] (f : F) : ⇑(f : A →ₐ[R] B) = f :=\n rfl\n#align alg_hom.coe_coe AlgHom.coe_coe\n\n@[simp]\ntheorem toFun_eq_coe (f : A →ₐ[R] B) : f.toFun = f :=\n rfl\n#align alg_hom.to_fun_eq_coe AlgHom.toFun_eq_coe\n\nattribute [coe] AlgHom.toRingHom\n\ninstance coeOutRingHom : CoeOut (A →ₐ[R] B) (A →+* B) :=\n ⟨AlgHom.toRingHom⟩\n#align alg_hom.coe_ring_hom AlgHom.coeOutRingHom\n\n-- Porting note: A new definition underlying a coercion `↑`.\n@[coe]\ndef toMonoidHom' (f : A →ₐ[R] B) : A →* B := (f : A →+* B)\n\ninstance coeOutMonoidHom : CoeOut (A →ₐ[R] B) (A →* B) :=\n ⟨AlgHom.toMonoidHom'⟩\n#align alg_hom.coe_monoid_hom AlgHom.coeOutMonoidHom\n\n-- Porting note: A new definition underlying a coercion `↑`.\n@[coe]\ndef toAddMonoidHom' (f : A →ₐ[R] B) : A →+ B := (f : A →+* B)\n\ninstance coeOutAddMonoidHom : CoeOut (A →ₐ[R] B) (A →+ B) :=\n ⟨AlgHom.toAddMonoidHom'⟩\n#align alg_hom.coe_add_monoid_hom AlgHom.coeOutAddMonoidHom\n\n-- Porting note: Lean 3: `@[simp, norm_cast] coe_mk`\n-- Lean 4: `@[simp] coe_mk` & `@[norm_cast] coe_mks`\n@[simp]\ntheorem coe_mk {f : A →+* B} (h) : ((⟨f, h⟩ : A →ₐ[R] B) : A → B) = f :=\n rfl\n\n@[norm_cast]\ntheorem coe_mks {f : A → B} (h₁ h₂ h₃ h₄ h₅) : ⇑(⟨⟨⟨⟨f, h₁⟩, h₂⟩, h₃, h₄⟩, h₅⟩ : A →ₐ[R] B) = f :=\n rfl\n#align alg_hom.coe_mk AlgHom.coe_mks\n\n-- Porting note: This theorem is new.\n@[simp, norm_cast]\ntheorem coe_ringHom_mk {f : A →+* B} (h) : ((⟨f, h⟩ : A →ₐ[R] B) : A →+* B) = f :=\n rfl\n\n-- make the coercion the simp-normal form\n@[simp]\ntheorem toRingHom_eq_coe (f : A →ₐ[R] B) : f.toRingHom = f :=\n rfl\n#align alg_hom.to_ring_hom_eq_coe AlgHom.toRingHom_eq_coe\n\n@[simp, norm_cast]\ntheorem coe_toRingHom (f : A →ₐ[R] B) : ⇑(f : A →+* B) = f :=\n rfl\n#align alg_hom.coe_to_ring_hom AlgHom.coe_toRingHom\n\n@[simp, norm_cast]\ntheorem coe_toMonoidHom (f : A →ₐ[R] B) : ⇑(f : A →* B) = f :=\n rfl\n#align alg_hom.coe_to_monoid_hom AlgHom.coe_toMonoidHom\n\n@[simp, norm_cast]\ntheorem coe_toAddMonoidHom (f : A →ₐ[R] B) : ⇑(f : A →+ B) = f :=\n rfl\n#align alg_hom.coe_to_add_monoid_hom AlgHom.coe_toAddMonoidHom\n\nvariable (φ : A →ₐ[R] B)\n\ntheorem coe_fn_injective : @Function.Injective (A →ₐ[R] B) (A → B) (↑) :=\n FunLike.coe_injective\n#align alg_hom.coe_fn_injective AlgHom.coe_fn_injective\n\ntheorem coe_fn_inj {φ₁ φ₂ : A →ₐ[R] B} : (φ₁ : A → B) = φ₂ ↔ φ₁ = φ₂ :=\n FunLike.coe_fn_eq\n#align alg_hom.coe_fn_inj AlgHom.coe_fn_inj\n\ntheorem coe_ringHom_injective : Function.Injective ((↑) : (A →ₐ[R] B) → A →+* B) := fun φ₁ φ₂ H =>\n coe_fn_injective <| show ((φ₁ : A →+* B) : A → B) = ((φ₂ : A →+* B) : A → B) from congr_arg _ H\n#align alg_hom.coe_ring_hom_injective AlgHom.coe_ringHom_injective\n\ntheorem coe_monoidHom_injective : Function.Injective ((↑) : (A →ₐ[R] B) → A →* B) :=\n RingHom.coe_monoidHom_injective.comp coe_ringHom_injective\n#align alg_hom.coe_monoid_hom_injective AlgHom.coe_monoidHom_injective\n\ntheorem coe_addMonoidHom_injective : Function.Injective ((↑) : (A →ₐ[R] B) → A →+ B) :=\n RingHom.coe_addMonoidHom_injective.comp coe_ringHom_injective\n#align alg_hom.coe_add_monoid_hom_injective AlgHom.coe_addMonoidHom_injective\n\nprotected theorem congr_fun {φ₁ φ₂ : A →ₐ[R] B} (H : φ₁ = φ₂) (x : A) : φ₁ x = φ₂ x :=\n FunLike.congr_fun H x\n#align alg_hom.congr_fun AlgHom.congr_fun\n\nprotected theorem congr_arg (φ : A →ₐ[R] B) {x y : A} (h : x = y) : φ x = φ y :=\n FunLike.congr_arg φ h\n#align alg_hom.congr_arg AlgHom.congr_arg\n\n@[ext]\ntheorem ext {φ₁ φ₂ : A →ₐ[R] B} (H : ∀ x, φ₁ x = φ₂ x) : φ₁ = φ₂ :=\n FunLike.ext _ _ H\n#align alg_hom.ext AlgHom.ext\n\ntheorem ext_iff {φ₁ φ₂ : A →ₐ[R] B} : φ₁ = φ₂ ↔ ∀ x, φ₁ x = φ₂ x :=\n FunLike.ext_iff\n#align alg_hom.ext_iff AlgHom.ext_iff\n\n@[simp]\ntheorem mk_coe {f : A →ₐ[R] B} (h₁ h₂ h₃ h₄ h₅) : (⟨⟨⟨⟨f, h₁⟩, h₂⟩, h₃, h₄⟩, h₅⟩ : A →ₐ[R] B) = f :=\n ext fun _ => rfl\n#align alg_hom.mk_coe AlgHom.mk_coe\n\n@[simp]\ntheorem commutes (r : R) : φ (algebraMap R A r) = algebraMap R B r :=\n φ.commutes' r\n#align alg_hom.commutes AlgHom.commutes\n\ntheorem comp_algebraMap : (φ : A →+* B).comp (algebraMap R A) = algebraMap R B :=\n RingHom.ext <| φ.commutes\n#align alg_hom.comp_algebra_map AlgHom.comp_algebraMap\n\nprotected theorem map_add (r s : A) : φ (r + s) = φ r + φ s :=\n map_add _ _ _\n#align alg_hom.map_add AlgHom.map_add\n\nprotected theorem map_zero : φ 0 = 0 :=\n map_zero _\n#align alg_hom.map_zero AlgHom.map_zero\n\nprotected theorem map_mul (x y) : φ (x * y) = φ x * φ y :=\n map_mul _ _ _\n#align alg_hom.map_mul AlgHom.map_mul\n\nprotected theorem map_one : φ 1 = 1 :=\n map_one _\n#align alg_hom.map_one AlgHom.map_one\n\nprotected theorem map_pow (x : A) (n : ℕ) : φ (x ^ n) = φ x ^ n :=\n map_pow _ _ _\n#align alg_hom.map_pow AlgHom.map_pow\n\n-- @[simp] -- Porting note: simp can prove this\nprotected theorem map_smul (r : R) (x : A) : φ (r • x) = r • φ x :=\n map_smul _ _ _\n#align alg_hom.map_smul AlgHom.map_smul\n\nprotected theorem map_sum {ι : Type*} (f : ι → A) (s : Finset ι) :\n φ (∑ x in s, f x) = ∑ x in s, φ (f x) :=\n map_sum _ _ _\n#align alg_hom.map_sum AlgHom.map_sum\n\nprotected theorem map_finsupp_sum {α : Type*} [Zero α] {ι : Type*} (f : ι →₀ α) (g : ι → α → A) :\n φ (f.sum g) = f.sum fun i a => φ (g i a) :=\n map_finsupp_sum _ _ _\n#align alg_hom.map_finsupp_sum AlgHom.map_finsupp_sum\n\nset_option linter.deprecated false in\nprotected theorem map_bit0 (x) : φ (bit0 x) = bit0 (φ x) :=\n map_bit0 _ _\n#align alg_hom.map_bit0 AlgHom.map_bit0\n\nset_option linter.deprecated false in\nprotected theorem map_bit1 (x) : φ (bit1 x) = bit1 (φ x) :=\n map_bit1 _ _\n#align alg_hom.map_bit1 AlgHom.map_bit1\n\n/-- If a `RingHom` is `R`-linear, then it is an `AlgHom`. -/\ndef mk' (f : A →+* B) (h : ∀ (c : R) (x), f (c • x) = c • f x) : A →ₐ[R] B :=\n { f with\n toFun := f\n commutes' := fun c => by simp only [Algebra.algebraMap_eq_smul_one, h, f.map_one] }\n#align alg_hom.mk' AlgHom.mk'\n\n@[simp]\ntheorem coe_mk' (f : A →+* B) (h : ∀ (c : R) (x), f (c • x) = c • f x) : ⇑(mk' f h) = f :=\n rfl\n#align alg_hom.coe_mk' AlgHom.coe_mk'\n\nsection\n\nvariable (R A)\n\n/-- Identity map as an `AlgHom`. -/\nprotected def id : A →ₐ[R] A :=\n { RingHom.id A with commutes' := fun _ => rfl }\n#align alg_hom.id AlgHom.id\n\n@[simp]\ntheorem coe_id : ⇑(AlgHom.id R A) = id :=\n rfl\n#align alg_hom.coe_id AlgHom.coe_id\n\n@[simp]\ntheorem id_toRingHom : (AlgHom.id R A : A →+* A) = RingHom.id _ :=\n rfl\n#align alg_hom.id_to_ring_hom AlgHom.id_toRingHom\n\nend\n\ntheorem id_apply (p : A) : AlgHom.id R A p = p :=\n rfl\n#align alg_hom.id_apply AlgHom.id_apply\n\n/-- Composition of algebra homeomorphisms. -/\ndef comp (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) : A →ₐ[R] C :=\n { φ₁.toRingHom.comp ↑φ₂ with\n commutes' := fun r : R => by ","nextTactic":"rw [← φ₁.commutes, ← φ₂.commutes]","declUpToTactic":"/-- Composition of algebra homeomorphisms. -/\ndef comp (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) : A →ₐ[R] C :=\n { φ₁.toRingHom.comp ↑φ₂ with\n commutes' := fun r : R => by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Hom.327_0.FxqJeqR3qcIVlrq","decl":"/-- Composition of algebra homeomorphisms. -/\ndef comp (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) : A →ₐ[R] C "} +{"state":"R : Type u\nA : Type v\nB : Type w\nC : Type u₁\nD : Type v₁\ninst✝⁸ : CommSemiring R\ninst✝⁷ : Semiring A\ninst✝⁶ : Semiring B\ninst✝⁵ : Semiring C\ninst✝⁴ : Semiring D\ninst✝³ : Algebra R A\ninst✝² : Algebra R B\ninst✝¹ : Algebra R C\ninst✝ : Algebra R D\nφ : A →ₐ[R] B\nφ₁ : B →ₐ[R] C\nφ₂ : A →ₐ[R] B\nsrc✝ : A →+* C := RingHom.comp ↑φ₁ ↑φ₂\nr : R\n⊢ OneHom.toFun\n (↑↑{ toMonoidHom := ↑src✝, map_zero' := (_ : OneHom.toFun (↑↑src✝) 0 = 0),\n map_add' :=\n (_ : ∀ (x y : A), OneHom.toFun (↑↑src✝) (x + y) = OneHom.toFun (↑↑src✝) x + OneHom.toFun (↑↑src✝) y) })\n ((algebraMap R A) r) =\n φ₁ (φ₂ ((algebraMap R A) r))","srcUpToTactic":"/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Yury Kudryashov\n-/\nimport Mathlib.Algebra.Algebra.Basic\nimport Mathlib.Algebra.BigOperators.Finsupp\n\n#align_import algebra.algebra.hom from \"leanprover-community/mathlib\"@\"e97cf15cd1aec9bd5c193b2ffac5a6dc9118912b\"\n\n/-!\n# Homomorphisms of `R`-algebras\n\nThis file defines bundled homomorphisms of `R`-algebras.\n\n## Main definitions\n\n* `AlgHom R A B`: the type of `R`-algebra morphisms from `A` to `B`.\n* `Algebra.ofId R A : R →ₐ[R] A`: the canonical map from `R` to `A`, as an `AlgHom`.\n\n## Notations\n\n* `A →ₐ[R] B` : `R`-algebra homomorphism from `A` to `B`.\n-/\n\nset_option autoImplicit true\n\n\nopen BigOperators\n\nuniverse u v w u₁ v₁\n\n/-- Defining the homomorphism in the category R-Alg. -/\n-- @[nolint has_nonempty_instance] -- Porting note: This linter does not exist yet.\nstructure AlgHom (R : Type u) (A : Type v) (B : Type w) [CommSemiring R] [Semiring A] [Semiring B]\n [Algebra R A] [Algebra R B] extends RingHom A B where\n commutes' : ∀ r : R, toFun (algebraMap R A r) = algebraMap R B r\n#align alg_hom AlgHom\n\n/-- Reinterpret an `AlgHom` as a `RingHom` -/\nadd_decl_doc AlgHom.toRingHom\n\n@[inherit_doc AlgHom]\ninfixr:25 \" →ₐ \" => AlgHom _\n\n@[inherit_doc]\nnotation:25 A \" →ₐ[\" R \"] \" B => AlgHom R A B\n\n/-- `AlgHomClass F R A B` asserts `F` is a type of bundled algebra homomorphisms\nfrom `A` to `B`. -/\nclass AlgHomClass (F : Type*) (R : outParam (Type*)) (A : outParam (Type*))\n (B : outParam (Type*)) [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A]\n [Algebra R B] extends RingHomClass F A B where\n commutes : ∀ (f : F) (r : R), f (algebraMap R A r) = algebraMap R B r\n#align alg_hom_class AlgHomClass\n\n-- Porting note: `dangerousInstance` linter has become smarter about `outParam`s\n-- attribute [nolint dangerousInstance] AlgHomClass.toRingHomClass\n\n-- Porting note: simp can prove this\n-- attribute [simp] AlgHomClass.commutes\n\nnamespace AlgHomClass\n\nvariable {R : Type*} {A : Type*} {B : Type*} [CommSemiring R] [Semiring A] [Semiring B]\n [Algebra R A] [Algebra R B]\n\n-- see Note [lower instance priority]\ninstance (priority := 100) linearMapClass [AlgHomClass F R A B] : LinearMapClass F R A B :=\n { ‹AlgHomClass F R A B› with\n map_smulₛₗ := fun f r x => by\n simp only [Algebra.smul_def, map_mul, commutes, RingHom.id_apply] }\n#align alg_hom_class.linear_map_class AlgHomClass.linearMapClass\n\n-- Porting note: A new definition underlying a coercion `↑`.\n/-- Turn an element of a type `F` satisfying `AlgHomClass F α β` into an actual\n`AlgHom`. This is declared as the default coercion from `F` to `α →+* β`. -/\n@[coe]\ndef toAlgHom {F : Type*} [AlgHomClass F R A B] (f : F) : A →ₐ[R] B :=\n { (f : A →+* B) with\n toFun := f\n commutes' := AlgHomClass.commutes f }\n\ninstance coeTC {F : Type*} [AlgHomClass F R A B] : CoeTC F (A →ₐ[R] B) :=\n ⟨AlgHomClass.toAlgHom⟩\n#align alg_hom_class.alg_hom.has_coe_t AlgHomClass.coeTC\n\nend AlgHomClass\n\nnamespace AlgHom\n\nvariable {R : Type u} {A : Type v} {B : Type w} {C : Type u₁} {D : Type v₁}\n\nsection Semiring\n\nvariable [CommSemiring R] [Semiring A] [Semiring B] [Semiring C] [Semiring D]\n\nvariable [Algebra R A] [Algebra R B] [Algebra R C] [Algebra R D]\n\n-- Porting note: we don't port specialized `CoeFun` instances if there is `FunLike` instead\n#noalign alg_hom.has_coe_to_fun\n\n-- Porting note: This instance is moved.\ninstance algHomClass : AlgHomClass (A →ₐ[R] B) R A B where\n coe f := f.toFun\n coe_injective' f g h := by\n rcases f with ⟨⟨⟨⟨_, _⟩, _⟩, _, _⟩, _⟩\n rcases g with ⟨⟨⟨⟨_, _⟩, _⟩, _, _⟩, _⟩\n congr\n map_add f := f.map_add'\n map_zero f := f.map_zero'\n map_mul f := f.map_mul'\n map_one f := f.map_one'\n commutes f := f.commutes'\n#align alg_hom.alg_hom_class AlgHom.algHomClass\n\n/-- See Note [custom simps projection] -/\ndef Simps.apply {R : Type u} {α : Type v} {β : Type w} [CommSemiring R]\n [Semiring α] [Semiring β] [Algebra R α] [Algebra R β] (f : α →ₐ[R] β) : α → β := f\n\ninitialize_simps_projections AlgHom (toFun → apply)\n\n@[simp]\nprotected theorem coe_coe {F : Type*} [AlgHomClass F R A B] (f : F) : ⇑(f : A →ₐ[R] B) = f :=\n rfl\n#align alg_hom.coe_coe AlgHom.coe_coe\n\n@[simp]\ntheorem toFun_eq_coe (f : A →ₐ[R] B) : f.toFun = f :=\n rfl\n#align alg_hom.to_fun_eq_coe AlgHom.toFun_eq_coe\n\nattribute [coe] AlgHom.toRingHom\n\ninstance coeOutRingHom : CoeOut (A →ₐ[R] B) (A →+* B) :=\n ⟨AlgHom.toRingHom⟩\n#align alg_hom.coe_ring_hom AlgHom.coeOutRingHom\n\n-- Porting note: A new definition underlying a coercion `↑`.\n@[coe]\ndef toMonoidHom' (f : A →ₐ[R] B) : A →* B := (f : A →+* B)\n\ninstance coeOutMonoidHom : CoeOut (A →ₐ[R] B) (A →* B) :=\n ⟨AlgHom.toMonoidHom'⟩\n#align alg_hom.coe_monoid_hom AlgHom.coeOutMonoidHom\n\n-- Porting note: A new definition underlying a coercion `↑`.\n@[coe]\ndef toAddMonoidHom' (f : A →ₐ[R] B) : A →+ B := (f : A →+* B)\n\ninstance coeOutAddMonoidHom : CoeOut (A →ₐ[R] B) (A →+ B) :=\n ⟨AlgHom.toAddMonoidHom'⟩\n#align alg_hom.coe_add_monoid_hom AlgHom.coeOutAddMonoidHom\n\n-- Porting note: Lean 3: `@[simp, norm_cast] coe_mk`\n-- Lean 4: `@[simp] coe_mk` & `@[norm_cast] coe_mks`\n@[simp]\ntheorem coe_mk {f : A →+* B} (h) : ((⟨f, h⟩ : A →ₐ[R] B) : A → B) = f :=\n rfl\n\n@[norm_cast]\ntheorem coe_mks {f : A → B} (h₁ h₂ h₃ h₄ h₅) : ⇑(⟨⟨⟨⟨f, h₁⟩, h₂⟩, h₃, h₄⟩, h₅⟩ : A →ₐ[R] B) = f :=\n rfl\n#align alg_hom.coe_mk AlgHom.coe_mks\n\n-- Porting note: This theorem is new.\n@[simp, norm_cast]\ntheorem coe_ringHom_mk {f : A →+* B} (h) : ((⟨f, h⟩ : A →ₐ[R] B) : A →+* B) = f :=\n rfl\n\n-- make the coercion the simp-normal form\n@[simp]\ntheorem toRingHom_eq_coe (f : A →ₐ[R] B) : f.toRingHom = f :=\n rfl\n#align alg_hom.to_ring_hom_eq_coe AlgHom.toRingHom_eq_coe\n\n@[simp, norm_cast]\ntheorem coe_toRingHom (f : A →ₐ[R] B) : ⇑(f : A →+* B) = f :=\n rfl\n#align alg_hom.coe_to_ring_hom AlgHom.coe_toRingHom\n\n@[simp, norm_cast]\ntheorem coe_toMonoidHom (f : A →ₐ[R] B) : ⇑(f : A →* B) = f :=\n rfl\n#align alg_hom.coe_to_monoid_hom AlgHom.coe_toMonoidHom\n\n@[simp, norm_cast]\ntheorem coe_toAddMonoidHom (f : A →ₐ[R] B) : ⇑(f : A →+ B) = f :=\n rfl\n#align alg_hom.coe_to_add_monoid_hom AlgHom.coe_toAddMonoidHom\n\nvariable (φ : A →ₐ[R] B)\n\ntheorem coe_fn_injective : @Function.Injective (A →ₐ[R] B) (A → B) (↑) :=\n FunLike.coe_injective\n#align alg_hom.coe_fn_injective AlgHom.coe_fn_injective\n\ntheorem coe_fn_inj {φ₁ φ₂ : A →ₐ[R] B} : (φ₁ : A → B) = φ₂ ↔ φ₁ = φ₂ :=\n FunLike.coe_fn_eq\n#align alg_hom.coe_fn_inj AlgHom.coe_fn_inj\n\ntheorem coe_ringHom_injective : Function.Injective ((↑) : (A →ₐ[R] B) → A →+* B) := fun φ₁ φ₂ H =>\n coe_fn_injective <| show ((φ₁ : A →+* B) : A → B) = ((φ₂ : A →+* B) : A → B) from congr_arg _ H\n#align alg_hom.coe_ring_hom_injective AlgHom.coe_ringHom_injective\n\ntheorem coe_monoidHom_injective : Function.Injective ((↑) : (A →ₐ[R] B) → A →* B) :=\n RingHom.coe_monoidHom_injective.comp coe_ringHom_injective\n#align alg_hom.coe_monoid_hom_injective AlgHom.coe_monoidHom_injective\n\ntheorem coe_addMonoidHom_injective : Function.Injective ((↑) : (A →ₐ[R] B) → A →+ B) :=\n RingHom.coe_addMonoidHom_injective.comp coe_ringHom_injective\n#align alg_hom.coe_add_monoid_hom_injective AlgHom.coe_addMonoidHom_injective\n\nprotected theorem congr_fun {φ₁ φ₂ : A →ₐ[R] B} (H : φ₁ = φ₂) (x : A) : φ₁ x = φ₂ x :=\n FunLike.congr_fun H x\n#align alg_hom.congr_fun AlgHom.congr_fun\n\nprotected theorem congr_arg (φ : A →ₐ[R] B) {x y : A} (h : x = y) : φ x = φ y :=\n FunLike.congr_arg φ h\n#align alg_hom.congr_arg AlgHom.congr_arg\n\n@[ext]\ntheorem ext {φ₁ φ₂ : A →ₐ[R] B} (H : ∀ x, φ₁ x = φ₂ x) : φ₁ = φ₂ :=\n FunLike.ext _ _ H\n#align alg_hom.ext AlgHom.ext\n\ntheorem ext_iff {φ₁ φ₂ : A →ₐ[R] B} : φ₁ = φ₂ ↔ ∀ x, φ₁ x = φ₂ x :=\n FunLike.ext_iff\n#align alg_hom.ext_iff AlgHom.ext_iff\n\n@[simp]\ntheorem mk_coe {f : A →ₐ[R] B} (h₁ h₂ h₃ h₄ h₅) : (⟨⟨⟨⟨f, h₁⟩, h₂⟩, h₃, h₄⟩, h₅⟩ : A →ₐ[R] B) = f :=\n ext fun _ => rfl\n#align alg_hom.mk_coe AlgHom.mk_coe\n\n@[simp]\ntheorem commutes (r : R) : φ (algebraMap R A r) = algebraMap R B r :=\n φ.commutes' r\n#align alg_hom.commutes AlgHom.commutes\n\ntheorem comp_algebraMap : (φ : A →+* B).comp (algebraMap R A) = algebraMap R B :=\n RingHom.ext <| φ.commutes\n#align alg_hom.comp_algebra_map AlgHom.comp_algebraMap\n\nprotected theorem map_add (r s : A) : φ (r + s) = φ r + φ s :=\n map_add _ _ _\n#align alg_hom.map_add AlgHom.map_add\n\nprotected theorem map_zero : φ 0 = 0 :=\n map_zero _\n#align alg_hom.map_zero AlgHom.map_zero\n\nprotected theorem map_mul (x y) : φ (x * y) = φ x * φ y :=\n map_mul _ _ _\n#align alg_hom.map_mul AlgHom.map_mul\n\nprotected theorem map_one : φ 1 = 1 :=\n map_one _\n#align alg_hom.map_one AlgHom.map_one\n\nprotected theorem map_pow (x : A) (n : ℕ) : φ (x ^ n) = φ x ^ n :=\n map_pow _ _ _\n#align alg_hom.map_pow AlgHom.map_pow\n\n-- @[simp] -- Porting note: simp can prove this\nprotected theorem map_smul (r : R) (x : A) : φ (r • x) = r • φ x :=\n map_smul _ _ _\n#align alg_hom.map_smul AlgHom.map_smul\n\nprotected theorem map_sum {ι : Type*} (f : ι → A) (s : Finset ι) :\n φ (∑ x in s, f x) = ∑ x in s, φ (f x) :=\n map_sum _ _ _\n#align alg_hom.map_sum AlgHom.map_sum\n\nprotected theorem map_finsupp_sum {α : Type*} [Zero α] {ι : Type*} (f : ι →₀ α) (g : ι → α → A) :\n φ (f.sum g) = f.sum fun i a => φ (g i a) :=\n map_finsupp_sum _ _ _\n#align alg_hom.map_finsupp_sum AlgHom.map_finsupp_sum\n\nset_option linter.deprecated false in\nprotected theorem map_bit0 (x) : φ (bit0 x) = bit0 (φ x) :=\n map_bit0 _ _\n#align alg_hom.map_bit0 AlgHom.map_bit0\n\nset_option linter.deprecated false in\nprotected theorem map_bit1 (x) : φ (bit1 x) = bit1 (φ x) :=\n map_bit1 _ _\n#align alg_hom.map_bit1 AlgHom.map_bit1\n\n/-- If a `RingHom` is `R`-linear, then it is an `AlgHom`. -/\ndef mk' (f : A →+* B) (h : ∀ (c : R) (x), f (c • x) = c • f x) : A →ₐ[R] B :=\n { f with\n toFun := f\n commutes' := fun c => by simp only [Algebra.algebraMap_eq_smul_one, h, f.map_one] }\n#align alg_hom.mk' AlgHom.mk'\n\n@[simp]\ntheorem coe_mk' (f : A →+* B) (h : ∀ (c : R) (x), f (c • x) = c • f x) : ⇑(mk' f h) = f :=\n rfl\n#align alg_hom.coe_mk' AlgHom.coe_mk'\n\nsection\n\nvariable (R A)\n\n/-- Identity map as an `AlgHom`. -/\nprotected def id : A →ₐ[R] A :=\n { RingHom.id A with commutes' := fun _ => rfl }\n#align alg_hom.id AlgHom.id\n\n@[simp]\ntheorem coe_id : ⇑(AlgHom.id R A) = id :=\n rfl\n#align alg_hom.coe_id AlgHom.coe_id\n\n@[simp]\ntheorem id_toRingHom : (AlgHom.id R A : A →+* A) = RingHom.id _ :=\n rfl\n#align alg_hom.id_to_ring_hom AlgHom.id_toRingHom\n\nend\n\ntheorem id_apply (p : A) : AlgHom.id R A p = p :=\n rfl\n#align alg_hom.id_apply AlgHom.id_apply\n\n/-- Composition of algebra homeomorphisms. -/\ndef comp (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) : A →ₐ[R] C :=\n { φ₁.toRingHom.comp ↑φ₂ with\n commutes' := fun r : R => by rw [← φ₁.commutes, ← φ₂.commutes]; ","nextTactic":"rfl","declUpToTactic":"/-- Composition of algebra homeomorphisms. -/\ndef comp (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) : A →ₐ[R] C :=\n { φ₁.toRingHom.comp ↑φ₂ with\n commutes' := fun r : R => by rw [← φ₁.commutes, ← φ₂.commutes]; ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Hom.327_0.FxqJeqR3qcIVlrq","decl":"/-- Composition of algebra homeomorphisms. -/\ndef comp (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) : A →ₐ[R] C "} +{"state":"R : Type u\nA : Type v\nB : Type w\nC : Type u₁\nD : Type v₁\ninst✝⁸ : CommSemiring R\ninst✝⁷ : Semiring A\ninst✝⁶ : Semiring B\ninst✝⁵ : Semiring C\ninst✝⁴ : Semiring D\ninst✝³ : Algebra R A\ninst✝² : Algebra R B\ninst✝¹ : Algebra R C\ninst✝ : Algebra R D\nφ : A →ₐ[R] B\nf : A →ₗ[R] B\nmap_one : f 1 = 1\nmap_mul : ∀ (x y : A), f (x * y) = f x * f y\nsrc✝ : A →+ B := LinearMap.toAddMonoidHom f\nc : R\n⊢ OneHom.toFun\n (↑↑{ toMonoidHom := { toOneHom := { toFun := ⇑f, map_one' := map_one }, map_mul' := map_mul },\n map_zero' := (_ : ZeroHom.toFun (↑src✝) 0 = 0),\n map_add' :=\n (_ : ∀ (x y : A), ZeroHom.toFun (↑src✝) (x + y) = ZeroHom.toFun (↑src✝) x + ZeroHom.toFun (↑src✝) y) })\n ((algebraMap R A) c) =\n (algebraMap R B) c","srcUpToTactic":"/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Yury Kudryashov\n-/\nimport Mathlib.Algebra.Algebra.Basic\nimport Mathlib.Algebra.BigOperators.Finsupp\n\n#align_import algebra.algebra.hom from \"leanprover-community/mathlib\"@\"e97cf15cd1aec9bd5c193b2ffac5a6dc9118912b\"\n\n/-!\n# Homomorphisms of `R`-algebras\n\nThis file defines bundled homomorphisms of `R`-algebras.\n\n## Main definitions\n\n* `AlgHom R A B`: the type of `R`-algebra morphisms from `A` to `B`.\n* `Algebra.ofId R A : R →ₐ[R] A`: the canonical map from `R` to `A`, as an `AlgHom`.\n\n## Notations\n\n* `A →ₐ[R] B` : `R`-algebra homomorphism from `A` to `B`.\n-/\n\nset_option autoImplicit true\n\n\nopen BigOperators\n\nuniverse u v w u₁ v₁\n\n/-- Defining the homomorphism in the category R-Alg. -/\n-- @[nolint has_nonempty_instance] -- Porting note: This linter does not exist yet.\nstructure AlgHom (R : Type u) (A : Type v) (B : Type w) [CommSemiring R] [Semiring A] [Semiring B]\n [Algebra R A] [Algebra R B] extends RingHom A B where\n commutes' : ∀ r : R, toFun (algebraMap R A r) = algebraMap R B r\n#align alg_hom AlgHom\n\n/-- Reinterpret an `AlgHom` as a `RingHom` -/\nadd_decl_doc AlgHom.toRingHom\n\n@[inherit_doc AlgHom]\ninfixr:25 \" →ₐ \" => AlgHom _\n\n@[inherit_doc]\nnotation:25 A \" →ₐ[\" R \"] \" B => AlgHom R A B\n\n/-- `AlgHomClass F R A B` asserts `F` is a type of bundled algebra homomorphisms\nfrom `A` to `B`. -/\nclass AlgHomClass (F : Type*) (R : outParam (Type*)) (A : outParam (Type*))\n (B : outParam (Type*)) [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A]\n [Algebra R B] extends RingHomClass F A B where\n commutes : ∀ (f : F) (r : R), f (algebraMap R A r) = algebraMap R B r\n#align alg_hom_class AlgHomClass\n\n-- Porting note: `dangerousInstance` linter has become smarter about `outParam`s\n-- attribute [nolint dangerousInstance] AlgHomClass.toRingHomClass\n\n-- Porting note: simp can prove this\n-- attribute [simp] AlgHomClass.commutes\n\nnamespace AlgHomClass\n\nvariable {R : Type*} {A : Type*} {B : Type*} [CommSemiring R] [Semiring A] [Semiring B]\n [Algebra R A] [Algebra R B]\n\n-- see Note [lower instance priority]\ninstance (priority := 100) linearMapClass [AlgHomClass F R A B] : LinearMapClass F R A B :=\n { ‹AlgHomClass F R A B› with\n map_smulₛₗ := fun f r x => by\n simp only [Algebra.smul_def, map_mul, commutes, RingHom.id_apply] }\n#align alg_hom_class.linear_map_class AlgHomClass.linearMapClass\n\n-- Porting note: A new definition underlying a coercion `↑`.\n/-- Turn an element of a type `F` satisfying `AlgHomClass F α β` into an actual\n`AlgHom`. This is declared as the default coercion from `F` to `α →+* β`. -/\n@[coe]\ndef toAlgHom {F : Type*} [AlgHomClass F R A B] (f : F) : A →ₐ[R] B :=\n { (f : A →+* B) with\n toFun := f\n commutes' := AlgHomClass.commutes f }\n\ninstance coeTC {F : Type*} [AlgHomClass F R A B] : CoeTC F (A →ₐ[R] B) :=\n ⟨AlgHomClass.toAlgHom⟩\n#align alg_hom_class.alg_hom.has_coe_t AlgHomClass.coeTC\n\nend AlgHomClass\n\nnamespace AlgHom\n\nvariable {R : Type u} {A : Type v} {B : Type w} {C : Type u₁} {D : Type v₁}\n\nsection Semiring\n\nvariable [CommSemiring R] [Semiring A] [Semiring B] [Semiring C] [Semiring D]\n\nvariable [Algebra R A] [Algebra R B] [Algebra R C] [Algebra R D]\n\n-- Porting note: we don't port specialized `CoeFun` instances if there is `FunLike` instead\n#noalign alg_hom.has_coe_to_fun\n\n-- Porting note: This instance is moved.\ninstance algHomClass : AlgHomClass (A →ₐ[R] B) R A B where\n coe f := f.toFun\n coe_injective' f g h := by\n rcases f with ⟨⟨⟨⟨_, _⟩, _⟩, _, _⟩, _⟩\n rcases g with ⟨⟨⟨⟨_, _⟩, _⟩, _, _⟩, _⟩\n congr\n map_add f := f.map_add'\n map_zero f := f.map_zero'\n map_mul f := f.map_mul'\n map_one f := f.map_one'\n commutes f := f.commutes'\n#align alg_hom.alg_hom_class AlgHom.algHomClass\n\n/-- See Note [custom simps projection] -/\ndef Simps.apply {R : Type u} {α : Type v} {β : Type w} [CommSemiring R]\n [Semiring α] [Semiring β] [Algebra R α] [Algebra R β] (f : α →ₐ[R] β) : α → β := f\n\ninitialize_simps_projections AlgHom (toFun → apply)\n\n@[simp]\nprotected theorem coe_coe {F : Type*} [AlgHomClass F R A B] (f : F) : ⇑(f : A →ₐ[R] B) = f :=\n rfl\n#align alg_hom.coe_coe AlgHom.coe_coe\n\n@[simp]\ntheorem toFun_eq_coe (f : A →ₐ[R] B) : f.toFun = f :=\n rfl\n#align alg_hom.to_fun_eq_coe AlgHom.toFun_eq_coe\n\nattribute [coe] AlgHom.toRingHom\n\ninstance coeOutRingHom : CoeOut (A →ₐ[R] B) (A →+* B) :=\n ⟨AlgHom.toRingHom⟩\n#align alg_hom.coe_ring_hom AlgHom.coeOutRingHom\n\n-- Porting note: A new definition underlying a coercion `↑`.\n@[coe]\ndef toMonoidHom' (f : A →ₐ[R] B) : A →* B := (f : A →+* B)\n\ninstance coeOutMonoidHom : CoeOut (A →ₐ[R] B) (A →* B) :=\n ⟨AlgHom.toMonoidHom'⟩\n#align alg_hom.coe_monoid_hom AlgHom.coeOutMonoidHom\n\n-- Porting note: A new definition underlying a coercion `↑`.\n@[coe]\ndef toAddMonoidHom' (f : A →ₐ[R] B) : A →+ B := (f : A →+* B)\n\ninstance coeOutAddMonoidHom : CoeOut (A →ₐ[R] B) (A →+ B) :=\n ⟨AlgHom.toAddMonoidHom'⟩\n#align alg_hom.coe_add_monoid_hom AlgHom.coeOutAddMonoidHom\n\n-- Porting note: Lean 3: `@[simp, norm_cast] coe_mk`\n-- Lean 4: `@[simp] coe_mk` & `@[norm_cast] coe_mks`\n@[simp]\ntheorem coe_mk {f : A →+* B} (h) : ((⟨f, h⟩ : A →ₐ[R] B) : A → B) = f :=\n rfl\n\n@[norm_cast]\ntheorem coe_mks {f : A → B} (h₁ h₂ h₃ h₄ h₅) : ⇑(⟨⟨⟨⟨f, h₁⟩, h₂⟩, h₃, h₄⟩, h₅⟩ : A →ₐ[R] B) = f :=\n rfl\n#align alg_hom.coe_mk AlgHom.coe_mks\n\n-- Porting note: This theorem is new.\n@[simp, norm_cast]\ntheorem coe_ringHom_mk {f : A →+* B} (h) : ((⟨f, h⟩ : A →ₐ[R] B) : A →+* B) = f :=\n rfl\n\n-- make the coercion the simp-normal form\n@[simp]\ntheorem toRingHom_eq_coe (f : A →ₐ[R] B) : f.toRingHom = f :=\n rfl\n#align alg_hom.to_ring_hom_eq_coe AlgHom.toRingHom_eq_coe\n\n@[simp, norm_cast]\ntheorem coe_toRingHom (f : A →ₐ[R] B) : ⇑(f : A →+* B) = f :=\n rfl\n#align alg_hom.coe_to_ring_hom AlgHom.coe_toRingHom\n\n@[simp, norm_cast]\ntheorem coe_toMonoidHom (f : A →ₐ[R] B) : ⇑(f : A →* B) = f :=\n rfl\n#align alg_hom.coe_to_monoid_hom AlgHom.coe_toMonoidHom\n\n@[simp, norm_cast]\ntheorem coe_toAddMonoidHom (f : A →ₐ[R] B) : ⇑(f : A →+ B) = f :=\n rfl\n#align alg_hom.coe_to_add_monoid_hom AlgHom.coe_toAddMonoidHom\n\nvariable (φ : A →ₐ[R] B)\n\ntheorem coe_fn_injective : @Function.Injective (A →ₐ[R] B) (A → B) (↑) :=\n FunLike.coe_injective\n#align alg_hom.coe_fn_injective AlgHom.coe_fn_injective\n\ntheorem coe_fn_inj {φ₁ φ₂ : A →ₐ[R] B} : (φ₁ : A → B) = φ₂ ↔ φ₁ = φ₂ :=\n FunLike.coe_fn_eq\n#align alg_hom.coe_fn_inj AlgHom.coe_fn_inj\n\ntheorem coe_ringHom_injective : Function.Injective ((↑) : (A →ₐ[R] B) → A →+* B) := fun φ₁ φ₂ H =>\n coe_fn_injective <| show ((φ₁ : A →+* B) : A → B) = ((φ₂ : A →+* B) : A → B) from congr_arg _ H\n#align alg_hom.coe_ring_hom_injective AlgHom.coe_ringHom_injective\n\ntheorem coe_monoidHom_injective : Function.Injective ((↑) : (A →ₐ[R] B) → A →* B) :=\n RingHom.coe_monoidHom_injective.comp coe_ringHom_injective\n#align alg_hom.coe_monoid_hom_injective AlgHom.coe_monoidHom_injective\n\ntheorem coe_addMonoidHom_injective : Function.Injective ((↑) : (A →ₐ[R] B) → A →+ B) :=\n RingHom.coe_addMonoidHom_injective.comp coe_ringHom_injective\n#align alg_hom.coe_add_monoid_hom_injective AlgHom.coe_addMonoidHom_injective\n\nprotected theorem congr_fun {φ₁ φ₂ : A →ₐ[R] B} (H : φ₁ = φ₂) (x : A) : φ₁ x = φ₂ x :=\n FunLike.congr_fun H x\n#align alg_hom.congr_fun AlgHom.congr_fun\n\nprotected theorem congr_arg (φ : A →ₐ[R] B) {x y : A} (h : x = y) : φ x = φ y :=\n FunLike.congr_arg φ h\n#align alg_hom.congr_arg AlgHom.congr_arg\n\n@[ext]\ntheorem ext {φ₁ φ₂ : A →ₐ[R] B} (H : ∀ x, φ₁ x = φ₂ x) : φ₁ = φ₂ :=\n FunLike.ext _ _ H\n#align alg_hom.ext AlgHom.ext\n\ntheorem ext_iff {φ₁ φ₂ : A →ₐ[R] B} : φ₁ = φ₂ ↔ ∀ x, φ₁ x = φ₂ x :=\n FunLike.ext_iff\n#align alg_hom.ext_iff AlgHom.ext_iff\n\n@[simp]\ntheorem mk_coe {f : A →ₐ[R] B} (h₁ h₂ h₃ h₄ h₅) : (⟨⟨⟨⟨f, h₁⟩, h₂⟩, h₃, h₄⟩, h₅⟩ : A →ₐ[R] B) = f :=\n ext fun _ => rfl\n#align alg_hom.mk_coe AlgHom.mk_coe\n\n@[simp]\ntheorem commutes (r : R) : φ (algebraMap R A r) = algebraMap R B r :=\n φ.commutes' r\n#align alg_hom.commutes AlgHom.commutes\n\ntheorem comp_algebraMap : (φ : A →+* B).comp (algebraMap R A) = algebraMap R B :=\n RingHom.ext <| φ.commutes\n#align alg_hom.comp_algebra_map AlgHom.comp_algebraMap\n\nprotected theorem map_add (r s : A) : φ (r + s) = φ r + φ s :=\n map_add _ _ _\n#align alg_hom.map_add AlgHom.map_add\n\nprotected theorem map_zero : φ 0 = 0 :=\n map_zero _\n#align alg_hom.map_zero AlgHom.map_zero\n\nprotected theorem map_mul (x y) : φ (x * y) = φ x * φ y :=\n map_mul _ _ _\n#align alg_hom.map_mul AlgHom.map_mul\n\nprotected theorem map_one : φ 1 = 1 :=\n map_one _\n#align alg_hom.map_one AlgHom.map_one\n\nprotected theorem map_pow (x : A) (n : ℕ) : φ (x ^ n) = φ x ^ n :=\n map_pow _ _ _\n#align alg_hom.map_pow AlgHom.map_pow\n\n-- @[simp] -- Porting note: simp can prove this\nprotected theorem map_smul (r : R) (x : A) : φ (r • x) = r • φ x :=\n map_smul _ _ _\n#align alg_hom.map_smul AlgHom.map_smul\n\nprotected theorem map_sum {ι : Type*} (f : ι → A) (s : Finset ι) :\n φ (∑ x in s, f x) = ∑ x in s, φ (f x) :=\n map_sum _ _ _\n#align alg_hom.map_sum AlgHom.map_sum\n\nprotected theorem map_finsupp_sum {α : Type*} [Zero α] {ι : Type*} (f : ι →₀ α) (g : ι → α → A) :\n φ (f.sum g) = f.sum fun i a => φ (g i a) :=\n map_finsupp_sum _ _ _\n#align alg_hom.map_finsupp_sum AlgHom.map_finsupp_sum\n\nset_option linter.deprecated false in\nprotected theorem map_bit0 (x) : φ (bit0 x) = bit0 (φ x) :=\n map_bit0 _ _\n#align alg_hom.map_bit0 AlgHom.map_bit0\n\nset_option linter.deprecated false in\nprotected theorem map_bit1 (x) : φ (bit1 x) = bit1 (φ x) :=\n map_bit1 _ _\n#align alg_hom.map_bit1 AlgHom.map_bit1\n\n/-- If a `RingHom` is `R`-linear, then it is an `AlgHom`. -/\ndef mk' (f : A →+* B) (h : ∀ (c : R) (x), f (c • x) = c • f x) : A →ₐ[R] B :=\n { f with\n toFun := f\n commutes' := fun c => by simp only [Algebra.algebraMap_eq_smul_one, h, f.map_one] }\n#align alg_hom.mk' AlgHom.mk'\n\n@[simp]\ntheorem coe_mk' (f : A →+* B) (h : ∀ (c : R) (x), f (c • x) = c • f x) : ⇑(mk' f h) = f :=\n rfl\n#align alg_hom.coe_mk' AlgHom.coe_mk'\n\nsection\n\nvariable (R A)\n\n/-- Identity map as an `AlgHom`. -/\nprotected def id : A →ₐ[R] A :=\n { RingHom.id A with commutes' := fun _ => rfl }\n#align alg_hom.id AlgHom.id\n\n@[simp]\ntheorem coe_id : ⇑(AlgHom.id R A) = id :=\n rfl\n#align alg_hom.coe_id AlgHom.coe_id\n\n@[simp]\ntheorem id_toRingHom : (AlgHom.id R A : A →+* A) = RingHom.id _ :=\n rfl\n#align alg_hom.id_to_ring_hom AlgHom.id_toRingHom\n\nend\n\ntheorem id_apply (p : A) : AlgHom.id R A p = p :=\n rfl\n#align alg_hom.id_apply AlgHom.id_apply\n\n/-- Composition of algebra homeomorphisms. -/\ndef comp (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) : A →ₐ[R] C :=\n { φ₁.toRingHom.comp ↑φ₂ with\n commutes' := fun r : R => by rw [← φ₁.commutes, ← φ₂.commutes]; rfl }\n#align alg_hom.comp AlgHom.comp\n\n@[simp]\ntheorem coe_comp (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) : ⇑(φ₁.comp φ₂) = φ₁ ∘ φ₂ :=\n rfl\n#align alg_hom.coe_comp AlgHom.coe_comp\n\ntheorem comp_apply (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) (p : A) : φ₁.comp φ₂ p = φ₁ (φ₂ p) :=\n rfl\n#align alg_hom.comp_apply AlgHom.comp_apply\n\ntheorem comp_toRingHom (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) :\n (φ₁.comp φ₂ : A →+* C) = (φ₁ : B →+* C).comp ↑φ₂ :=\n rfl\n#align alg_hom.comp_to_ring_hom AlgHom.comp_toRingHom\n\n@[simp]\ntheorem comp_id : φ.comp (AlgHom.id R A) = φ :=\n ext fun _x => rfl\n#align alg_hom.comp_id AlgHom.comp_id\n\n@[simp]\ntheorem id_comp : (AlgHom.id R B).comp φ = φ :=\n ext fun _x => rfl\n#align alg_hom.id_comp AlgHom.id_comp\n\ntheorem comp_assoc (φ₁ : C →ₐ[R] D) (φ₂ : B →ₐ[R] C) (φ₃ : A →ₐ[R] B) :\n (φ₁.comp φ₂).comp φ₃ = φ₁.comp (φ₂.comp φ₃) :=\n ext fun _x => rfl\n#align alg_hom.comp_assoc AlgHom.comp_assoc\n\n/-- R-Alg ⥤ R-Mod -/\ndef toLinearMap : A →ₗ[R] B where\n toFun := φ\n map_add' := map_add _\n map_smul' := map_smul _\n#align alg_hom.to_linear_map AlgHom.toLinearMap\n\n@[simp]\ntheorem toLinearMap_apply (p : A) : φ.toLinearMap p = φ p :=\n rfl\n#align alg_hom.to_linear_map_apply AlgHom.toLinearMap_apply\n\ntheorem toLinearMap_injective :\n Function.Injective (toLinearMap : _ → A →ₗ[R] B) := fun _φ₁ _φ₂ h =>\n ext <| LinearMap.congr_fun h\n#align alg_hom.to_linear_map_injective AlgHom.toLinearMap_injective\n\n@[simp]\ntheorem comp_toLinearMap (f : A →ₐ[R] B) (g : B →ₐ[R] C) :\n (g.comp f).toLinearMap = g.toLinearMap.comp f.toLinearMap :=\n rfl\n#align alg_hom.comp_to_linear_map AlgHom.comp_toLinearMap\n\n@[simp]\ntheorem toLinearMap_id : toLinearMap (AlgHom.id R A) = LinearMap.id :=\n LinearMap.ext fun _ => rfl\n#align alg_hom.to_linear_map_id AlgHom.toLinearMap_id\n\n/-- Promote a `LinearMap` to an `AlgHom` by supplying proofs about the behavior on `1` and `*`. -/\n@[simps]\ndef ofLinearMap (f : A →ₗ[R] B) (map_one : f 1 = 1) (map_mul : ∀ x y, f (x * y) = f x * f y) :\n A →ₐ[R] B :=\n { f.toAddMonoidHom with\n toFun := f\n map_one' := map_one\n map_mul' := map_mul\n commutes' := fun c => by ","nextTactic":"simp only [Algebra.algebraMap_eq_smul_one, f.map_smul, map_one]","declUpToTactic":"/-- Promote a `LinearMap` to an `AlgHom` by supplying proofs about the behavior on `1` and `*`. -/\n@[simps]\ndef ofLinearMap (f : A →ₗ[R] B) (map_one : f 1 = 1) (map_mul : ∀ x y, f (x * y) = f x * f y) :\n A →ₐ[R] B :=\n { f.toAddMonoidHom with\n toFun := f\n map_one' := map_one\n map_mul' := map_mul\n commutes' := fun c => by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Hom.390_0.FxqJeqR3qcIVlrq","decl":"/-- Promote a `LinearMap` to an `AlgHom` by supplying proofs about the behavior on `1` and `*`. -/\n@[simps]\ndef ofLinearMap (f : A →ₗ[R] B) (map_one : f 1 = 1) (map_mul : ∀ x y, f (x * y) = f x * f y) :\n A →ₐ[R] B "} +{"state":"R : Type u\nA : Type v\nB : Type w\nC : Type u₁\nD : Type v₁\ninst✝⁸ : CommSemiring R\ninst✝⁷ : Semiring A\ninst✝⁶ : Semiring B\ninst✝⁵ : Semiring C\ninst✝⁴ : Semiring D\ninst✝³ : Algebra R A\ninst✝² : Algebra R B\ninst✝¹ : Algebra R C\ninst✝ : Algebra R D\nφ : A →ₐ[R] B\nmap_one : (toLinearMap φ) 1 = 1\nmap_mul : ∀ (x y : A), (toLinearMap φ) (x * y) = (toLinearMap φ) x * (toLinearMap φ) y\n⊢ ofLinearMap (toLinearMap φ) map_one map_mul = φ","srcUpToTactic":"/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Yury Kudryashov\n-/\nimport Mathlib.Algebra.Algebra.Basic\nimport Mathlib.Algebra.BigOperators.Finsupp\n\n#align_import algebra.algebra.hom from \"leanprover-community/mathlib\"@\"e97cf15cd1aec9bd5c193b2ffac5a6dc9118912b\"\n\n/-!\n# Homomorphisms of `R`-algebras\n\nThis file defines bundled homomorphisms of `R`-algebras.\n\n## Main definitions\n\n* `AlgHom R A B`: the type of `R`-algebra morphisms from `A` to `B`.\n* `Algebra.ofId R A : R →ₐ[R] A`: the canonical map from `R` to `A`, as an `AlgHom`.\n\n## Notations\n\n* `A →ₐ[R] B` : `R`-algebra homomorphism from `A` to `B`.\n-/\n\nset_option autoImplicit true\n\n\nopen BigOperators\n\nuniverse u v w u₁ v₁\n\n/-- Defining the homomorphism in the category R-Alg. -/\n-- @[nolint has_nonempty_instance] -- Porting note: This linter does not exist yet.\nstructure AlgHom (R : Type u) (A : Type v) (B : Type w) [CommSemiring R] [Semiring A] [Semiring B]\n [Algebra R A] [Algebra R B] extends RingHom A B where\n commutes' : ∀ r : R, toFun (algebraMap R A r) = algebraMap R B r\n#align alg_hom AlgHom\n\n/-- Reinterpret an `AlgHom` as a `RingHom` -/\nadd_decl_doc AlgHom.toRingHom\n\n@[inherit_doc AlgHom]\ninfixr:25 \" →ₐ \" => AlgHom _\n\n@[inherit_doc]\nnotation:25 A \" →ₐ[\" R \"] \" B => AlgHom R A B\n\n/-- `AlgHomClass F R A B` asserts `F` is a type of bundled algebra homomorphisms\nfrom `A` to `B`. -/\nclass AlgHomClass (F : Type*) (R : outParam (Type*)) (A : outParam (Type*))\n (B : outParam (Type*)) [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A]\n [Algebra R B] extends RingHomClass F A B where\n commutes : ∀ (f : F) (r : R), f (algebraMap R A r) = algebraMap R B r\n#align alg_hom_class AlgHomClass\n\n-- Porting note: `dangerousInstance` linter has become smarter about `outParam`s\n-- attribute [nolint dangerousInstance] AlgHomClass.toRingHomClass\n\n-- Porting note: simp can prove this\n-- attribute [simp] AlgHomClass.commutes\n\nnamespace AlgHomClass\n\nvariable {R : Type*} {A : Type*} {B : Type*} [CommSemiring R] [Semiring A] [Semiring B]\n [Algebra R A] [Algebra R B]\n\n-- see Note [lower instance priority]\ninstance (priority := 100) linearMapClass [AlgHomClass F R A B] : LinearMapClass F R A B :=\n { ‹AlgHomClass F R A B› with\n map_smulₛₗ := fun f r x => by\n simp only [Algebra.smul_def, map_mul, commutes, RingHom.id_apply] }\n#align alg_hom_class.linear_map_class AlgHomClass.linearMapClass\n\n-- Porting note: A new definition underlying a coercion `↑`.\n/-- Turn an element of a type `F` satisfying `AlgHomClass F α β` into an actual\n`AlgHom`. This is declared as the default coercion from `F` to `α →+* β`. -/\n@[coe]\ndef toAlgHom {F : Type*} [AlgHomClass F R A B] (f : F) : A →ₐ[R] B :=\n { (f : A →+* B) with\n toFun := f\n commutes' := AlgHomClass.commutes f }\n\ninstance coeTC {F : Type*} [AlgHomClass F R A B] : CoeTC F (A →ₐ[R] B) :=\n ⟨AlgHomClass.toAlgHom⟩\n#align alg_hom_class.alg_hom.has_coe_t AlgHomClass.coeTC\n\nend AlgHomClass\n\nnamespace AlgHom\n\nvariable {R : Type u} {A : Type v} {B : Type w} {C : Type u₁} {D : Type v₁}\n\nsection Semiring\n\nvariable [CommSemiring R] [Semiring A] [Semiring B] [Semiring C] [Semiring D]\n\nvariable [Algebra R A] [Algebra R B] [Algebra R C] [Algebra R D]\n\n-- Porting note: we don't port specialized `CoeFun` instances if there is `FunLike` instead\n#noalign alg_hom.has_coe_to_fun\n\n-- Porting note: This instance is moved.\ninstance algHomClass : AlgHomClass (A →ₐ[R] B) R A B where\n coe f := f.toFun\n coe_injective' f g h := by\n rcases f with ⟨⟨⟨⟨_, _⟩, _⟩, _, _⟩, _⟩\n rcases g with ⟨⟨⟨⟨_, _⟩, _⟩, _, _⟩, _⟩\n congr\n map_add f := f.map_add'\n map_zero f := f.map_zero'\n map_mul f := f.map_mul'\n map_one f := f.map_one'\n commutes f := f.commutes'\n#align alg_hom.alg_hom_class AlgHom.algHomClass\n\n/-- See Note [custom simps projection] -/\ndef Simps.apply {R : Type u} {α : Type v} {β : Type w} [CommSemiring R]\n [Semiring α] [Semiring β] [Algebra R α] [Algebra R β] (f : α →ₐ[R] β) : α → β := f\n\ninitialize_simps_projections AlgHom (toFun → apply)\n\n@[simp]\nprotected theorem coe_coe {F : Type*} [AlgHomClass F R A B] (f : F) : ⇑(f : A →ₐ[R] B) = f :=\n rfl\n#align alg_hom.coe_coe AlgHom.coe_coe\n\n@[simp]\ntheorem toFun_eq_coe (f : A →ₐ[R] B) : f.toFun = f :=\n rfl\n#align alg_hom.to_fun_eq_coe AlgHom.toFun_eq_coe\n\nattribute [coe] AlgHom.toRingHom\n\ninstance coeOutRingHom : CoeOut (A →ₐ[R] B) (A →+* B) :=\n ⟨AlgHom.toRingHom⟩\n#align alg_hom.coe_ring_hom AlgHom.coeOutRingHom\n\n-- Porting note: A new definition underlying a coercion `↑`.\n@[coe]\ndef toMonoidHom' (f : A →ₐ[R] B) : A →* B := (f : A →+* B)\n\ninstance coeOutMonoidHom : CoeOut (A →ₐ[R] B) (A →* B) :=\n ⟨AlgHom.toMonoidHom'⟩\n#align alg_hom.coe_monoid_hom AlgHom.coeOutMonoidHom\n\n-- Porting note: A new definition underlying a coercion `↑`.\n@[coe]\ndef toAddMonoidHom' (f : A →ₐ[R] B) : A →+ B := (f : A →+* B)\n\ninstance coeOutAddMonoidHom : CoeOut (A →ₐ[R] B) (A →+ B) :=\n ⟨AlgHom.toAddMonoidHom'⟩\n#align alg_hom.coe_add_monoid_hom AlgHom.coeOutAddMonoidHom\n\n-- Porting note: Lean 3: `@[simp, norm_cast] coe_mk`\n-- Lean 4: `@[simp] coe_mk` & `@[norm_cast] coe_mks`\n@[simp]\ntheorem coe_mk {f : A →+* B} (h) : ((⟨f, h⟩ : A →ₐ[R] B) : A → B) = f :=\n rfl\n\n@[norm_cast]\ntheorem coe_mks {f : A → B} (h₁ h₂ h₃ h₄ h₅) : ⇑(⟨⟨⟨⟨f, h₁⟩, h₂⟩, h₃, h₄⟩, h₅⟩ : A →ₐ[R] B) = f :=\n rfl\n#align alg_hom.coe_mk AlgHom.coe_mks\n\n-- Porting note: This theorem is new.\n@[simp, norm_cast]\ntheorem coe_ringHom_mk {f : A →+* B} (h) : ((⟨f, h⟩ : A →ₐ[R] B) : A →+* B) = f :=\n rfl\n\n-- make the coercion the simp-normal form\n@[simp]\ntheorem toRingHom_eq_coe (f : A →ₐ[R] B) : f.toRingHom = f :=\n rfl\n#align alg_hom.to_ring_hom_eq_coe AlgHom.toRingHom_eq_coe\n\n@[simp, norm_cast]\ntheorem coe_toRingHom (f : A →ₐ[R] B) : ⇑(f : A →+* B) = f :=\n rfl\n#align alg_hom.coe_to_ring_hom AlgHom.coe_toRingHom\n\n@[simp, norm_cast]\ntheorem coe_toMonoidHom (f : A →ₐ[R] B) : ⇑(f : A →* B) = f :=\n rfl\n#align alg_hom.coe_to_monoid_hom AlgHom.coe_toMonoidHom\n\n@[simp, norm_cast]\ntheorem coe_toAddMonoidHom (f : A →ₐ[R] B) : ⇑(f : A →+ B) = f :=\n rfl\n#align alg_hom.coe_to_add_monoid_hom AlgHom.coe_toAddMonoidHom\n\nvariable (φ : A →ₐ[R] B)\n\ntheorem coe_fn_injective : @Function.Injective (A →ₐ[R] B) (A → B) (↑) :=\n FunLike.coe_injective\n#align alg_hom.coe_fn_injective AlgHom.coe_fn_injective\n\ntheorem coe_fn_inj {φ₁ φ₂ : A →ₐ[R] B} : (φ₁ : A → B) = φ₂ ↔ φ₁ = φ₂ :=\n FunLike.coe_fn_eq\n#align alg_hom.coe_fn_inj AlgHom.coe_fn_inj\n\ntheorem coe_ringHom_injective : Function.Injective ((↑) : (A →ₐ[R] B) → A →+* B) := fun φ₁ φ₂ H =>\n coe_fn_injective <| show ((φ₁ : A →+* B) : A → B) = ((φ₂ : A →+* B) : A → B) from congr_arg _ H\n#align alg_hom.coe_ring_hom_injective AlgHom.coe_ringHom_injective\n\ntheorem coe_monoidHom_injective : Function.Injective ((↑) : (A →ₐ[R] B) → A →* B) :=\n RingHom.coe_monoidHom_injective.comp coe_ringHom_injective\n#align alg_hom.coe_monoid_hom_injective AlgHom.coe_monoidHom_injective\n\ntheorem coe_addMonoidHom_injective : Function.Injective ((↑) : (A →ₐ[R] B) → A →+ B) :=\n RingHom.coe_addMonoidHom_injective.comp coe_ringHom_injective\n#align alg_hom.coe_add_monoid_hom_injective AlgHom.coe_addMonoidHom_injective\n\nprotected theorem congr_fun {φ₁ φ₂ : A →ₐ[R] B} (H : φ₁ = φ₂) (x : A) : φ₁ x = φ₂ x :=\n FunLike.congr_fun H x\n#align alg_hom.congr_fun AlgHom.congr_fun\n\nprotected theorem congr_arg (φ : A →ₐ[R] B) {x y : A} (h : x = y) : φ x = φ y :=\n FunLike.congr_arg φ h\n#align alg_hom.congr_arg AlgHom.congr_arg\n\n@[ext]\ntheorem ext {φ₁ φ₂ : A →ₐ[R] B} (H : ∀ x, φ₁ x = φ₂ x) : φ₁ = φ₂ :=\n FunLike.ext _ _ H\n#align alg_hom.ext AlgHom.ext\n\ntheorem ext_iff {φ₁ φ₂ : A →ₐ[R] B} : φ₁ = φ₂ ↔ ∀ x, φ₁ x = φ₂ x :=\n FunLike.ext_iff\n#align alg_hom.ext_iff AlgHom.ext_iff\n\n@[simp]\ntheorem mk_coe {f : A →ₐ[R] B} (h₁ h₂ h₃ h₄ h₅) : (⟨⟨⟨⟨f, h₁⟩, h₂⟩, h₃, h₄⟩, h₅⟩ : A →ₐ[R] B) = f :=\n ext fun _ => rfl\n#align alg_hom.mk_coe AlgHom.mk_coe\n\n@[simp]\ntheorem commutes (r : R) : φ (algebraMap R A r) = algebraMap R B r :=\n φ.commutes' r\n#align alg_hom.commutes AlgHom.commutes\n\ntheorem comp_algebraMap : (φ : A →+* B).comp (algebraMap R A) = algebraMap R B :=\n RingHom.ext <| φ.commutes\n#align alg_hom.comp_algebra_map AlgHom.comp_algebraMap\n\nprotected theorem map_add (r s : A) : φ (r + s) = φ r + φ s :=\n map_add _ _ _\n#align alg_hom.map_add AlgHom.map_add\n\nprotected theorem map_zero : φ 0 = 0 :=\n map_zero _\n#align alg_hom.map_zero AlgHom.map_zero\n\nprotected theorem map_mul (x y) : φ (x * y) = φ x * φ y :=\n map_mul _ _ _\n#align alg_hom.map_mul AlgHom.map_mul\n\nprotected theorem map_one : φ 1 = 1 :=\n map_one _\n#align alg_hom.map_one AlgHom.map_one\n\nprotected theorem map_pow (x : A) (n : ℕ) : φ (x ^ n) = φ x ^ n :=\n map_pow _ _ _\n#align alg_hom.map_pow AlgHom.map_pow\n\n-- @[simp] -- Porting note: simp can prove this\nprotected theorem map_smul (r : R) (x : A) : φ (r • x) = r • φ x :=\n map_smul _ _ _\n#align alg_hom.map_smul AlgHom.map_smul\n\nprotected theorem map_sum {ι : Type*} (f : ι → A) (s : Finset ι) :\n φ (∑ x in s, f x) = ∑ x in s, φ (f x) :=\n map_sum _ _ _\n#align alg_hom.map_sum AlgHom.map_sum\n\nprotected theorem map_finsupp_sum {α : Type*} [Zero α] {ι : Type*} (f : ι →₀ α) (g : ι → α → A) :\n φ (f.sum g) = f.sum fun i a => φ (g i a) :=\n map_finsupp_sum _ _ _\n#align alg_hom.map_finsupp_sum AlgHom.map_finsupp_sum\n\nset_option linter.deprecated false in\nprotected theorem map_bit0 (x) : φ (bit0 x) = bit0 (φ x) :=\n map_bit0 _ _\n#align alg_hom.map_bit0 AlgHom.map_bit0\n\nset_option linter.deprecated false in\nprotected theorem map_bit1 (x) : φ (bit1 x) = bit1 (φ x) :=\n map_bit1 _ _\n#align alg_hom.map_bit1 AlgHom.map_bit1\n\n/-- If a `RingHom` is `R`-linear, then it is an `AlgHom`. -/\ndef mk' (f : A →+* B) (h : ∀ (c : R) (x), f (c • x) = c • f x) : A →ₐ[R] B :=\n { f with\n toFun := f\n commutes' := fun c => by simp only [Algebra.algebraMap_eq_smul_one, h, f.map_one] }\n#align alg_hom.mk' AlgHom.mk'\n\n@[simp]\ntheorem coe_mk' (f : A →+* B) (h : ∀ (c : R) (x), f (c • x) = c • f x) : ⇑(mk' f h) = f :=\n rfl\n#align alg_hom.coe_mk' AlgHom.coe_mk'\n\nsection\n\nvariable (R A)\n\n/-- Identity map as an `AlgHom`. -/\nprotected def id : A →ₐ[R] A :=\n { RingHom.id A with commutes' := fun _ => rfl }\n#align alg_hom.id AlgHom.id\n\n@[simp]\ntheorem coe_id : ⇑(AlgHom.id R A) = id :=\n rfl\n#align alg_hom.coe_id AlgHom.coe_id\n\n@[simp]\ntheorem id_toRingHom : (AlgHom.id R A : A →+* A) = RingHom.id _ :=\n rfl\n#align alg_hom.id_to_ring_hom AlgHom.id_toRingHom\n\nend\n\ntheorem id_apply (p : A) : AlgHom.id R A p = p :=\n rfl\n#align alg_hom.id_apply AlgHom.id_apply\n\n/-- Composition of algebra homeomorphisms. -/\ndef comp (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) : A →ₐ[R] C :=\n { φ₁.toRingHom.comp ↑φ₂ with\n commutes' := fun r : R => by rw [← φ₁.commutes, ← φ₂.commutes]; rfl }\n#align alg_hom.comp AlgHom.comp\n\n@[simp]\ntheorem coe_comp (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) : ⇑(φ₁.comp φ₂) = φ₁ ∘ φ₂ :=\n rfl\n#align alg_hom.coe_comp AlgHom.coe_comp\n\ntheorem comp_apply (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) (p : A) : φ₁.comp φ₂ p = φ₁ (φ₂ p) :=\n rfl\n#align alg_hom.comp_apply AlgHom.comp_apply\n\ntheorem comp_toRingHom (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) :\n (φ₁.comp φ₂ : A →+* C) = (φ₁ : B →+* C).comp ↑φ₂ :=\n rfl\n#align alg_hom.comp_to_ring_hom AlgHom.comp_toRingHom\n\n@[simp]\ntheorem comp_id : φ.comp (AlgHom.id R A) = φ :=\n ext fun _x => rfl\n#align alg_hom.comp_id AlgHom.comp_id\n\n@[simp]\ntheorem id_comp : (AlgHom.id R B).comp φ = φ :=\n ext fun _x => rfl\n#align alg_hom.id_comp AlgHom.id_comp\n\ntheorem comp_assoc (φ₁ : C →ₐ[R] D) (φ₂ : B →ₐ[R] C) (φ₃ : A →ₐ[R] B) :\n (φ₁.comp φ₂).comp φ₃ = φ₁.comp (φ₂.comp φ₃) :=\n ext fun _x => rfl\n#align alg_hom.comp_assoc AlgHom.comp_assoc\n\n/-- R-Alg ⥤ R-Mod -/\ndef toLinearMap : A →ₗ[R] B where\n toFun := φ\n map_add' := map_add _\n map_smul' := map_smul _\n#align alg_hom.to_linear_map AlgHom.toLinearMap\n\n@[simp]\ntheorem toLinearMap_apply (p : A) : φ.toLinearMap p = φ p :=\n rfl\n#align alg_hom.to_linear_map_apply AlgHom.toLinearMap_apply\n\ntheorem toLinearMap_injective :\n Function.Injective (toLinearMap : _ → A →ₗ[R] B) := fun _φ₁ _φ₂ h =>\n ext <| LinearMap.congr_fun h\n#align alg_hom.to_linear_map_injective AlgHom.toLinearMap_injective\n\n@[simp]\ntheorem comp_toLinearMap (f : A →ₐ[R] B) (g : B →ₐ[R] C) :\n (g.comp f).toLinearMap = g.toLinearMap.comp f.toLinearMap :=\n rfl\n#align alg_hom.comp_to_linear_map AlgHom.comp_toLinearMap\n\n@[simp]\ntheorem toLinearMap_id : toLinearMap (AlgHom.id R A) = LinearMap.id :=\n LinearMap.ext fun _ => rfl\n#align alg_hom.to_linear_map_id AlgHom.toLinearMap_id\n\n/-- Promote a `LinearMap` to an `AlgHom` by supplying proofs about the behavior on `1` and `*`. -/\n@[simps]\ndef ofLinearMap (f : A →ₗ[R] B) (map_one : f 1 = 1) (map_mul : ∀ x y, f (x * y) = f x * f y) :\n A →ₐ[R] B :=\n { f.toAddMonoidHom with\n toFun := f\n map_one' := map_one\n map_mul' := map_mul\n commutes' := fun c => by simp only [Algebra.algebraMap_eq_smul_one, f.map_smul, map_one] }\n#align alg_hom.of_linear_map AlgHom.ofLinearMap\n\n@[simp]\ntheorem ofLinearMap_toLinearMap (map_one) (map_mul) :\n ofLinearMap φ.toLinearMap map_one map_mul = φ := by\n ","nextTactic":"ext","declUpToTactic":"@[simp]\ntheorem ofLinearMap_toLinearMap (map_one) (map_mul) :\n ofLinearMap φ.toLinearMap map_one map_mul = φ := by\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Hom.401_0.FxqJeqR3qcIVlrq","decl":"@[simp]\ntheorem ofLinearMap_toLinearMap (map_one) (map_mul) :\n ofLinearMap φ.toLinearMap map_one map_mul = φ "} +{"state":"case H\nR : Type u\nA : Type v\nB : Type w\nC : Type u₁\nD : Type v₁\ninst✝⁸ : CommSemiring R\ninst✝⁷ : Semiring A\ninst✝⁶ : Semiring B\ninst✝⁵ : Semiring C\ninst✝⁴ : Semiring D\ninst✝³ : Algebra R A\ninst✝² : Algebra R B\ninst✝¹ : Algebra R C\ninst✝ : Algebra R D\nφ : A →ₐ[R] B\nmap_one : (toLinearMap φ) 1 = 1\nmap_mul : ∀ (x y : A), (toLinearMap φ) (x * y) = (toLinearMap φ) x * (toLinearMap φ) y\nx✝ : A\n⊢ (ofLinearMap (toLinearMap φ) map_one map_mul) x✝ = φ x✝","srcUpToTactic":"/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Yury Kudryashov\n-/\nimport Mathlib.Algebra.Algebra.Basic\nimport Mathlib.Algebra.BigOperators.Finsupp\n\n#align_import algebra.algebra.hom from \"leanprover-community/mathlib\"@\"e97cf15cd1aec9bd5c193b2ffac5a6dc9118912b\"\n\n/-!\n# Homomorphisms of `R`-algebras\n\nThis file defines bundled homomorphisms of `R`-algebras.\n\n## Main definitions\n\n* `AlgHom R A B`: the type of `R`-algebra morphisms from `A` to `B`.\n* `Algebra.ofId R A : R →ₐ[R] A`: the canonical map from `R` to `A`, as an `AlgHom`.\n\n## Notations\n\n* `A →ₐ[R] B` : `R`-algebra homomorphism from `A` to `B`.\n-/\n\nset_option autoImplicit true\n\n\nopen BigOperators\n\nuniverse u v w u₁ v₁\n\n/-- Defining the homomorphism in the category R-Alg. -/\n-- @[nolint has_nonempty_instance] -- Porting note: This linter does not exist yet.\nstructure AlgHom (R : Type u) (A : Type v) (B : Type w) [CommSemiring R] [Semiring A] [Semiring B]\n [Algebra R A] [Algebra R B] extends RingHom A B where\n commutes' : ∀ r : R, toFun (algebraMap R A r) = algebraMap R B r\n#align alg_hom AlgHom\n\n/-- Reinterpret an `AlgHom` as a `RingHom` -/\nadd_decl_doc AlgHom.toRingHom\n\n@[inherit_doc AlgHom]\ninfixr:25 \" →ₐ \" => AlgHom _\n\n@[inherit_doc]\nnotation:25 A \" →ₐ[\" R \"] \" B => AlgHom R A B\n\n/-- `AlgHomClass F R A B` asserts `F` is a type of bundled algebra homomorphisms\nfrom `A` to `B`. -/\nclass AlgHomClass (F : Type*) (R : outParam (Type*)) (A : outParam (Type*))\n (B : outParam (Type*)) [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A]\n [Algebra R B] extends RingHomClass F A B where\n commutes : ∀ (f : F) (r : R), f (algebraMap R A r) = algebraMap R B r\n#align alg_hom_class AlgHomClass\n\n-- Porting note: `dangerousInstance` linter has become smarter about `outParam`s\n-- attribute [nolint dangerousInstance] AlgHomClass.toRingHomClass\n\n-- Porting note: simp can prove this\n-- attribute [simp] AlgHomClass.commutes\n\nnamespace AlgHomClass\n\nvariable {R : Type*} {A : Type*} {B : Type*} [CommSemiring R] [Semiring A] [Semiring B]\n [Algebra R A] [Algebra R B]\n\n-- see Note [lower instance priority]\ninstance (priority := 100) linearMapClass [AlgHomClass F R A B] : LinearMapClass F R A B :=\n { ‹AlgHomClass F R A B› with\n map_smulₛₗ := fun f r x => by\n simp only [Algebra.smul_def, map_mul, commutes, RingHom.id_apply] }\n#align alg_hom_class.linear_map_class AlgHomClass.linearMapClass\n\n-- Porting note: A new definition underlying a coercion `↑`.\n/-- Turn an element of a type `F` satisfying `AlgHomClass F α β` into an actual\n`AlgHom`. This is declared as the default coercion from `F` to `α →+* β`. -/\n@[coe]\ndef toAlgHom {F : Type*} [AlgHomClass F R A B] (f : F) : A →ₐ[R] B :=\n { (f : A →+* B) with\n toFun := f\n commutes' := AlgHomClass.commutes f }\n\ninstance coeTC {F : Type*} [AlgHomClass F R A B] : CoeTC F (A →ₐ[R] B) :=\n ⟨AlgHomClass.toAlgHom⟩\n#align alg_hom_class.alg_hom.has_coe_t AlgHomClass.coeTC\n\nend AlgHomClass\n\nnamespace AlgHom\n\nvariable {R : Type u} {A : Type v} {B : Type w} {C : Type u₁} {D : Type v₁}\n\nsection Semiring\n\nvariable [CommSemiring R] [Semiring A] [Semiring B] [Semiring C] [Semiring D]\n\nvariable [Algebra R A] [Algebra R B] [Algebra R C] [Algebra R D]\n\n-- Porting note: we don't port specialized `CoeFun` instances if there is `FunLike` instead\n#noalign alg_hom.has_coe_to_fun\n\n-- Porting note: This instance is moved.\ninstance algHomClass : AlgHomClass (A →ₐ[R] B) R A B where\n coe f := f.toFun\n coe_injective' f g h := by\n rcases f with ⟨⟨⟨⟨_, _⟩, _⟩, _, _⟩, _⟩\n rcases g with ⟨⟨⟨⟨_, _⟩, _⟩, _, _⟩, _⟩\n congr\n map_add f := f.map_add'\n map_zero f := f.map_zero'\n map_mul f := f.map_mul'\n map_one f := f.map_one'\n commutes f := f.commutes'\n#align alg_hom.alg_hom_class AlgHom.algHomClass\n\n/-- See Note [custom simps projection] -/\ndef Simps.apply {R : Type u} {α : Type v} {β : Type w} [CommSemiring R]\n [Semiring α] [Semiring β] [Algebra R α] [Algebra R β] (f : α →ₐ[R] β) : α → β := f\n\ninitialize_simps_projections AlgHom (toFun → apply)\n\n@[simp]\nprotected theorem coe_coe {F : Type*} [AlgHomClass F R A B] (f : F) : ⇑(f : A →ₐ[R] B) = f :=\n rfl\n#align alg_hom.coe_coe AlgHom.coe_coe\n\n@[simp]\ntheorem toFun_eq_coe (f : A →ₐ[R] B) : f.toFun = f :=\n rfl\n#align alg_hom.to_fun_eq_coe AlgHom.toFun_eq_coe\n\nattribute [coe] AlgHom.toRingHom\n\ninstance coeOutRingHom : CoeOut (A →ₐ[R] B) (A →+* B) :=\n ⟨AlgHom.toRingHom⟩\n#align alg_hom.coe_ring_hom AlgHom.coeOutRingHom\n\n-- Porting note: A new definition underlying a coercion `↑`.\n@[coe]\ndef toMonoidHom' (f : A →ₐ[R] B) : A →* B := (f : A →+* B)\n\ninstance coeOutMonoidHom : CoeOut (A →ₐ[R] B) (A →* B) :=\n ⟨AlgHom.toMonoidHom'⟩\n#align alg_hom.coe_monoid_hom AlgHom.coeOutMonoidHom\n\n-- Porting note: A new definition underlying a coercion `↑`.\n@[coe]\ndef toAddMonoidHom' (f : A →ₐ[R] B) : A →+ B := (f : A →+* B)\n\ninstance coeOutAddMonoidHom : CoeOut (A →ₐ[R] B) (A →+ B) :=\n ⟨AlgHom.toAddMonoidHom'⟩\n#align alg_hom.coe_add_monoid_hom AlgHom.coeOutAddMonoidHom\n\n-- Porting note: Lean 3: `@[simp, norm_cast] coe_mk`\n-- Lean 4: `@[simp] coe_mk` & `@[norm_cast] coe_mks`\n@[simp]\ntheorem coe_mk {f : A →+* B} (h) : ((⟨f, h⟩ : A →ₐ[R] B) : A → B) = f :=\n rfl\n\n@[norm_cast]\ntheorem coe_mks {f : A → B} (h₁ h₂ h₃ h₄ h₅) : ⇑(⟨⟨⟨⟨f, h₁⟩, h₂⟩, h₃, h₄⟩, h₅⟩ : A →ₐ[R] B) = f :=\n rfl\n#align alg_hom.coe_mk AlgHom.coe_mks\n\n-- Porting note: This theorem is new.\n@[simp, norm_cast]\ntheorem coe_ringHom_mk {f : A →+* B} (h) : ((⟨f, h⟩ : A →ₐ[R] B) : A →+* B) = f :=\n rfl\n\n-- make the coercion the simp-normal form\n@[simp]\ntheorem toRingHom_eq_coe (f : A →ₐ[R] B) : f.toRingHom = f :=\n rfl\n#align alg_hom.to_ring_hom_eq_coe AlgHom.toRingHom_eq_coe\n\n@[simp, norm_cast]\ntheorem coe_toRingHom (f : A →ₐ[R] B) : ⇑(f : A →+* B) = f :=\n rfl\n#align alg_hom.coe_to_ring_hom AlgHom.coe_toRingHom\n\n@[simp, norm_cast]\ntheorem coe_toMonoidHom (f : A →ₐ[R] B) : ⇑(f : A →* B) = f :=\n rfl\n#align alg_hom.coe_to_monoid_hom AlgHom.coe_toMonoidHom\n\n@[simp, norm_cast]\ntheorem coe_toAddMonoidHom (f : A →ₐ[R] B) : ⇑(f : A →+ B) = f :=\n rfl\n#align alg_hom.coe_to_add_monoid_hom AlgHom.coe_toAddMonoidHom\n\nvariable (φ : A →ₐ[R] B)\n\ntheorem coe_fn_injective : @Function.Injective (A →ₐ[R] B) (A → B) (↑) :=\n FunLike.coe_injective\n#align alg_hom.coe_fn_injective AlgHom.coe_fn_injective\n\ntheorem coe_fn_inj {φ₁ φ₂ : A →ₐ[R] B} : (φ₁ : A → B) = φ₂ ↔ φ₁ = φ₂ :=\n FunLike.coe_fn_eq\n#align alg_hom.coe_fn_inj AlgHom.coe_fn_inj\n\ntheorem coe_ringHom_injective : Function.Injective ((↑) : (A →ₐ[R] B) → A →+* B) := fun φ₁ φ₂ H =>\n coe_fn_injective <| show ((φ₁ : A →+* B) : A → B) = ((φ₂ : A →+* B) : A → B) from congr_arg _ H\n#align alg_hom.coe_ring_hom_injective AlgHom.coe_ringHom_injective\n\ntheorem coe_monoidHom_injective : Function.Injective ((↑) : (A →ₐ[R] B) → A →* B) :=\n RingHom.coe_monoidHom_injective.comp coe_ringHom_injective\n#align alg_hom.coe_monoid_hom_injective AlgHom.coe_monoidHom_injective\n\ntheorem coe_addMonoidHom_injective : Function.Injective ((↑) : (A →ₐ[R] B) → A →+ B) :=\n RingHom.coe_addMonoidHom_injective.comp coe_ringHom_injective\n#align alg_hom.coe_add_monoid_hom_injective AlgHom.coe_addMonoidHom_injective\n\nprotected theorem congr_fun {φ₁ φ₂ : A →ₐ[R] B} (H : φ₁ = φ₂) (x : A) : φ₁ x = φ₂ x :=\n FunLike.congr_fun H x\n#align alg_hom.congr_fun AlgHom.congr_fun\n\nprotected theorem congr_arg (φ : A →ₐ[R] B) {x y : A} (h : x = y) : φ x = φ y :=\n FunLike.congr_arg φ h\n#align alg_hom.congr_arg AlgHom.congr_arg\n\n@[ext]\ntheorem ext {φ₁ φ₂ : A →ₐ[R] B} (H : ∀ x, φ₁ x = φ₂ x) : φ₁ = φ₂ :=\n FunLike.ext _ _ H\n#align alg_hom.ext AlgHom.ext\n\ntheorem ext_iff {φ₁ φ₂ : A →ₐ[R] B} : φ₁ = φ₂ ↔ ∀ x, φ₁ x = φ₂ x :=\n FunLike.ext_iff\n#align alg_hom.ext_iff AlgHom.ext_iff\n\n@[simp]\ntheorem mk_coe {f : A →ₐ[R] B} (h₁ h₂ h₃ h₄ h₅) : (⟨⟨⟨⟨f, h₁⟩, h₂⟩, h₃, h₄⟩, h₅⟩ : A →ₐ[R] B) = f :=\n ext fun _ => rfl\n#align alg_hom.mk_coe AlgHom.mk_coe\n\n@[simp]\ntheorem commutes (r : R) : φ (algebraMap R A r) = algebraMap R B r :=\n φ.commutes' r\n#align alg_hom.commutes AlgHom.commutes\n\ntheorem comp_algebraMap : (φ : A →+* B).comp (algebraMap R A) = algebraMap R B :=\n RingHom.ext <| φ.commutes\n#align alg_hom.comp_algebra_map AlgHom.comp_algebraMap\n\nprotected theorem map_add (r s : A) : φ (r + s) = φ r + φ s :=\n map_add _ _ _\n#align alg_hom.map_add AlgHom.map_add\n\nprotected theorem map_zero : φ 0 = 0 :=\n map_zero _\n#align alg_hom.map_zero AlgHom.map_zero\n\nprotected theorem map_mul (x y) : φ (x * y) = φ x * φ y :=\n map_mul _ _ _\n#align alg_hom.map_mul AlgHom.map_mul\n\nprotected theorem map_one : φ 1 = 1 :=\n map_one _\n#align alg_hom.map_one AlgHom.map_one\n\nprotected theorem map_pow (x : A) (n : ℕ) : φ (x ^ n) = φ x ^ n :=\n map_pow _ _ _\n#align alg_hom.map_pow AlgHom.map_pow\n\n-- @[simp] -- Porting note: simp can prove this\nprotected theorem map_smul (r : R) (x : A) : φ (r • x) = r • φ x :=\n map_smul _ _ _\n#align alg_hom.map_smul AlgHom.map_smul\n\nprotected theorem map_sum {ι : Type*} (f : ι → A) (s : Finset ι) :\n φ (∑ x in s, f x) = ∑ x in s, φ (f x) :=\n map_sum _ _ _\n#align alg_hom.map_sum AlgHom.map_sum\n\nprotected theorem map_finsupp_sum {α : Type*} [Zero α] {ι : Type*} (f : ι →₀ α) (g : ι → α → A) :\n φ (f.sum g) = f.sum fun i a => φ (g i a) :=\n map_finsupp_sum _ _ _\n#align alg_hom.map_finsupp_sum AlgHom.map_finsupp_sum\n\nset_option linter.deprecated false in\nprotected theorem map_bit0 (x) : φ (bit0 x) = bit0 (φ x) :=\n map_bit0 _ _\n#align alg_hom.map_bit0 AlgHom.map_bit0\n\nset_option linter.deprecated false in\nprotected theorem map_bit1 (x) : φ (bit1 x) = bit1 (φ x) :=\n map_bit1 _ _\n#align alg_hom.map_bit1 AlgHom.map_bit1\n\n/-- If a `RingHom` is `R`-linear, then it is an `AlgHom`. -/\ndef mk' (f : A →+* B) (h : ∀ (c : R) (x), f (c • x) = c • f x) : A →ₐ[R] B :=\n { f with\n toFun := f\n commutes' := fun c => by simp only [Algebra.algebraMap_eq_smul_one, h, f.map_one] }\n#align alg_hom.mk' AlgHom.mk'\n\n@[simp]\ntheorem coe_mk' (f : A →+* B) (h : ∀ (c : R) (x), f (c • x) = c • f x) : ⇑(mk' f h) = f :=\n rfl\n#align alg_hom.coe_mk' AlgHom.coe_mk'\n\nsection\n\nvariable (R A)\n\n/-- Identity map as an `AlgHom`. -/\nprotected def id : A →ₐ[R] A :=\n { RingHom.id A with commutes' := fun _ => rfl }\n#align alg_hom.id AlgHom.id\n\n@[simp]\ntheorem coe_id : ⇑(AlgHom.id R A) = id :=\n rfl\n#align alg_hom.coe_id AlgHom.coe_id\n\n@[simp]\ntheorem id_toRingHom : (AlgHom.id R A : A →+* A) = RingHom.id _ :=\n rfl\n#align alg_hom.id_to_ring_hom AlgHom.id_toRingHom\n\nend\n\ntheorem id_apply (p : A) : AlgHom.id R A p = p :=\n rfl\n#align alg_hom.id_apply AlgHom.id_apply\n\n/-- Composition of algebra homeomorphisms. -/\ndef comp (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) : A →ₐ[R] C :=\n { φ₁.toRingHom.comp ↑φ₂ with\n commutes' := fun r : R => by rw [← φ₁.commutes, ← φ₂.commutes]; rfl }\n#align alg_hom.comp AlgHom.comp\n\n@[simp]\ntheorem coe_comp (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) : ⇑(φ₁.comp φ₂) = φ₁ ∘ φ₂ :=\n rfl\n#align alg_hom.coe_comp AlgHom.coe_comp\n\ntheorem comp_apply (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) (p : A) : φ₁.comp φ₂ p = φ₁ (φ₂ p) :=\n rfl\n#align alg_hom.comp_apply AlgHom.comp_apply\n\ntheorem comp_toRingHom (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) :\n (φ₁.comp φ₂ : A →+* C) = (φ₁ : B →+* C).comp ↑φ₂ :=\n rfl\n#align alg_hom.comp_to_ring_hom AlgHom.comp_toRingHom\n\n@[simp]\ntheorem comp_id : φ.comp (AlgHom.id R A) = φ :=\n ext fun _x => rfl\n#align alg_hom.comp_id AlgHom.comp_id\n\n@[simp]\ntheorem id_comp : (AlgHom.id R B).comp φ = φ :=\n ext fun _x => rfl\n#align alg_hom.id_comp AlgHom.id_comp\n\ntheorem comp_assoc (φ₁ : C →ₐ[R] D) (φ₂ : B →ₐ[R] C) (φ₃ : A →ₐ[R] B) :\n (φ₁.comp φ₂).comp φ₃ = φ₁.comp (φ₂.comp φ₃) :=\n ext fun _x => rfl\n#align alg_hom.comp_assoc AlgHom.comp_assoc\n\n/-- R-Alg ⥤ R-Mod -/\ndef toLinearMap : A →ₗ[R] B where\n toFun := φ\n map_add' := map_add _\n map_smul' := map_smul _\n#align alg_hom.to_linear_map AlgHom.toLinearMap\n\n@[simp]\ntheorem toLinearMap_apply (p : A) : φ.toLinearMap p = φ p :=\n rfl\n#align alg_hom.to_linear_map_apply AlgHom.toLinearMap_apply\n\ntheorem toLinearMap_injective :\n Function.Injective (toLinearMap : _ → A →ₗ[R] B) := fun _φ₁ _φ₂ h =>\n ext <| LinearMap.congr_fun h\n#align alg_hom.to_linear_map_injective AlgHom.toLinearMap_injective\n\n@[simp]\ntheorem comp_toLinearMap (f : A →ₐ[R] B) (g : B →ₐ[R] C) :\n (g.comp f).toLinearMap = g.toLinearMap.comp f.toLinearMap :=\n rfl\n#align alg_hom.comp_to_linear_map AlgHom.comp_toLinearMap\n\n@[simp]\ntheorem toLinearMap_id : toLinearMap (AlgHom.id R A) = LinearMap.id :=\n LinearMap.ext fun _ => rfl\n#align alg_hom.to_linear_map_id AlgHom.toLinearMap_id\n\n/-- Promote a `LinearMap` to an `AlgHom` by supplying proofs about the behavior on `1` and `*`. -/\n@[simps]\ndef ofLinearMap (f : A →ₗ[R] B) (map_one : f 1 = 1) (map_mul : ∀ x y, f (x * y) = f x * f y) :\n A →ₐ[R] B :=\n { f.toAddMonoidHom with\n toFun := f\n map_one' := map_one\n map_mul' := map_mul\n commutes' := fun c => by simp only [Algebra.algebraMap_eq_smul_one, f.map_smul, map_one] }\n#align alg_hom.of_linear_map AlgHom.ofLinearMap\n\n@[simp]\ntheorem ofLinearMap_toLinearMap (map_one) (map_mul) :\n ofLinearMap φ.toLinearMap map_one map_mul = φ := by\n ext\n ","nextTactic":"rfl","declUpToTactic":"@[simp]\ntheorem ofLinearMap_toLinearMap (map_one) (map_mul) :\n ofLinearMap φ.toLinearMap map_one map_mul = φ := by\n ext\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Hom.401_0.FxqJeqR3qcIVlrq","decl":"@[simp]\ntheorem ofLinearMap_toLinearMap (map_one) (map_mul) :\n ofLinearMap φ.toLinearMap map_one map_mul = φ "} +{"state":"R : Type u\nA : Type v\nB : Type w\nC : Type u₁\nD : Type v₁\ninst✝⁸ : CommSemiring R\ninst✝⁷ : Semiring A\ninst✝⁶ : Semiring B\ninst✝⁵ : Semiring C\ninst✝⁴ : Semiring D\ninst✝³ : Algebra R A\ninst✝² : Algebra R B\ninst✝¹ : Algebra R C\ninst✝ : Algebra R D\nφ : A →ₐ[R] B\nf : A →ₗ[R] B\nmap_one : f 1 = 1\nmap_mul : ∀ (x y : A), f (x * y) = f x * f y\n⊢ toLinearMap (ofLinearMap f map_one map_mul) = f","srcUpToTactic":"/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Yury Kudryashov\n-/\nimport Mathlib.Algebra.Algebra.Basic\nimport Mathlib.Algebra.BigOperators.Finsupp\n\n#align_import algebra.algebra.hom from \"leanprover-community/mathlib\"@\"e97cf15cd1aec9bd5c193b2ffac5a6dc9118912b\"\n\n/-!\n# Homomorphisms of `R`-algebras\n\nThis file defines bundled homomorphisms of `R`-algebras.\n\n## Main definitions\n\n* `AlgHom R A B`: the type of `R`-algebra morphisms from `A` to `B`.\n* `Algebra.ofId R A : R →ₐ[R] A`: the canonical map from `R` to `A`, as an `AlgHom`.\n\n## Notations\n\n* `A →ₐ[R] B` : `R`-algebra homomorphism from `A` to `B`.\n-/\n\nset_option autoImplicit true\n\n\nopen BigOperators\n\nuniverse u v w u₁ v₁\n\n/-- Defining the homomorphism in the category R-Alg. -/\n-- @[nolint has_nonempty_instance] -- Porting note: This linter does not exist yet.\nstructure AlgHom (R : Type u) (A : Type v) (B : Type w) [CommSemiring R] [Semiring A] [Semiring B]\n [Algebra R A] [Algebra R B] extends RingHom A B where\n commutes' : ∀ r : R, toFun (algebraMap R A r) = algebraMap R B r\n#align alg_hom AlgHom\n\n/-- Reinterpret an `AlgHom` as a `RingHom` -/\nadd_decl_doc AlgHom.toRingHom\n\n@[inherit_doc AlgHom]\ninfixr:25 \" →ₐ \" => AlgHom _\n\n@[inherit_doc]\nnotation:25 A \" →ₐ[\" R \"] \" B => AlgHom R A B\n\n/-- `AlgHomClass F R A B` asserts `F` is a type of bundled algebra homomorphisms\nfrom `A` to `B`. -/\nclass AlgHomClass (F : Type*) (R : outParam (Type*)) (A : outParam (Type*))\n (B : outParam (Type*)) [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A]\n [Algebra R B] extends RingHomClass F A B where\n commutes : ∀ (f : F) (r : R), f (algebraMap R A r) = algebraMap R B r\n#align alg_hom_class AlgHomClass\n\n-- Porting note: `dangerousInstance` linter has become smarter about `outParam`s\n-- attribute [nolint dangerousInstance] AlgHomClass.toRingHomClass\n\n-- Porting note: simp can prove this\n-- attribute [simp] AlgHomClass.commutes\n\nnamespace AlgHomClass\n\nvariable {R : Type*} {A : Type*} {B : Type*} [CommSemiring R] [Semiring A] [Semiring B]\n [Algebra R A] [Algebra R B]\n\n-- see Note [lower instance priority]\ninstance (priority := 100) linearMapClass [AlgHomClass F R A B] : LinearMapClass F R A B :=\n { ‹AlgHomClass F R A B› with\n map_smulₛₗ := fun f r x => by\n simp only [Algebra.smul_def, map_mul, commutes, RingHom.id_apply] }\n#align alg_hom_class.linear_map_class AlgHomClass.linearMapClass\n\n-- Porting note: A new definition underlying a coercion `↑`.\n/-- Turn an element of a type `F` satisfying `AlgHomClass F α β` into an actual\n`AlgHom`. This is declared as the default coercion from `F` to `α →+* β`. -/\n@[coe]\ndef toAlgHom {F : Type*} [AlgHomClass F R A B] (f : F) : A →ₐ[R] B :=\n { (f : A →+* B) with\n toFun := f\n commutes' := AlgHomClass.commutes f }\n\ninstance coeTC {F : Type*} [AlgHomClass F R A B] : CoeTC F (A →ₐ[R] B) :=\n ⟨AlgHomClass.toAlgHom⟩\n#align alg_hom_class.alg_hom.has_coe_t AlgHomClass.coeTC\n\nend AlgHomClass\n\nnamespace AlgHom\n\nvariable {R : Type u} {A : Type v} {B : Type w} {C : Type u₁} {D : Type v₁}\n\nsection Semiring\n\nvariable [CommSemiring R] [Semiring A] [Semiring B] [Semiring C] [Semiring D]\n\nvariable [Algebra R A] [Algebra R B] [Algebra R C] [Algebra R D]\n\n-- Porting note: we don't port specialized `CoeFun` instances if there is `FunLike` instead\n#noalign alg_hom.has_coe_to_fun\n\n-- Porting note: This instance is moved.\ninstance algHomClass : AlgHomClass (A →ₐ[R] B) R A B where\n coe f := f.toFun\n coe_injective' f g h := by\n rcases f with ⟨⟨⟨⟨_, _⟩, _⟩, _, _⟩, _⟩\n rcases g with ⟨⟨⟨⟨_, _⟩, _⟩, _, _⟩, _⟩\n congr\n map_add f := f.map_add'\n map_zero f := f.map_zero'\n map_mul f := f.map_mul'\n map_one f := f.map_one'\n commutes f := f.commutes'\n#align alg_hom.alg_hom_class AlgHom.algHomClass\n\n/-- See Note [custom simps projection] -/\ndef Simps.apply {R : Type u} {α : Type v} {β : Type w} [CommSemiring R]\n [Semiring α] [Semiring β] [Algebra R α] [Algebra R β] (f : α →ₐ[R] β) : α → β := f\n\ninitialize_simps_projections AlgHom (toFun → apply)\n\n@[simp]\nprotected theorem coe_coe {F : Type*} [AlgHomClass F R A B] (f : F) : ⇑(f : A →ₐ[R] B) = f :=\n rfl\n#align alg_hom.coe_coe AlgHom.coe_coe\n\n@[simp]\ntheorem toFun_eq_coe (f : A →ₐ[R] B) : f.toFun = f :=\n rfl\n#align alg_hom.to_fun_eq_coe AlgHom.toFun_eq_coe\n\nattribute [coe] AlgHom.toRingHom\n\ninstance coeOutRingHom : CoeOut (A →ₐ[R] B) (A →+* B) :=\n ⟨AlgHom.toRingHom⟩\n#align alg_hom.coe_ring_hom AlgHom.coeOutRingHom\n\n-- Porting note: A new definition underlying a coercion `↑`.\n@[coe]\ndef toMonoidHom' (f : A →ₐ[R] B) : A →* B := (f : A →+* B)\n\ninstance coeOutMonoidHom : CoeOut (A →ₐ[R] B) (A →* B) :=\n ⟨AlgHom.toMonoidHom'⟩\n#align alg_hom.coe_monoid_hom AlgHom.coeOutMonoidHom\n\n-- Porting note: A new definition underlying a coercion `↑`.\n@[coe]\ndef toAddMonoidHom' (f : A →ₐ[R] B) : A →+ B := (f : A →+* B)\n\ninstance coeOutAddMonoidHom : CoeOut (A →ₐ[R] B) (A →+ B) :=\n ⟨AlgHom.toAddMonoidHom'⟩\n#align alg_hom.coe_add_monoid_hom AlgHom.coeOutAddMonoidHom\n\n-- Porting note: Lean 3: `@[simp, norm_cast] coe_mk`\n-- Lean 4: `@[simp] coe_mk` & `@[norm_cast] coe_mks`\n@[simp]\ntheorem coe_mk {f : A →+* B} (h) : ((⟨f, h⟩ : A →ₐ[R] B) : A → B) = f :=\n rfl\n\n@[norm_cast]\ntheorem coe_mks {f : A → B} (h₁ h₂ h₃ h₄ h₅) : ⇑(⟨⟨⟨⟨f, h₁⟩, h₂⟩, h₃, h₄⟩, h₅⟩ : A →ₐ[R] B) = f :=\n rfl\n#align alg_hom.coe_mk AlgHom.coe_mks\n\n-- Porting note: This theorem is new.\n@[simp, norm_cast]\ntheorem coe_ringHom_mk {f : A →+* B} (h) : ((⟨f, h⟩ : A →ₐ[R] B) : A →+* B) = f :=\n rfl\n\n-- make the coercion the simp-normal form\n@[simp]\ntheorem toRingHom_eq_coe (f : A →ₐ[R] B) : f.toRingHom = f :=\n rfl\n#align alg_hom.to_ring_hom_eq_coe AlgHom.toRingHom_eq_coe\n\n@[simp, norm_cast]\ntheorem coe_toRingHom (f : A →ₐ[R] B) : ⇑(f : A →+* B) = f :=\n rfl\n#align alg_hom.coe_to_ring_hom AlgHom.coe_toRingHom\n\n@[simp, norm_cast]\ntheorem coe_toMonoidHom (f : A →ₐ[R] B) : ⇑(f : A →* B) = f :=\n rfl\n#align alg_hom.coe_to_monoid_hom AlgHom.coe_toMonoidHom\n\n@[simp, norm_cast]\ntheorem coe_toAddMonoidHom (f : A →ₐ[R] B) : ⇑(f : A →+ B) = f :=\n rfl\n#align alg_hom.coe_to_add_monoid_hom AlgHom.coe_toAddMonoidHom\n\nvariable (φ : A →ₐ[R] B)\n\ntheorem coe_fn_injective : @Function.Injective (A →ₐ[R] B) (A → B) (↑) :=\n FunLike.coe_injective\n#align alg_hom.coe_fn_injective AlgHom.coe_fn_injective\n\ntheorem coe_fn_inj {φ₁ φ₂ : A →ₐ[R] B} : (φ₁ : A → B) = φ₂ ↔ φ₁ = φ₂ :=\n FunLike.coe_fn_eq\n#align alg_hom.coe_fn_inj AlgHom.coe_fn_inj\n\ntheorem coe_ringHom_injective : Function.Injective ((↑) : (A →ₐ[R] B) → A →+* B) := fun φ₁ φ₂ H =>\n coe_fn_injective <| show ((φ₁ : A →+* B) : A → B) = ((φ₂ : A →+* B) : A → B) from congr_arg _ H\n#align alg_hom.coe_ring_hom_injective AlgHom.coe_ringHom_injective\n\ntheorem coe_monoidHom_injective : Function.Injective ((↑) : (A →ₐ[R] B) → A →* B) :=\n RingHom.coe_monoidHom_injective.comp coe_ringHom_injective\n#align alg_hom.coe_monoid_hom_injective AlgHom.coe_monoidHom_injective\n\ntheorem coe_addMonoidHom_injective : Function.Injective ((↑) : (A →ₐ[R] B) → A →+ B) :=\n RingHom.coe_addMonoidHom_injective.comp coe_ringHom_injective\n#align alg_hom.coe_add_monoid_hom_injective AlgHom.coe_addMonoidHom_injective\n\nprotected theorem congr_fun {φ₁ φ₂ : A →ₐ[R] B} (H : φ₁ = φ₂) (x : A) : φ₁ x = φ₂ x :=\n FunLike.congr_fun H x\n#align alg_hom.congr_fun AlgHom.congr_fun\n\nprotected theorem congr_arg (φ : A →ₐ[R] B) {x y : A} (h : x = y) : φ x = φ y :=\n FunLike.congr_arg φ h\n#align alg_hom.congr_arg AlgHom.congr_arg\n\n@[ext]\ntheorem ext {φ₁ φ₂ : A →ₐ[R] B} (H : ∀ x, φ₁ x = φ₂ x) : φ₁ = φ₂ :=\n FunLike.ext _ _ H\n#align alg_hom.ext AlgHom.ext\n\ntheorem ext_iff {φ₁ φ₂ : A →ₐ[R] B} : φ₁ = φ₂ ↔ ∀ x, φ₁ x = φ₂ x :=\n FunLike.ext_iff\n#align alg_hom.ext_iff AlgHom.ext_iff\n\n@[simp]\ntheorem mk_coe {f : A →ₐ[R] B} (h₁ h₂ h₃ h₄ h₅) : (⟨⟨⟨⟨f, h₁⟩, h₂⟩, h₃, h₄⟩, h₅⟩ : A →ₐ[R] B) = f :=\n ext fun _ => rfl\n#align alg_hom.mk_coe AlgHom.mk_coe\n\n@[simp]\ntheorem commutes (r : R) : φ (algebraMap R A r) = algebraMap R B r :=\n φ.commutes' r\n#align alg_hom.commutes AlgHom.commutes\n\ntheorem comp_algebraMap : (φ : A →+* B).comp (algebraMap R A) = algebraMap R B :=\n RingHom.ext <| φ.commutes\n#align alg_hom.comp_algebra_map AlgHom.comp_algebraMap\n\nprotected theorem map_add (r s : A) : φ (r + s) = φ r + φ s :=\n map_add _ _ _\n#align alg_hom.map_add AlgHom.map_add\n\nprotected theorem map_zero : φ 0 = 0 :=\n map_zero _\n#align alg_hom.map_zero AlgHom.map_zero\n\nprotected theorem map_mul (x y) : φ (x * y) = φ x * φ y :=\n map_mul _ _ _\n#align alg_hom.map_mul AlgHom.map_mul\n\nprotected theorem map_one : φ 1 = 1 :=\n map_one _\n#align alg_hom.map_one AlgHom.map_one\n\nprotected theorem map_pow (x : A) (n : ℕ) : φ (x ^ n) = φ x ^ n :=\n map_pow _ _ _\n#align alg_hom.map_pow AlgHom.map_pow\n\n-- @[simp] -- Porting note: simp can prove this\nprotected theorem map_smul (r : R) (x : A) : φ (r • x) = r • φ x :=\n map_smul _ _ _\n#align alg_hom.map_smul AlgHom.map_smul\n\nprotected theorem map_sum {ι : Type*} (f : ι → A) (s : Finset ι) :\n φ (∑ x in s, f x) = ∑ x in s, φ (f x) :=\n map_sum _ _ _\n#align alg_hom.map_sum AlgHom.map_sum\n\nprotected theorem map_finsupp_sum {α : Type*} [Zero α] {ι : Type*} (f : ι →₀ α) (g : ι → α → A) :\n φ (f.sum g) = f.sum fun i a => φ (g i a) :=\n map_finsupp_sum _ _ _\n#align alg_hom.map_finsupp_sum AlgHom.map_finsupp_sum\n\nset_option linter.deprecated false in\nprotected theorem map_bit0 (x) : φ (bit0 x) = bit0 (φ x) :=\n map_bit0 _ _\n#align alg_hom.map_bit0 AlgHom.map_bit0\n\nset_option linter.deprecated false in\nprotected theorem map_bit1 (x) : φ (bit1 x) = bit1 (φ x) :=\n map_bit1 _ _\n#align alg_hom.map_bit1 AlgHom.map_bit1\n\n/-- If a `RingHom` is `R`-linear, then it is an `AlgHom`. -/\ndef mk' (f : A →+* B) (h : ∀ (c : R) (x), f (c • x) = c • f x) : A →ₐ[R] B :=\n { f with\n toFun := f\n commutes' := fun c => by simp only [Algebra.algebraMap_eq_smul_one, h, f.map_one] }\n#align alg_hom.mk' AlgHom.mk'\n\n@[simp]\ntheorem coe_mk' (f : A →+* B) (h : ∀ (c : R) (x), f (c • x) = c • f x) : ⇑(mk' f h) = f :=\n rfl\n#align alg_hom.coe_mk' AlgHom.coe_mk'\n\nsection\n\nvariable (R A)\n\n/-- Identity map as an `AlgHom`. -/\nprotected def id : A →ₐ[R] A :=\n { RingHom.id A with commutes' := fun _ => rfl }\n#align alg_hom.id AlgHom.id\n\n@[simp]\ntheorem coe_id : ⇑(AlgHom.id R A) = id :=\n rfl\n#align alg_hom.coe_id AlgHom.coe_id\n\n@[simp]\ntheorem id_toRingHom : (AlgHom.id R A : A →+* A) = RingHom.id _ :=\n rfl\n#align alg_hom.id_to_ring_hom AlgHom.id_toRingHom\n\nend\n\ntheorem id_apply (p : A) : AlgHom.id R A p = p :=\n rfl\n#align alg_hom.id_apply AlgHom.id_apply\n\n/-- Composition of algebra homeomorphisms. -/\ndef comp (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) : A →ₐ[R] C :=\n { φ₁.toRingHom.comp ↑φ₂ with\n commutes' := fun r : R => by rw [← φ₁.commutes, ← φ₂.commutes]; rfl }\n#align alg_hom.comp AlgHom.comp\n\n@[simp]\ntheorem coe_comp (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) : ⇑(φ₁.comp φ₂) = φ₁ ∘ φ₂ :=\n rfl\n#align alg_hom.coe_comp AlgHom.coe_comp\n\ntheorem comp_apply (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) (p : A) : φ₁.comp φ₂ p = φ₁ (φ₂ p) :=\n rfl\n#align alg_hom.comp_apply AlgHom.comp_apply\n\ntheorem comp_toRingHom (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) :\n (φ₁.comp φ₂ : A →+* C) = (φ₁ : B →+* C).comp ↑φ₂ :=\n rfl\n#align alg_hom.comp_to_ring_hom AlgHom.comp_toRingHom\n\n@[simp]\ntheorem comp_id : φ.comp (AlgHom.id R A) = φ :=\n ext fun _x => rfl\n#align alg_hom.comp_id AlgHom.comp_id\n\n@[simp]\ntheorem id_comp : (AlgHom.id R B).comp φ = φ :=\n ext fun _x => rfl\n#align alg_hom.id_comp AlgHom.id_comp\n\ntheorem comp_assoc (φ₁ : C →ₐ[R] D) (φ₂ : B →ₐ[R] C) (φ₃ : A →ₐ[R] B) :\n (φ₁.comp φ₂).comp φ₃ = φ₁.comp (φ₂.comp φ₃) :=\n ext fun _x => rfl\n#align alg_hom.comp_assoc AlgHom.comp_assoc\n\n/-- R-Alg ⥤ R-Mod -/\ndef toLinearMap : A →ₗ[R] B where\n toFun := φ\n map_add' := map_add _\n map_smul' := map_smul _\n#align alg_hom.to_linear_map AlgHom.toLinearMap\n\n@[simp]\ntheorem toLinearMap_apply (p : A) : φ.toLinearMap p = φ p :=\n rfl\n#align alg_hom.to_linear_map_apply AlgHom.toLinearMap_apply\n\ntheorem toLinearMap_injective :\n Function.Injective (toLinearMap : _ → A →ₗ[R] B) := fun _φ₁ _φ₂ h =>\n ext <| LinearMap.congr_fun h\n#align alg_hom.to_linear_map_injective AlgHom.toLinearMap_injective\n\n@[simp]\ntheorem comp_toLinearMap (f : A →ₐ[R] B) (g : B →ₐ[R] C) :\n (g.comp f).toLinearMap = g.toLinearMap.comp f.toLinearMap :=\n rfl\n#align alg_hom.comp_to_linear_map AlgHom.comp_toLinearMap\n\n@[simp]\ntheorem toLinearMap_id : toLinearMap (AlgHom.id R A) = LinearMap.id :=\n LinearMap.ext fun _ => rfl\n#align alg_hom.to_linear_map_id AlgHom.toLinearMap_id\n\n/-- Promote a `LinearMap` to an `AlgHom` by supplying proofs about the behavior on `1` and `*`. -/\n@[simps]\ndef ofLinearMap (f : A →ₗ[R] B) (map_one : f 1 = 1) (map_mul : ∀ x y, f (x * y) = f x * f y) :\n A →ₐ[R] B :=\n { f.toAddMonoidHom with\n toFun := f\n map_one' := map_one\n map_mul' := map_mul\n commutes' := fun c => by simp only [Algebra.algebraMap_eq_smul_one, f.map_smul, map_one] }\n#align alg_hom.of_linear_map AlgHom.ofLinearMap\n\n@[simp]\ntheorem ofLinearMap_toLinearMap (map_one) (map_mul) :\n ofLinearMap φ.toLinearMap map_one map_mul = φ := by\n ext\n rfl\n#align alg_hom.of_linear_map_to_linear_map AlgHom.ofLinearMap_toLinearMap\n\n@[simp]\ntheorem toLinearMap_ofLinearMap (f : A →ₗ[R] B) (map_one) (map_mul) :\n toLinearMap (ofLinearMap f map_one map_mul) = f := by\n ","nextTactic":"ext","declUpToTactic":"@[simp]\ntheorem toLinearMap_ofLinearMap (f : A →ₗ[R] B) (map_one) (map_mul) :\n toLinearMap (ofLinearMap f map_one map_mul) = f := by\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Hom.408_0.FxqJeqR3qcIVlrq","decl":"@[simp]\ntheorem toLinearMap_ofLinearMap (f : A →ₗ[R] B) (map_one) (map_mul) :\n toLinearMap (ofLinearMap f map_one map_mul) = f "} +{"state":"case h\nR : Type u\nA : Type v\nB : Type w\nC : Type u₁\nD : Type v₁\ninst✝⁸ : CommSemiring R\ninst✝⁷ : Semiring A\ninst✝⁶ : Semiring B\ninst✝⁵ : Semiring C\ninst✝⁴ : Semiring D\ninst✝³ : Algebra R A\ninst✝² : Algebra R B\ninst✝¹ : Algebra R C\ninst✝ : Algebra R D\nφ : A →ₐ[R] B\nf : A →ₗ[R] B\nmap_one : f 1 = 1\nmap_mul : ∀ (x y : A), f (x * y) = f x * f y\nx✝ : A\n⊢ (toLinearMap (ofLinearMap f map_one map_mul)) x✝ = f x✝","srcUpToTactic":"/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Yury Kudryashov\n-/\nimport Mathlib.Algebra.Algebra.Basic\nimport Mathlib.Algebra.BigOperators.Finsupp\n\n#align_import algebra.algebra.hom from \"leanprover-community/mathlib\"@\"e97cf15cd1aec9bd5c193b2ffac5a6dc9118912b\"\n\n/-!\n# Homomorphisms of `R`-algebras\n\nThis file defines bundled homomorphisms of `R`-algebras.\n\n## Main definitions\n\n* `AlgHom R A B`: the type of `R`-algebra morphisms from `A` to `B`.\n* `Algebra.ofId R A : R →ₐ[R] A`: the canonical map from `R` to `A`, as an `AlgHom`.\n\n## Notations\n\n* `A →ₐ[R] B` : `R`-algebra homomorphism from `A` to `B`.\n-/\n\nset_option autoImplicit true\n\n\nopen BigOperators\n\nuniverse u v w u₁ v₁\n\n/-- Defining the homomorphism in the category R-Alg. -/\n-- @[nolint has_nonempty_instance] -- Porting note: This linter does not exist yet.\nstructure AlgHom (R : Type u) (A : Type v) (B : Type w) [CommSemiring R] [Semiring A] [Semiring B]\n [Algebra R A] [Algebra R B] extends RingHom A B where\n commutes' : ∀ r : R, toFun (algebraMap R A r) = algebraMap R B r\n#align alg_hom AlgHom\n\n/-- Reinterpret an `AlgHom` as a `RingHom` -/\nadd_decl_doc AlgHom.toRingHom\n\n@[inherit_doc AlgHom]\ninfixr:25 \" →ₐ \" => AlgHom _\n\n@[inherit_doc]\nnotation:25 A \" →ₐ[\" R \"] \" B => AlgHom R A B\n\n/-- `AlgHomClass F R A B` asserts `F` is a type of bundled algebra homomorphisms\nfrom `A` to `B`. -/\nclass AlgHomClass (F : Type*) (R : outParam (Type*)) (A : outParam (Type*))\n (B : outParam (Type*)) [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A]\n [Algebra R B] extends RingHomClass F A B where\n commutes : ∀ (f : F) (r : R), f (algebraMap R A r) = algebraMap R B r\n#align alg_hom_class AlgHomClass\n\n-- Porting note: `dangerousInstance` linter has become smarter about `outParam`s\n-- attribute [nolint dangerousInstance] AlgHomClass.toRingHomClass\n\n-- Porting note: simp can prove this\n-- attribute [simp] AlgHomClass.commutes\n\nnamespace AlgHomClass\n\nvariable {R : Type*} {A : Type*} {B : Type*} [CommSemiring R] [Semiring A] [Semiring B]\n [Algebra R A] [Algebra R B]\n\n-- see Note [lower instance priority]\ninstance (priority := 100) linearMapClass [AlgHomClass F R A B] : LinearMapClass F R A B :=\n { ‹AlgHomClass F R A B› with\n map_smulₛₗ := fun f r x => by\n simp only [Algebra.smul_def, map_mul, commutes, RingHom.id_apply] }\n#align alg_hom_class.linear_map_class AlgHomClass.linearMapClass\n\n-- Porting note: A new definition underlying a coercion `↑`.\n/-- Turn an element of a type `F` satisfying `AlgHomClass F α β` into an actual\n`AlgHom`. This is declared as the default coercion from `F` to `α →+* β`. -/\n@[coe]\ndef toAlgHom {F : Type*} [AlgHomClass F R A B] (f : F) : A →ₐ[R] B :=\n { (f : A →+* B) with\n toFun := f\n commutes' := AlgHomClass.commutes f }\n\ninstance coeTC {F : Type*} [AlgHomClass F R A B] : CoeTC F (A →ₐ[R] B) :=\n ⟨AlgHomClass.toAlgHom⟩\n#align alg_hom_class.alg_hom.has_coe_t AlgHomClass.coeTC\n\nend AlgHomClass\n\nnamespace AlgHom\n\nvariable {R : Type u} {A : Type v} {B : Type w} {C : Type u₁} {D : Type v₁}\n\nsection Semiring\n\nvariable [CommSemiring R] [Semiring A] [Semiring B] [Semiring C] [Semiring D]\n\nvariable [Algebra R A] [Algebra R B] [Algebra R C] [Algebra R D]\n\n-- Porting note: we don't port specialized `CoeFun` instances if there is `FunLike` instead\n#noalign alg_hom.has_coe_to_fun\n\n-- Porting note: This instance is moved.\ninstance algHomClass : AlgHomClass (A →ₐ[R] B) R A B where\n coe f := f.toFun\n coe_injective' f g h := by\n rcases f with ⟨⟨⟨⟨_, _⟩, _⟩, _, _⟩, _⟩\n rcases g with ⟨⟨⟨⟨_, _⟩, _⟩, _, _⟩, _⟩\n congr\n map_add f := f.map_add'\n map_zero f := f.map_zero'\n map_mul f := f.map_mul'\n map_one f := f.map_one'\n commutes f := f.commutes'\n#align alg_hom.alg_hom_class AlgHom.algHomClass\n\n/-- See Note [custom simps projection] -/\ndef Simps.apply {R : Type u} {α : Type v} {β : Type w} [CommSemiring R]\n [Semiring α] [Semiring β] [Algebra R α] [Algebra R β] (f : α →ₐ[R] β) : α → β := f\n\ninitialize_simps_projections AlgHom (toFun → apply)\n\n@[simp]\nprotected theorem coe_coe {F : Type*} [AlgHomClass F R A B] (f : F) : ⇑(f : A →ₐ[R] B) = f :=\n rfl\n#align alg_hom.coe_coe AlgHom.coe_coe\n\n@[simp]\ntheorem toFun_eq_coe (f : A →ₐ[R] B) : f.toFun = f :=\n rfl\n#align alg_hom.to_fun_eq_coe AlgHom.toFun_eq_coe\n\nattribute [coe] AlgHom.toRingHom\n\ninstance coeOutRingHom : CoeOut (A →ₐ[R] B) (A →+* B) :=\n ⟨AlgHom.toRingHom⟩\n#align alg_hom.coe_ring_hom AlgHom.coeOutRingHom\n\n-- Porting note: A new definition underlying a coercion `↑`.\n@[coe]\ndef toMonoidHom' (f : A →ₐ[R] B) : A →* B := (f : A →+* B)\n\ninstance coeOutMonoidHom : CoeOut (A →ₐ[R] B) (A →* B) :=\n ⟨AlgHom.toMonoidHom'⟩\n#align alg_hom.coe_monoid_hom AlgHom.coeOutMonoidHom\n\n-- Porting note: A new definition underlying a coercion `↑`.\n@[coe]\ndef toAddMonoidHom' (f : A →ₐ[R] B) : A →+ B := (f : A →+* B)\n\ninstance coeOutAddMonoidHom : CoeOut (A →ₐ[R] B) (A →+ B) :=\n ⟨AlgHom.toAddMonoidHom'⟩\n#align alg_hom.coe_add_monoid_hom AlgHom.coeOutAddMonoidHom\n\n-- Porting note: Lean 3: `@[simp, norm_cast] coe_mk`\n-- Lean 4: `@[simp] coe_mk` & `@[norm_cast] coe_mks`\n@[simp]\ntheorem coe_mk {f : A →+* B} (h) : ((⟨f, h⟩ : A →ₐ[R] B) : A → B) = f :=\n rfl\n\n@[norm_cast]\ntheorem coe_mks {f : A → B} (h₁ h₂ h₃ h₄ h₅) : ⇑(⟨⟨⟨⟨f, h₁⟩, h₂⟩, h₃, h₄⟩, h₅⟩ : A →ₐ[R] B) = f :=\n rfl\n#align alg_hom.coe_mk AlgHom.coe_mks\n\n-- Porting note: This theorem is new.\n@[simp, norm_cast]\ntheorem coe_ringHom_mk {f : A →+* B} (h) : ((⟨f, h⟩ : A →ₐ[R] B) : A →+* B) = f :=\n rfl\n\n-- make the coercion the simp-normal form\n@[simp]\ntheorem toRingHom_eq_coe (f : A →ₐ[R] B) : f.toRingHom = f :=\n rfl\n#align alg_hom.to_ring_hom_eq_coe AlgHom.toRingHom_eq_coe\n\n@[simp, norm_cast]\ntheorem coe_toRingHom (f : A →ₐ[R] B) : ⇑(f : A →+* B) = f :=\n rfl\n#align alg_hom.coe_to_ring_hom AlgHom.coe_toRingHom\n\n@[simp, norm_cast]\ntheorem coe_toMonoidHom (f : A →ₐ[R] B) : ⇑(f : A →* B) = f :=\n rfl\n#align alg_hom.coe_to_monoid_hom AlgHom.coe_toMonoidHom\n\n@[simp, norm_cast]\ntheorem coe_toAddMonoidHom (f : A →ₐ[R] B) : ⇑(f : A →+ B) = f :=\n rfl\n#align alg_hom.coe_to_add_monoid_hom AlgHom.coe_toAddMonoidHom\n\nvariable (φ : A →ₐ[R] B)\n\ntheorem coe_fn_injective : @Function.Injective (A →ₐ[R] B) (A → B) (↑) :=\n FunLike.coe_injective\n#align alg_hom.coe_fn_injective AlgHom.coe_fn_injective\n\ntheorem coe_fn_inj {φ₁ φ₂ : A →ₐ[R] B} : (φ₁ : A → B) = φ₂ ↔ φ₁ = φ₂ :=\n FunLike.coe_fn_eq\n#align alg_hom.coe_fn_inj AlgHom.coe_fn_inj\n\ntheorem coe_ringHom_injective : Function.Injective ((↑) : (A →ₐ[R] B) → A →+* B) := fun φ₁ φ₂ H =>\n coe_fn_injective <| show ((φ₁ : A →+* B) : A → B) = ((φ₂ : A →+* B) : A → B) from congr_arg _ H\n#align alg_hom.coe_ring_hom_injective AlgHom.coe_ringHom_injective\n\ntheorem coe_monoidHom_injective : Function.Injective ((↑) : (A →ₐ[R] B) → A →* B) :=\n RingHom.coe_monoidHom_injective.comp coe_ringHom_injective\n#align alg_hom.coe_monoid_hom_injective AlgHom.coe_monoidHom_injective\n\ntheorem coe_addMonoidHom_injective : Function.Injective ((↑) : (A →ₐ[R] B) → A →+ B) :=\n RingHom.coe_addMonoidHom_injective.comp coe_ringHom_injective\n#align alg_hom.coe_add_monoid_hom_injective AlgHom.coe_addMonoidHom_injective\n\nprotected theorem congr_fun {φ₁ φ₂ : A →ₐ[R] B} (H : φ₁ = φ₂) (x : A) : φ₁ x = φ₂ x :=\n FunLike.congr_fun H x\n#align alg_hom.congr_fun AlgHom.congr_fun\n\nprotected theorem congr_arg (φ : A →ₐ[R] B) {x y : A} (h : x = y) : φ x = φ y :=\n FunLike.congr_arg φ h\n#align alg_hom.congr_arg AlgHom.congr_arg\n\n@[ext]\ntheorem ext {φ₁ φ₂ : A →ₐ[R] B} (H : ∀ x, φ₁ x = φ₂ x) : φ₁ = φ₂ :=\n FunLike.ext _ _ H\n#align alg_hom.ext AlgHom.ext\n\ntheorem ext_iff {φ₁ φ₂ : A →ₐ[R] B} : φ₁ = φ₂ ↔ ∀ x, φ₁ x = φ₂ x :=\n FunLike.ext_iff\n#align alg_hom.ext_iff AlgHom.ext_iff\n\n@[simp]\ntheorem mk_coe {f : A →ₐ[R] B} (h₁ h₂ h₃ h₄ h₅) : (⟨⟨⟨⟨f, h₁⟩, h₂⟩, h₃, h₄⟩, h₅⟩ : A →ₐ[R] B) = f :=\n ext fun _ => rfl\n#align alg_hom.mk_coe AlgHom.mk_coe\n\n@[simp]\ntheorem commutes (r : R) : φ (algebraMap R A r) = algebraMap R B r :=\n φ.commutes' r\n#align alg_hom.commutes AlgHom.commutes\n\ntheorem comp_algebraMap : (φ : A →+* B).comp (algebraMap R A) = algebraMap R B :=\n RingHom.ext <| φ.commutes\n#align alg_hom.comp_algebra_map AlgHom.comp_algebraMap\n\nprotected theorem map_add (r s : A) : φ (r + s) = φ r + φ s :=\n map_add _ _ _\n#align alg_hom.map_add AlgHom.map_add\n\nprotected theorem map_zero : φ 0 = 0 :=\n map_zero _\n#align alg_hom.map_zero AlgHom.map_zero\n\nprotected theorem map_mul (x y) : φ (x * y) = φ x * φ y :=\n map_mul _ _ _\n#align alg_hom.map_mul AlgHom.map_mul\n\nprotected theorem map_one : φ 1 = 1 :=\n map_one _\n#align alg_hom.map_one AlgHom.map_one\n\nprotected theorem map_pow (x : A) (n : ℕ) : φ (x ^ n) = φ x ^ n :=\n map_pow _ _ _\n#align alg_hom.map_pow AlgHom.map_pow\n\n-- @[simp] -- Porting note: simp can prove this\nprotected theorem map_smul (r : R) (x : A) : φ (r • x) = r • φ x :=\n map_smul _ _ _\n#align alg_hom.map_smul AlgHom.map_smul\n\nprotected theorem map_sum {ι : Type*} (f : ι → A) (s : Finset ι) :\n φ (∑ x in s, f x) = ∑ x in s, φ (f x) :=\n map_sum _ _ _\n#align alg_hom.map_sum AlgHom.map_sum\n\nprotected theorem map_finsupp_sum {α : Type*} [Zero α] {ι : Type*} (f : ι →₀ α) (g : ι → α → A) :\n φ (f.sum g) = f.sum fun i a => φ (g i a) :=\n map_finsupp_sum _ _ _\n#align alg_hom.map_finsupp_sum AlgHom.map_finsupp_sum\n\nset_option linter.deprecated false in\nprotected theorem map_bit0 (x) : φ (bit0 x) = bit0 (φ x) :=\n map_bit0 _ _\n#align alg_hom.map_bit0 AlgHom.map_bit0\n\nset_option linter.deprecated false in\nprotected theorem map_bit1 (x) : φ (bit1 x) = bit1 (φ x) :=\n map_bit1 _ _\n#align alg_hom.map_bit1 AlgHom.map_bit1\n\n/-- If a `RingHom` is `R`-linear, then it is an `AlgHom`. -/\ndef mk' (f : A →+* B) (h : ∀ (c : R) (x), f (c • x) = c • f x) : A →ₐ[R] B :=\n { f with\n toFun := f\n commutes' := fun c => by simp only [Algebra.algebraMap_eq_smul_one, h, f.map_one] }\n#align alg_hom.mk' AlgHom.mk'\n\n@[simp]\ntheorem coe_mk' (f : A →+* B) (h : ∀ (c : R) (x), f (c • x) = c • f x) : ⇑(mk' f h) = f :=\n rfl\n#align alg_hom.coe_mk' AlgHom.coe_mk'\n\nsection\n\nvariable (R A)\n\n/-- Identity map as an `AlgHom`. -/\nprotected def id : A →ₐ[R] A :=\n { RingHom.id A with commutes' := fun _ => rfl }\n#align alg_hom.id AlgHom.id\n\n@[simp]\ntheorem coe_id : ⇑(AlgHom.id R A) = id :=\n rfl\n#align alg_hom.coe_id AlgHom.coe_id\n\n@[simp]\ntheorem id_toRingHom : (AlgHom.id R A : A →+* A) = RingHom.id _ :=\n rfl\n#align alg_hom.id_to_ring_hom AlgHom.id_toRingHom\n\nend\n\ntheorem id_apply (p : A) : AlgHom.id R A p = p :=\n rfl\n#align alg_hom.id_apply AlgHom.id_apply\n\n/-- Composition of algebra homeomorphisms. -/\ndef comp (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) : A →ₐ[R] C :=\n { φ₁.toRingHom.comp ↑φ₂ with\n commutes' := fun r : R => by rw [← φ₁.commutes, ← φ₂.commutes]; rfl }\n#align alg_hom.comp AlgHom.comp\n\n@[simp]\ntheorem coe_comp (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) : ⇑(φ₁.comp φ₂) = φ₁ ∘ φ₂ :=\n rfl\n#align alg_hom.coe_comp AlgHom.coe_comp\n\ntheorem comp_apply (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) (p : A) : φ₁.comp φ₂ p = φ₁ (φ₂ p) :=\n rfl\n#align alg_hom.comp_apply AlgHom.comp_apply\n\ntheorem comp_toRingHom (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) :\n (φ₁.comp φ₂ : A →+* C) = (φ₁ : B →+* C).comp ↑φ₂ :=\n rfl\n#align alg_hom.comp_to_ring_hom AlgHom.comp_toRingHom\n\n@[simp]\ntheorem comp_id : φ.comp (AlgHom.id R A) = φ :=\n ext fun _x => rfl\n#align alg_hom.comp_id AlgHom.comp_id\n\n@[simp]\ntheorem id_comp : (AlgHom.id R B).comp φ = φ :=\n ext fun _x => rfl\n#align alg_hom.id_comp AlgHom.id_comp\n\ntheorem comp_assoc (φ₁ : C →ₐ[R] D) (φ₂ : B →ₐ[R] C) (φ₃ : A →ₐ[R] B) :\n (φ₁.comp φ₂).comp φ₃ = φ₁.comp (φ₂.comp φ₃) :=\n ext fun _x => rfl\n#align alg_hom.comp_assoc AlgHom.comp_assoc\n\n/-- R-Alg ⥤ R-Mod -/\ndef toLinearMap : A →ₗ[R] B where\n toFun := φ\n map_add' := map_add _\n map_smul' := map_smul _\n#align alg_hom.to_linear_map AlgHom.toLinearMap\n\n@[simp]\ntheorem toLinearMap_apply (p : A) : φ.toLinearMap p = φ p :=\n rfl\n#align alg_hom.to_linear_map_apply AlgHom.toLinearMap_apply\n\ntheorem toLinearMap_injective :\n Function.Injective (toLinearMap : _ → A →ₗ[R] B) := fun _φ₁ _φ₂ h =>\n ext <| LinearMap.congr_fun h\n#align alg_hom.to_linear_map_injective AlgHom.toLinearMap_injective\n\n@[simp]\ntheorem comp_toLinearMap (f : A →ₐ[R] B) (g : B →ₐ[R] C) :\n (g.comp f).toLinearMap = g.toLinearMap.comp f.toLinearMap :=\n rfl\n#align alg_hom.comp_to_linear_map AlgHom.comp_toLinearMap\n\n@[simp]\ntheorem toLinearMap_id : toLinearMap (AlgHom.id R A) = LinearMap.id :=\n LinearMap.ext fun _ => rfl\n#align alg_hom.to_linear_map_id AlgHom.toLinearMap_id\n\n/-- Promote a `LinearMap` to an `AlgHom` by supplying proofs about the behavior on `1` and `*`. -/\n@[simps]\ndef ofLinearMap (f : A →ₗ[R] B) (map_one : f 1 = 1) (map_mul : ∀ x y, f (x * y) = f x * f y) :\n A →ₐ[R] B :=\n { f.toAddMonoidHom with\n toFun := f\n map_one' := map_one\n map_mul' := map_mul\n commutes' := fun c => by simp only [Algebra.algebraMap_eq_smul_one, f.map_smul, map_one] }\n#align alg_hom.of_linear_map AlgHom.ofLinearMap\n\n@[simp]\ntheorem ofLinearMap_toLinearMap (map_one) (map_mul) :\n ofLinearMap φ.toLinearMap map_one map_mul = φ := by\n ext\n rfl\n#align alg_hom.of_linear_map_to_linear_map AlgHom.ofLinearMap_toLinearMap\n\n@[simp]\ntheorem toLinearMap_ofLinearMap (f : A →ₗ[R] B) (map_one) (map_mul) :\n toLinearMap (ofLinearMap f map_one map_mul) = f := by\n ext\n ","nextTactic":"rfl","declUpToTactic":"@[simp]\ntheorem toLinearMap_ofLinearMap (f : A →ₗ[R] B) (map_one) (map_mul) :\n toLinearMap (ofLinearMap f map_one map_mul) = f := by\n ext\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Hom.408_0.FxqJeqR3qcIVlrq","decl":"@[simp]\ntheorem toLinearMap_ofLinearMap (f : A →ₗ[R] B) (map_one) (map_mul) :\n toLinearMap (ofLinearMap f map_one map_mul) = f "} +{"state":"R : Type u_1\nS : Type u_2\ninst✝¹ : Semiring R\ninst✝ : Semiring S\nf : R →+* S\nn : ℕ\n⊢ OneHom.toFun\n (↑↑{\n toMonoidHom :=\n { toOneHom := { toFun := ⇑f, map_one' := (_ : OneHom.toFun (↑↑f) 1 = 1) },\n map_mul' :=\n (_ : ∀ (x y : R), OneHom.toFun (↑↑f) (x * y) = OneHom.toFun (↑↑f) x * OneHom.toFun (↑↑f) y) },\n map_zero' := (_ : OneHom.toFun (↑↑f) 0 = 0),\n map_add' := (_ : ∀ (x y : R), OneHom.toFun (↑↑f) (x + y) = OneHom.toFun (↑↑f) x + OneHom.toFun (↑↑f) y) })\n ((algebraMap ℕ R) n) =\n (algebraMap ℕ S) n","srcUpToTactic":"/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Yury Kudryashov\n-/\nimport Mathlib.Algebra.Algebra.Basic\nimport Mathlib.Algebra.BigOperators.Finsupp\n\n#align_import algebra.algebra.hom from \"leanprover-community/mathlib\"@\"e97cf15cd1aec9bd5c193b2ffac5a6dc9118912b\"\n\n/-!\n# Homomorphisms of `R`-algebras\n\nThis file defines bundled homomorphisms of `R`-algebras.\n\n## Main definitions\n\n* `AlgHom R A B`: the type of `R`-algebra morphisms from `A` to `B`.\n* `Algebra.ofId R A : R →ₐ[R] A`: the canonical map from `R` to `A`, as an `AlgHom`.\n\n## Notations\n\n* `A →ₐ[R] B` : `R`-algebra homomorphism from `A` to `B`.\n-/\n\nset_option autoImplicit true\n\n\nopen BigOperators\n\nuniverse u v w u₁ v₁\n\n/-- Defining the homomorphism in the category R-Alg. -/\n-- @[nolint has_nonempty_instance] -- Porting note: This linter does not exist yet.\nstructure AlgHom (R : Type u) (A : Type v) (B : Type w) [CommSemiring R] [Semiring A] [Semiring B]\n [Algebra R A] [Algebra R B] extends RingHom A B where\n commutes' : ∀ r : R, toFun (algebraMap R A r) = algebraMap R B r\n#align alg_hom AlgHom\n\n/-- Reinterpret an `AlgHom` as a `RingHom` -/\nadd_decl_doc AlgHom.toRingHom\n\n@[inherit_doc AlgHom]\ninfixr:25 \" →ₐ \" => AlgHom _\n\n@[inherit_doc]\nnotation:25 A \" →ₐ[\" R \"] \" B => AlgHom R A B\n\n/-- `AlgHomClass F R A B` asserts `F` is a type of bundled algebra homomorphisms\nfrom `A` to `B`. -/\nclass AlgHomClass (F : Type*) (R : outParam (Type*)) (A : outParam (Type*))\n (B : outParam (Type*)) [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A]\n [Algebra R B] extends RingHomClass F A B where\n commutes : ∀ (f : F) (r : R), f (algebraMap R A r) = algebraMap R B r\n#align alg_hom_class AlgHomClass\n\n-- Porting note: `dangerousInstance` linter has become smarter about `outParam`s\n-- attribute [nolint dangerousInstance] AlgHomClass.toRingHomClass\n\n-- Porting note: simp can prove this\n-- attribute [simp] AlgHomClass.commutes\n\nnamespace AlgHomClass\n\nvariable {R : Type*} {A : Type*} {B : Type*} [CommSemiring R] [Semiring A] [Semiring B]\n [Algebra R A] [Algebra R B]\n\n-- see Note [lower instance priority]\ninstance (priority := 100) linearMapClass [AlgHomClass F R A B] : LinearMapClass F R A B :=\n { ‹AlgHomClass F R A B› with\n map_smulₛₗ := fun f r x => by\n simp only [Algebra.smul_def, map_mul, commutes, RingHom.id_apply] }\n#align alg_hom_class.linear_map_class AlgHomClass.linearMapClass\n\n-- Porting note: A new definition underlying a coercion `↑`.\n/-- Turn an element of a type `F` satisfying `AlgHomClass F α β` into an actual\n`AlgHom`. This is declared as the default coercion from `F` to `α →+* β`. -/\n@[coe]\ndef toAlgHom {F : Type*} [AlgHomClass F R A B] (f : F) : A →ₐ[R] B :=\n { (f : A →+* B) with\n toFun := f\n commutes' := AlgHomClass.commutes f }\n\ninstance coeTC {F : Type*} [AlgHomClass F R A B] : CoeTC F (A →ₐ[R] B) :=\n ⟨AlgHomClass.toAlgHom⟩\n#align alg_hom_class.alg_hom.has_coe_t AlgHomClass.coeTC\n\nend AlgHomClass\n\nnamespace AlgHom\n\nvariable {R : Type u} {A : Type v} {B : Type w} {C : Type u₁} {D : Type v₁}\n\nsection Semiring\n\nvariable [CommSemiring R] [Semiring A] [Semiring B] [Semiring C] [Semiring D]\n\nvariable [Algebra R A] [Algebra R B] [Algebra R C] [Algebra R D]\n\n-- Porting note: we don't port specialized `CoeFun` instances if there is `FunLike` instead\n#noalign alg_hom.has_coe_to_fun\n\n-- Porting note: This instance is moved.\ninstance algHomClass : AlgHomClass (A →ₐ[R] B) R A B where\n coe f := f.toFun\n coe_injective' f g h := by\n rcases f with ⟨⟨⟨⟨_, _⟩, _⟩, _, _⟩, _⟩\n rcases g with ⟨⟨⟨⟨_, _⟩, _⟩, _, _⟩, _⟩\n congr\n map_add f := f.map_add'\n map_zero f := f.map_zero'\n map_mul f := f.map_mul'\n map_one f := f.map_one'\n commutes f := f.commutes'\n#align alg_hom.alg_hom_class AlgHom.algHomClass\n\n/-- See Note [custom simps projection] -/\ndef Simps.apply {R : Type u} {α : Type v} {β : Type w} [CommSemiring R]\n [Semiring α] [Semiring β] [Algebra R α] [Algebra R β] (f : α →ₐ[R] β) : α → β := f\n\ninitialize_simps_projections AlgHom (toFun → apply)\n\n@[simp]\nprotected theorem coe_coe {F : Type*} [AlgHomClass F R A B] (f : F) : ⇑(f : A →ₐ[R] B) = f :=\n rfl\n#align alg_hom.coe_coe AlgHom.coe_coe\n\n@[simp]\ntheorem toFun_eq_coe (f : A →ₐ[R] B) : f.toFun = f :=\n rfl\n#align alg_hom.to_fun_eq_coe AlgHom.toFun_eq_coe\n\nattribute [coe] AlgHom.toRingHom\n\ninstance coeOutRingHom : CoeOut (A →ₐ[R] B) (A →+* B) :=\n ⟨AlgHom.toRingHom⟩\n#align alg_hom.coe_ring_hom AlgHom.coeOutRingHom\n\n-- Porting note: A new definition underlying a coercion `↑`.\n@[coe]\ndef toMonoidHom' (f : A →ₐ[R] B) : A →* B := (f : A →+* B)\n\ninstance coeOutMonoidHom : CoeOut (A →ₐ[R] B) (A →* B) :=\n ⟨AlgHom.toMonoidHom'⟩\n#align alg_hom.coe_monoid_hom AlgHom.coeOutMonoidHom\n\n-- Porting note: A new definition underlying a coercion `↑`.\n@[coe]\ndef toAddMonoidHom' (f : A →ₐ[R] B) : A →+ B := (f : A →+* B)\n\ninstance coeOutAddMonoidHom : CoeOut (A →ₐ[R] B) (A →+ B) :=\n ⟨AlgHom.toAddMonoidHom'⟩\n#align alg_hom.coe_add_monoid_hom AlgHom.coeOutAddMonoidHom\n\n-- Porting note: Lean 3: `@[simp, norm_cast] coe_mk`\n-- Lean 4: `@[simp] coe_mk` & `@[norm_cast] coe_mks`\n@[simp]\ntheorem coe_mk {f : A →+* B} (h) : ((⟨f, h⟩ : A →ₐ[R] B) : A → B) = f :=\n rfl\n\n@[norm_cast]\ntheorem coe_mks {f : A → B} (h₁ h₂ h₃ h₄ h₅) : ⇑(⟨⟨⟨⟨f, h₁⟩, h₂⟩, h₃, h₄⟩, h₅⟩ : A →ₐ[R] B) = f :=\n rfl\n#align alg_hom.coe_mk AlgHom.coe_mks\n\n-- Porting note: This theorem is new.\n@[simp, norm_cast]\ntheorem coe_ringHom_mk {f : A →+* B} (h) : ((⟨f, h⟩ : A →ₐ[R] B) : A →+* B) = f :=\n rfl\n\n-- make the coercion the simp-normal form\n@[simp]\ntheorem toRingHom_eq_coe (f : A →ₐ[R] B) : f.toRingHom = f :=\n rfl\n#align alg_hom.to_ring_hom_eq_coe AlgHom.toRingHom_eq_coe\n\n@[simp, norm_cast]\ntheorem coe_toRingHom (f : A →ₐ[R] B) : ⇑(f : A →+* B) = f :=\n rfl\n#align alg_hom.coe_to_ring_hom AlgHom.coe_toRingHom\n\n@[simp, norm_cast]\ntheorem coe_toMonoidHom (f : A →ₐ[R] B) : ⇑(f : A →* B) = f :=\n rfl\n#align alg_hom.coe_to_monoid_hom AlgHom.coe_toMonoidHom\n\n@[simp, norm_cast]\ntheorem coe_toAddMonoidHom (f : A →ₐ[R] B) : ⇑(f : A →+ B) = f :=\n rfl\n#align alg_hom.coe_to_add_monoid_hom AlgHom.coe_toAddMonoidHom\n\nvariable (φ : A →ₐ[R] B)\n\ntheorem coe_fn_injective : @Function.Injective (A →ₐ[R] B) (A → B) (↑) :=\n FunLike.coe_injective\n#align alg_hom.coe_fn_injective AlgHom.coe_fn_injective\n\ntheorem coe_fn_inj {φ₁ φ₂ : A →ₐ[R] B} : (φ₁ : A → B) = φ₂ ↔ φ₁ = φ₂ :=\n FunLike.coe_fn_eq\n#align alg_hom.coe_fn_inj AlgHom.coe_fn_inj\n\ntheorem coe_ringHom_injective : Function.Injective ((↑) : (A →ₐ[R] B) → A →+* B) := fun φ₁ φ₂ H =>\n coe_fn_injective <| show ((φ₁ : A →+* B) : A → B) = ((φ₂ : A →+* B) : A → B) from congr_arg _ H\n#align alg_hom.coe_ring_hom_injective AlgHom.coe_ringHom_injective\n\ntheorem coe_monoidHom_injective : Function.Injective ((↑) : (A →ₐ[R] B) → A →* B) :=\n RingHom.coe_monoidHom_injective.comp coe_ringHom_injective\n#align alg_hom.coe_monoid_hom_injective AlgHom.coe_monoidHom_injective\n\ntheorem coe_addMonoidHom_injective : Function.Injective ((↑) : (A →ₐ[R] B) → A →+ B) :=\n RingHom.coe_addMonoidHom_injective.comp coe_ringHom_injective\n#align alg_hom.coe_add_monoid_hom_injective AlgHom.coe_addMonoidHom_injective\n\nprotected theorem congr_fun {φ₁ φ₂ : A →ₐ[R] B} (H : φ₁ = φ₂) (x : A) : φ₁ x = φ₂ x :=\n FunLike.congr_fun H x\n#align alg_hom.congr_fun AlgHom.congr_fun\n\nprotected theorem congr_arg (φ : A →ₐ[R] B) {x y : A} (h : x = y) : φ x = φ y :=\n FunLike.congr_arg φ h\n#align alg_hom.congr_arg AlgHom.congr_arg\n\n@[ext]\ntheorem ext {φ₁ φ₂ : A →ₐ[R] B} (H : ∀ x, φ₁ x = φ₂ x) : φ₁ = φ₂ :=\n FunLike.ext _ _ H\n#align alg_hom.ext AlgHom.ext\n\ntheorem ext_iff {φ₁ φ₂ : A →ₐ[R] B} : φ₁ = φ₂ ↔ ∀ x, φ₁ x = φ₂ x :=\n FunLike.ext_iff\n#align alg_hom.ext_iff AlgHom.ext_iff\n\n@[simp]\ntheorem mk_coe {f : A →ₐ[R] B} (h₁ h₂ h₃ h₄ h₅) : (⟨⟨⟨⟨f, h₁⟩, h₂⟩, h₃, h₄⟩, h₅⟩ : A →ₐ[R] B) = f :=\n ext fun _ => rfl\n#align alg_hom.mk_coe AlgHom.mk_coe\n\n@[simp]\ntheorem commutes (r : R) : φ (algebraMap R A r) = algebraMap R B r :=\n φ.commutes' r\n#align alg_hom.commutes AlgHom.commutes\n\ntheorem comp_algebraMap : (φ : A →+* B).comp (algebraMap R A) = algebraMap R B :=\n RingHom.ext <| φ.commutes\n#align alg_hom.comp_algebra_map AlgHom.comp_algebraMap\n\nprotected theorem map_add (r s : A) : φ (r + s) = φ r + φ s :=\n map_add _ _ _\n#align alg_hom.map_add AlgHom.map_add\n\nprotected theorem map_zero : φ 0 = 0 :=\n map_zero _\n#align alg_hom.map_zero AlgHom.map_zero\n\nprotected theorem map_mul (x y) : φ (x * y) = φ x * φ y :=\n map_mul _ _ _\n#align alg_hom.map_mul AlgHom.map_mul\n\nprotected theorem map_one : φ 1 = 1 :=\n map_one _\n#align alg_hom.map_one AlgHom.map_one\n\nprotected theorem map_pow (x : A) (n : ℕ) : φ (x ^ n) = φ x ^ n :=\n map_pow _ _ _\n#align alg_hom.map_pow AlgHom.map_pow\n\n-- @[simp] -- Porting note: simp can prove this\nprotected theorem map_smul (r : R) (x : A) : φ (r • x) = r • φ x :=\n map_smul _ _ _\n#align alg_hom.map_smul AlgHom.map_smul\n\nprotected theorem map_sum {ι : Type*} (f : ι → A) (s : Finset ι) :\n φ (∑ x in s, f x) = ∑ x in s, φ (f x) :=\n map_sum _ _ _\n#align alg_hom.map_sum AlgHom.map_sum\n\nprotected theorem map_finsupp_sum {α : Type*} [Zero α] {ι : Type*} (f : ι →₀ α) (g : ι → α → A) :\n φ (f.sum g) = f.sum fun i a => φ (g i a) :=\n map_finsupp_sum _ _ _\n#align alg_hom.map_finsupp_sum AlgHom.map_finsupp_sum\n\nset_option linter.deprecated false in\nprotected theorem map_bit0 (x) : φ (bit0 x) = bit0 (φ x) :=\n map_bit0 _ _\n#align alg_hom.map_bit0 AlgHom.map_bit0\n\nset_option linter.deprecated false in\nprotected theorem map_bit1 (x) : φ (bit1 x) = bit1 (φ x) :=\n map_bit1 _ _\n#align alg_hom.map_bit1 AlgHom.map_bit1\n\n/-- If a `RingHom` is `R`-linear, then it is an `AlgHom`. -/\ndef mk' (f : A →+* B) (h : ∀ (c : R) (x), f (c • x) = c • f x) : A →ₐ[R] B :=\n { f with\n toFun := f\n commutes' := fun c => by simp only [Algebra.algebraMap_eq_smul_one, h, f.map_one] }\n#align alg_hom.mk' AlgHom.mk'\n\n@[simp]\ntheorem coe_mk' (f : A →+* B) (h : ∀ (c : R) (x), f (c • x) = c • f x) : ⇑(mk' f h) = f :=\n rfl\n#align alg_hom.coe_mk' AlgHom.coe_mk'\n\nsection\n\nvariable (R A)\n\n/-- Identity map as an `AlgHom`. -/\nprotected def id : A →ₐ[R] A :=\n { RingHom.id A with commutes' := fun _ => rfl }\n#align alg_hom.id AlgHom.id\n\n@[simp]\ntheorem coe_id : ⇑(AlgHom.id R A) = id :=\n rfl\n#align alg_hom.coe_id AlgHom.coe_id\n\n@[simp]\ntheorem id_toRingHom : (AlgHom.id R A : A →+* A) = RingHom.id _ :=\n rfl\n#align alg_hom.id_to_ring_hom AlgHom.id_toRingHom\n\nend\n\ntheorem id_apply (p : A) : AlgHom.id R A p = p :=\n rfl\n#align alg_hom.id_apply AlgHom.id_apply\n\n/-- Composition of algebra homeomorphisms. -/\ndef comp (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) : A →ₐ[R] C :=\n { φ₁.toRingHom.comp ↑φ₂ with\n commutes' := fun r : R => by rw [← φ₁.commutes, ← φ₂.commutes]; rfl }\n#align alg_hom.comp AlgHom.comp\n\n@[simp]\ntheorem coe_comp (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) : ⇑(φ₁.comp φ₂) = φ₁ ∘ φ₂ :=\n rfl\n#align alg_hom.coe_comp AlgHom.coe_comp\n\ntheorem comp_apply (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) (p : A) : φ₁.comp φ₂ p = φ₁ (φ₂ p) :=\n rfl\n#align alg_hom.comp_apply AlgHom.comp_apply\n\ntheorem comp_toRingHom (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) :\n (φ₁.comp φ₂ : A →+* C) = (φ₁ : B →+* C).comp ↑φ₂ :=\n rfl\n#align alg_hom.comp_to_ring_hom AlgHom.comp_toRingHom\n\n@[simp]\ntheorem comp_id : φ.comp (AlgHom.id R A) = φ :=\n ext fun _x => rfl\n#align alg_hom.comp_id AlgHom.comp_id\n\n@[simp]\ntheorem id_comp : (AlgHom.id R B).comp φ = φ :=\n ext fun _x => rfl\n#align alg_hom.id_comp AlgHom.id_comp\n\ntheorem comp_assoc (φ₁ : C →ₐ[R] D) (φ₂ : B →ₐ[R] C) (φ₃ : A →ₐ[R] B) :\n (φ₁.comp φ₂).comp φ₃ = φ₁.comp (φ₂.comp φ₃) :=\n ext fun _x => rfl\n#align alg_hom.comp_assoc AlgHom.comp_assoc\n\n/-- R-Alg ⥤ R-Mod -/\ndef toLinearMap : A →ₗ[R] B where\n toFun := φ\n map_add' := map_add _\n map_smul' := map_smul _\n#align alg_hom.to_linear_map AlgHom.toLinearMap\n\n@[simp]\ntheorem toLinearMap_apply (p : A) : φ.toLinearMap p = φ p :=\n rfl\n#align alg_hom.to_linear_map_apply AlgHom.toLinearMap_apply\n\ntheorem toLinearMap_injective :\n Function.Injective (toLinearMap : _ → A →ₗ[R] B) := fun _φ₁ _φ₂ h =>\n ext <| LinearMap.congr_fun h\n#align alg_hom.to_linear_map_injective AlgHom.toLinearMap_injective\n\n@[simp]\ntheorem comp_toLinearMap (f : A →ₐ[R] B) (g : B →ₐ[R] C) :\n (g.comp f).toLinearMap = g.toLinearMap.comp f.toLinearMap :=\n rfl\n#align alg_hom.comp_to_linear_map AlgHom.comp_toLinearMap\n\n@[simp]\ntheorem toLinearMap_id : toLinearMap (AlgHom.id R A) = LinearMap.id :=\n LinearMap.ext fun _ => rfl\n#align alg_hom.to_linear_map_id AlgHom.toLinearMap_id\n\n/-- Promote a `LinearMap` to an `AlgHom` by supplying proofs about the behavior on `1` and `*`. -/\n@[simps]\ndef ofLinearMap (f : A →ₗ[R] B) (map_one : f 1 = 1) (map_mul : ∀ x y, f (x * y) = f x * f y) :\n A →ₐ[R] B :=\n { f.toAddMonoidHom with\n toFun := f\n map_one' := map_one\n map_mul' := map_mul\n commutes' := fun c => by simp only [Algebra.algebraMap_eq_smul_one, f.map_smul, map_one] }\n#align alg_hom.of_linear_map AlgHom.ofLinearMap\n\n@[simp]\ntheorem ofLinearMap_toLinearMap (map_one) (map_mul) :\n ofLinearMap φ.toLinearMap map_one map_mul = φ := by\n ext\n rfl\n#align alg_hom.of_linear_map_to_linear_map AlgHom.ofLinearMap_toLinearMap\n\n@[simp]\ntheorem toLinearMap_ofLinearMap (f : A →ₗ[R] B) (map_one) (map_mul) :\n toLinearMap (ofLinearMap f map_one map_mul) = f := by\n ext\n rfl\n#align alg_hom.to_linear_map_of_linear_map AlgHom.toLinearMap_ofLinearMap\n\n@[simp]\ntheorem ofLinearMap_id (map_one) (map_mul) :\n ofLinearMap LinearMap.id map_one map_mul = AlgHom.id R A :=\n ext fun _ => rfl\n#align alg_hom.of_linear_map_id AlgHom.ofLinearMap_id\n\ntheorem map_smul_of_tower {R'} [SMul R' A] [SMul R' B] [LinearMap.CompatibleSMul A B R' R] (r : R')\n (x : A) : φ (r • x) = r • φ x :=\n φ.toLinearMap.map_smul_of_tower r x\n#align alg_hom.map_smul_of_tower AlgHom.map_smul_of_tower\n\ntheorem map_list_prod (s : List A) : φ s.prod = (s.map φ).prod :=\n φ.toRingHom.map_list_prod s\n#align alg_hom.map_list_prod AlgHom.map_list_prod\n\n@[simps (config := .lemmasOnly) toSemigroup_toMul_mul toOne_one]\ninstance End : Monoid (A →ₐ[R] A) where\n mul := comp\n mul_assoc ϕ ψ χ := rfl\n one := AlgHom.id R A\n one_mul ϕ := ext fun x => rfl\n mul_one ϕ := ext fun x => rfl\n#align alg_hom.End AlgHom.End\n\n@[simp]\ntheorem one_apply (x : A) : (1 : A →ₐ[R] A) x = x :=\n rfl\n#align alg_hom.one_apply AlgHom.one_apply\n\n@[simp]\ntheorem mul_apply (φ ψ : A →ₐ[R] A) (x : A) : (φ * ψ) x = φ (ψ x) :=\n rfl\n#align alg_hom.mul_apply AlgHom.mul_apply\n\ntheorem algebraMap_eq_apply (f : A →ₐ[R] B) {y : R} {x : A} (h : algebraMap R A y = x) :\n algebraMap R B y = f x :=\n h ▸ (f.commutes _).symm\n#align alg_hom.algebra_map_eq_apply AlgHom.algebraMap_eq_apply\n\nend Semiring\n\nsection CommSemiring\n\nvariable [CommSemiring R] [CommSemiring A] [CommSemiring B]\n\nvariable [Algebra R A] [Algebra R B] (φ : A →ₐ[R] B)\n\nprotected theorem map_multiset_prod (s : Multiset A) : φ s.prod = (s.map φ).prod :=\n map_multiset_prod _ _\n#align alg_hom.map_multiset_prod AlgHom.map_multiset_prod\n\nprotected theorem map_prod {ι : Type*} (f : ι → A) (s : Finset ι) :\n φ (∏ x in s, f x) = ∏ x in s, φ (f x) :=\n map_prod _ _ _\n#align alg_hom.map_prod AlgHom.map_prod\n\nprotected theorem map_finsupp_prod {α : Type*} [Zero α] {ι : Type*} (f : ι →₀ α) (g : ι → α → A) :\n φ (f.prod g) = f.prod fun i a => φ (g i a) :=\n map_finsupp_prod _ _ _\n#align alg_hom.map_finsupp_prod AlgHom.map_finsupp_prod\n\nend CommSemiring\n\nsection Ring\n\nvariable [CommSemiring R] [Ring A] [Ring B]\n\nvariable [Algebra R A] [Algebra R B] (φ : A →ₐ[R] B)\n\nprotected theorem map_neg (x) : φ (-x) = -φ x :=\n map_neg _ _\n#align alg_hom.map_neg AlgHom.map_neg\n\nprotected theorem map_sub (x y) : φ (x - y) = φ x - φ y :=\n map_sub _ _ _\n#align alg_hom.map_sub AlgHom.map_sub\n\nend Ring\n\nend AlgHom\n\nnamespace RingHom\n\nvariable {R S : Type*}\n\n/-- Reinterpret a `RingHom` as an `ℕ`-algebra homomorphism. -/\ndef toNatAlgHom [Semiring R] [Semiring S] (f : R →+* S) : R →ₐ[ℕ] S :=\n { f with\n toFun := f\n commutes' := fun n => by ","nextTactic":"simp","declUpToTactic":"/-- Reinterpret a `RingHom` as an `ℕ`-algebra homomorphism. -/\ndef toNatAlgHom [Semiring R] [Semiring S] (f : R →+* S) : R →ₐ[ℕ] S :=\n { f with\n toFun := f\n commutes' := fun n => by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Hom.500_0.FxqJeqR3qcIVlrq","decl":"/-- Reinterpret a `RingHom` as an `ℕ`-algebra homomorphism. -/\ndef toNatAlgHom [Semiring R] [Semiring S] (f : R →+* S) : R →ₐ[ℕ] S "} +{"state":"R : Type u_1\nS : Type u_2\ninst✝³ : Ring R\ninst✝² : Ring S\ninst✝¹ : Algebra ℤ R\ninst✝ : Algebra ℤ S\nf : R →+* S\nn : ℤ\n⊢ OneHom.toFun\n (↑↑{ toMonoidHom := ↑f, map_zero' := (_ : OneHom.toFun (↑↑f) 0 = 0),\n map_add' := (_ : ∀ (x y : R), OneHom.toFun (↑↑f) (x + y) = OneHom.toFun (↑↑f) x + OneHom.toFun (↑↑f) y) })\n ((algebraMap ℤ R) n) =\n (algebraMap ℤ S) n","srcUpToTactic":"/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Yury Kudryashov\n-/\nimport Mathlib.Algebra.Algebra.Basic\nimport Mathlib.Algebra.BigOperators.Finsupp\n\n#align_import algebra.algebra.hom from \"leanprover-community/mathlib\"@\"e97cf15cd1aec9bd5c193b2ffac5a6dc9118912b\"\n\n/-!\n# Homomorphisms of `R`-algebras\n\nThis file defines bundled homomorphisms of `R`-algebras.\n\n## Main definitions\n\n* `AlgHom R A B`: the type of `R`-algebra morphisms from `A` to `B`.\n* `Algebra.ofId R A : R →ₐ[R] A`: the canonical map from `R` to `A`, as an `AlgHom`.\n\n## Notations\n\n* `A →ₐ[R] B` : `R`-algebra homomorphism from `A` to `B`.\n-/\n\nset_option autoImplicit true\n\n\nopen BigOperators\n\nuniverse u v w u₁ v₁\n\n/-- Defining the homomorphism in the category R-Alg. -/\n-- @[nolint has_nonempty_instance] -- Porting note: This linter does not exist yet.\nstructure AlgHom (R : Type u) (A : Type v) (B : Type w) [CommSemiring R] [Semiring A] [Semiring B]\n [Algebra R A] [Algebra R B] extends RingHom A B where\n commutes' : ∀ r : R, toFun (algebraMap R A r) = algebraMap R B r\n#align alg_hom AlgHom\n\n/-- Reinterpret an `AlgHom` as a `RingHom` -/\nadd_decl_doc AlgHom.toRingHom\n\n@[inherit_doc AlgHom]\ninfixr:25 \" →ₐ \" => AlgHom _\n\n@[inherit_doc]\nnotation:25 A \" →ₐ[\" R \"] \" B => AlgHom R A B\n\n/-- `AlgHomClass F R A B` asserts `F` is a type of bundled algebra homomorphisms\nfrom `A` to `B`. -/\nclass AlgHomClass (F : Type*) (R : outParam (Type*)) (A : outParam (Type*))\n (B : outParam (Type*)) [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A]\n [Algebra R B] extends RingHomClass F A B where\n commutes : ∀ (f : F) (r : R), f (algebraMap R A r) = algebraMap R B r\n#align alg_hom_class AlgHomClass\n\n-- Porting note: `dangerousInstance` linter has become smarter about `outParam`s\n-- attribute [nolint dangerousInstance] AlgHomClass.toRingHomClass\n\n-- Porting note: simp can prove this\n-- attribute [simp] AlgHomClass.commutes\n\nnamespace AlgHomClass\n\nvariable {R : Type*} {A : Type*} {B : Type*} [CommSemiring R] [Semiring A] [Semiring B]\n [Algebra R A] [Algebra R B]\n\n-- see Note [lower instance priority]\ninstance (priority := 100) linearMapClass [AlgHomClass F R A B] : LinearMapClass F R A B :=\n { ‹AlgHomClass F R A B› with\n map_smulₛₗ := fun f r x => by\n simp only [Algebra.smul_def, map_mul, commutes, RingHom.id_apply] }\n#align alg_hom_class.linear_map_class AlgHomClass.linearMapClass\n\n-- Porting note: A new definition underlying a coercion `↑`.\n/-- Turn an element of a type `F` satisfying `AlgHomClass F α β` into an actual\n`AlgHom`. This is declared as the default coercion from `F` to `α →+* β`. -/\n@[coe]\ndef toAlgHom {F : Type*} [AlgHomClass F R A B] (f : F) : A →ₐ[R] B :=\n { (f : A →+* B) with\n toFun := f\n commutes' := AlgHomClass.commutes f }\n\ninstance coeTC {F : Type*} [AlgHomClass F R A B] : CoeTC F (A →ₐ[R] B) :=\n ⟨AlgHomClass.toAlgHom⟩\n#align alg_hom_class.alg_hom.has_coe_t AlgHomClass.coeTC\n\nend AlgHomClass\n\nnamespace AlgHom\n\nvariable {R : Type u} {A : Type v} {B : Type w} {C : Type u₁} {D : Type v₁}\n\nsection Semiring\n\nvariable [CommSemiring R] [Semiring A] [Semiring B] [Semiring C] [Semiring D]\n\nvariable [Algebra R A] [Algebra R B] [Algebra R C] [Algebra R D]\n\n-- Porting note: we don't port specialized `CoeFun` instances if there is `FunLike` instead\n#noalign alg_hom.has_coe_to_fun\n\n-- Porting note: This instance is moved.\ninstance algHomClass : AlgHomClass (A →ₐ[R] B) R A B where\n coe f := f.toFun\n coe_injective' f g h := by\n rcases f with ⟨⟨⟨⟨_, _⟩, _⟩, _, _⟩, _⟩\n rcases g with ⟨⟨⟨⟨_, _⟩, _⟩, _, _⟩, _⟩\n congr\n map_add f := f.map_add'\n map_zero f := f.map_zero'\n map_mul f := f.map_mul'\n map_one f := f.map_one'\n commutes f := f.commutes'\n#align alg_hom.alg_hom_class AlgHom.algHomClass\n\n/-- See Note [custom simps projection] -/\ndef Simps.apply {R : Type u} {α : Type v} {β : Type w} [CommSemiring R]\n [Semiring α] [Semiring β] [Algebra R α] [Algebra R β] (f : α →ₐ[R] β) : α → β := f\n\ninitialize_simps_projections AlgHom (toFun → apply)\n\n@[simp]\nprotected theorem coe_coe {F : Type*} [AlgHomClass F R A B] (f : F) : ⇑(f : A →ₐ[R] B) = f :=\n rfl\n#align alg_hom.coe_coe AlgHom.coe_coe\n\n@[simp]\ntheorem toFun_eq_coe (f : A →ₐ[R] B) : f.toFun = f :=\n rfl\n#align alg_hom.to_fun_eq_coe AlgHom.toFun_eq_coe\n\nattribute [coe] AlgHom.toRingHom\n\ninstance coeOutRingHom : CoeOut (A →ₐ[R] B) (A →+* B) :=\n ⟨AlgHom.toRingHom⟩\n#align alg_hom.coe_ring_hom AlgHom.coeOutRingHom\n\n-- Porting note: A new definition underlying a coercion `↑`.\n@[coe]\ndef toMonoidHom' (f : A →ₐ[R] B) : A →* B := (f : A →+* B)\n\ninstance coeOutMonoidHom : CoeOut (A →ₐ[R] B) (A →* B) :=\n ⟨AlgHom.toMonoidHom'⟩\n#align alg_hom.coe_monoid_hom AlgHom.coeOutMonoidHom\n\n-- Porting note: A new definition underlying a coercion `↑`.\n@[coe]\ndef toAddMonoidHom' (f : A →ₐ[R] B) : A →+ B := (f : A →+* B)\n\ninstance coeOutAddMonoidHom : CoeOut (A →ₐ[R] B) (A →+ B) :=\n ⟨AlgHom.toAddMonoidHom'⟩\n#align alg_hom.coe_add_monoid_hom AlgHom.coeOutAddMonoidHom\n\n-- Porting note: Lean 3: `@[simp, norm_cast] coe_mk`\n-- Lean 4: `@[simp] coe_mk` & `@[norm_cast] coe_mks`\n@[simp]\ntheorem coe_mk {f : A →+* B} (h) : ((⟨f, h⟩ : A →ₐ[R] B) : A → B) = f :=\n rfl\n\n@[norm_cast]\ntheorem coe_mks {f : A → B} (h₁ h₂ h₃ h₄ h₅) : ⇑(⟨⟨⟨⟨f, h₁⟩, h₂⟩, h₃, h₄⟩, h₅⟩ : A →ₐ[R] B) = f :=\n rfl\n#align alg_hom.coe_mk AlgHom.coe_mks\n\n-- Porting note: This theorem is new.\n@[simp, norm_cast]\ntheorem coe_ringHom_mk {f : A →+* B} (h) : ((⟨f, h⟩ : A →ₐ[R] B) : A →+* B) = f :=\n rfl\n\n-- make the coercion the simp-normal form\n@[simp]\ntheorem toRingHom_eq_coe (f : A →ₐ[R] B) : f.toRingHom = f :=\n rfl\n#align alg_hom.to_ring_hom_eq_coe AlgHom.toRingHom_eq_coe\n\n@[simp, norm_cast]\ntheorem coe_toRingHom (f : A →ₐ[R] B) : ⇑(f : A →+* B) = f :=\n rfl\n#align alg_hom.coe_to_ring_hom AlgHom.coe_toRingHom\n\n@[simp, norm_cast]\ntheorem coe_toMonoidHom (f : A →ₐ[R] B) : ⇑(f : A →* B) = f :=\n rfl\n#align alg_hom.coe_to_monoid_hom AlgHom.coe_toMonoidHom\n\n@[simp, norm_cast]\ntheorem coe_toAddMonoidHom (f : A →ₐ[R] B) : ⇑(f : A →+ B) = f :=\n rfl\n#align alg_hom.coe_to_add_monoid_hom AlgHom.coe_toAddMonoidHom\n\nvariable (φ : A →ₐ[R] B)\n\ntheorem coe_fn_injective : @Function.Injective (A →ₐ[R] B) (A → B) (↑) :=\n FunLike.coe_injective\n#align alg_hom.coe_fn_injective AlgHom.coe_fn_injective\n\ntheorem coe_fn_inj {φ₁ φ₂ : A →ₐ[R] B} : (φ₁ : A → B) = φ₂ ↔ φ₁ = φ₂ :=\n FunLike.coe_fn_eq\n#align alg_hom.coe_fn_inj AlgHom.coe_fn_inj\n\ntheorem coe_ringHom_injective : Function.Injective ((↑) : (A →ₐ[R] B) → A →+* B) := fun φ₁ φ₂ H =>\n coe_fn_injective <| show ((φ₁ : A →+* B) : A → B) = ((φ₂ : A →+* B) : A → B) from congr_arg _ H\n#align alg_hom.coe_ring_hom_injective AlgHom.coe_ringHom_injective\n\ntheorem coe_monoidHom_injective : Function.Injective ((↑) : (A →ₐ[R] B) → A →* B) :=\n RingHom.coe_monoidHom_injective.comp coe_ringHom_injective\n#align alg_hom.coe_monoid_hom_injective AlgHom.coe_monoidHom_injective\n\ntheorem coe_addMonoidHom_injective : Function.Injective ((↑) : (A →ₐ[R] B) → A →+ B) :=\n RingHom.coe_addMonoidHom_injective.comp coe_ringHom_injective\n#align alg_hom.coe_add_monoid_hom_injective AlgHom.coe_addMonoidHom_injective\n\nprotected theorem congr_fun {φ₁ φ₂ : A →ₐ[R] B} (H : φ₁ = φ₂) (x : A) : φ₁ x = φ₂ x :=\n FunLike.congr_fun H x\n#align alg_hom.congr_fun AlgHom.congr_fun\n\nprotected theorem congr_arg (φ : A →ₐ[R] B) {x y : A} (h : x = y) : φ x = φ y :=\n FunLike.congr_arg φ h\n#align alg_hom.congr_arg AlgHom.congr_arg\n\n@[ext]\ntheorem ext {φ₁ φ₂ : A →ₐ[R] B} (H : ∀ x, φ₁ x = φ₂ x) : φ₁ = φ₂ :=\n FunLike.ext _ _ H\n#align alg_hom.ext AlgHom.ext\n\ntheorem ext_iff {φ₁ φ₂ : A →ₐ[R] B} : φ₁ = φ₂ ↔ ∀ x, φ₁ x = φ₂ x :=\n FunLike.ext_iff\n#align alg_hom.ext_iff AlgHom.ext_iff\n\n@[simp]\ntheorem mk_coe {f : A →ₐ[R] B} (h₁ h₂ h₃ h₄ h₅) : (⟨⟨⟨⟨f, h₁⟩, h₂⟩, h₃, h₄⟩, h₅⟩ : A →ₐ[R] B) = f :=\n ext fun _ => rfl\n#align alg_hom.mk_coe AlgHom.mk_coe\n\n@[simp]\ntheorem commutes (r : R) : φ (algebraMap R A r) = algebraMap R B r :=\n φ.commutes' r\n#align alg_hom.commutes AlgHom.commutes\n\ntheorem comp_algebraMap : (φ : A →+* B).comp (algebraMap R A) = algebraMap R B :=\n RingHom.ext <| φ.commutes\n#align alg_hom.comp_algebra_map AlgHom.comp_algebraMap\n\nprotected theorem map_add (r s : A) : φ (r + s) = φ r + φ s :=\n map_add _ _ _\n#align alg_hom.map_add AlgHom.map_add\n\nprotected theorem map_zero : φ 0 = 0 :=\n map_zero _\n#align alg_hom.map_zero AlgHom.map_zero\n\nprotected theorem map_mul (x y) : φ (x * y) = φ x * φ y :=\n map_mul _ _ _\n#align alg_hom.map_mul AlgHom.map_mul\n\nprotected theorem map_one : φ 1 = 1 :=\n map_one _\n#align alg_hom.map_one AlgHom.map_one\n\nprotected theorem map_pow (x : A) (n : ℕ) : φ (x ^ n) = φ x ^ n :=\n map_pow _ _ _\n#align alg_hom.map_pow AlgHom.map_pow\n\n-- @[simp] -- Porting note: simp can prove this\nprotected theorem map_smul (r : R) (x : A) : φ (r • x) = r • φ x :=\n map_smul _ _ _\n#align alg_hom.map_smul AlgHom.map_smul\n\nprotected theorem map_sum {ι : Type*} (f : ι → A) (s : Finset ι) :\n φ (∑ x in s, f x) = ∑ x in s, φ (f x) :=\n map_sum _ _ _\n#align alg_hom.map_sum AlgHom.map_sum\n\nprotected theorem map_finsupp_sum {α : Type*} [Zero α] {ι : Type*} (f : ι →₀ α) (g : ι → α → A) :\n φ (f.sum g) = f.sum fun i a => φ (g i a) :=\n map_finsupp_sum _ _ _\n#align alg_hom.map_finsupp_sum AlgHom.map_finsupp_sum\n\nset_option linter.deprecated false in\nprotected theorem map_bit0 (x) : φ (bit0 x) = bit0 (φ x) :=\n map_bit0 _ _\n#align alg_hom.map_bit0 AlgHom.map_bit0\n\nset_option linter.deprecated false in\nprotected theorem map_bit1 (x) : φ (bit1 x) = bit1 (φ x) :=\n map_bit1 _ _\n#align alg_hom.map_bit1 AlgHom.map_bit1\n\n/-- If a `RingHom` is `R`-linear, then it is an `AlgHom`. -/\ndef mk' (f : A →+* B) (h : ∀ (c : R) (x), f (c • x) = c • f x) : A →ₐ[R] B :=\n { f with\n toFun := f\n commutes' := fun c => by simp only [Algebra.algebraMap_eq_smul_one, h, f.map_one] }\n#align alg_hom.mk' AlgHom.mk'\n\n@[simp]\ntheorem coe_mk' (f : A →+* B) (h : ∀ (c : R) (x), f (c • x) = c • f x) : ⇑(mk' f h) = f :=\n rfl\n#align alg_hom.coe_mk' AlgHom.coe_mk'\n\nsection\n\nvariable (R A)\n\n/-- Identity map as an `AlgHom`. -/\nprotected def id : A →ₐ[R] A :=\n { RingHom.id A with commutes' := fun _ => rfl }\n#align alg_hom.id AlgHom.id\n\n@[simp]\ntheorem coe_id : ⇑(AlgHom.id R A) = id :=\n rfl\n#align alg_hom.coe_id AlgHom.coe_id\n\n@[simp]\ntheorem id_toRingHom : (AlgHom.id R A : A →+* A) = RingHom.id _ :=\n rfl\n#align alg_hom.id_to_ring_hom AlgHom.id_toRingHom\n\nend\n\ntheorem id_apply (p : A) : AlgHom.id R A p = p :=\n rfl\n#align alg_hom.id_apply AlgHom.id_apply\n\n/-- Composition of algebra homeomorphisms. -/\ndef comp (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) : A →ₐ[R] C :=\n { φ₁.toRingHom.comp ↑φ₂ with\n commutes' := fun r : R => by rw [← φ₁.commutes, ← φ₂.commutes]; rfl }\n#align alg_hom.comp AlgHom.comp\n\n@[simp]\ntheorem coe_comp (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) : ⇑(φ₁.comp φ₂) = φ₁ ∘ φ₂ :=\n rfl\n#align alg_hom.coe_comp AlgHom.coe_comp\n\ntheorem comp_apply (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) (p : A) : φ₁.comp φ₂ p = φ₁ (φ₂ p) :=\n rfl\n#align alg_hom.comp_apply AlgHom.comp_apply\n\ntheorem comp_toRingHom (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) :\n (φ₁.comp φ₂ : A →+* C) = (φ₁ : B →+* C).comp ↑φ₂ :=\n rfl\n#align alg_hom.comp_to_ring_hom AlgHom.comp_toRingHom\n\n@[simp]\ntheorem comp_id : φ.comp (AlgHom.id R A) = φ :=\n ext fun _x => rfl\n#align alg_hom.comp_id AlgHom.comp_id\n\n@[simp]\ntheorem id_comp : (AlgHom.id R B).comp φ = φ :=\n ext fun _x => rfl\n#align alg_hom.id_comp AlgHom.id_comp\n\ntheorem comp_assoc (φ₁ : C →ₐ[R] D) (φ₂ : B →ₐ[R] C) (φ₃ : A →ₐ[R] B) :\n (φ₁.comp φ₂).comp φ₃ = φ₁.comp (φ₂.comp φ₃) :=\n ext fun _x => rfl\n#align alg_hom.comp_assoc AlgHom.comp_assoc\n\n/-- R-Alg ⥤ R-Mod -/\ndef toLinearMap : A →ₗ[R] B where\n toFun := φ\n map_add' := map_add _\n map_smul' := map_smul _\n#align alg_hom.to_linear_map AlgHom.toLinearMap\n\n@[simp]\ntheorem toLinearMap_apply (p : A) : φ.toLinearMap p = φ p :=\n rfl\n#align alg_hom.to_linear_map_apply AlgHom.toLinearMap_apply\n\ntheorem toLinearMap_injective :\n Function.Injective (toLinearMap : _ → A →ₗ[R] B) := fun _φ₁ _φ₂ h =>\n ext <| LinearMap.congr_fun h\n#align alg_hom.to_linear_map_injective AlgHom.toLinearMap_injective\n\n@[simp]\ntheorem comp_toLinearMap (f : A →ₐ[R] B) (g : B →ₐ[R] C) :\n (g.comp f).toLinearMap = g.toLinearMap.comp f.toLinearMap :=\n rfl\n#align alg_hom.comp_to_linear_map AlgHom.comp_toLinearMap\n\n@[simp]\ntheorem toLinearMap_id : toLinearMap (AlgHom.id R A) = LinearMap.id :=\n LinearMap.ext fun _ => rfl\n#align alg_hom.to_linear_map_id AlgHom.toLinearMap_id\n\n/-- Promote a `LinearMap` to an `AlgHom` by supplying proofs about the behavior on `1` and `*`. -/\n@[simps]\ndef ofLinearMap (f : A →ₗ[R] B) (map_one : f 1 = 1) (map_mul : ∀ x y, f (x * y) = f x * f y) :\n A →ₐ[R] B :=\n { f.toAddMonoidHom with\n toFun := f\n map_one' := map_one\n map_mul' := map_mul\n commutes' := fun c => by simp only [Algebra.algebraMap_eq_smul_one, f.map_smul, map_one] }\n#align alg_hom.of_linear_map AlgHom.ofLinearMap\n\n@[simp]\ntheorem ofLinearMap_toLinearMap (map_one) (map_mul) :\n ofLinearMap φ.toLinearMap map_one map_mul = φ := by\n ext\n rfl\n#align alg_hom.of_linear_map_to_linear_map AlgHom.ofLinearMap_toLinearMap\n\n@[simp]\ntheorem toLinearMap_ofLinearMap (f : A →ₗ[R] B) (map_one) (map_mul) :\n toLinearMap (ofLinearMap f map_one map_mul) = f := by\n ext\n rfl\n#align alg_hom.to_linear_map_of_linear_map AlgHom.toLinearMap_ofLinearMap\n\n@[simp]\ntheorem ofLinearMap_id (map_one) (map_mul) :\n ofLinearMap LinearMap.id map_one map_mul = AlgHom.id R A :=\n ext fun _ => rfl\n#align alg_hom.of_linear_map_id AlgHom.ofLinearMap_id\n\ntheorem map_smul_of_tower {R'} [SMul R' A] [SMul R' B] [LinearMap.CompatibleSMul A B R' R] (r : R')\n (x : A) : φ (r • x) = r • φ x :=\n φ.toLinearMap.map_smul_of_tower r x\n#align alg_hom.map_smul_of_tower AlgHom.map_smul_of_tower\n\ntheorem map_list_prod (s : List A) : φ s.prod = (s.map φ).prod :=\n φ.toRingHom.map_list_prod s\n#align alg_hom.map_list_prod AlgHom.map_list_prod\n\n@[simps (config := .lemmasOnly) toSemigroup_toMul_mul toOne_one]\ninstance End : Monoid (A →ₐ[R] A) where\n mul := comp\n mul_assoc ϕ ψ χ := rfl\n one := AlgHom.id R A\n one_mul ϕ := ext fun x => rfl\n mul_one ϕ := ext fun x => rfl\n#align alg_hom.End AlgHom.End\n\n@[simp]\ntheorem one_apply (x : A) : (1 : A →ₐ[R] A) x = x :=\n rfl\n#align alg_hom.one_apply AlgHom.one_apply\n\n@[simp]\ntheorem mul_apply (φ ψ : A →ₐ[R] A) (x : A) : (φ * ψ) x = φ (ψ x) :=\n rfl\n#align alg_hom.mul_apply AlgHom.mul_apply\n\ntheorem algebraMap_eq_apply (f : A →ₐ[R] B) {y : R} {x : A} (h : algebraMap R A y = x) :\n algebraMap R B y = f x :=\n h ▸ (f.commutes _).symm\n#align alg_hom.algebra_map_eq_apply AlgHom.algebraMap_eq_apply\n\nend Semiring\n\nsection CommSemiring\n\nvariable [CommSemiring R] [CommSemiring A] [CommSemiring B]\n\nvariable [Algebra R A] [Algebra R B] (φ : A →ₐ[R] B)\n\nprotected theorem map_multiset_prod (s : Multiset A) : φ s.prod = (s.map φ).prod :=\n map_multiset_prod _ _\n#align alg_hom.map_multiset_prod AlgHom.map_multiset_prod\n\nprotected theorem map_prod {ι : Type*} (f : ι → A) (s : Finset ι) :\n φ (∏ x in s, f x) = ∏ x in s, φ (f x) :=\n map_prod _ _ _\n#align alg_hom.map_prod AlgHom.map_prod\n\nprotected theorem map_finsupp_prod {α : Type*} [Zero α] {ι : Type*} (f : ι →₀ α) (g : ι → α → A) :\n φ (f.prod g) = f.prod fun i a => φ (g i a) :=\n map_finsupp_prod _ _ _\n#align alg_hom.map_finsupp_prod AlgHom.map_finsupp_prod\n\nend CommSemiring\n\nsection Ring\n\nvariable [CommSemiring R] [Ring A] [Ring B]\n\nvariable [Algebra R A] [Algebra R B] (φ : A →ₐ[R] B)\n\nprotected theorem map_neg (x) : φ (-x) = -φ x :=\n map_neg _ _\n#align alg_hom.map_neg AlgHom.map_neg\n\nprotected theorem map_sub (x y) : φ (x - y) = φ x - φ y :=\n map_sub _ _ _\n#align alg_hom.map_sub AlgHom.map_sub\n\nend Ring\n\nend AlgHom\n\nnamespace RingHom\n\nvariable {R S : Type*}\n\n/-- Reinterpret a `RingHom` as an `ℕ`-algebra homomorphism. -/\ndef toNatAlgHom [Semiring R] [Semiring S] (f : R →+* S) : R →ₐ[ℕ] S :=\n { f with\n toFun := f\n commutes' := fun n => by simp }\n#align ring_hom.to_nat_alg_hom RingHom.toNatAlgHom\n\n/-- Reinterpret a `RingHom` as a `ℤ`-algebra homomorphism. -/\ndef toIntAlgHom [Ring R] [Ring S] [Algebra ℤ R] [Algebra ℤ S] (f : R →+* S) : R →ₐ[ℤ] S :=\n { f with commutes' := fun n => by ","nextTactic":"simp","declUpToTactic":"/-- Reinterpret a `RingHom` as a `ℤ`-algebra homomorphism. -/\ndef toIntAlgHom [Ring R] [Ring S] [Algebra ℤ R] [Algebra ℤ S] (f : R →+* S) : R →ₐ[ℤ] S :=\n { f with commutes' := fun n => by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Hom.507_0.FxqJeqR3qcIVlrq","decl":"/-- Reinterpret a `RingHom` as a `ℤ`-algebra homomorphism. -/\ndef toIntAlgHom [Ring R] [Ring S] [Algebra ℤ R] [Algebra ℤ S] (f : R →+* S) : R →ₐ[ℤ] S "} +{"state":"R : Type u\nA : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nx : Aˣ\n⊢ 1 • x = x","srcUpToTactic":"/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Yury Kudryashov\n-/\nimport Mathlib.Algebra.Algebra.Basic\nimport Mathlib.Algebra.BigOperators.Finsupp\n\n#align_import algebra.algebra.hom from \"leanprover-community/mathlib\"@\"e97cf15cd1aec9bd5c193b2ffac5a6dc9118912b\"\n\n/-!\n# Homomorphisms of `R`-algebras\n\nThis file defines bundled homomorphisms of `R`-algebras.\n\n## Main definitions\n\n* `AlgHom R A B`: the type of `R`-algebra morphisms from `A` to `B`.\n* `Algebra.ofId R A : R →ₐ[R] A`: the canonical map from `R` to `A`, as an `AlgHom`.\n\n## Notations\n\n* `A →ₐ[R] B` : `R`-algebra homomorphism from `A` to `B`.\n-/\n\nset_option autoImplicit true\n\n\nopen BigOperators\n\nuniverse u v w u₁ v₁\n\n/-- Defining the homomorphism in the category R-Alg. -/\n-- @[nolint has_nonempty_instance] -- Porting note: This linter does not exist yet.\nstructure AlgHom (R : Type u) (A : Type v) (B : Type w) [CommSemiring R] [Semiring A] [Semiring B]\n [Algebra R A] [Algebra R B] extends RingHom A B where\n commutes' : ∀ r : R, toFun (algebraMap R A r) = algebraMap R B r\n#align alg_hom AlgHom\n\n/-- Reinterpret an `AlgHom` as a `RingHom` -/\nadd_decl_doc AlgHom.toRingHom\n\n@[inherit_doc AlgHom]\ninfixr:25 \" →ₐ \" => AlgHom _\n\n@[inherit_doc]\nnotation:25 A \" →ₐ[\" R \"] \" B => AlgHom R A B\n\n/-- `AlgHomClass F R A B` asserts `F` is a type of bundled algebra homomorphisms\nfrom `A` to `B`. -/\nclass AlgHomClass (F : Type*) (R : outParam (Type*)) (A : outParam (Type*))\n (B : outParam (Type*)) [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A]\n [Algebra R B] extends RingHomClass F A B where\n commutes : ∀ (f : F) (r : R), f (algebraMap R A r) = algebraMap R B r\n#align alg_hom_class AlgHomClass\n\n-- Porting note: `dangerousInstance` linter has become smarter about `outParam`s\n-- attribute [nolint dangerousInstance] AlgHomClass.toRingHomClass\n\n-- Porting note: simp can prove this\n-- attribute [simp] AlgHomClass.commutes\n\nnamespace AlgHomClass\n\nvariable {R : Type*} {A : Type*} {B : Type*} [CommSemiring R] [Semiring A] [Semiring B]\n [Algebra R A] [Algebra R B]\n\n-- see Note [lower instance priority]\ninstance (priority := 100) linearMapClass [AlgHomClass F R A B] : LinearMapClass F R A B :=\n { ‹AlgHomClass F R A B› with\n map_smulₛₗ := fun f r x => by\n simp only [Algebra.smul_def, map_mul, commutes, RingHom.id_apply] }\n#align alg_hom_class.linear_map_class AlgHomClass.linearMapClass\n\n-- Porting note: A new definition underlying a coercion `↑`.\n/-- Turn an element of a type `F` satisfying `AlgHomClass F α β` into an actual\n`AlgHom`. This is declared as the default coercion from `F` to `α →+* β`. -/\n@[coe]\ndef toAlgHom {F : Type*} [AlgHomClass F R A B] (f : F) : A →ₐ[R] B :=\n { (f : A →+* B) with\n toFun := f\n commutes' := AlgHomClass.commutes f }\n\ninstance coeTC {F : Type*} [AlgHomClass F R A B] : CoeTC F (A →ₐ[R] B) :=\n ⟨AlgHomClass.toAlgHom⟩\n#align alg_hom_class.alg_hom.has_coe_t AlgHomClass.coeTC\n\nend AlgHomClass\n\nnamespace AlgHom\n\nvariable {R : Type u} {A : Type v} {B : Type w} {C : Type u₁} {D : Type v₁}\n\nsection Semiring\n\nvariable [CommSemiring R] [Semiring A] [Semiring B] [Semiring C] [Semiring D]\n\nvariable [Algebra R A] [Algebra R B] [Algebra R C] [Algebra R D]\n\n-- Porting note: we don't port specialized `CoeFun` instances if there is `FunLike` instead\n#noalign alg_hom.has_coe_to_fun\n\n-- Porting note: This instance is moved.\ninstance algHomClass : AlgHomClass (A →ₐ[R] B) R A B where\n coe f := f.toFun\n coe_injective' f g h := by\n rcases f with ⟨⟨⟨⟨_, _⟩, _⟩, _, _⟩, _⟩\n rcases g with ⟨⟨⟨⟨_, _⟩, _⟩, _, _⟩, _⟩\n congr\n map_add f := f.map_add'\n map_zero f := f.map_zero'\n map_mul f := f.map_mul'\n map_one f := f.map_one'\n commutes f := f.commutes'\n#align alg_hom.alg_hom_class AlgHom.algHomClass\n\n/-- See Note [custom simps projection] -/\ndef Simps.apply {R : Type u} {α : Type v} {β : Type w} [CommSemiring R]\n [Semiring α] [Semiring β] [Algebra R α] [Algebra R β] (f : α →ₐ[R] β) : α → β := f\n\ninitialize_simps_projections AlgHom (toFun → apply)\n\n@[simp]\nprotected theorem coe_coe {F : Type*} [AlgHomClass F R A B] (f : F) : ⇑(f : A →ₐ[R] B) = f :=\n rfl\n#align alg_hom.coe_coe AlgHom.coe_coe\n\n@[simp]\ntheorem toFun_eq_coe (f : A →ₐ[R] B) : f.toFun = f :=\n rfl\n#align alg_hom.to_fun_eq_coe AlgHom.toFun_eq_coe\n\nattribute [coe] AlgHom.toRingHom\n\ninstance coeOutRingHom : CoeOut (A →ₐ[R] B) (A →+* B) :=\n ⟨AlgHom.toRingHom⟩\n#align alg_hom.coe_ring_hom AlgHom.coeOutRingHom\n\n-- Porting note: A new definition underlying a coercion `↑`.\n@[coe]\ndef toMonoidHom' (f : A →ₐ[R] B) : A →* B := (f : A →+* B)\n\ninstance coeOutMonoidHom : CoeOut (A →ₐ[R] B) (A →* B) :=\n ⟨AlgHom.toMonoidHom'⟩\n#align alg_hom.coe_monoid_hom AlgHom.coeOutMonoidHom\n\n-- Porting note: A new definition underlying a coercion `↑`.\n@[coe]\ndef toAddMonoidHom' (f : A →ₐ[R] B) : A →+ B := (f : A →+* B)\n\ninstance coeOutAddMonoidHom : CoeOut (A →ₐ[R] B) (A →+ B) :=\n ⟨AlgHom.toAddMonoidHom'⟩\n#align alg_hom.coe_add_monoid_hom AlgHom.coeOutAddMonoidHom\n\n-- Porting note: Lean 3: `@[simp, norm_cast] coe_mk`\n-- Lean 4: `@[simp] coe_mk` & `@[norm_cast] coe_mks`\n@[simp]\ntheorem coe_mk {f : A →+* B} (h) : ((⟨f, h⟩ : A →ₐ[R] B) : A → B) = f :=\n rfl\n\n@[norm_cast]\ntheorem coe_mks {f : A → B} (h₁ h₂ h₃ h₄ h₅) : ⇑(⟨⟨⟨⟨f, h₁⟩, h₂⟩, h₃, h₄⟩, h₅⟩ : A →ₐ[R] B) = f :=\n rfl\n#align alg_hom.coe_mk AlgHom.coe_mks\n\n-- Porting note: This theorem is new.\n@[simp, norm_cast]\ntheorem coe_ringHom_mk {f : A →+* B} (h) : ((⟨f, h⟩ : A →ₐ[R] B) : A →+* B) = f :=\n rfl\n\n-- make the coercion the simp-normal form\n@[simp]\ntheorem toRingHom_eq_coe (f : A →ₐ[R] B) : f.toRingHom = f :=\n rfl\n#align alg_hom.to_ring_hom_eq_coe AlgHom.toRingHom_eq_coe\n\n@[simp, norm_cast]\ntheorem coe_toRingHom (f : A →ₐ[R] B) : ⇑(f : A →+* B) = f :=\n rfl\n#align alg_hom.coe_to_ring_hom AlgHom.coe_toRingHom\n\n@[simp, norm_cast]\ntheorem coe_toMonoidHom (f : A →ₐ[R] B) : ⇑(f : A →* B) = f :=\n rfl\n#align alg_hom.coe_to_monoid_hom AlgHom.coe_toMonoidHom\n\n@[simp, norm_cast]\ntheorem coe_toAddMonoidHom (f : A →ₐ[R] B) : ⇑(f : A →+ B) = f :=\n rfl\n#align alg_hom.coe_to_add_monoid_hom AlgHom.coe_toAddMonoidHom\n\nvariable (φ : A →ₐ[R] B)\n\ntheorem coe_fn_injective : @Function.Injective (A →ₐ[R] B) (A → B) (↑) :=\n FunLike.coe_injective\n#align alg_hom.coe_fn_injective AlgHom.coe_fn_injective\n\ntheorem coe_fn_inj {φ₁ φ₂ : A →ₐ[R] B} : (φ₁ : A → B) = φ₂ ↔ φ₁ = φ₂ :=\n FunLike.coe_fn_eq\n#align alg_hom.coe_fn_inj AlgHom.coe_fn_inj\n\ntheorem coe_ringHom_injective : Function.Injective ((↑) : (A →ₐ[R] B) → A →+* B) := fun φ₁ φ₂ H =>\n coe_fn_injective <| show ((φ₁ : A →+* B) : A → B) = ((φ₂ : A →+* B) : A → B) from congr_arg _ H\n#align alg_hom.coe_ring_hom_injective AlgHom.coe_ringHom_injective\n\ntheorem coe_monoidHom_injective : Function.Injective ((↑) : (A →ₐ[R] B) → A →* B) :=\n RingHom.coe_monoidHom_injective.comp coe_ringHom_injective\n#align alg_hom.coe_monoid_hom_injective AlgHom.coe_monoidHom_injective\n\ntheorem coe_addMonoidHom_injective : Function.Injective ((↑) : (A →ₐ[R] B) → A →+ B) :=\n RingHom.coe_addMonoidHom_injective.comp coe_ringHom_injective\n#align alg_hom.coe_add_monoid_hom_injective AlgHom.coe_addMonoidHom_injective\n\nprotected theorem congr_fun {φ₁ φ₂ : A →ₐ[R] B} (H : φ₁ = φ₂) (x : A) : φ₁ x = φ₂ x :=\n FunLike.congr_fun H x\n#align alg_hom.congr_fun AlgHom.congr_fun\n\nprotected theorem congr_arg (φ : A →ₐ[R] B) {x y : A} (h : x = y) : φ x = φ y :=\n FunLike.congr_arg φ h\n#align alg_hom.congr_arg AlgHom.congr_arg\n\n@[ext]\ntheorem ext {φ₁ φ₂ : A →ₐ[R] B} (H : ∀ x, φ₁ x = φ₂ x) : φ₁ = φ₂ :=\n FunLike.ext _ _ H\n#align alg_hom.ext AlgHom.ext\n\ntheorem ext_iff {φ₁ φ₂ : A →ₐ[R] B} : φ₁ = φ₂ ↔ ∀ x, φ₁ x = φ₂ x :=\n FunLike.ext_iff\n#align alg_hom.ext_iff AlgHom.ext_iff\n\n@[simp]\ntheorem mk_coe {f : A →ₐ[R] B} (h₁ h₂ h₃ h₄ h₅) : (⟨⟨⟨⟨f, h₁⟩, h₂⟩, h₃, h₄⟩, h₅⟩ : A →ₐ[R] B) = f :=\n ext fun _ => rfl\n#align alg_hom.mk_coe AlgHom.mk_coe\n\n@[simp]\ntheorem commutes (r : R) : φ (algebraMap R A r) = algebraMap R B r :=\n φ.commutes' r\n#align alg_hom.commutes AlgHom.commutes\n\ntheorem comp_algebraMap : (φ : A →+* B).comp (algebraMap R A) = algebraMap R B :=\n RingHom.ext <| φ.commutes\n#align alg_hom.comp_algebra_map AlgHom.comp_algebraMap\n\nprotected theorem map_add (r s : A) : φ (r + s) = φ r + φ s :=\n map_add _ _ _\n#align alg_hom.map_add AlgHom.map_add\n\nprotected theorem map_zero : φ 0 = 0 :=\n map_zero _\n#align alg_hom.map_zero AlgHom.map_zero\n\nprotected theorem map_mul (x y) : φ (x * y) = φ x * φ y :=\n map_mul _ _ _\n#align alg_hom.map_mul AlgHom.map_mul\n\nprotected theorem map_one : φ 1 = 1 :=\n map_one _\n#align alg_hom.map_one AlgHom.map_one\n\nprotected theorem map_pow (x : A) (n : ℕ) : φ (x ^ n) = φ x ^ n :=\n map_pow _ _ _\n#align alg_hom.map_pow AlgHom.map_pow\n\n-- @[simp] -- Porting note: simp can prove this\nprotected theorem map_smul (r : R) (x : A) : φ (r • x) = r • φ x :=\n map_smul _ _ _\n#align alg_hom.map_smul AlgHom.map_smul\n\nprotected theorem map_sum {ι : Type*} (f : ι → A) (s : Finset ι) :\n φ (∑ x in s, f x) = ∑ x in s, φ (f x) :=\n map_sum _ _ _\n#align alg_hom.map_sum AlgHom.map_sum\n\nprotected theorem map_finsupp_sum {α : Type*} [Zero α] {ι : Type*} (f : ι →₀ α) (g : ι → α → A) :\n φ (f.sum g) = f.sum fun i a => φ (g i a) :=\n map_finsupp_sum _ _ _\n#align alg_hom.map_finsupp_sum AlgHom.map_finsupp_sum\n\nset_option linter.deprecated false in\nprotected theorem map_bit0 (x) : φ (bit0 x) = bit0 (φ x) :=\n map_bit0 _ _\n#align alg_hom.map_bit0 AlgHom.map_bit0\n\nset_option linter.deprecated false in\nprotected theorem map_bit1 (x) : φ (bit1 x) = bit1 (φ x) :=\n map_bit1 _ _\n#align alg_hom.map_bit1 AlgHom.map_bit1\n\n/-- If a `RingHom` is `R`-linear, then it is an `AlgHom`. -/\ndef mk' (f : A →+* B) (h : ∀ (c : R) (x), f (c • x) = c • f x) : A →ₐ[R] B :=\n { f with\n toFun := f\n commutes' := fun c => by simp only [Algebra.algebraMap_eq_smul_one, h, f.map_one] }\n#align alg_hom.mk' AlgHom.mk'\n\n@[simp]\ntheorem coe_mk' (f : A →+* B) (h : ∀ (c : R) (x), f (c • x) = c • f x) : ⇑(mk' f h) = f :=\n rfl\n#align alg_hom.coe_mk' AlgHom.coe_mk'\n\nsection\n\nvariable (R A)\n\n/-- Identity map as an `AlgHom`. -/\nprotected def id : A →ₐ[R] A :=\n { RingHom.id A with commutes' := fun _ => rfl }\n#align alg_hom.id AlgHom.id\n\n@[simp]\ntheorem coe_id : ⇑(AlgHom.id R A) = id :=\n rfl\n#align alg_hom.coe_id AlgHom.coe_id\n\n@[simp]\ntheorem id_toRingHom : (AlgHom.id R A : A →+* A) = RingHom.id _ :=\n rfl\n#align alg_hom.id_to_ring_hom AlgHom.id_toRingHom\n\nend\n\ntheorem id_apply (p : A) : AlgHom.id R A p = p :=\n rfl\n#align alg_hom.id_apply AlgHom.id_apply\n\n/-- Composition of algebra homeomorphisms. -/\ndef comp (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) : A →ₐ[R] C :=\n { φ₁.toRingHom.comp ↑φ₂ with\n commutes' := fun r : R => by rw [← φ₁.commutes, ← φ₂.commutes]; rfl }\n#align alg_hom.comp AlgHom.comp\n\n@[simp]\ntheorem coe_comp (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) : ⇑(φ₁.comp φ₂) = φ₁ ∘ φ₂ :=\n rfl\n#align alg_hom.coe_comp AlgHom.coe_comp\n\ntheorem comp_apply (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) (p : A) : φ₁.comp φ₂ p = φ₁ (φ₂ p) :=\n rfl\n#align alg_hom.comp_apply AlgHom.comp_apply\n\ntheorem comp_toRingHom (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) :\n (φ₁.comp φ₂ : A →+* C) = (φ₁ : B →+* C).comp ↑φ₂ :=\n rfl\n#align alg_hom.comp_to_ring_hom AlgHom.comp_toRingHom\n\n@[simp]\ntheorem comp_id : φ.comp (AlgHom.id R A) = φ :=\n ext fun _x => rfl\n#align alg_hom.comp_id AlgHom.comp_id\n\n@[simp]\ntheorem id_comp : (AlgHom.id R B).comp φ = φ :=\n ext fun _x => rfl\n#align alg_hom.id_comp AlgHom.id_comp\n\ntheorem comp_assoc (φ₁ : C →ₐ[R] D) (φ₂ : B →ₐ[R] C) (φ₃ : A →ₐ[R] B) :\n (φ₁.comp φ₂).comp φ₃ = φ₁.comp (φ₂.comp φ₃) :=\n ext fun _x => rfl\n#align alg_hom.comp_assoc AlgHom.comp_assoc\n\n/-- R-Alg ⥤ R-Mod -/\ndef toLinearMap : A →ₗ[R] B where\n toFun := φ\n map_add' := map_add _\n map_smul' := map_smul _\n#align alg_hom.to_linear_map AlgHom.toLinearMap\n\n@[simp]\ntheorem toLinearMap_apply (p : A) : φ.toLinearMap p = φ p :=\n rfl\n#align alg_hom.to_linear_map_apply AlgHom.toLinearMap_apply\n\ntheorem toLinearMap_injective :\n Function.Injective (toLinearMap : _ → A →ₗ[R] B) := fun _φ₁ _φ₂ h =>\n ext <| LinearMap.congr_fun h\n#align alg_hom.to_linear_map_injective AlgHom.toLinearMap_injective\n\n@[simp]\ntheorem comp_toLinearMap (f : A →ₐ[R] B) (g : B →ₐ[R] C) :\n (g.comp f).toLinearMap = g.toLinearMap.comp f.toLinearMap :=\n rfl\n#align alg_hom.comp_to_linear_map AlgHom.comp_toLinearMap\n\n@[simp]\ntheorem toLinearMap_id : toLinearMap (AlgHom.id R A) = LinearMap.id :=\n LinearMap.ext fun _ => rfl\n#align alg_hom.to_linear_map_id AlgHom.toLinearMap_id\n\n/-- Promote a `LinearMap` to an `AlgHom` by supplying proofs about the behavior on `1` and `*`. -/\n@[simps]\ndef ofLinearMap (f : A →ₗ[R] B) (map_one : f 1 = 1) (map_mul : ∀ x y, f (x * y) = f x * f y) :\n A →ₐ[R] B :=\n { f.toAddMonoidHom with\n toFun := f\n map_one' := map_one\n map_mul' := map_mul\n commutes' := fun c => by simp only [Algebra.algebraMap_eq_smul_one, f.map_smul, map_one] }\n#align alg_hom.of_linear_map AlgHom.ofLinearMap\n\n@[simp]\ntheorem ofLinearMap_toLinearMap (map_one) (map_mul) :\n ofLinearMap φ.toLinearMap map_one map_mul = φ := by\n ext\n rfl\n#align alg_hom.of_linear_map_to_linear_map AlgHom.ofLinearMap_toLinearMap\n\n@[simp]\ntheorem toLinearMap_ofLinearMap (f : A →ₗ[R] B) (map_one) (map_mul) :\n toLinearMap (ofLinearMap f map_one map_mul) = f := by\n ext\n rfl\n#align alg_hom.to_linear_map_of_linear_map AlgHom.toLinearMap_ofLinearMap\n\n@[simp]\ntheorem ofLinearMap_id (map_one) (map_mul) :\n ofLinearMap LinearMap.id map_one map_mul = AlgHom.id R A :=\n ext fun _ => rfl\n#align alg_hom.of_linear_map_id AlgHom.ofLinearMap_id\n\ntheorem map_smul_of_tower {R'} [SMul R' A] [SMul R' B] [LinearMap.CompatibleSMul A B R' R] (r : R')\n (x : A) : φ (r • x) = r • φ x :=\n φ.toLinearMap.map_smul_of_tower r x\n#align alg_hom.map_smul_of_tower AlgHom.map_smul_of_tower\n\ntheorem map_list_prod (s : List A) : φ s.prod = (s.map φ).prod :=\n φ.toRingHom.map_list_prod s\n#align alg_hom.map_list_prod AlgHom.map_list_prod\n\n@[simps (config := .lemmasOnly) toSemigroup_toMul_mul toOne_one]\ninstance End : Monoid (A →ₐ[R] A) where\n mul := comp\n mul_assoc ϕ ψ χ := rfl\n one := AlgHom.id R A\n one_mul ϕ := ext fun x => rfl\n mul_one ϕ := ext fun x => rfl\n#align alg_hom.End AlgHom.End\n\n@[simp]\ntheorem one_apply (x : A) : (1 : A →ₐ[R] A) x = x :=\n rfl\n#align alg_hom.one_apply AlgHom.one_apply\n\n@[simp]\ntheorem mul_apply (φ ψ : A →ₐ[R] A) (x : A) : (φ * ψ) x = φ (ψ x) :=\n rfl\n#align alg_hom.mul_apply AlgHom.mul_apply\n\ntheorem algebraMap_eq_apply (f : A →ₐ[R] B) {y : R} {x : A} (h : algebraMap R A y = x) :\n algebraMap R B y = f x :=\n h ▸ (f.commutes _).symm\n#align alg_hom.algebra_map_eq_apply AlgHom.algebraMap_eq_apply\n\nend Semiring\n\nsection CommSemiring\n\nvariable [CommSemiring R] [CommSemiring A] [CommSemiring B]\n\nvariable [Algebra R A] [Algebra R B] (φ : A →ₐ[R] B)\n\nprotected theorem map_multiset_prod (s : Multiset A) : φ s.prod = (s.map φ).prod :=\n map_multiset_prod _ _\n#align alg_hom.map_multiset_prod AlgHom.map_multiset_prod\n\nprotected theorem map_prod {ι : Type*} (f : ι → A) (s : Finset ι) :\n φ (∏ x in s, f x) = ∏ x in s, φ (f x) :=\n map_prod _ _ _\n#align alg_hom.map_prod AlgHom.map_prod\n\nprotected theorem map_finsupp_prod {α : Type*} [Zero α] {ι : Type*} (f : ι →₀ α) (g : ι → α → A) :\n φ (f.prod g) = f.prod fun i a => φ (g i a) :=\n map_finsupp_prod _ _ _\n#align alg_hom.map_finsupp_prod AlgHom.map_finsupp_prod\n\nend CommSemiring\n\nsection Ring\n\nvariable [CommSemiring R] [Ring A] [Ring B]\n\nvariable [Algebra R A] [Algebra R B] (φ : A →ₐ[R] B)\n\nprotected theorem map_neg (x) : φ (-x) = -φ x :=\n map_neg _ _\n#align alg_hom.map_neg AlgHom.map_neg\n\nprotected theorem map_sub (x y) : φ (x - y) = φ x - φ y :=\n map_sub _ _ _\n#align alg_hom.map_sub AlgHom.map_sub\n\nend Ring\n\nend AlgHom\n\nnamespace RingHom\n\nvariable {R S : Type*}\n\n/-- Reinterpret a `RingHom` as an `ℕ`-algebra homomorphism. -/\ndef toNatAlgHom [Semiring R] [Semiring S] (f : R →+* S) : R →ₐ[ℕ] S :=\n { f with\n toFun := f\n commutes' := fun n => by simp }\n#align ring_hom.to_nat_alg_hom RingHom.toNatAlgHom\n\n/-- Reinterpret a `RingHom` as a `ℤ`-algebra homomorphism. -/\ndef toIntAlgHom [Ring R] [Ring S] [Algebra ℤ R] [Algebra ℤ S] (f : R →+* S) : R →ₐ[ℤ] S :=\n { f with commutes' := fun n => by simp }\n#align ring_hom.to_int_alg_hom RingHom.toIntAlgHom\n\nlemma toIntAlgHom_injective [Ring R] [Ring S] [Algebra ℤ R] [Algebra ℤ S] :\n Function.Injective (RingHom.toIntAlgHom : (R →+* S) → _) :=\n fun _ _ e ↦ FunLike.ext _ _ (fun x ↦ FunLike.congr_fun e x)\n\n/-- Reinterpret a `RingHom` as a `ℚ`-algebra homomorphism. This actually yields an equivalence,\nsee `RingHom.equivRatAlgHom`. -/\ndef toRatAlgHom [Ring R] [Ring S] [Algebra ℚ R] [Algebra ℚ S] (f : R →+* S) : R →ₐ[ℚ] S :=\n { f with commutes' := f.map_rat_algebraMap }\n#align ring_hom.to_rat_alg_hom RingHom.toRatAlgHom\n\n@[simp]\ntheorem toRatAlgHom_toRingHom [Ring R] [Ring S] [Algebra ℚ R] [Algebra ℚ S] (f : R →+* S) :\n ↑f.toRatAlgHom = f :=\n RingHom.ext fun _x => rfl\n#align ring_hom.to_rat_alg_hom_to_ring_hom RingHom.toRatAlgHom_toRingHom\n\nend RingHom\n\nsection\n\nvariable {R S : Type*}\n\n@[simp]\ntheorem AlgHom.toRingHom_toRatAlgHom [Ring R] [Ring S] [Algebra ℚ R] [Algebra ℚ S]\n (f : R →ₐ[ℚ] S) : (f : R →+* S).toRatAlgHom = f :=\n AlgHom.ext fun _x => rfl\n#align alg_hom.to_ring_hom_to_rat_alg_hom AlgHom.toRingHom_toRatAlgHom\n\n/-- The equivalence between `RingHom` and `ℚ`-algebra homomorphisms. -/\n@[simps]\ndef RingHom.equivRatAlgHom [Ring R] [Ring S] [Algebra ℚ R] [Algebra ℚ S] : (R →+* S) ≃ (R →ₐ[ℚ] S)\n where\n toFun := RingHom.toRatAlgHom\n invFun := AlgHom.toRingHom\n left_inv f := RingHom.toRatAlgHom_toRingHom f\n right_inv f := AlgHom.toRingHom_toRatAlgHom f\n#align ring_hom.equiv_rat_alg_hom RingHom.equivRatAlgHom\n\nend\n\nnamespace Algebra\n\nvariable (R : Type u) (A : Type v)\n\nvariable [CommSemiring R] [Semiring A] [Algebra R A]\n\n/-- `AlgebraMap` as an `AlgHom`. -/\ndef ofId : R →ₐ[R] A :=\n { algebraMap R A with commutes' := fun _ => rfl }\n#align algebra.of_id Algebra.ofId\n\nvariable {R}\n\ntheorem ofId_apply (r) : ofId R A r = algebraMap R A r :=\n rfl\n#align algebra.of_id_apply Algebra.ofId_apply\n\n/-- This is a special case of a more general instance that we define in a later file. -/\ninstance subsingleton_id : Subsingleton (R →ₐ[R] A) :=\n ⟨fun f g => AlgHom.ext fun _ => (f.commutes _).trans (g.commutes _).symm⟩\n\n/-- This ext lemma closes trivial subgoals create when chaining heterobasic ext lemmas. -/\n@[ext high]\ntheorem ext_id (f g : R →ₐ[R] A) : f = g := Subsingleton.elim _ _\n\nsection MulDistribMulAction\n\ninstance : MulDistribMulAction (A →ₐ[R] A) Aˣ where\n smul := fun f => Units.map f\n one_smul := fun x => by ","nextTactic":"ext","declUpToTactic":"instance : MulDistribMulAction (A →ₐ[R] A) Aˣ where\n smul := fun f => Units.map f\n one_smul := fun x => by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Hom.579_0.FxqJeqR3qcIVlrq","decl":"instance : MulDistribMulAction (A →ₐ[R] A) Aˣ where\n smul "} +{"state":"case a\nR : Type u\nA : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nx : Aˣ\n⊢ ↑(1 • x) = ↑x","srcUpToTactic":"/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Yury Kudryashov\n-/\nimport Mathlib.Algebra.Algebra.Basic\nimport Mathlib.Algebra.BigOperators.Finsupp\n\n#align_import algebra.algebra.hom from \"leanprover-community/mathlib\"@\"e97cf15cd1aec9bd5c193b2ffac5a6dc9118912b\"\n\n/-!\n# Homomorphisms of `R`-algebras\n\nThis file defines bundled homomorphisms of `R`-algebras.\n\n## Main definitions\n\n* `AlgHom R A B`: the type of `R`-algebra morphisms from `A` to `B`.\n* `Algebra.ofId R A : R →ₐ[R] A`: the canonical map from `R` to `A`, as an `AlgHom`.\n\n## Notations\n\n* `A →ₐ[R] B` : `R`-algebra homomorphism from `A` to `B`.\n-/\n\nset_option autoImplicit true\n\n\nopen BigOperators\n\nuniverse u v w u₁ v₁\n\n/-- Defining the homomorphism in the category R-Alg. -/\n-- @[nolint has_nonempty_instance] -- Porting note: This linter does not exist yet.\nstructure AlgHom (R : Type u) (A : Type v) (B : Type w) [CommSemiring R] [Semiring A] [Semiring B]\n [Algebra R A] [Algebra R B] extends RingHom A B where\n commutes' : ∀ r : R, toFun (algebraMap R A r) = algebraMap R B r\n#align alg_hom AlgHom\n\n/-- Reinterpret an `AlgHom` as a `RingHom` -/\nadd_decl_doc AlgHom.toRingHom\n\n@[inherit_doc AlgHom]\ninfixr:25 \" →ₐ \" => AlgHom _\n\n@[inherit_doc]\nnotation:25 A \" →ₐ[\" R \"] \" B => AlgHom R A B\n\n/-- `AlgHomClass F R A B` asserts `F` is a type of bundled algebra homomorphisms\nfrom `A` to `B`. -/\nclass AlgHomClass (F : Type*) (R : outParam (Type*)) (A : outParam (Type*))\n (B : outParam (Type*)) [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A]\n [Algebra R B] extends RingHomClass F A B where\n commutes : ∀ (f : F) (r : R), f (algebraMap R A r) = algebraMap R B r\n#align alg_hom_class AlgHomClass\n\n-- Porting note: `dangerousInstance` linter has become smarter about `outParam`s\n-- attribute [nolint dangerousInstance] AlgHomClass.toRingHomClass\n\n-- Porting note: simp can prove this\n-- attribute [simp] AlgHomClass.commutes\n\nnamespace AlgHomClass\n\nvariable {R : Type*} {A : Type*} {B : Type*} [CommSemiring R] [Semiring A] [Semiring B]\n [Algebra R A] [Algebra R B]\n\n-- see Note [lower instance priority]\ninstance (priority := 100) linearMapClass [AlgHomClass F R A B] : LinearMapClass F R A B :=\n { ‹AlgHomClass F R A B› with\n map_smulₛₗ := fun f r x => by\n simp only [Algebra.smul_def, map_mul, commutes, RingHom.id_apply] }\n#align alg_hom_class.linear_map_class AlgHomClass.linearMapClass\n\n-- Porting note: A new definition underlying a coercion `↑`.\n/-- Turn an element of a type `F` satisfying `AlgHomClass F α β` into an actual\n`AlgHom`. This is declared as the default coercion from `F` to `α →+* β`. -/\n@[coe]\ndef toAlgHom {F : Type*} [AlgHomClass F R A B] (f : F) : A →ₐ[R] B :=\n { (f : A →+* B) with\n toFun := f\n commutes' := AlgHomClass.commutes f }\n\ninstance coeTC {F : Type*} [AlgHomClass F R A B] : CoeTC F (A →ₐ[R] B) :=\n ⟨AlgHomClass.toAlgHom⟩\n#align alg_hom_class.alg_hom.has_coe_t AlgHomClass.coeTC\n\nend AlgHomClass\n\nnamespace AlgHom\n\nvariable {R : Type u} {A : Type v} {B : Type w} {C : Type u₁} {D : Type v₁}\n\nsection Semiring\n\nvariable [CommSemiring R] [Semiring A] [Semiring B] [Semiring C] [Semiring D]\n\nvariable [Algebra R A] [Algebra R B] [Algebra R C] [Algebra R D]\n\n-- Porting note: we don't port specialized `CoeFun` instances if there is `FunLike` instead\n#noalign alg_hom.has_coe_to_fun\n\n-- Porting note: This instance is moved.\ninstance algHomClass : AlgHomClass (A →ₐ[R] B) R A B where\n coe f := f.toFun\n coe_injective' f g h := by\n rcases f with ⟨⟨⟨⟨_, _⟩, _⟩, _, _⟩, _⟩\n rcases g with ⟨⟨⟨⟨_, _⟩, _⟩, _, _⟩, _⟩\n congr\n map_add f := f.map_add'\n map_zero f := f.map_zero'\n map_mul f := f.map_mul'\n map_one f := f.map_one'\n commutes f := f.commutes'\n#align alg_hom.alg_hom_class AlgHom.algHomClass\n\n/-- See Note [custom simps projection] -/\ndef Simps.apply {R : Type u} {α : Type v} {β : Type w} [CommSemiring R]\n [Semiring α] [Semiring β] [Algebra R α] [Algebra R β] (f : α →ₐ[R] β) : α → β := f\n\ninitialize_simps_projections AlgHom (toFun → apply)\n\n@[simp]\nprotected theorem coe_coe {F : Type*} [AlgHomClass F R A B] (f : F) : ⇑(f : A →ₐ[R] B) = f :=\n rfl\n#align alg_hom.coe_coe AlgHom.coe_coe\n\n@[simp]\ntheorem toFun_eq_coe (f : A →ₐ[R] B) : f.toFun = f :=\n rfl\n#align alg_hom.to_fun_eq_coe AlgHom.toFun_eq_coe\n\nattribute [coe] AlgHom.toRingHom\n\ninstance coeOutRingHom : CoeOut (A →ₐ[R] B) (A →+* B) :=\n ⟨AlgHom.toRingHom⟩\n#align alg_hom.coe_ring_hom AlgHom.coeOutRingHom\n\n-- Porting note: A new definition underlying a coercion `↑`.\n@[coe]\ndef toMonoidHom' (f : A →ₐ[R] B) : A →* B := (f : A →+* B)\n\ninstance coeOutMonoidHom : CoeOut (A →ₐ[R] B) (A →* B) :=\n ⟨AlgHom.toMonoidHom'⟩\n#align alg_hom.coe_monoid_hom AlgHom.coeOutMonoidHom\n\n-- Porting note: A new definition underlying a coercion `↑`.\n@[coe]\ndef toAddMonoidHom' (f : A →ₐ[R] B) : A →+ B := (f : A →+* B)\n\ninstance coeOutAddMonoidHom : CoeOut (A →ₐ[R] B) (A →+ B) :=\n ⟨AlgHom.toAddMonoidHom'⟩\n#align alg_hom.coe_add_monoid_hom AlgHom.coeOutAddMonoidHom\n\n-- Porting note: Lean 3: `@[simp, norm_cast] coe_mk`\n-- Lean 4: `@[simp] coe_mk` & `@[norm_cast] coe_mks`\n@[simp]\ntheorem coe_mk {f : A →+* B} (h) : ((⟨f, h⟩ : A →ₐ[R] B) : A → B) = f :=\n rfl\n\n@[norm_cast]\ntheorem coe_mks {f : A → B} (h₁ h₂ h₃ h₄ h₅) : ⇑(⟨⟨⟨⟨f, h₁⟩, h₂⟩, h₃, h₄⟩, h₅⟩ : A →ₐ[R] B) = f :=\n rfl\n#align alg_hom.coe_mk AlgHom.coe_mks\n\n-- Porting note: This theorem is new.\n@[simp, norm_cast]\ntheorem coe_ringHom_mk {f : A →+* B} (h) : ((⟨f, h⟩ : A →ₐ[R] B) : A →+* B) = f :=\n rfl\n\n-- make the coercion the simp-normal form\n@[simp]\ntheorem toRingHom_eq_coe (f : A →ₐ[R] B) : f.toRingHom = f :=\n rfl\n#align alg_hom.to_ring_hom_eq_coe AlgHom.toRingHom_eq_coe\n\n@[simp, norm_cast]\ntheorem coe_toRingHom (f : A →ₐ[R] B) : ⇑(f : A →+* B) = f :=\n rfl\n#align alg_hom.coe_to_ring_hom AlgHom.coe_toRingHom\n\n@[simp, norm_cast]\ntheorem coe_toMonoidHom (f : A →ₐ[R] B) : ⇑(f : A →* B) = f :=\n rfl\n#align alg_hom.coe_to_monoid_hom AlgHom.coe_toMonoidHom\n\n@[simp, norm_cast]\ntheorem coe_toAddMonoidHom (f : A →ₐ[R] B) : ⇑(f : A →+ B) = f :=\n rfl\n#align alg_hom.coe_to_add_monoid_hom AlgHom.coe_toAddMonoidHom\n\nvariable (φ : A →ₐ[R] B)\n\ntheorem coe_fn_injective : @Function.Injective (A →ₐ[R] B) (A → B) (↑) :=\n FunLike.coe_injective\n#align alg_hom.coe_fn_injective AlgHom.coe_fn_injective\n\ntheorem coe_fn_inj {φ₁ φ₂ : A →ₐ[R] B} : (φ₁ : A → B) = φ₂ ↔ φ₁ = φ₂ :=\n FunLike.coe_fn_eq\n#align alg_hom.coe_fn_inj AlgHom.coe_fn_inj\n\ntheorem coe_ringHom_injective : Function.Injective ((↑) : (A →ₐ[R] B) → A →+* B) := fun φ₁ φ₂ H =>\n coe_fn_injective <| show ((φ₁ : A →+* B) : A → B) = ((φ₂ : A →+* B) : A → B) from congr_arg _ H\n#align alg_hom.coe_ring_hom_injective AlgHom.coe_ringHom_injective\n\ntheorem coe_monoidHom_injective : Function.Injective ((↑) : (A →ₐ[R] B) → A →* B) :=\n RingHom.coe_monoidHom_injective.comp coe_ringHom_injective\n#align alg_hom.coe_monoid_hom_injective AlgHom.coe_monoidHom_injective\n\ntheorem coe_addMonoidHom_injective : Function.Injective ((↑) : (A →ₐ[R] B) → A →+ B) :=\n RingHom.coe_addMonoidHom_injective.comp coe_ringHom_injective\n#align alg_hom.coe_add_monoid_hom_injective AlgHom.coe_addMonoidHom_injective\n\nprotected theorem congr_fun {φ₁ φ₂ : A →ₐ[R] B} (H : φ₁ = φ₂) (x : A) : φ₁ x = φ₂ x :=\n FunLike.congr_fun H x\n#align alg_hom.congr_fun AlgHom.congr_fun\n\nprotected theorem congr_arg (φ : A →ₐ[R] B) {x y : A} (h : x = y) : φ x = φ y :=\n FunLike.congr_arg φ h\n#align alg_hom.congr_arg AlgHom.congr_arg\n\n@[ext]\ntheorem ext {φ₁ φ₂ : A →ₐ[R] B} (H : ∀ x, φ₁ x = φ₂ x) : φ₁ = φ₂ :=\n FunLike.ext _ _ H\n#align alg_hom.ext AlgHom.ext\n\ntheorem ext_iff {φ₁ φ₂ : A →ₐ[R] B} : φ₁ = φ₂ ↔ ∀ x, φ₁ x = φ₂ x :=\n FunLike.ext_iff\n#align alg_hom.ext_iff AlgHom.ext_iff\n\n@[simp]\ntheorem mk_coe {f : A →ₐ[R] B} (h₁ h₂ h₃ h₄ h₅) : (⟨⟨⟨⟨f, h₁⟩, h₂⟩, h₃, h₄⟩, h₅⟩ : A →ₐ[R] B) = f :=\n ext fun _ => rfl\n#align alg_hom.mk_coe AlgHom.mk_coe\n\n@[simp]\ntheorem commutes (r : R) : φ (algebraMap R A r) = algebraMap R B r :=\n φ.commutes' r\n#align alg_hom.commutes AlgHom.commutes\n\ntheorem comp_algebraMap : (φ : A →+* B).comp (algebraMap R A) = algebraMap R B :=\n RingHom.ext <| φ.commutes\n#align alg_hom.comp_algebra_map AlgHom.comp_algebraMap\n\nprotected theorem map_add (r s : A) : φ (r + s) = φ r + φ s :=\n map_add _ _ _\n#align alg_hom.map_add AlgHom.map_add\n\nprotected theorem map_zero : φ 0 = 0 :=\n map_zero _\n#align alg_hom.map_zero AlgHom.map_zero\n\nprotected theorem map_mul (x y) : φ (x * y) = φ x * φ y :=\n map_mul _ _ _\n#align alg_hom.map_mul AlgHom.map_mul\n\nprotected theorem map_one : φ 1 = 1 :=\n map_one _\n#align alg_hom.map_one AlgHom.map_one\n\nprotected theorem map_pow (x : A) (n : ℕ) : φ (x ^ n) = φ x ^ n :=\n map_pow _ _ _\n#align alg_hom.map_pow AlgHom.map_pow\n\n-- @[simp] -- Porting note: simp can prove this\nprotected theorem map_smul (r : R) (x : A) : φ (r • x) = r • φ x :=\n map_smul _ _ _\n#align alg_hom.map_smul AlgHom.map_smul\n\nprotected theorem map_sum {ι : Type*} (f : ι → A) (s : Finset ι) :\n φ (∑ x in s, f x) = ∑ x in s, φ (f x) :=\n map_sum _ _ _\n#align alg_hom.map_sum AlgHom.map_sum\n\nprotected theorem map_finsupp_sum {α : Type*} [Zero α] {ι : Type*} (f : ι →₀ α) (g : ι → α → A) :\n φ (f.sum g) = f.sum fun i a => φ (g i a) :=\n map_finsupp_sum _ _ _\n#align alg_hom.map_finsupp_sum AlgHom.map_finsupp_sum\n\nset_option linter.deprecated false in\nprotected theorem map_bit0 (x) : φ (bit0 x) = bit0 (φ x) :=\n map_bit0 _ _\n#align alg_hom.map_bit0 AlgHom.map_bit0\n\nset_option linter.deprecated false in\nprotected theorem map_bit1 (x) : φ (bit1 x) = bit1 (φ x) :=\n map_bit1 _ _\n#align alg_hom.map_bit1 AlgHom.map_bit1\n\n/-- If a `RingHom` is `R`-linear, then it is an `AlgHom`. -/\ndef mk' (f : A →+* B) (h : ∀ (c : R) (x), f (c • x) = c • f x) : A →ₐ[R] B :=\n { f with\n toFun := f\n commutes' := fun c => by simp only [Algebra.algebraMap_eq_smul_one, h, f.map_one] }\n#align alg_hom.mk' AlgHom.mk'\n\n@[simp]\ntheorem coe_mk' (f : A →+* B) (h : ∀ (c : R) (x), f (c • x) = c • f x) : ⇑(mk' f h) = f :=\n rfl\n#align alg_hom.coe_mk' AlgHom.coe_mk'\n\nsection\n\nvariable (R A)\n\n/-- Identity map as an `AlgHom`. -/\nprotected def id : A →ₐ[R] A :=\n { RingHom.id A with commutes' := fun _ => rfl }\n#align alg_hom.id AlgHom.id\n\n@[simp]\ntheorem coe_id : ⇑(AlgHom.id R A) = id :=\n rfl\n#align alg_hom.coe_id AlgHom.coe_id\n\n@[simp]\ntheorem id_toRingHom : (AlgHom.id R A : A →+* A) = RingHom.id _ :=\n rfl\n#align alg_hom.id_to_ring_hom AlgHom.id_toRingHom\n\nend\n\ntheorem id_apply (p : A) : AlgHom.id R A p = p :=\n rfl\n#align alg_hom.id_apply AlgHom.id_apply\n\n/-- Composition of algebra homeomorphisms. -/\ndef comp (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) : A →ₐ[R] C :=\n { φ₁.toRingHom.comp ↑φ₂ with\n commutes' := fun r : R => by rw [← φ₁.commutes, ← φ₂.commutes]; rfl }\n#align alg_hom.comp AlgHom.comp\n\n@[simp]\ntheorem coe_comp (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) : ⇑(φ₁.comp φ₂) = φ₁ ∘ φ₂ :=\n rfl\n#align alg_hom.coe_comp AlgHom.coe_comp\n\ntheorem comp_apply (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) (p : A) : φ₁.comp φ₂ p = φ₁ (φ₂ p) :=\n rfl\n#align alg_hom.comp_apply AlgHom.comp_apply\n\ntheorem comp_toRingHom (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) :\n (φ₁.comp φ₂ : A →+* C) = (φ₁ : B →+* C).comp ↑φ₂ :=\n rfl\n#align alg_hom.comp_to_ring_hom AlgHom.comp_toRingHom\n\n@[simp]\ntheorem comp_id : φ.comp (AlgHom.id R A) = φ :=\n ext fun _x => rfl\n#align alg_hom.comp_id AlgHom.comp_id\n\n@[simp]\ntheorem id_comp : (AlgHom.id R B).comp φ = φ :=\n ext fun _x => rfl\n#align alg_hom.id_comp AlgHom.id_comp\n\ntheorem comp_assoc (φ₁ : C →ₐ[R] D) (φ₂ : B →ₐ[R] C) (φ₃ : A →ₐ[R] B) :\n (φ₁.comp φ₂).comp φ₃ = φ₁.comp (φ₂.comp φ₃) :=\n ext fun _x => rfl\n#align alg_hom.comp_assoc AlgHom.comp_assoc\n\n/-- R-Alg ⥤ R-Mod -/\ndef toLinearMap : A →ₗ[R] B where\n toFun := φ\n map_add' := map_add _\n map_smul' := map_smul _\n#align alg_hom.to_linear_map AlgHom.toLinearMap\n\n@[simp]\ntheorem toLinearMap_apply (p : A) : φ.toLinearMap p = φ p :=\n rfl\n#align alg_hom.to_linear_map_apply AlgHom.toLinearMap_apply\n\ntheorem toLinearMap_injective :\n Function.Injective (toLinearMap : _ → A →ₗ[R] B) := fun _φ₁ _φ₂ h =>\n ext <| LinearMap.congr_fun h\n#align alg_hom.to_linear_map_injective AlgHom.toLinearMap_injective\n\n@[simp]\ntheorem comp_toLinearMap (f : A →ₐ[R] B) (g : B →ₐ[R] C) :\n (g.comp f).toLinearMap = g.toLinearMap.comp f.toLinearMap :=\n rfl\n#align alg_hom.comp_to_linear_map AlgHom.comp_toLinearMap\n\n@[simp]\ntheorem toLinearMap_id : toLinearMap (AlgHom.id R A) = LinearMap.id :=\n LinearMap.ext fun _ => rfl\n#align alg_hom.to_linear_map_id AlgHom.toLinearMap_id\n\n/-- Promote a `LinearMap` to an `AlgHom` by supplying proofs about the behavior on `1` and `*`. -/\n@[simps]\ndef ofLinearMap (f : A →ₗ[R] B) (map_one : f 1 = 1) (map_mul : ∀ x y, f (x * y) = f x * f y) :\n A →ₐ[R] B :=\n { f.toAddMonoidHom with\n toFun := f\n map_one' := map_one\n map_mul' := map_mul\n commutes' := fun c => by simp only [Algebra.algebraMap_eq_smul_one, f.map_smul, map_one] }\n#align alg_hom.of_linear_map AlgHom.ofLinearMap\n\n@[simp]\ntheorem ofLinearMap_toLinearMap (map_one) (map_mul) :\n ofLinearMap φ.toLinearMap map_one map_mul = φ := by\n ext\n rfl\n#align alg_hom.of_linear_map_to_linear_map AlgHom.ofLinearMap_toLinearMap\n\n@[simp]\ntheorem toLinearMap_ofLinearMap (f : A →ₗ[R] B) (map_one) (map_mul) :\n toLinearMap (ofLinearMap f map_one map_mul) = f := by\n ext\n rfl\n#align alg_hom.to_linear_map_of_linear_map AlgHom.toLinearMap_ofLinearMap\n\n@[simp]\ntheorem ofLinearMap_id (map_one) (map_mul) :\n ofLinearMap LinearMap.id map_one map_mul = AlgHom.id R A :=\n ext fun _ => rfl\n#align alg_hom.of_linear_map_id AlgHom.ofLinearMap_id\n\ntheorem map_smul_of_tower {R'} [SMul R' A] [SMul R' B] [LinearMap.CompatibleSMul A B R' R] (r : R')\n (x : A) : φ (r • x) = r • φ x :=\n φ.toLinearMap.map_smul_of_tower r x\n#align alg_hom.map_smul_of_tower AlgHom.map_smul_of_tower\n\ntheorem map_list_prod (s : List A) : φ s.prod = (s.map φ).prod :=\n φ.toRingHom.map_list_prod s\n#align alg_hom.map_list_prod AlgHom.map_list_prod\n\n@[simps (config := .lemmasOnly) toSemigroup_toMul_mul toOne_one]\ninstance End : Monoid (A →ₐ[R] A) where\n mul := comp\n mul_assoc ϕ ψ χ := rfl\n one := AlgHom.id R A\n one_mul ϕ := ext fun x => rfl\n mul_one ϕ := ext fun x => rfl\n#align alg_hom.End AlgHom.End\n\n@[simp]\ntheorem one_apply (x : A) : (1 : A →ₐ[R] A) x = x :=\n rfl\n#align alg_hom.one_apply AlgHom.one_apply\n\n@[simp]\ntheorem mul_apply (φ ψ : A →ₐ[R] A) (x : A) : (φ * ψ) x = φ (ψ x) :=\n rfl\n#align alg_hom.mul_apply AlgHom.mul_apply\n\ntheorem algebraMap_eq_apply (f : A →ₐ[R] B) {y : R} {x : A} (h : algebraMap R A y = x) :\n algebraMap R B y = f x :=\n h ▸ (f.commutes _).symm\n#align alg_hom.algebra_map_eq_apply AlgHom.algebraMap_eq_apply\n\nend Semiring\n\nsection CommSemiring\n\nvariable [CommSemiring R] [CommSemiring A] [CommSemiring B]\n\nvariable [Algebra R A] [Algebra R B] (φ : A →ₐ[R] B)\n\nprotected theorem map_multiset_prod (s : Multiset A) : φ s.prod = (s.map φ).prod :=\n map_multiset_prod _ _\n#align alg_hom.map_multiset_prod AlgHom.map_multiset_prod\n\nprotected theorem map_prod {ι : Type*} (f : ι → A) (s : Finset ι) :\n φ (∏ x in s, f x) = ∏ x in s, φ (f x) :=\n map_prod _ _ _\n#align alg_hom.map_prod AlgHom.map_prod\n\nprotected theorem map_finsupp_prod {α : Type*} [Zero α] {ι : Type*} (f : ι →₀ α) (g : ι → α → A) :\n φ (f.prod g) = f.prod fun i a => φ (g i a) :=\n map_finsupp_prod _ _ _\n#align alg_hom.map_finsupp_prod AlgHom.map_finsupp_prod\n\nend CommSemiring\n\nsection Ring\n\nvariable [CommSemiring R] [Ring A] [Ring B]\n\nvariable [Algebra R A] [Algebra R B] (φ : A →ₐ[R] B)\n\nprotected theorem map_neg (x) : φ (-x) = -φ x :=\n map_neg _ _\n#align alg_hom.map_neg AlgHom.map_neg\n\nprotected theorem map_sub (x y) : φ (x - y) = φ x - φ y :=\n map_sub _ _ _\n#align alg_hom.map_sub AlgHom.map_sub\n\nend Ring\n\nend AlgHom\n\nnamespace RingHom\n\nvariable {R S : Type*}\n\n/-- Reinterpret a `RingHom` as an `ℕ`-algebra homomorphism. -/\ndef toNatAlgHom [Semiring R] [Semiring S] (f : R →+* S) : R →ₐ[ℕ] S :=\n { f with\n toFun := f\n commutes' := fun n => by simp }\n#align ring_hom.to_nat_alg_hom RingHom.toNatAlgHom\n\n/-- Reinterpret a `RingHom` as a `ℤ`-algebra homomorphism. -/\ndef toIntAlgHom [Ring R] [Ring S] [Algebra ℤ R] [Algebra ℤ S] (f : R →+* S) : R →ₐ[ℤ] S :=\n { f with commutes' := fun n => by simp }\n#align ring_hom.to_int_alg_hom RingHom.toIntAlgHom\n\nlemma toIntAlgHom_injective [Ring R] [Ring S] [Algebra ℤ R] [Algebra ℤ S] :\n Function.Injective (RingHom.toIntAlgHom : (R →+* S) → _) :=\n fun _ _ e ↦ FunLike.ext _ _ (fun x ↦ FunLike.congr_fun e x)\n\n/-- Reinterpret a `RingHom` as a `ℚ`-algebra homomorphism. This actually yields an equivalence,\nsee `RingHom.equivRatAlgHom`. -/\ndef toRatAlgHom [Ring R] [Ring S] [Algebra ℚ R] [Algebra ℚ S] (f : R →+* S) : R →ₐ[ℚ] S :=\n { f with commutes' := f.map_rat_algebraMap }\n#align ring_hom.to_rat_alg_hom RingHom.toRatAlgHom\n\n@[simp]\ntheorem toRatAlgHom_toRingHom [Ring R] [Ring S] [Algebra ℚ R] [Algebra ℚ S] (f : R →+* S) :\n ↑f.toRatAlgHom = f :=\n RingHom.ext fun _x => rfl\n#align ring_hom.to_rat_alg_hom_to_ring_hom RingHom.toRatAlgHom_toRingHom\n\nend RingHom\n\nsection\n\nvariable {R S : Type*}\n\n@[simp]\ntheorem AlgHom.toRingHom_toRatAlgHom [Ring R] [Ring S] [Algebra ℚ R] [Algebra ℚ S]\n (f : R →ₐ[ℚ] S) : (f : R →+* S).toRatAlgHom = f :=\n AlgHom.ext fun _x => rfl\n#align alg_hom.to_ring_hom_to_rat_alg_hom AlgHom.toRingHom_toRatAlgHom\n\n/-- The equivalence between `RingHom` and `ℚ`-algebra homomorphisms. -/\n@[simps]\ndef RingHom.equivRatAlgHom [Ring R] [Ring S] [Algebra ℚ R] [Algebra ℚ S] : (R →+* S) ≃ (R →ₐ[ℚ] S)\n where\n toFun := RingHom.toRatAlgHom\n invFun := AlgHom.toRingHom\n left_inv f := RingHom.toRatAlgHom_toRingHom f\n right_inv f := AlgHom.toRingHom_toRatAlgHom f\n#align ring_hom.equiv_rat_alg_hom RingHom.equivRatAlgHom\n\nend\n\nnamespace Algebra\n\nvariable (R : Type u) (A : Type v)\n\nvariable [CommSemiring R] [Semiring A] [Algebra R A]\n\n/-- `AlgebraMap` as an `AlgHom`. -/\ndef ofId : R →ₐ[R] A :=\n { algebraMap R A with commutes' := fun _ => rfl }\n#align algebra.of_id Algebra.ofId\n\nvariable {R}\n\ntheorem ofId_apply (r) : ofId R A r = algebraMap R A r :=\n rfl\n#align algebra.of_id_apply Algebra.ofId_apply\n\n/-- This is a special case of a more general instance that we define in a later file. -/\ninstance subsingleton_id : Subsingleton (R →ₐ[R] A) :=\n ⟨fun f g => AlgHom.ext fun _ => (f.commutes _).trans (g.commutes _).symm⟩\n\n/-- This ext lemma closes trivial subgoals create when chaining heterobasic ext lemmas. -/\n@[ext high]\ntheorem ext_id (f g : R →ₐ[R] A) : f = g := Subsingleton.elim _ _\n\nsection MulDistribMulAction\n\ninstance : MulDistribMulAction (A →ₐ[R] A) Aˣ where\n smul := fun f => Units.map f\n one_smul := fun x => by ext; ","nextTactic":"rfl","declUpToTactic":"instance : MulDistribMulAction (A →ₐ[R] A) Aˣ where\n smul := fun f => Units.map f\n one_smul := fun x => by ext; ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Hom.579_0.FxqJeqR3qcIVlrq","decl":"instance : MulDistribMulAction (A →ₐ[R] A) Aˣ where\n smul "} +{"state":"R : Type u\nA : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nx y : A →ₐ[R] A\nz : Aˣ\n⊢ (x * y) • z = x • y • z","srcUpToTactic":"/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Yury Kudryashov\n-/\nimport Mathlib.Algebra.Algebra.Basic\nimport Mathlib.Algebra.BigOperators.Finsupp\n\n#align_import algebra.algebra.hom from \"leanprover-community/mathlib\"@\"e97cf15cd1aec9bd5c193b2ffac5a6dc9118912b\"\n\n/-!\n# Homomorphisms of `R`-algebras\n\nThis file defines bundled homomorphisms of `R`-algebras.\n\n## Main definitions\n\n* `AlgHom R A B`: the type of `R`-algebra morphisms from `A` to `B`.\n* `Algebra.ofId R A : R →ₐ[R] A`: the canonical map from `R` to `A`, as an `AlgHom`.\n\n## Notations\n\n* `A →ₐ[R] B` : `R`-algebra homomorphism from `A` to `B`.\n-/\n\nset_option autoImplicit true\n\n\nopen BigOperators\n\nuniverse u v w u₁ v₁\n\n/-- Defining the homomorphism in the category R-Alg. -/\n-- @[nolint has_nonempty_instance] -- Porting note: This linter does not exist yet.\nstructure AlgHom (R : Type u) (A : Type v) (B : Type w) [CommSemiring R] [Semiring A] [Semiring B]\n [Algebra R A] [Algebra R B] extends RingHom A B where\n commutes' : ∀ r : R, toFun (algebraMap R A r) = algebraMap R B r\n#align alg_hom AlgHom\n\n/-- Reinterpret an `AlgHom` as a `RingHom` -/\nadd_decl_doc AlgHom.toRingHom\n\n@[inherit_doc AlgHom]\ninfixr:25 \" →ₐ \" => AlgHom _\n\n@[inherit_doc]\nnotation:25 A \" →ₐ[\" R \"] \" B => AlgHom R A B\n\n/-- `AlgHomClass F R A B` asserts `F` is a type of bundled algebra homomorphisms\nfrom `A` to `B`. -/\nclass AlgHomClass (F : Type*) (R : outParam (Type*)) (A : outParam (Type*))\n (B : outParam (Type*)) [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A]\n [Algebra R B] extends RingHomClass F A B where\n commutes : ∀ (f : F) (r : R), f (algebraMap R A r) = algebraMap R B r\n#align alg_hom_class AlgHomClass\n\n-- Porting note: `dangerousInstance` linter has become smarter about `outParam`s\n-- attribute [nolint dangerousInstance] AlgHomClass.toRingHomClass\n\n-- Porting note: simp can prove this\n-- attribute [simp] AlgHomClass.commutes\n\nnamespace AlgHomClass\n\nvariable {R : Type*} {A : Type*} {B : Type*} [CommSemiring R] [Semiring A] [Semiring B]\n [Algebra R A] [Algebra R B]\n\n-- see Note [lower instance priority]\ninstance (priority := 100) linearMapClass [AlgHomClass F R A B] : LinearMapClass F R A B :=\n { ‹AlgHomClass F R A B› with\n map_smulₛₗ := fun f r x => by\n simp only [Algebra.smul_def, map_mul, commutes, RingHom.id_apply] }\n#align alg_hom_class.linear_map_class AlgHomClass.linearMapClass\n\n-- Porting note: A new definition underlying a coercion `↑`.\n/-- Turn an element of a type `F` satisfying `AlgHomClass F α β` into an actual\n`AlgHom`. This is declared as the default coercion from `F` to `α →+* β`. -/\n@[coe]\ndef toAlgHom {F : Type*} [AlgHomClass F R A B] (f : F) : A →ₐ[R] B :=\n { (f : A →+* B) with\n toFun := f\n commutes' := AlgHomClass.commutes f }\n\ninstance coeTC {F : Type*} [AlgHomClass F R A B] : CoeTC F (A →ₐ[R] B) :=\n ⟨AlgHomClass.toAlgHom⟩\n#align alg_hom_class.alg_hom.has_coe_t AlgHomClass.coeTC\n\nend AlgHomClass\n\nnamespace AlgHom\n\nvariable {R : Type u} {A : Type v} {B : Type w} {C : Type u₁} {D : Type v₁}\n\nsection Semiring\n\nvariable [CommSemiring R] [Semiring A] [Semiring B] [Semiring C] [Semiring D]\n\nvariable [Algebra R A] [Algebra R B] [Algebra R C] [Algebra R D]\n\n-- Porting note: we don't port specialized `CoeFun` instances if there is `FunLike` instead\n#noalign alg_hom.has_coe_to_fun\n\n-- Porting note: This instance is moved.\ninstance algHomClass : AlgHomClass (A →ₐ[R] B) R A B where\n coe f := f.toFun\n coe_injective' f g h := by\n rcases f with ⟨⟨⟨���_, _⟩, _⟩, _, _⟩, _⟩\n rcases g with ⟨⟨⟨⟨_, _⟩, _⟩, _, _⟩, _⟩\n congr\n map_add f := f.map_add'\n map_zero f := f.map_zero'\n map_mul f := f.map_mul'\n map_one f := f.map_one'\n commutes f := f.commutes'\n#align alg_hom.alg_hom_class AlgHom.algHomClass\n\n/-- See Note [custom simps projection] -/\ndef Simps.apply {R : Type u} {α : Type v} {β : Type w} [CommSemiring R]\n [Semiring α] [Semiring β] [Algebra R α] [Algebra R β] (f : α →ₐ[R] β) : α → β := f\n\ninitialize_simps_projections AlgHom (toFun → apply)\n\n@[simp]\nprotected theorem coe_coe {F : Type*} [AlgHomClass F R A B] (f : F) : ⇑(f : A →ₐ[R] B) = f :=\n rfl\n#align alg_hom.coe_coe AlgHom.coe_coe\n\n@[simp]\ntheorem toFun_eq_coe (f : A →ₐ[R] B) : f.toFun = f :=\n rfl\n#align alg_hom.to_fun_eq_coe AlgHom.toFun_eq_coe\n\nattribute [coe] AlgHom.toRingHom\n\ninstance coeOutRingHom : CoeOut (A →ₐ[R] B) (A →+* B) :=\n ⟨AlgHom.toRingHom⟩\n#align alg_hom.coe_ring_hom AlgHom.coeOutRingHom\n\n-- Porting note: A new definition underlying a coercion `↑`.\n@[coe]\ndef toMonoidHom' (f : A →ₐ[R] B) : A →* B := (f : A →+* B)\n\ninstance coeOutMonoidHom : CoeOut (A →ₐ[R] B) (A →* B) :=\n ⟨AlgHom.toMonoidHom'⟩\n#align alg_hom.coe_monoid_hom AlgHom.coeOutMonoidHom\n\n-- Porting note: A new definition underlying a coercion `↑`.\n@[coe]\ndef toAddMonoidHom' (f : A →ₐ[R] B) : A →+ B := (f : A →+* B)\n\ninstance coeOutAddMonoidHom : CoeOut (A →ₐ[R] B) (A →+ B) :=\n ⟨AlgHom.toAddMonoidHom'⟩\n#align alg_hom.coe_add_monoid_hom AlgHom.coeOutAddMonoidHom\n\n-- Porting note: Lean 3: `@[simp, norm_cast] coe_mk`\n-- Lean 4: `@[simp] coe_mk` & `@[norm_cast] coe_mks`\n@[simp]\ntheorem coe_mk {f : A →+* B} (h) : ((⟨f, h⟩ : A →ₐ[R] B) : A → B) = f :=\n rfl\n\n@[norm_cast]\ntheorem coe_mks {f : A → B} (h₁ h₂ h₃ h₄ h₅) : ⇑(⟨⟨⟨⟨f, h₁⟩, h₂⟩, h₃, h₄⟩, h₅⟩ : A →ₐ[R] B) = f :=\n rfl\n#align alg_hom.coe_mk AlgHom.coe_mks\n\n-- Porting note: This theorem is new.\n@[simp, norm_cast]\ntheorem coe_ringHom_mk {f : A →+* B} (h) : ((⟨f, h⟩ : A →ₐ[R] B) : A →+* B) = f :=\n rfl\n\n-- make the coercion the simp-normal form\n@[simp]\ntheorem toRingHom_eq_coe (f : A →ₐ[R] B) : f.toRingHom = f :=\n rfl\n#align alg_hom.to_ring_hom_eq_coe AlgHom.toRingHom_eq_coe\n\n@[simp, norm_cast]\ntheorem coe_toRingHom (f : A →ₐ[R] B) : ⇑(f : A →+* B) = f :=\n rfl\n#align alg_hom.coe_to_ring_hom AlgHom.coe_toRingHom\n\n@[simp, norm_cast]\ntheorem coe_toMonoidHom (f : A →ₐ[R] B) : ⇑(f : A →* B) = f :=\n rfl\n#align alg_hom.coe_to_monoid_hom AlgHom.coe_toMonoidHom\n\n@[simp, norm_cast]\ntheorem coe_toAddMonoidHom (f : A →ₐ[R] B) : ⇑(f : A →+ B) = f :=\n rfl\n#align alg_hom.coe_to_add_monoid_hom AlgHom.coe_toAddMonoidHom\n\nvariable (φ : A →ₐ[R] B)\n\ntheorem coe_fn_injective : @Function.Injective (A →ₐ[R] B) (A → B) (↑) :=\n FunLike.coe_injective\n#align alg_hom.coe_fn_injective AlgHom.coe_fn_injective\n\ntheorem coe_fn_inj {φ₁ φ₂ : A →ₐ[R] B} : (φ₁ : A → B) = φ₂ ↔ φ₁ = φ₂ :=\n FunLike.coe_fn_eq\n#align alg_hom.coe_fn_inj AlgHom.coe_fn_inj\n\ntheorem coe_ringHom_injective : Function.Injective ((↑) : (A →ₐ[R] B) → A →+* B) := fun φ₁ φ₂ H =>\n coe_fn_injective <| show ((φ₁ : A →+* B) : A → B) = ((φ₂ : A →+* B) : A → B) from congr_arg _ H\n#align alg_hom.coe_ring_hom_injective AlgHom.coe_ringHom_injective\n\ntheorem coe_monoidHom_injective : Function.Injective ((↑) : (A →ₐ[R] B) → A →* B) :=\n RingHom.coe_monoidHom_injective.comp coe_ringHom_injective\n#align alg_hom.coe_monoid_hom_injective AlgHom.coe_monoidHom_injective\n\ntheorem coe_addMonoidHom_injective : Function.Injective ((↑) : (A →ₐ[R] B) → A →+ B) :=\n RingHom.coe_addMonoidHom_injective.comp coe_ringHom_injective\n#align alg_hom.coe_add_monoid_hom_injective AlgHom.coe_addMonoidHom_injective\n\nprotected theorem congr_fun {φ₁ φ₂ : A →ₐ[R] B} (H : φ₁ = φ₂) (x : A) : φ₁ x = φ₂ x :=\n FunLike.congr_fun H x\n#align alg_hom.congr_fun AlgHom.congr_fun\n\nprotected theorem congr_arg (φ : A →ₐ[R] B) {x y : A} (h : x = y) : φ x = φ y :=\n FunLike.congr_arg φ h\n#align alg_hom.congr_arg AlgHom.congr_arg\n\n@[ext]\ntheorem ext {φ₁ φ₂ : A →ₐ[R] B} (H : ∀ x, φ₁ x = φ₂ x) : φ₁ = φ₂ :=\n FunLike.ext _ _ H\n#align alg_hom.ext AlgHom.ext\n\ntheorem ext_iff {φ₁ φ₂ : A →ₐ[R] B} : φ₁ = φ₂ ↔ ∀ x, φ₁ x = φ₂ x :=\n FunLike.ext_iff\n#align alg_hom.ext_iff AlgHom.ext_iff\n\n@[simp]\ntheorem mk_coe {f : A →ₐ[R] B} (h₁ h₂ h₃ h₄ h₅) : (⟨⟨⟨⟨f, h₁⟩, h₂⟩, h₃, h₄⟩, h₅⟩ : A →ₐ[R] B) = f :=\n ext fun _ => rfl\n#align alg_hom.mk_coe AlgHom.mk_coe\n\n@[simp]\ntheorem commutes (r : R) : φ (algebraMap R A r) = algebraMap R B r :=\n φ.commutes' r\n#align alg_hom.commutes AlgHom.commutes\n\ntheorem comp_algebraMap : (φ : A →+* B).comp (algebraMap R A) = algebraMap R B :=\n RingHom.ext <| φ.commutes\n#align alg_hom.comp_algebra_map AlgHom.comp_algebraMap\n\nprotected theorem map_add (r s : A) : φ (r + s) = φ r + φ s :=\n map_add _ _ _\n#align alg_hom.map_add AlgHom.map_add\n\nprotected theorem map_zero : φ 0 = 0 :=\n map_zero _\n#align alg_hom.map_zero AlgHom.map_zero\n\nprotected theorem map_mul (x y) : φ (x * y) = φ x * φ y :=\n map_mul _ _ _\n#align alg_hom.map_mul AlgHom.map_mul\n\nprotected theorem map_one : φ 1 = 1 :=\n map_one _\n#align alg_hom.map_one AlgHom.map_one\n\nprotected theorem map_pow (x : A) (n : ℕ) : φ (x ^ n) = φ x ^ n :=\n map_pow _ _ _\n#align alg_hom.map_pow AlgHom.map_pow\n\n-- @[simp] -- Porting note: simp can prove this\nprotected theorem map_smul (r : R) (x : A) : φ (r • x) = r • φ x :=\n map_smul _ _ _\n#align alg_hom.map_smul AlgHom.map_smul\n\nprotected theorem map_sum {ι : Type*} (f : ι → A) (s : Finset ι) :\n φ (∑ x in s, f x) = ∑ x in s, φ (f x) :=\n map_sum _ _ _\n#align alg_hom.map_sum AlgHom.map_sum\n\nprotected theorem map_finsupp_sum {α : Type*} [Zero α] {ι : Type*} (f : ι →₀ α) (g : ι → α → A) :\n φ (f.sum g) = f.sum fun i a => φ (g i a) :=\n map_finsupp_sum _ _ _\n#align alg_hom.map_finsupp_sum AlgHom.map_finsupp_sum\n\nset_option linter.deprecated false in\nprotected theorem map_bit0 (x) : φ (bit0 x) = bit0 (φ x) :=\n map_bit0 _ _\n#align alg_hom.map_bit0 AlgHom.map_bit0\n\nset_option linter.deprecated false in\nprotected theorem map_bit1 (x) : φ (bit1 x) = bit1 (φ x) :=\n map_bit1 _ _\n#align alg_hom.map_bit1 AlgHom.map_bit1\n\n/-- If a `RingHom` is `R`-linear, then it is an `AlgHom`. -/\ndef mk' (f : A →+* B) (h : ∀ (c : R) (x), f (c • x) = c • f x) : A →ₐ[R] B :=\n { f with\n toFun := f\n commutes' := fun c => by simp only [Algebra.algebraMap_eq_smul_one, h, f.map_one] }\n#align alg_hom.mk' AlgHom.mk'\n\n@[simp]\ntheorem coe_mk' (f : A →+* B) (h : ∀ (c : R) (x), f (c • x) = c • f x) : ⇑(mk' f h) = f :=\n rfl\n#align alg_hom.coe_mk' AlgHom.coe_mk'\n\nsection\n\nvariable (R A)\n\n/-- Identity map as an `AlgHom`. -/\nprotected def id : A →ₐ[R] A :=\n { RingHom.id A with commutes' := fun _ => rfl }\n#align alg_hom.id AlgHom.id\n\n@[simp]\ntheorem coe_id : ⇑(AlgHom.id R A) = id :=\n rfl\n#align alg_hom.coe_id AlgHom.coe_id\n\n@[simp]\ntheorem id_toRingHom : (AlgHom.id R A : A →+* A) = RingHom.id _ :=\n rfl\n#align alg_hom.id_to_ring_hom AlgHom.id_toRingHom\n\nend\n\ntheorem id_apply (p : A) : AlgHom.id R A p = p :=\n rfl\n#align alg_hom.id_apply AlgHom.id_apply\n\n/-- Composition of algebra homeomorphisms. -/\ndef comp (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) : A →ₐ[R] C :=\n { φ₁.toRingHom.comp ↑φ₂ with\n commutes' := fun r : R => by rw [← φ₁.commutes, ← φ₂.commutes]; rfl }\n#align alg_hom.comp AlgHom.comp\n\n@[simp]\ntheorem coe_comp (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) : ⇑(φ₁.comp φ₂) = φ₁ ∘ φ₂ :=\n rfl\n#align alg_hom.coe_comp AlgHom.coe_comp\n\ntheorem comp_apply (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) (p : A) : φ₁.comp φ₂ p = φ₁ (φ₂ p) :=\n rfl\n#align alg_hom.comp_apply AlgHom.comp_apply\n\ntheorem comp_toRingHom (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) :\n (φ₁.comp φ₂ : A →+* C) = (φ₁ : B →+* C).comp ↑φ₂ :=\n rfl\n#align alg_hom.comp_to_ring_hom AlgHom.comp_toRingHom\n\n@[simp]\ntheorem comp_id : φ.comp (AlgHom.id R A) = φ :=\n ext fun _x => rfl\n#align alg_hom.comp_id AlgHom.comp_id\n\n@[simp]\ntheorem id_comp : (AlgHom.id R B).comp φ = φ :=\n ext fun _x => rfl\n#align alg_hom.id_comp AlgHom.id_comp\n\ntheorem comp_assoc (φ₁ : C →ₐ[R] D) (φ₂ : B →ₐ[R] C) (φ₃ : A →ₐ[R] B) :\n (φ₁.comp φ₂).comp φ₃ = φ₁.comp (φ₂.comp φ₃) :=\n ext fun _x => rfl\n#align alg_hom.comp_assoc AlgHom.comp_assoc\n\n/-- R-Alg ⥤ R-Mod -/\ndef toLinearMap : A →ₗ[R] B where\n toFun := φ\n map_add' := map_add _\n map_smul' := map_smul _\n#align alg_hom.to_linear_map AlgHom.toLinearMap\n\n@[simp]\ntheorem toLinearMap_apply (p : A) : φ.toLinearMap p = φ p :=\n rfl\n#align alg_hom.to_linear_map_apply AlgHom.toLinearMap_apply\n\ntheorem toLinearMap_injective :\n Function.Injective (toLinearMap : _ → A →ₗ[R] B) := fun _φ₁ _φ₂ h =>\n ext <| LinearMap.congr_fun h\n#align alg_hom.to_linear_map_injective AlgHom.toLinearMap_injective\n\n@[simp]\ntheorem comp_toLinearMap (f : A →ₐ[R] B) (g : B →ₐ[R] C) :\n (g.comp f).toLinearMap = g.toLinearMap.comp f.toLinearMap :=\n rfl\n#align alg_hom.comp_to_linear_map AlgHom.comp_toLinearMap\n\n@[simp]\ntheorem toLinearMap_id : toLinearMap (AlgHom.id R A) = LinearMap.id :=\n LinearMap.ext fun _ => rfl\n#align alg_hom.to_linear_map_id AlgHom.toLinearMap_id\n\n/-- Promote a `LinearMap` to an `AlgHom` by supplying proofs about the behavior on `1` and `*`. -/\n@[simps]\ndef ofLinearMap (f : A →ₗ[R] B) (map_one : f 1 = 1) (map_mul : ∀ x y, f (x * y) = f x * f y) :\n A →ₐ[R] B :=\n { f.toAddMonoidHom with\n toFun := f\n map_one' := map_one\n map_mul' := map_mul\n commutes' := fun c => by simp only [Algebra.algebraMap_eq_smul_one, f.map_smul, map_one] }\n#align alg_hom.of_linear_map AlgHom.ofLinearMap\n\n@[simp]\ntheorem ofLinearMap_toLinearMap (map_one) (map_mul) :\n ofLinearMap φ.toLinearMap map_one map_mul = φ := by\n ext\n rfl\n#align alg_hom.of_linear_map_to_linear_map AlgHom.ofLinearMap_toLinearMap\n\n@[simp]\ntheorem toLinearMap_ofLinearMap (f : A →ₗ[R] B) (map_one) (map_mul) :\n toLinearMap (ofLinearMap f map_one map_mul) = f := by\n ext\n rfl\n#align alg_hom.to_linear_map_of_linear_map AlgHom.toLinearMap_ofLinearMap\n\n@[simp]\ntheorem ofLinearMap_id (map_one) (map_mul) :\n ofLinearMap LinearMap.id map_one map_mul = AlgHom.id R A :=\n ext fun _ => rfl\n#align alg_hom.of_linear_map_id AlgHom.ofLinearMap_id\n\ntheorem map_smul_of_tower {R'} [SMul R' A] [SMul R' B] [LinearMap.CompatibleSMul A B R' R] (r : R')\n (x : A) : φ (r • x) = r • φ x :=\n φ.toLinearMap.map_smul_of_tower r x\n#align alg_hom.map_smul_of_tower AlgHom.map_smul_of_tower\n\ntheorem map_list_prod (s : List A) : φ s.prod = (s.map φ).prod :=\n φ.toRingHom.map_list_prod s\n#align alg_hom.map_list_prod AlgHom.map_list_prod\n\n@[simps (config := .lemmasOnly) toSemigroup_toMul_mul toOne_one]\ninstance End : Monoid (A →ₐ[R] A) where\n mul := comp\n mul_assoc ϕ ψ χ := rfl\n one := AlgHom.id R A\n one_mul ϕ := ext fun x => rfl\n mul_one ϕ := ext fun x => rfl\n#align alg_hom.End AlgHom.End\n\n@[simp]\ntheorem one_apply (x : A) : (1 : A →ₐ[R] A) x = x :=\n rfl\n#align alg_hom.one_apply AlgHom.one_apply\n\n@[simp]\ntheorem mul_apply (φ ψ : A →ₐ[R] A) (x : A) : (φ * ψ) x = φ (ψ x) :=\n rfl\n#align alg_hom.mul_apply AlgHom.mul_apply\n\ntheorem algebraMap_eq_apply (f : A →ₐ[R] B) {y : R} {x : A} (h : algebraMap R A y = x) :\n algebraMap R B y = f x :=\n h ▸ (f.commutes _).symm\n#align alg_hom.algebra_map_eq_apply AlgHom.algebraMap_eq_apply\n\nend Semiring\n\nsection CommSemiring\n\nvariable [CommSemiring R] [CommSemiring A] [CommSemiring B]\n\nvariable [Algebra R A] [Algebra R B] (φ : A →ₐ[R] B)\n\nprotected theorem map_multiset_prod (s : Multiset A) : φ s.prod = (s.map φ).prod :=\n map_multiset_prod _ _\n#align alg_hom.map_multiset_prod AlgHom.map_multiset_prod\n\nprotected theorem map_prod {ι : Type*} (f : ι → A) (s : Finset ι) :\n φ (∏ x in s, f x) = ∏ x in s, φ (f x) :=\n map_prod _ _ _\n#align alg_hom.map_prod AlgHom.map_prod\n\nprotected theorem map_finsupp_prod {α : Type*} [Zero α] {ι : Type*} (f : ι →₀ α) (g : ι → α → A) :\n φ (f.prod g) = f.prod fun i a => φ (g i a) :=\n map_finsupp_prod _ _ _\n#align alg_hom.map_finsupp_prod AlgHom.map_finsupp_prod\n\nend CommSemiring\n\nsection Ring\n\nvariable [CommSemiring R] [Ring A] [Ring B]\n\nvariable [Algebra R A] [Algebra R B] (φ : A →ₐ[R] B)\n\nprotected theorem map_neg (x) : φ (-x) = -φ x :=\n map_neg _ _\n#align alg_hom.map_neg AlgHom.map_neg\n\nprotected theorem map_sub (x y) : φ (x - y) = φ x - φ y :=\n map_sub _ _ _\n#align alg_hom.map_sub AlgHom.map_sub\n\nend Ring\n\nend AlgHom\n\nnamespace RingHom\n\nvariable {R S : Type*}\n\n/-- Reinterpret a `RingHom` as an `ℕ`-algebra homomorphism. -/\ndef toNatAlgHom [Semiring R] [Semiring S] (f : R →+* S) : R →ₐ[ℕ] S :=\n { f with\n toFun := f\n commutes' := fun n => by simp }\n#align ring_hom.to_nat_alg_hom RingHom.toNatAlgHom\n\n/-- Reinterpret a `RingHom` as a `ℤ`-algebra homomorphism. -/\ndef toIntAlgHom [Ring R] [Ring S] [Algebra ℤ R] [Algebra ℤ S] (f : R →+* S) : R →ₐ[ℤ] S :=\n { f with commutes' := fun n => by simp }\n#align ring_hom.to_int_alg_hom RingHom.toIntAlgHom\n\nlemma toIntAlgHom_injective [Ring R] [Ring S] [Algebra ℤ R] [Algebra ℤ S] :\n Function.Injective (RingHom.toIntAlgHom : (R →+* S) → _) :=\n fun _ _ e ↦ FunLike.ext _ _ (fun x ↦ FunLike.congr_fun e x)\n\n/-- Reinterpret a `RingHom` as a `ℚ`-algebra homomorphism. This actually yields an equivalence,\nsee `RingHom.equivRatAlgHom`. -/\ndef toRatAlgHom [Ring R] [Ring S] [Algebra ℚ R] [Algebra ℚ S] (f : R →+* S) : R →ₐ[ℚ] S :=\n { f with commutes' := f.map_rat_algebraMap }\n#align ring_hom.to_rat_alg_hom RingHom.toRatAlgHom\n\n@[simp]\ntheorem toRatAlgHom_toRingHom [Ring R] [Ring S] [Algebra ℚ R] [Algebra ℚ S] (f : R →+* S) :\n ↑f.toRatAlgHom = f :=\n RingHom.ext fun _x => rfl\n#align ring_hom.to_rat_alg_hom_to_ring_hom RingHom.toRatAlgHom_toRingHom\n\nend RingHom\n\nsection\n\nvariable {R S : Type*}\n\n@[simp]\ntheorem AlgHom.toRingHom_toRatAlgHom [Ring R] [Ring S] [Algebra ℚ R] [Algebra ℚ S]\n (f : R →ₐ[ℚ] S) : (f : R →+* S).toRatAlgHom = f :=\n AlgHom.ext fun _x => rfl\n#align alg_hom.to_ring_hom_to_rat_alg_hom AlgHom.toRingHom_toRatAlgHom\n\n/-- The equivalence between `RingHom` and `ℚ`-algebra homomorphisms. -/\n@[simps]\ndef RingHom.equivRatAlgHom [Ring R] [Ring S] [Algebra ℚ R] [Algebra ℚ S] : (R →+* S) ≃ (R →ₐ[ℚ] S)\n where\n toFun := RingHom.toRatAlgHom\n invFun := AlgHom.toRingHom\n left_inv f := RingHom.toRatAlgHom_toRingHom f\n right_inv f := AlgHom.toRingHom_toRatAlgHom f\n#align ring_hom.equiv_rat_alg_hom RingHom.equivRatAlgHom\n\nend\n\nnamespace Algebra\n\nvariable (R : Type u) (A : Type v)\n\nvariable [CommSemiring R] [Semiring A] [Algebra R A]\n\n/-- `AlgebraMap` as an `AlgHom`. -/\ndef ofId : R →ₐ[R] A :=\n { algebraMap R A with commutes' := fun _ => rfl }\n#align algebra.of_id Algebra.ofId\n\nvariable {R}\n\ntheorem ofId_apply (r) : ofId R A r = algebraMap R A r :=\n rfl\n#align algebra.of_id_apply Algebra.ofId_apply\n\n/-- This is a special case of a more general instance that we define in a later file. -/\ninstance subsingleton_id : Subsingleton (R →ₐ[R] A) :=\n ⟨fun f g => AlgHom.ext fun _ => (f.commutes _).trans (g.commutes _).symm⟩\n\n/-- This ext lemma closes trivial subgoals create when chaining heterobasic ext lemmas. -/\n@[ext high]\ntheorem ext_id (f g : R →ₐ[R] A) : f = g := Subsingleton.elim _ _\n\nsection MulDistribMulAction\n\ninstance : MulDistribMulAction (A →ₐ[R] A) Aˣ where\n smul := fun f => Units.map f\n one_smul := fun x => by ext; rfl\n mul_smul := fun x y z => by ","nextTactic":"ext","declUpToTactic":"instance : MulDistribMulAction (A →ₐ[R] A) Aˣ where\n smul := fun f => Units.map f\n one_smul := fun x => by ext; rfl\n mul_smul := fun x y z => by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Hom.579_0.FxqJeqR3qcIVlrq","decl":"instance : MulDistribMulAction (A →ₐ[R] A) Aˣ where\n smul "} +{"state":"case a\nR : Type u\nA : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nx y : A →ₐ[R] A\nz : Aˣ\n⊢ ↑((x * y) • z) = ↑(x • y • z)","srcUpToTactic":"/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Yury Kudryashov\n-/\nimport Mathlib.Algebra.Algebra.Basic\nimport Mathlib.Algebra.BigOperators.Finsupp\n\n#align_import algebra.algebra.hom from \"leanprover-community/mathlib\"@\"e97cf15cd1aec9bd5c193b2ffac5a6dc9118912b\"\n\n/-!\n# Homomorphisms of `R`-algebras\n\nThis file defines bundled homomorphisms of `R`-algebras.\n\n## Main definitions\n\n* `AlgHom R A B`: the type of `R`-algebra morphisms from `A` to `B`.\n* `Algebra.ofId R A : R →ₐ[R] A`: the canonical map from `R` to `A`, as an `AlgHom`.\n\n## Notations\n\n* `A →ₐ[R] B` : `R`-algebra homomorphism from `A` to `B`.\n-/\n\nset_option autoImplicit true\n\n\nopen BigOperators\n\nuniverse u v w u₁ v₁\n\n/-- Defining the homomorphism in the category R-Alg. -/\n-- @[nolint has_nonempty_instance] -- Porting note: This linter does not exist yet.\nstructure AlgHom (R : Type u) (A : Type v) (B : Type w) [CommSemiring R] [Semiring A] [Semiring B]\n [Algebra R A] [Algebra R B] extends RingHom A B where\n commutes' : ∀ r : R, toFun (algebraMap R A r) = algebraMap R B r\n#align alg_hom AlgHom\n\n/-- Reinterpret an `AlgHom` as a `RingHom` -/\nadd_decl_doc AlgHom.toRingHom\n\n@[inherit_doc AlgHom]\ninfixr:25 \" →ₐ \" => AlgHom _\n\n@[inherit_doc]\nnotation:25 A \" →ₐ[\" R \"] \" B => AlgHom R A B\n\n/-- `AlgHomClass F R A B` asserts `F` is a type of bundled algebra homomorphisms\nfrom `A` to `B`. -/\nclass AlgHomClass (F : Type*) (R : outParam (Type*)) (A : outParam (Type*))\n (B : outParam (Type*)) [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A]\n [Algebra R B] extends RingHomClass F A B where\n commutes : ∀ (f : F) (r : R), f (algebraMap R A r) = algebraMap R B r\n#align alg_hom_class AlgHomClass\n\n-- Porting note: `dangerousInstance` linter has become smarter about `outParam`s\n-- attribute [nolint dangerousInstance] AlgHomClass.toRingHomClass\n\n-- Porting note: simp can prove this\n-- attribute [simp] AlgHomClass.commutes\n\nnamespace AlgHomClass\n\nvariable {R : Type*} {A : Type*} {B : Type*} [CommSemiring R] [Semiring A] [Semiring B]\n [Algebra R A] [Algebra R B]\n\n-- see Note [lower instance priority]\ninstance (priority := 100) linearMapClass [AlgHomClass F R A B] : LinearMapClass F R A B :=\n { ‹AlgHomClass F R A B› with\n map_smulₛₗ := fun f r x => by\n simp only [Algebra.smul_def, map_mul, commutes, RingHom.id_apply] }\n#align alg_hom_class.linear_map_class AlgHomClass.linearMapClass\n\n-- Porting note: A new definition underlying a coercion `↑`.\n/-- Turn an element of a type `F` satisfying `AlgHomClass F α β` into an actual\n`AlgHom`. This is declared as the default coercion from `F` to `α →+* β`. -/\n@[coe]\ndef toAlgHom {F : Type*} [AlgHomClass F R A B] (f : F) : A →ₐ[R] B :=\n { (f : A →+* B) with\n toFun := f\n commutes' := AlgHomClass.commutes f }\n\ninstance coeTC {F : Type*} [AlgHomClass F R A B] : CoeTC F (A →ₐ[R] B) :=\n ⟨AlgHomClass.toAlgHom⟩\n#align alg_hom_class.alg_hom.has_coe_t AlgHomClass.coeTC\n\nend AlgHomClass\n\nnamespace AlgHom\n\nvariable {R : Type u} {A : Type v} {B : Type w} {C : Type u₁} {D : Type v₁}\n\nsection Semiring\n\nvariable [CommSemiring R] [Semiring A] [Semiring B] [Semiring C] [Semiring D]\n\nvariable [Algebra R A] [Algebra R B] [Algebra R C] [Algebra R D]\n\n-- Porting note: we don't port specialized `CoeFun` instances if there is `FunLike` instead\n#noalign alg_hom.has_coe_to_fun\n\n-- Porting note: This instance is moved.\ninstance algHomClass : AlgHomClass (A →ₐ[R] B) R A B where\n coe f := f.toFun\n coe_injective' f g h := by\n rcases f with ⟨⟨⟨⟨_, _⟩, _⟩, _, _⟩, _⟩\n rcases g with ⟨⟨⟨⟨_, _⟩, _⟩, _, _⟩, _⟩\n congr\n map_add f := f.map_add'\n map_zero f := f.map_zero'\n map_mul f := f.map_mul'\n map_one f := f.map_one'\n commutes f := f.commutes'\n#align alg_hom.alg_hom_class AlgHom.algHomClass\n\n/-- See Note [custom simps projection] -/\ndef Simps.apply {R : Type u} {α : Type v} {β : Type w} [CommSemiring R]\n [Semiring α] [Semiring β] [Algebra R α] [Algebra R β] (f : α →ₐ[R] β) : α → β := f\n\ninitialize_simps_projections AlgHom (toFun → apply)\n\n@[simp]\nprotected theorem coe_coe {F : Type*} [AlgHomClass F R A B] (f : F) : ⇑(f : A →ₐ[R] B) = f :=\n rfl\n#align alg_hom.coe_coe AlgHom.coe_coe\n\n@[simp]\ntheorem toFun_eq_coe (f : A →ₐ[R] B) : f.toFun = f :=\n rfl\n#align alg_hom.to_fun_eq_coe AlgHom.toFun_eq_coe\n\nattribute [coe] AlgHom.toRingHom\n\ninstance coeOutRingHom : CoeOut (A →ₐ[R] B) (A →+* B) :=\n ⟨AlgHom.toRingHom⟩\n#align alg_hom.coe_ring_hom AlgHom.coeOutRingHom\n\n-- Porting note: A new definition underlying a coercion `↑`.\n@[coe]\ndef toMonoidHom' (f : A →ₐ[R] B) : A →* B := (f : A →+* B)\n\ninstance coeOutMonoidHom : CoeOut (A →ₐ[R] B) (A →* B) :=\n ⟨AlgHom.toMonoidHom'⟩\n#align alg_hom.coe_monoid_hom AlgHom.coeOutMonoidHom\n\n-- Porting note: A new definition underlying a coercion `↑`.\n@[coe]\ndef toAddMonoidHom' (f : A →ₐ[R] B) : A →+ B := (f : A →+* B)\n\ninstance coeOutAddMonoidHom : CoeOut (A →ₐ[R] B) (A →+ B) :=\n ⟨AlgHom.toAddMonoidHom'⟩\n#align alg_hom.coe_add_monoid_hom AlgHom.coeOutAddMonoidHom\n\n-- Porting note: Lean 3: `@[simp, norm_cast] coe_mk`\n-- Lean 4: `@[simp] coe_mk` & `@[norm_cast] coe_mks`\n@[simp]\ntheorem coe_mk {f : A →+* B} (h) : ((⟨f, h⟩ : A →ₐ[R] B) : A → B) = f :=\n rfl\n\n@[norm_cast]\ntheorem coe_mks {f : A → B} (h₁ h₂ h₃ h₄ h₅) : ⇑(⟨⟨⟨⟨f, h₁⟩, h₂⟩, h₃, h₄⟩, h₅⟩ : A →ₐ[R] B) = f :=\n rfl\n#align alg_hom.coe_mk AlgHom.coe_mks\n\n-- Porting note: This theorem is new.\n@[simp, norm_cast]\ntheorem coe_ringHom_mk {f : A →+* B} (h) : ((⟨f, h⟩ : A →ₐ[R] B) : A →+* B) = f :=\n rfl\n\n-- make the coercion the simp-normal form\n@[simp]\ntheorem toRingHom_eq_coe (f : A →ₐ[R] B) : f.toRingHom = f :=\n rfl\n#align alg_hom.to_ring_hom_eq_coe AlgHom.toRingHom_eq_coe\n\n@[simp, norm_cast]\ntheorem coe_toRingHom (f : A →ₐ[R] B) : ⇑(f : A →+* B) = f :=\n rfl\n#align alg_hom.coe_to_ring_hom AlgHom.coe_toRingHom\n\n@[simp, norm_cast]\ntheorem coe_toMonoidHom (f : A →ₐ[R] B) : ⇑(f : A →* B) = f :=\n rfl\n#align alg_hom.coe_to_monoid_hom AlgHom.coe_toMonoidHom\n\n@[simp, norm_cast]\ntheorem coe_toAddMonoidHom (f : A →ₐ[R] B) : ⇑(f : A →+ B) = f :=\n rfl\n#align alg_hom.coe_to_add_monoid_hom AlgHom.coe_toAddMonoidHom\n\nvariable (φ : A →ₐ[R] B)\n\ntheorem coe_fn_injective : @Function.Injective (A →ₐ[R] B) (A → B) (↑) :=\n FunLike.coe_injective\n#align alg_hom.coe_fn_injective AlgHom.coe_fn_injective\n\ntheorem coe_fn_inj {φ₁ φ₂ : A →ₐ[R] B} : (φ₁ : A → B) = φ₂ ↔ φ₁ = φ₂ :=\n FunLike.coe_fn_eq\n#align alg_hom.coe_fn_inj AlgHom.coe_fn_inj\n\ntheorem coe_ringHom_injective : Function.Injective ((↑) : (A →ₐ[R] B) → A →+* B) := fun φ₁ φ₂ H =>\n coe_fn_injective <| show ((φ₁ : A →+* B) : A → B) = ((φ₂ : A →+* B) : A → B) from congr_arg _ H\n#align alg_hom.coe_ring_hom_injective AlgHom.coe_ringHom_injective\n\ntheorem coe_monoidHom_injective : Function.Injective ((↑) : (A →ₐ[R] B) → A →* B) :=\n RingHom.coe_monoidHom_injective.comp coe_ringHom_injective\n#align alg_hom.coe_monoid_hom_injective AlgHom.coe_monoidHom_injective\n\ntheorem coe_addMonoidHom_injective : Function.Injective ((↑) : (A →ₐ[R] B) → A →+ B) :=\n RingHom.coe_addMonoidHom_injective.comp coe_ringHom_injective\n#align alg_hom.coe_add_monoid_hom_injective AlgHom.coe_addMonoidHom_injective\n\nprotected theorem congr_fun {φ₁ φ₂ : A →ₐ[R] B} (H : φ₁ = φ₂) (x : A) : φ₁ x = φ₂ x :=\n FunLike.congr_fun H x\n#align alg_hom.congr_fun AlgHom.congr_fun\n\nprotected theorem congr_arg (φ : A →ₐ[R] B) {x y : A} (h : x = y) : φ x = φ y :=\n FunLike.congr_arg φ h\n#align alg_hom.congr_arg AlgHom.congr_arg\n\n@[ext]\ntheorem ext {φ₁ φ₂ : A →ₐ[R] B} (H : ∀ x, φ₁ x = φ₂ x) : φ₁ = φ₂ :=\n FunLike.ext _ _ H\n#align alg_hom.ext AlgHom.ext\n\ntheorem ext_iff {φ₁ φ₂ : A →ₐ[R] B} : φ₁ = φ₂ ↔ ∀ x, φ₁ x = φ₂ x :=\n FunLike.ext_iff\n#align alg_hom.ext_iff AlgHom.ext_iff\n\n@[simp]\ntheorem mk_coe {f : A ��ₐ[R] B} (h₁ h₂ h₃ h₄ h₅) : (⟨⟨⟨⟨f, h₁⟩, h₂⟩, h₃, h₄⟩, h₅⟩ : A →ₐ[R] B) = f :=\n ext fun _ => rfl\n#align alg_hom.mk_coe AlgHom.mk_coe\n\n@[simp]\ntheorem commutes (r : R) : φ (algebraMap R A r) = algebraMap R B r :=\n φ.commutes' r\n#align alg_hom.commutes AlgHom.commutes\n\ntheorem comp_algebraMap : (φ : A →+* B).comp (algebraMap R A) = algebraMap R B :=\n RingHom.ext <| φ.commutes\n#align alg_hom.comp_algebra_map AlgHom.comp_algebraMap\n\nprotected theorem map_add (r s : A) : φ (r + s) = φ r + φ s :=\n map_add _ _ _\n#align alg_hom.map_add AlgHom.map_add\n\nprotected theorem map_zero : φ 0 = 0 :=\n map_zero _\n#align alg_hom.map_zero AlgHom.map_zero\n\nprotected theorem map_mul (x y) : φ (x * y) = φ x * φ y :=\n map_mul _ _ _\n#align alg_hom.map_mul AlgHom.map_mul\n\nprotected theorem map_one : φ 1 = 1 :=\n map_one _\n#align alg_hom.map_one AlgHom.map_one\n\nprotected theorem map_pow (x : A) (n : ℕ) : φ (x ^ n) = φ x ^ n :=\n map_pow _ _ _\n#align alg_hom.map_pow AlgHom.map_pow\n\n-- @[simp] -- Porting note: simp can prove this\nprotected theorem map_smul (r : R) (x : A) : φ (r • x) = r • φ x :=\n map_smul _ _ _\n#align alg_hom.map_smul AlgHom.map_smul\n\nprotected theorem map_sum {ι : Type*} (f : ι → A) (s : Finset ι) :\n φ (∑ x in s, f x) = ∑ x in s, φ (f x) :=\n map_sum _ _ _\n#align alg_hom.map_sum AlgHom.map_sum\n\nprotected theorem map_finsupp_sum {α : Type*} [Zero α] {ι : Type*} (f : ι →₀ α) (g : ι → α → A) :\n φ (f.sum g) = f.sum fun i a => φ (g i a) :=\n map_finsupp_sum _ _ _\n#align alg_hom.map_finsupp_sum AlgHom.map_finsupp_sum\n\nset_option linter.deprecated false in\nprotected theorem map_bit0 (x) : φ (bit0 x) = bit0 (φ x) :=\n map_bit0 _ _\n#align alg_hom.map_bit0 AlgHom.map_bit0\n\nset_option linter.deprecated false in\nprotected theorem map_bit1 (x) : φ (bit1 x) = bit1 (φ x) :=\n map_bit1 _ _\n#align alg_hom.map_bit1 AlgHom.map_bit1\n\n/-- If a `RingHom` is `R`-linear, then it is an `AlgHom`. -/\ndef mk' (f : A →+* B) (h : ∀ (c : R) (x), f (c • x) = c • f x) : A →ₐ[R] B :=\n { f with\n toFun := f\n commutes' := fun c => by simp only [Algebra.algebraMap_eq_smul_one, h, f.map_one] }\n#align alg_hom.mk' AlgHom.mk'\n\n@[simp]\ntheorem coe_mk' (f : A →+* B) (h : ∀ (c : R) (x), f (c • x) = c • f x) : ⇑(mk' f h) = f :=\n rfl\n#align alg_hom.coe_mk' AlgHom.coe_mk'\n\nsection\n\nvariable (R A)\n\n/-- Identity map as an `AlgHom`. -/\nprotected def id : A →ₐ[R] A :=\n { RingHom.id A with commutes' := fun _ => rfl }\n#align alg_hom.id AlgHom.id\n\n@[simp]\ntheorem coe_id : ⇑(AlgHom.id R A) = id :=\n rfl\n#align alg_hom.coe_id AlgHom.coe_id\n\n@[simp]\ntheorem id_toRingHom : (AlgHom.id R A : A →+* A) = RingHom.id _ :=\n rfl\n#align alg_hom.id_to_ring_hom AlgHom.id_toRingHom\n\nend\n\ntheorem id_apply (p : A) : AlgHom.id R A p = p :=\n rfl\n#align alg_hom.id_apply AlgHom.id_apply\n\n/-- Composition of algebra homeomorphisms. -/\ndef comp (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) : A →ₐ[R] C :=\n { φ₁.toRingHom.comp ↑φ₂ with\n commutes' := fun r : R => by rw [← φ₁.commutes, ← φ₂.commutes]; rfl }\n#align alg_hom.comp AlgHom.comp\n\n@[simp]\ntheorem coe_comp (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) : ⇑(φ₁.comp φ₂) = φ₁ ∘ φ₂ :=\n rfl\n#align alg_hom.coe_comp AlgHom.coe_comp\n\ntheorem comp_apply (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) (p : A) : φ₁.comp φ₂ p = φ₁ (φ₂ p) :=\n rfl\n#align alg_hom.comp_apply AlgHom.comp_apply\n\ntheorem comp_toRingHom (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) :\n (φ₁.comp φ₂ : A →+* C) = (φ₁ : B →+* C).comp ↑φ₂ :=\n rfl\n#align alg_hom.comp_to_ring_hom AlgHom.comp_toRingHom\n\n@[simp]\ntheorem comp_id : φ.comp (AlgHom.id R A) = φ :=\n ext fun _x => rfl\n#align alg_hom.comp_id AlgHom.comp_id\n\n@[simp]\ntheorem id_comp : (AlgHom.id R B).comp φ = φ :=\n ext fun _x => rfl\n#align alg_hom.id_comp AlgHom.id_comp\n\ntheorem comp_assoc (φ₁ : C →ₐ[R] D) (φ₂ : B →ₐ[R] C) (φ₃ : A →ₐ[R] B) :\n (φ₁.comp φ₂).comp φ₃ = φ₁.comp (φ₂.comp φ₃) :=\n ext fun _x => rfl\n#align alg_hom.comp_assoc AlgHom.comp_assoc\n\n/-- R-Alg ⥤ R-Mod -/\ndef toLinearMap : A →ₗ[R] B where\n toFun := φ\n map_add' := map_add _\n map_smul' := map_smul _\n#align alg_hom.to_linear_map AlgHom.toLinearMap\n\n@[simp]\ntheorem toLinearMap_apply (p : A) : φ.toLinearMap p = φ p :=\n rfl\n#align alg_hom.to_linear_map_apply AlgHom.toLinearMap_apply\n\ntheorem toLinearMap_injective :\n Function.Injective (toLinearMap : _ → A →ₗ[R] B) := fun _φ₁ _φ₂ h =>\n ext <| LinearMap.congr_fun h\n#align alg_hom.to_linear_map_injective AlgHom.toLinearMap_injective\n\n@[simp]\ntheorem comp_toLinearMap (f : A →ₐ[R] B) (g : B →ₐ[R] C) :\n (g.comp f).toLinearMap = g.toLinearMap.comp f.toLinearMap :=\n rfl\n#align alg_hom.comp_to_linear_map AlgHom.comp_toLinearMap\n\n@[simp]\ntheorem toLinearMap_id : toLinearMap (AlgHom.id R A) = LinearMap.id :=\n LinearMap.ext fun _ => rfl\n#align alg_hom.to_linear_map_id AlgHom.toLinearMap_id\n\n/-- Promote a `LinearMap` to an `AlgHom` by supplying proofs about the behavior on `1` and `*`. -/\n@[simps]\ndef ofLinearMap (f : A →ₗ[R] B) (map_one : f 1 = 1) (map_mul : ∀ x y, f (x * y) = f x * f y) :\n A →ₐ[R] B :=\n { f.toAddMonoidHom with\n toFun := f\n map_one' := map_one\n map_mul' := map_mul\n commutes' := fun c => by simp only [Algebra.algebraMap_eq_smul_one, f.map_smul, map_one] }\n#align alg_hom.of_linear_map AlgHom.ofLinearMap\n\n@[simp]\ntheorem ofLinearMap_toLinearMap (map_one) (map_mul) :\n ofLinearMap φ.toLinearMap map_one map_mul = φ := by\n ext\n rfl\n#align alg_hom.of_linear_map_to_linear_map AlgHom.ofLinearMap_toLinearMap\n\n@[simp]\ntheorem toLinearMap_ofLinearMap (f : A →ₗ[R] B) (map_one) (map_mul) :\n toLinearMap (ofLinearMap f map_one map_mul) = f := by\n ext\n rfl\n#align alg_hom.to_linear_map_of_linear_map AlgHom.toLinearMap_ofLinearMap\n\n@[simp]\ntheorem ofLinearMap_id (map_one) (map_mul) :\n ofLinearMap LinearMap.id map_one map_mul = AlgHom.id R A :=\n ext fun _ => rfl\n#align alg_hom.of_linear_map_id AlgHom.ofLinearMap_id\n\ntheorem map_smul_of_tower {R'} [SMul R' A] [SMul R' B] [LinearMap.CompatibleSMul A B R' R] (r : R')\n (x : A) : φ (r • x) = r • φ x :=\n φ.toLinearMap.map_smul_of_tower r x\n#align alg_hom.map_smul_of_tower AlgHom.map_smul_of_tower\n\ntheorem map_list_prod (s : List A) : φ s.prod = (s.map φ).prod :=\n φ.toRingHom.map_list_prod s\n#align alg_hom.map_list_prod AlgHom.map_list_prod\n\n@[simps (config := .lemmasOnly) toSemigroup_toMul_mul toOne_one]\ninstance End : Monoid (A →ₐ[R] A) where\n mul := comp\n mul_assoc ϕ ψ χ := rfl\n one := AlgHom.id R A\n one_mul ϕ := ext fun x => rfl\n mul_one ϕ := ext fun x => rfl\n#align alg_hom.End AlgHom.End\n\n@[simp]\ntheorem one_apply (x : A) : (1 : A →ₐ[R] A) x = x :=\n rfl\n#align alg_hom.one_apply AlgHom.one_apply\n\n@[simp]\ntheorem mul_apply (φ ψ : A →ₐ[R] A) (x : A) : (φ * ψ) x = φ (ψ x) :=\n rfl\n#align alg_hom.mul_apply AlgHom.mul_apply\n\ntheorem algebraMap_eq_apply (f : A →ₐ[R] B) {y : R} {x : A} (h : algebraMap R A y = x) :\n algebraMap R B y = f x :=\n h ▸ (f.commutes _).symm\n#align alg_hom.algebra_map_eq_apply AlgHom.algebraMap_eq_apply\n\nend Semiring\n\nsection CommSemiring\n\nvariable [CommSemiring R] [CommSemiring A] [CommSemiring B]\n\nvariable [Algebra R A] [Algebra R B] (φ : A →ₐ[R] B)\n\nprotected theorem map_multiset_prod (s : Multiset A) : φ s.prod = (s.map φ).prod :=\n map_multiset_prod _ _\n#align alg_hom.map_multiset_prod AlgHom.map_multiset_prod\n\nprotected theorem map_prod {ι : Type*} (f : ι → A) (s : Finset ι) :\n φ (∏ x in s, f x) = ∏ x in s, φ (f x) :=\n map_prod _ _ _\n#align alg_hom.map_prod AlgHom.map_prod\n\nprotected theorem map_finsupp_prod {α : Type*} [Zero α] {ι : Type*} (f : ι →₀ α) (g : ι → α → A) :\n φ (f.prod g) = f.prod fun i a => φ (g i a) :=\n map_finsupp_prod _ _ _\n#align alg_hom.map_finsupp_prod AlgHom.map_finsupp_prod\n\nend CommSemiring\n\nsection Ring\n\nvariable [CommSemiring R] [Ring A] [Ring B]\n\nvariable [Algebra R A] [Algebra R B] (φ : A →ₐ[R] B)\n\nprotected theorem map_neg (x) : φ (-x) = -φ x :=\n map_neg _ _\n#align alg_hom.map_neg AlgHom.map_neg\n\nprotected theorem map_sub (x y) : φ (x - y) = φ x - φ y :=\n map_sub _ _ _\n#align alg_hom.map_sub AlgHom.map_sub\n\nend Ring\n\nend AlgHom\n\nnamespace RingHom\n\nvariable {R S : Type*}\n\n/-- Reinterpret a `RingHom` as an `ℕ`-algebra homomorphism. -/\ndef toNatAlgHom [Semiring R] [Semiring S] (f : R →+* S) : R →ₐ[ℕ] S :=\n { f with\n toFun := f\n commutes' := fun n => by simp }\n#align ring_hom.to_nat_alg_hom RingHom.toNatAlgHom\n\n/-- Reinterpret a `RingHom` as a `ℤ`-algebra homomorphism. -/\ndef toIntAlgHom [Ring R] [Ring S] [Algebra ℤ R] [Algebra ℤ S] (f : R →+* S) : R →ₐ[ℤ] S :=\n { f with commutes' := fun n => by simp }\n#align ring_hom.to_int_alg_hom RingHom.toIntAlgHom\n\nlemma toIntAlgHom_injective [Ring R] [Ring S] [Algebra ℤ R] [Algebra ℤ S] :\n Function.Injective (RingHom.toIntAlgHom : (R →+* S) → _) :=\n fun _ _ e ↦ FunLike.ext _ _ (fun x ↦ FunLike.congr_fun e x)\n\n/-- Reinterpret a `RingHom` as a `ℚ`-algebra homomorphism. This actually yields an equivalence,\nsee `RingHom.equivRatAlgHom`. -/\ndef toRatAlgHom [Ring R] [Ring S] [Algebra ℚ R] [Algebra ℚ S] (f : R →+* S) : R →ₐ[ℚ] S :=\n { f with commutes' := f.map_rat_algebraMap }\n#align ring_hom.to_rat_alg_hom RingHom.toRatAlgHom\n\n@[simp]\ntheorem toRatAlgHom_toRingHom [Ring R] [Ring S] [Algebra ℚ R] [Algebra ℚ S] (f : R →+* S) :\n ↑f.toRatAlgHom = f :=\n RingHom.ext fun _x => rfl\n#align ring_hom.to_rat_alg_hom_to_ring_hom RingHom.toRatAlgHom_toRingHom\n\nend RingHom\n\nsection\n\nvariable {R S : Type*}\n\n@[simp]\ntheorem AlgHom.toRingHom_toRatAlgHom [Ring R] [Ring S] [Algebra ℚ R] [Algebra ℚ S]\n (f : R →ₐ[���] S) : (f : R →+* S).toRatAlgHom = f :=\n AlgHom.ext fun _x => rfl\n#align alg_hom.to_ring_hom_to_rat_alg_hom AlgHom.toRingHom_toRatAlgHom\n\n/-- The equivalence between `RingHom` and `ℚ`-algebra homomorphisms. -/\n@[simps]\ndef RingHom.equivRatAlgHom [Ring R] [Ring S] [Algebra ℚ R] [Algebra ℚ S] : (R →+* S) ≃ (R →ₐ[ℚ] S)\n where\n toFun := RingHom.toRatAlgHom\n invFun := AlgHom.toRingHom\n left_inv f := RingHom.toRatAlgHom_toRingHom f\n right_inv f := AlgHom.toRingHom_toRatAlgHom f\n#align ring_hom.equiv_rat_alg_hom RingHom.equivRatAlgHom\n\nend\n\nnamespace Algebra\n\nvariable (R : Type u) (A : Type v)\n\nvariable [CommSemiring R] [Semiring A] [Algebra R A]\n\n/-- `AlgebraMap` as an `AlgHom`. -/\ndef ofId : R →ₐ[R] A :=\n { algebraMap R A with commutes' := fun _ => rfl }\n#align algebra.of_id Algebra.ofId\n\nvariable {R}\n\ntheorem ofId_apply (r) : ofId R A r = algebraMap R A r :=\n rfl\n#align algebra.of_id_apply Algebra.ofId_apply\n\n/-- This is a special case of a more general instance that we define in a later file. -/\ninstance subsingleton_id : Subsingleton (R →ₐ[R] A) :=\n ⟨fun f g => AlgHom.ext fun _ => (f.commutes _).trans (g.commutes _).symm⟩\n\n/-- This ext lemma closes trivial subgoals create when chaining heterobasic ext lemmas. -/\n@[ext high]\ntheorem ext_id (f g : R →ₐ[R] A) : f = g := Subsingleton.elim _ _\n\nsection MulDistribMulAction\n\ninstance : MulDistribMulAction (A →ₐ[R] A) Aˣ where\n smul := fun f => Units.map f\n one_smul := fun x => by ext; rfl\n mul_smul := fun x y z => by ext; ","nextTactic":"rfl","declUpToTactic":"instance : MulDistribMulAction (A →ₐ[R] A) Aˣ where\n smul := fun f => Units.map f\n one_smul := fun x => by ext; rfl\n mul_smul := fun x y z => by ext; ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Hom.579_0.FxqJeqR3qcIVlrq","decl":"instance : MulDistribMulAction (A →ₐ[R] A) Aˣ where\n smul "} +{"state":"R : Type u\nA : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nx : A →ₐ[R] A\ny z : Aˣ\n⊢ x • (y * z) = x • y * x • z","srcUpToTactic":"/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Yury Kudryashov\n-/\nimport Mathlib.Algebra.Algebra.Basic\nimport Mathlib.Algebra.BigOperators.Finsupp\n\n#align_import algebra.algebra.hom from \"leanprover-community/mathlib\"@\"e97cf15cd1aec9bd5c193b2ffac5a6dc9118912b\"\n\n/-!\n# Homomorphisms of `R`-algebras\n\nThis file defines bundled homomorphisms of `R`-algebras.\n\n## Main definitions\n\n* `AlgHom R A B`: the type of `R`-algebra morphisms from `A` to `B`.\n* `Algebra.ofId R A : R →ₐ[R] A`: the canonical map from `R` to `A`, as an `AlgHom`.\n\n## Notations\n\n* `A →ₐ[R] B` : `R`-algebra homomorphism from `A` to `B`.\n-/\n\nset_option autoImplicit true\n\n\nopen BigOperators\n\nuniverse u v w u₁ v₁\n\n/-- Defining the homomorphism in the category R-Alg. -/\n-- @[nolint has_nonempty_instance] -- Porting note: This linter does not exist yet.\nstructure AlgHom (R : Type u) (A : Type v) (B : Type w) [CommSemiring R] [Semiring A] [Semiring B]\n [Algebra R A] [Algebra R B] extends RingHom A B where\n commutes' : ∀ r : R, toFun (algebraMap R A r) = algebraMap R B r\n#align alg_hom AlgHom\n\n/-- Reinterpret an `AlgHom` as a `RingHom` -/\nadd_decl_doc AlgHom.toRingHom\n\n@[inherit_doc AlgHom]\ninfixr:25 \" →ₐ \" => AlgHom _\n\n@[inherit_doc]\nnotation:25 A \" →ₐ[\" R \"] \" B => AlgHom R A B\n\n/-- `AlgHomClass F R A B` asserts `F` is a type of bundled algebra homomorphisms\nfrom `A` to `B`. -/\nclass AlgHomClass (F : Type*) (R : outParam (Type*)) (A : outParam (Type*))\n (B : outParam (Type*)) [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A]\n [Algebra R B] extends RingHomClass F A B where\n commutes : ∀ (f : F) (r : R), f (algebraMap R A r) = algebraMap R B r\n#align alg_hom_class AlgHomClass\n\n-- Porting note: `dangerousInstance` linter has become smarter about `outParam`s\n-- attribute [nolint dangerousInstance] AlgHomClass.toRingHomClass\n\n-- Porting note: simp can prove this\n-- attribute [simp] AlgHomClass.commutes\n\nnamespace AlgHomClass\n\nvariable {R : Type*} {A : Type*} {B : Type*} [CommSemiring R] [Semiring A] [Semiring B]\n [Algebra R A] [Algebra R B]\n\n-- see Note [lower instance priority]\ninstance (priority := 100) linearMapClass [AlgHomClass F R A B] : LinearMapClass F R A B :=\n { ‹AlgHomClass F R A B› with\n map_smulₛₗ := fun f r x => by\n simp only [Algebra.smul_def, map_mul, commutes, RingHom.id_apply] }\n#align alg_hom_class.linear_map_class AlgHomClass.linearMapClass\n\n-- Porting note: A new definition underlying a coercion `↑`.\n/-- Turn an element of a type `F` satisfying `AlgHomClass F α β` into an actual\n`AlgHom`. This is declared as the default coercion from `F` to `α →+* β`. -/\n@[coe]\ndef toAlgHom {F : Type*} [AlgHomClass F R A B] (f : F) : A →ₐ[R] B :=\n { (f : A →+* B) with\n toFun := f\n commutes' := AlgHomClass.commutes f }\n\ninstance coeTC {F : Type*} [AlgHomClass F R A B] : CoeTC F (A →ₐ[R] B) :=\n ⟨AlgHomClass.toAlgHom⟩\n#align alg_hom_class.alg_hom.has_coe_t AlgHomClass.coeTC\n\nend AlgHomClass\n\nnamespace AlgHom\n\nvariable {R : Type u} {A : Type v} {B : Type w} {C : Type u₁} {D : Type v₁}\n\nsection Semiring\n\nvariable [CommSemiring R] [Semiring A] [Semiring B] [Semiring C] [Semiring D]\n\nvariable [Algebra R A] [Algebra R B] [Algebra R C] [Algebra R D]\n\n-- Porting note: we don't port specialized `CoeFun` instances if there is `FunLike` instead\n#noalign alg_hom.has_coe_to_fun\n\n-- Porting note: This instance is moved.\ninstance algHomClass : AlgHomClass (A →ₐ[R] B) R A B where\n coe f := f.toFun\n coe_injective' f g h := by\n rcases f with ⟨⟨⟨⟨_, _⟩, _⟩, _, _⟩, _⟩\n rcases g with ⟨⟨⟨⟨_, _⟩, _⟩, _, _⟩, _⟩\n congr\n map_add f := f.map_add'\n map_zero f := f.map_zero'\n map_mul f := f.map_mul'\n map_one f := f.map_one'\n commutes f := f.commutes'\n#align alg_hom.alg_hom_class AlgHom.algHomClass\n\n/-- See Note [custom simps projection] -/\ndef Simps.apply {R : Type u} {α : Type v} {β : Type w} [CommSemiring R]\n [Semiring α] [Semiring β] [Algebra R α] [Algebra R β] (f : α →ₐ[R] β) : α → β := f\n\ninitialize_simps_projections AlgHom (toFun → apply)\n\n@[simp]\nprotected theorem coe_coe {F : Type*} [AlgHomClass F R A B] (f : F) : ⇑(f : A →ₐ[R] B) = f :=\n rfl\n#align alg_hom.coe_coe AlgHom.coe_coe\n\n@[simp]\ntheorem toFun_eq_coe (f : A →ₐ[R] B) : f.toFun = f :=\n rfl\n#align alg_hom.to_fun_eq_coe AlgHom.toFun_eq_coe\n\nattribute [coe] AlgHom.toRingHom\n\ninstance coeOutRingHom : CoeOut (A →ₐ[R] B) (A →+* B) :=\n ⟨AlgHom.toRingHom⟩\n#align alg_hom.coe_ring_hom AlgHom.coeOutRingHom\n\n-- Porting note: A new definition underlying a coercion `↑`.\n@[coe]\ndef toMonoidHom' (f : A →ₐ[R] B) : A →* B := (f : A →+* B)\n\ninstance coeOutMonoidHom : CoeOut (A →ₐ[R] B) (A →* B) :=\n ⟨AlgHom.toMonoidHom'⟩\n#align alg_hom.coe_monoid_hom AlgHom.coeOutMonoidHom\n\n-- Porting note: A new definition underlying a coercion `↑`.\n@[coe]\ndef toAddMonoidHom' (f : A →ₐ[R] B) : A →+ B := (f : A →+* B)\n\ninstance coeOutAddMonoidHom : CoeOut (A →ₐ[R] B) (A →+ B) :=\n ⟨AlgHom.toAddMonoidHom'⟩\n#align alg_hom.coe_add_monoid_hom AlgHom.coeOutAddMonoidHom\n\n-- Porting note: Lean 3: `@[simp, norm_cast] coe_mk`\n-- Lean 4: `@[simp] coe_mk` & `@[norm_cast] coe_mks`\n@[simp]\ntheorem coe_mk {f : A →+* B} (h) : ((⟨f, h⟩ : A →ₐ[R] B) : A → B) = f :=\n rfl\n\n@[norm_cast]\ntheorem coe_mks {f : A → B} (h₁ h₂ h₃ h₄ h₅) : ⇑(⟨⟨⟨⟨f, h₁⟩, h₂⟩, h₃, h₄⟩, h₅⟩ : A →ₐ[R] B) = f :=\n rfl\n#align alg_hom.coe_mk AlgHom.coe_mks\n\n-- Porting note: This theorem is new.\n@[simp, norm_cast]\ntheorem coe_ringHom_mk {f : A →+* B} (h) : ((⟨f, h⟩ : A →ₐ[R] B) : A →+* B) = f :=\n rfl\n\n-- make the coercion the simp-normal form\n@[simp]\ntheorem toRingHom_eq_coe (f : A →ₐ[R] B) : f.toRingHom = f :=\n rfl\n#align alg_hom.to_ring_hom_eq_coe AlgHom.toRingHom_eq_coe\n\n@[simp, norm_cast]\ntheorem coe_toRingHom (f : A →ₐ[R] B) : ⇑(f : A →+* B) = f :=\n rfl\n#align alg_hom.coe_to_ring_hom AlgHom.coe_toRingHom\n\n@[simp, norm_cast]\ntheorem coe_toMonoidHom (f : A →ₐ[R] B) : ⇑(f : A →* B) = f :=\n rfl\n#align alg_hom.coe_to_monoid_hom AlgHom.coe_toMonoidHom\n\n@[simp, norm_cast]\ntheorem coe_toAddMonoidHom (f : A →ₐ[R] B) : ⇑(f : A →+ B) = f :=\n rfl\n#align alg_hom.coe_to_add_monoid_hom AlgHom.coe_toAddMonoidHom\n\nvariable (φ : A →ₐ[R] B)\n\ntheorem coe_fn_injective : @Function.Injective (A →ₐ[R] B) (A → B) (↑) :=\n FunLike.coe_injective\n#align alg_hom.coe_fn_injective AlgHom.coe_fn_injective\n\ntheorem coe_fn_inj {φ₁ φ₂ : A →ₐ[R] B} : (φ₁ : A → B) = φ₂ ↔ φ₁ = φ₂ :=\n FunLike.coe_fn_eq\n#align alg_hom.coe_fn_inj AlgHom.coe_fn_inj\n\ntheorem coe_ringHom_injective : Function.Injective ((↑) : (A →ₐ[R] B) → A →+* B) := fun φ₁ φ₂ H =>\n coe_fn_injective <| show ((φ₁ : A →+* B) : A → B) = ((φ₂ : A →+* B) : A → B) from congr_arg _ H\n#align alg_hom.coe_ring_hom_injective AlgHom.coe_ringHom_injective\n\ntheorem coe_monoidHom_injective : Function.Injective ((↑) : (A →ₐ[R] B) → A →* B) :=\n RingHom.coe_monoidHom_injective.comp coe_ringHom_injective\n#align alg_hom.coe_monoid_hom_injective AlgHom.coe_monoidHom_injective\n\ntheorem coe_addMonoidHom_injective : Function.Injective ((↑) : (A →ₐ[R] B) → A →+ B) :=\n RingHom.coe_addMonoidHom_injective.comp coe_ringHom_injective\n#align alg_hom.coe_add_monoid_hom_injective AlgHom.coe_addMonoidHom_injective\n\nprotected theorem congr_fun {φ₁ φ₂ : A →ₐ[R] B} (H : φ₁ = φ₂) (x : A) : φ₁ x = φ₂ x :=\n FunLike.congr_fun H x\n#align alg_hom.congr_fun AlgHom.congr_fun\n\nprotected theorem congr_arg (φ : A →ₐ[R] B) {x y : A} (h : x = y) : φ x = φ y :=\n FunLike.congr_arg φ h\n#align alg_hom.congr_arg AlgHom.congr_arg\n\n@[ext]\ntheorem ext {φ₁ φ₂ : A →ₐ[R] B} (H : ∀ x, φ₁ x = φ₂ x) : φ₁ = φ₂ :=\n FunLike.ext _ _ H\n#align alg_hom.ext AlgHom.ext\n\ntheorem ext_iff {φ₁ φ₂ : A →ₐ[R] B} : φ₁ = φ₂ ↔ ∀ x, φ₁ x = φ₂ x :=\n FunLike.ext_iff\n#align alg_hom.ext_iff AlgHom.ext_iff\n\n@[simp]\ntheorem mk_coe {f : A →ₐ[R] B} (h₁ h₂ h₃ h₄ h₅) : (⟨⟨⟨⟨f, h₁⟩, h₂⟩, h₃, h₄⟩, h₅⟩ : A →ₐ[R] B) = f :=\n ext fun _ => rfl\n#align alg_hom.mk_coe AlgHom.mk_coe\n\n@[simp]\ntheorem commutes (r : R) : φ (algebraMap R A r) = algebraMap R B r :=\n φ.commutes' r\n#align alg_hom.commutes AlgHom.commutes\n\ntheorem comp_algebraMap : (φ : A →+* B).comp (algebraMap R A) = algebraMap R B :=\n RingHom.ext <| φ.commutes\n#align alg_hom.comp_algebra_map AlgHom.comp_algebraMap\n\nprotected theorem map_add (r s : A) : φ (r + s) = φ r + φ s :=\n map_add _ _ _\n#align alg_hom.map_add AlgHom.map_add\n\nprotected theorem map_zero : φ 0 = 0 :=\n map_zero _\n#align alg_hom.map_zero AlgHom.map_zero\n\nprotected theorem map_mul (x y) : φ (x * y) = φ x * φ y :=\n map_mul _ _ _\n#align alg_hom.map_mul AlgHom.map_mul\n\nprotected theorem map_one : φ 1 = 1 :=\n map_one _\n#align alg_hom.map_one AlgHom.map_one\n\nprotected theorem map_pow (x : A) (n : ℕ) : φ (x ^ n) = φ x ^ n :=\n map_pow _ _ _\n#align alg_hom.map_pow AlgHom.map_pow\n\n-- @[simp] -- Porting note: simp can prove this\nprotected theorem map_smul (r : R) (x : A) : φ (r • x) = r • φ x :=\n map_smul _ _ _\n#align alg_hom.map_smul AlgHom.map_smul\n\nprotected theorem map_sum {ι : Type*} (f : ι → A) (s : Finset ι) :\n φ (∑ x in s, f x) = ∑ x in s, φ (f x) :=\n map_sum _ _ _\n#align alg_hom.map_sum AlgHom.map_sum\n\nprotected theorem map_finsupp_sum {α : Type*} [Zero α] {ι : Type*} (f : ι →₀ α) (g : ι → α → A) :\n φ (f.sum g) = f.sum fun i a => φ (g i a) :=\n map_finsupp_sum _ _ _\n#align alg_hom.map_finsupp_sum AlgHom.map_finsupp_sum\n\nset_option linter.deprecated false in\nprotected theorem map_bit0 (x) : φ (bit0 x) = bit0 (φ x) :=\n map_bit0 _ _\n#align alg_hom.map_bit0 AlgHom.map_bit0\n\nset_option linter.deprecated false in\nprotected theorem map_bit1 (x) : φ (bit1 x) = bit1 (φ x) :=\n map_bit1 _ _\n#align alg_hom.map_bit1 AlgHom.map_bit1\n\n/-- If a `RingHom` is `R`-linear, then it is an `AlgHom`. -/\ndef mk' (f : A →+* B) (h : ∀ (c : R) (x), f (c • x) = c • f x) : A →ₐ[R] B :=\n { f with\n toFun := f\n commutes' := fun c => by simp only [Algebra.algebraMap_eq_smul_one, h, f.map_one] }\n#align alg_hom.mk' AlgHom.mk'\n\n@[simp]\ntheorem coe_mk' (f : A →+* B) (h : ∀ (c : R) (x), f (c • x) = c • f x) : ⇑(mk' f h) = f :=\n rfl\n#align alg_hom.coe_mk' AlgHom.coe_mk'\n\nsection\n\nvariable (R A)\n\n/-- Identity map as an `AlgHom`. -/\nprotected def id : A →ₐ[R] A :=\n { RingHom.id A with commutes' := fun _ => rfl }\n#align alg_hom.id AlgHom.id\n\n@[simp]\ntheorem coe_id : ⇑(AlgHom.id R A) = id :=\n rfl\n#align alg_hom.coe_id AlgHom.coe_id\n\n@[simp]\ntheorem id_toRingHom : (AlgHom.id R A : A →+* A) = RingHom.id _ :=\n rfl\n#align alg_hom.id_to_ring_hom AlgHom.id_toRingHom\n\nend\n\ntheorem id_apply (p : A) : AlgHom.id R A p = p :=\n rfl\n#align alg_hom.id_apply AlgHom.id_apply\n\n/-- Composition of algebra homeomorphisms. -/\ndef comp (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) : A →ₐ[R] C :=\n { φ₁.toRingHom.comp ↑φ₂ with\n commutes' := fun r : R => by rw [← φ₁.commutes, ← φ₂.commutes]; rfl }\n#align alg_hom.comp AlgHom.comp\n\n@[simp]\ntheorem coe_comp (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) : ⇑(φ₁.comp φ₂) = φ₁ ∘ φ₂ :=\n rfl\n#align alg_hom.coe_comp AlgHom.coe_comp\n\ntheorem comp_apply (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) (p : A) : φ₁.comp φ₂ p = φ₁ (φ₂ p) :=\n rfl\n#align alg_hom.comp_apply AlgHom.comp_apply\n\ntheorem comp_toRingHom (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) :\n (φ₁.comp φ₂ : A →+* C) = (φ₁ : B →+* C).comp ↑φ₂ :=\n rfl\n#align alg_hom.comp_to_ring_hom AlgHom.comp_toRingHom\n\n@[simp]\ntheorem comp_id : φ.comp (AlgHom.id R A) = φ :=\n ext fun _x => rfl\n#align alg_hom.comp_id AlgHom.comp_id\n\n@[simp]\ntheorem id_comp : (AlgHom.id R B).comp φ = φ :=\n ext fun _x => rfl\n#align alg_hom.id_comp AlgHom.id_comp\n\ntheorem comp_assoc (φ₁ : C →ₐ[R] D) (φ₂ : B →ₐ[R] C) (φ₃ : A →ₐ[R] B) :\n (φ₁.comp φ₂).comp φ₃ = φ₁.comp (φ₂.comp φ₃) :=\n ext fun _x => rfl\n#align alg_hom.comp_assoc AlgHom.comp_assoc\n\n/-- R-Alg ⥤ R-Mod -/\ndef toLinearMap : A →ₗ[R] B where\n toFun := φ\n map_add' := map_add _\n map_smul' := map_smul _\n#align alg_hom.to_linear_map AlgHom.toLinearMap\n\n@[simp]\ntheorem toLinearMap_apply (p : A) : φ.toLinearMap p = φ p :=\n rfl\n#align alg_hom.to_linear_map_apply AlgHom.toLinearMap_apply\n\ntheorem toLinearMap_injective :\n Function.Injective (toLinearMap : _ → A →ₗ[R] B) := fun _φ₁ _φ₂ h =>\n ext <| LinearMap.congr_fun h\n#align alg_hom.to_linear_map_injective AlgHom.toLinearMap_injective\n\n@[simp]\ntheorem comp_toLinearMap (f : A →ₐ[R] B) (g : B →ₐ[R] C) :\n (g.comp f).toLinearMap = g.toLinearMap.comp f.toLinearMap :=\n rfl\n#align alg_hom.comp_to_linear_map AlgHom.comp_toLinearMap\n\n@[simp]\ntheorem toLinearMap_id : toLinearMap (AlgHom.id R A) = LinearMap.id :=\n LinearMap.ext fun _ => rfl\n#align alg_hom.to_linear_map_id AlgHom.toLinearMap_id\n\n/-- Promote a `LinearMap` to an `AlgHom` by supplying proofs about the behavior on `1` and `*`. -/\n@[simps]\ndef ofLinearMap (f : A →ₗ[R] B) (map_one : f 1 = 1) (map_mul : ∀ x y, f (x * y) = f x * f y) :\n A →ₐ[R] B :=\n { f.toAddMonoidHom with\n toFun := f\n map_one' := map_one\n map_mul' := map_mul\n commutes' := fun c => by simp only [Algebra.algebraMap_eq_smul_one, f.map_smul, map_one] }\n#align alg_hom.of_linear_map AlgHom.ofLinearMap\n\n@[simp]\ntheorem ofLinearMap_toLinearMap (map_one) (map_mul) :\n ofLinearMap φ.toLinearMap map_one map_mul = φ := by\n ext\n rfl\n#align alg_hom.of_linear_map_to_linear_map AlgHom.ofLinearMap_toLinearMap\n\n@[simp]\ntheorem toLinearMap_ofLinearMap (f : A →ₗ[R] B) (map_one) (map_mul) :\n toLinearMap (ofLinearMap f map_one map_mul) = f := by\n ext\n rfl\n#align alg_hom.to_linear_map_of_linear_map AlgHom.toLinearMap_ofLinearMap\n\n@[simp]\ntheorem ofLinearMap_id (map_one) (map_mul) :\n ofLinearMap LinearMap.id map_one map_mul = AlgHom.id R A :=\n ext fun _ => rfl\n#align alg_hom.of_linear_map_id AlgHom.ofLinearMap_id\n\ntheorem map_smul_of_tower {R'} [SMul R' A] [SMul R' B] [LinearMap.CompatibleSMul A B R' R] (r : R')\n (x : A) : φ (r • x) = r • φ x :=\n φ.toLinearMap.map_smul_of_tower r x\n#align alg_hom.map_smul_of_tower AlgHom.map_smul_of_tower\n\ntheorem map_list_prod (s : List A) : φ s.prod = (s.map φ).prod :=\n φ.toRingHom.map_list_prod s\n#align alg_hom.map_list_prod AlgHom.map_list_prod\n\n@[simps (config := .lemmasOnly) toSemigroup_toMul_mul toOne_one]\ninstance End : Monoid (A →ₐ[R] A) where\n mul := comp\n mul_assoc ϕ ψ χ := rfl\n one := AlgHom.id R A\n one_mul ϕ := ext fun x => rfl\n mul_one ϕ := ext fun x => rfl\n#align alg_hom.End AlgHom.End\n\n@[simp]\ntheorem one_apply (x : A) : (1 : A →ₐ[R] A) x = x :=\n rfl\n#align alg_hom.one_apply AlgHom.one_apply\n\n@[simp]\ntheorem mul_apply (φ ψ : A →ₐ[R] A) (x : A) : (φ * ψ) x = φ (ψ x) :=\n rfl\n#align alg_hom.mul_apply AlgHom.mul_apply\n\ntheorem algebraMap_eq_apply (f : A →ₐ[R] B) {y : R} {x : A} (h : algebraMap R A y = x) :\n algebraMap R B y = f x :=\n h ▸ (f.commutes _).symm\n#align alg_hom.algebra_map_eq_apply AlgHom.algebraMap_eq_apply\n\nend Semiring\n\nsection CommSemiring\n\nvariable [CommSemiring R] [CommSemiring A] [CommSemiring B]\n\nvariable [Algebra R A] [Algebra R B] (φ : A →ₐ[R] B)\n\nprotected theorem map_multiset_prod (s : Multiset A) : φ s.prod = (s.map φ).prod :=\n map_multiset_prod _ _\n#align alg_hom.map_multiset_prod AlgHom.map_multiset_prod\n\nprotected theorem map_prod {ι : Type*} (f : ι → A) (s : Finset ι) :\n φ (∏ x in s, f x) = ∏ x in s, φ (f x) :=\n map_prod _ _ _\n#align alg_hom.map_prod AlgHom.map_prod\n\nprotected theorem map_finsupp_prod {α : Type*} [Zero α] {ι : Type*} (f : ι →₀ α) (g : ι → α → A) :\n φ (f.prod g) = f.prod fun i a => φ (g i a) :=\n map_finsupp_prod _ _ _\n#align alg_hom.map_finsupp_prod AlgHom.map_finsupp_prod\n\nend CommSemiring\n\nsection Ring\n\nvariable [CommSemiring R] [Ring A] [Ring B]\n\nvariable [Algebra R A] [Algebra R B] (φ : A →ₐ[R] B)\n\nprotected theorem map_neg (x) : φ (-x) = -φ x :=\n map_neg _ _\n#align alg_hom.map_neg AlgHom.map_neg\n\nprotected theorem map_sub (x y) : φ (x - y) = φ x - φ y :=\n map_sub _ _ _\n#align alg_hom.map_sub AlgHom.map_sub\n\nend Ring\n\nend AlgHom\n\nnamespace RingHom\n\nvariable {R S : Type*}\n\n/-- Reinterpret a `RingHom` as an `ℕ`-algebra homomorphism. -/\ndef toNatAlgHom [Semiring R] [Semiring S] (f : R →+* S) : R →ₐ[ℕ] S :=\n { f with\n toFun := f\n commutes' := fun n => by simp }\n#align ring_hom.to_nat_alg_hom RingHom.toNatAlgHom\n\n/-- Reinterpret a `RingHom` as a `ℤ`-algebra homomorphism. -/\ndef toIntAlgHom [Ring R] [Ring S] [Algebra ℤ R] [Algebra ℤ S] (f : R →+* S) : R →ₐ[ℤ] S :=\n { f with commutes' := fun n => by simp }\n#align ring_hom.to_int_alg_hom RingHom.toIntAlgHom\n\nlemma toIntAlgHom_injective [Ring R] [Ring S] [Algebra ℤ R] [Algebra ℤ S] :\n Function.Injective (RingHom.toIntAlgHom : (R →+* S) → _) :=\n fun _ _ e ↦ FunLike.ext _ _ (fun x ↦ FunLike.congr_fun e x)\n\n/-- Reinterpret a `RingHom` as a `ℚ`-algebra homomorphism. This actually yields an equivalence,\nsee `RingHom.equivRatAlgHom`. -/\ndef toRatAlgHom [Ring R] [Ring S] [Algebra ℚ R] [Algebra ℚ S] (f : R →+* S) : R →ₐ[ℚ] S :=\n { f with commutes' := f.map_rat_algebraMap }\n#align ring_hom.to_rat_alg_hom RingHom.toRatAlgHom\n\n@[simp]\ntheorem toRatAlgHom_toRingHom [Ring R] [Ring S] [Algebra ℚ R] [Algebra ℚ S] (f : R →+* S) :\n ↑f.toRatAlgHom = f :=\n RingHom.ext fun _x => rfl\n#align ring_hom.to_rat_alg_hom_to_ring_hom RingHom.toRatAlgHom_toRingHom\n\nend RingHom\n\nsection\n\nvariable {R S : Type*}\n\n@[simp]\ntheorem AlgHom.toRingHom_toRatAlgHom [Ring R] [Ring S] [Algebra ℚ R] [Algebra ℚ S]\n (f : R →ₐ[ℚ] S) : (f : R →+* S).toRatAlgHom = f :=\n AlgHom.ext fun _x => rfl\n#align alg_hom.to_ring_hom_to_rat_alg_hom AlgHom.toRingHom_toRatAlgHom\n\n/-- The equivalence between `RingHom` and `ℚ`-algebra homomorphisms. -/\n@[simps]\ndef RingHom.equivRatAlgHom [Ring R] [Ring S] [Algebra ℚ R] [Algebra ℚ S] : (R →+* S) ≃ (R →ₐ[ℚ] S)\n where\n toFun := RingHom.toRatAlgHom\n invFun := AlgHom.toRingHom\n left_inv f := RingHom.toRatAlgHom_toRingHom f\n right_inv f := AlgHom.toRingHom_toRatAlgHom f\n#align ring_hom.equiv_rat_alg_hom RingHom.equivRatAlgHom\n\nend\n\nnamespace Algebra\n\nvariable (R : Type u) (A : Type v)\n\nvariable [CommSemiring R] [Semiring A] [Algebra R A]\n\n/-- `AlgebraMap` as an `AlgHom`. -/\ndef ofId : R →ₐ[R] A :=\n { algebraMap R A with commutes' := fun _ => rfl }\n#align algebra.of_id Algebra.ofId\n\nvariable {R}\n\ntheorem ofId_apply (r) : ofId R A r = algebraMap R A r :=\n rfl\n#align algebra.of_id_apply Algebra.ofId_apply\n\n/-- This is a special case of a more general instance that we define in a later file. -/\ninstance subsingleton_id : Subsingleton (R →ₐ[R] A) :=\n ⟨fun f g => AlgHom.ext fun _ => (f.commutes _).trans (g.commutes _).symm⟩\n\n/-- This ext lemma closes trivial subgoals create when chaining heterobasic ext lemmas. -/\n@[ext high]\ntheorem ext_id (f g : R →ₐ[R] A) : f = g := Subsingleton.elim _ _\n\nsection MulDistribMulAction\n\ninstance : MulDistribMulAction (A →ₐ[R] A) Aˣ where\n smul := fun f => Units.map f\n one_smul := fun x => by ext; rfl\n mul_smul := fun x y z => by ext; rfl\n smul_mul := fun x y z => by ","nextTactic":"ext","declUpToTactic":"instance : MulDistribMulAction (A →ₐ[R] A) Aˣ where\n smul := fun f => Units.map f\n one_smul := fun x => by ext; rfl\n mul_smul := fun x y z => by ext; rfl\n smul_mul := fun x y z => by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Hom.579_0.FxqJeqR3qcIVlrq","decl":"instance : MulDistribMulAction (A →ₐ[R] A) Aˣ where\n smul "} +{"state":"case a\nR : Type u\nA : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nx : A →ₐ[R] A\ny z : Aˣ\n⊢ ↑(x • (y * z)) = ↑(x • y * x • z)","srcUpToTactic":"/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Yury Kudryashov\n-/\nimport Mathlib.Algebra.Algebra.Basic\nimport Mathlib.Algebra.BigOperators.Finsupp\n\n#align_import algebra.algebra.hom from \"leanprover-community/mathlib\"@\"e97cf15cd1aec9bd5c193b2ffac5a6dc9118912b\"\n\n/-!\n# Homomorphisms of `R`-algebras\n\nThis file defines bundled homomorphisms of `R`-algebras.\n\n## Main definitions\n\n* `AlgHom R A B`: the type of `R`-algebra morphisms from `A` to `B`.\n* `Algebra.ofId R A : R →ₐ[R] A`: the canonical map from `R` to `A`, as an `AlgHom`.\n\n## Notations\n\n* `A →ₐ[R] B` : `R`-algebra homomorphism from `A` to `B`.\n-/\n\nset_option autoImplicit true\n\n\nopen BigOperators\n\nuniverse u v w u₁ v₁\n\n/-- Defining the homomorphism in the category R-Alg. -/\n-- @[nolint has_nonempty_instance] -- Porting note: This linter does not exist yet.\nstructure AlgHom (R : Type u) (A : Type v) (B : Type w) [CommSemiring R] [Semiring A] [Semiring B]\n [Algebra R A] [Algebra R B] extends RingHom A B where\n commutes' : ∀ r : R, toFun (algebraMap R A r) = algebraMap R B r\n#align alg_hom AlgHom\n\n/-- Reinterpret an `AlgHom` as a `RingHom` -/\nadd_decl_doc AlgHom.toRingHom\n\n@[inherit_doc AlgHom]\ninfixr:25 \" →ₐ \" => AlgHom _\n\n@[inherit_doc]\nnotation:25 A \" →ₐ[\" R \"] \" B => AlgHom R A B\n\n/-- `AlgHomClass F R A B` asserts `F` is a type of bundled algebra homomorphisms\nfrom `A` to `B`. -/\nclass AlgHomClass (F : Type*) (R : outParam (Type*)) (A : outParam (Type*))\n (B : outParam (Type*)) [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A]\n [Algebra R B] extends RingHomClass F A B where\n commutes : ∀ (f : F) (r : R), f (algebraMap R A r) = algebraMap R B r\n#align alg_hom_class AlgHomClass\n\n-- Porting note: `dangerousInstance` linter has become smarter about `outParam`s\n-- attribute [nolint dangerousInstance] AlgHomClass.toRingHomClass\n\n-- Porting note: simp can prove this\n-- attribute [simp] AlgHomClass.commutes\n\nnamespace AlgHomClass\n\nvariable {R : Type*} {A : Type*} {B : Type*} [CommSemiring R] [Semiring A] [Semiring B]\n [Algebra R A] [Algebra R B]\n\n-- see Note [lower instance priority]\ninstance (priority := 100) linearMapClass [AlgHomClass F R A B] : LinearMapClass F R A B :=\n { ‹AlgHomClass F R A B› with\n map_smulₛₗ := fun f r x => by\n simp only [Algebra.smul_def, map_mul, commutes, RingHom.id_apply] }\n#align alg_hom_class.linear_map_class AlgHomClass.linearMapClass\n\n-- Porting note: A new definition underlying a coercion `↑`.\n/-- Turn an element of a type `F` satisfying `AlgHomClass F α β` into an actual\n`AlgHom`. This is declared as the default coercion from `F` to `α →+* β`. -/\n@[coe]\ndef toAlgHom {F : Type*} [AlgHomClass F R A B] (f : F) : A →ₐ[R] B :=\n { (f : A →+* B) with\n toFun := f\n commutes' := AlgHomClass.commutes f }\n\ninstance coeTC {F : Type*} [AlgHomClass F R A B] : CoeTC F (A →ₐ[R] B) :=\n ⟨AlgHomClass.toAlgHom⟩\n#align alg_hom_class.alg_hom.has_coe_t AlgHomClass.coeTC\n\nend AlgHomClass\n\nnamespace AlgHom\n\nvariable {R : Type u} {A : Type v} {B : Type w} {C : Type u₁} {D : Type v₁}\n\nsection Semiring\n\nvariable [CommSemiring R] [Semiring A] [Semiring B] [Semiring C] [Semiring D]\n\nvariable [Algebra R A] [Algebra R B] [Algebra R C] [Algebra R D]\n\n-- Porting note: we don't port specialized `CoeFun` instances if there is `FunLike` instead\n#noalign alg_hom.has_coe_to_fun\n\n-- Porting note: This instance is moved.\ninstance algHomClass : AlgHomClass (A →ₐ[R] B) R A B where\n coe f := f.toFun\n coe_injective' f g h := by\n rcases f with ⟨⟨⟨⟨_, _⟩, _⟩, _, _⟩, _⟩\n rcases g with ⟨⟨⟨⟨_, _⟩, _⟩, _, _⟩, _⟩\n congr\n map_add f := f.map_add'\n map_zero f := f.map_zero'\n map_mul f := f.map_mul'\n map_one f := f.map_one'\n commutes f := f.commutes'\n#align alg_hom.alg_hom_class AlgHom.algHomClass\n\n/-- See Note [custom simps projection] -/\ndef Simps.apply {R : Type u} {α : Type v} {β : Type w} [CommSemiring R]\n [Semiring α] [Semiring β] [Algebra R α] [Algebra R β] (f : α →ₐ[R] β) : α → β := f\n\ninitialize_simps_projections AlgHom (toFun → apply)\n\n@[simp]\nprotected theorem coe_coe {F : Type*} [AlgHomClass F R A B] (f : F) : ⇑(f : A →ₐ[R] B) = f :=\n rfl\n#align alg_hom.coe_coe AlgHom.coe_coe\n\n@[simp]\ntheorem toFun_eq_coe (f : A →ₐ[R] B) : f.toFun = f :=\n rfl\n#align alg_hom.to_fun_eq_coe AlgHom.toFun_eq_coe\n\nattribute [coe] AlgHom.toRingHom\n\ninstance coeOutRingHom : CoeOut (A →ₐ[R] B) (A →+* B) :=\n ⟨AlgHom.toRingHom⟩\n#align alg_hom.coe_ring_hom AlgHom.coeOutRingHom\n\n-- Porting note: A new definition underlying a coercion `↑`.\n@[coe]\ndef toMonoidHom' (f : A →ₐ[R] B) : A →* B := (f : A →+* B)\n\ninstance coeOutMonoidHom : CoeOut (A →ₐ[R] B) (A →* B) :=\n ⟨AlgHom.toMonoidHom'⟩\n#align alg_hom.coe_monoid_hom AlgHom.coeOutMonoidHom\n\n-- Porting note: A new definition underlying a coercion `↑`.\n@[coe]\ndef toAddMonoidHom' (f : A →ₐ[R] B) : A →+ B := (f : A →+* B)\n\ninstance coeOutAddMonoidHom : CoeOut (A →ₐ[R] B) (A →+ B) :=\n ⟨AlgHom.toAddMonoidHom'⟩\n#align alg_hom.coe_add_monoid_hom AlgHom.coeOutAddMonoidHom\n\n-- Porting note: Lean 3: `@[simp, norm_cast] coe_mk`\n-- Lean 4: `@[simp] coe_mk` & `@[norm_cast] coe_mks`\n@[simp]\ntheorem coe_mk {f : A →+* B} (h) : ((⟨f, h⟩ : A →ₐ[R] B) : A → B) = f :=\n rfl\n\n@[norm_cast]\ntheorem coe_mks {f : A → B} (h₁ h₂ h₃ h₄ h₅) : ⇑(⟨⟨⟨⟨f, h₁⟩, h₂⟩, h₃, h₄⟩, h₅⟩ : A →ₐ[R] B) = f :=\n rfl\n#align alg_hom.coe_mk AlgHom.coe_mks\n\n-- Porting note: This theorem is new.\n@[simp, norm_cast]\ntheorem coe_ringHom_mk {f : A →+* B} (h) : ((⟨f, h⟩ : A →ₐ[R] B) : A →+* B) = f :=\n rfl\n\n-- make the coercion the simp-normal form\n@[simp]\ntheorem toRingHom_eq_coe (f : A →ₐ[R] B) : f.toRingHom = f :=\n rfl\n#align alg_hom.to_ring_hom_eq_coe AlgHom.toRingHom_eq_coe\n\n@[simp, norm_cast]\ntheorem coe_toRingHom (f : A →ₐ[R] B) : ⇑(f : A →+* B) = f :=\n rfl\n#align alg_hom.coe_to_ring_hom AlgHom.coe_toRingHom\n\n@[simp, norm_cast]\ntheorem coe_toMonoidHom (f : A →ₐ[R] B) : ⇑(f : A →* B) = f :=\n rfl\n#align alg_hom.coe_to_monoid_hom AlgHom.coe_toMonoidHom\n\n@[simp, norm_cast]\ntheorem coe_toAddMonoidHom (f : A →ₐ[R] B) : ⇑(f : A →+ B) = f :=\n rfl\n#align alg_hom.coe_to_add_monoid_hom AlgHom.coe_toAddMonoidHom\n\nvariable (φ : A →ₐ[R] B)\n\ntheorem coe_fn_injective : @Function.Injective (A →ₐ[R] B) (A → B) (↑) :=\n FunLike.coe_injective\n#align alg_hom.coe_fn_injective AlgHom.coe_fn_injective\n\ntheorem coe_fn_inj {φ₁ φ₂ : A →ₐ[R] B} : (φ₁ : A → B) = φ₂ ↔ φ₁ = φ₂ :=\n FunLike.coe_fn_eq\n#align alg_hom.coe_fn_inj AlgHom.coe_fn_inj\n\ntheorem coe_ringHom_injective : Function.Injective ((↑) : (A →ₐ[R] B) → A →+* B) := fun φ₁ φ₂ H =>\n coe_fn_injective <| show ((φ₁ : A →+* B) : A → B) = ((φ₂ : A →+* B) : A → B) from congr_arg _ H\n#align alg_hom.coe_ring_hom_injective AlgHom.coe_ringHom_injective\n\ntheorem coe_monoidHom_injective : Function.Injective ((↑) : (A →ₐ[R] B) → A →* B) :=\n RingHom.coe_monoidHom_injective.comp coe_ringHom_injective\n#align alg_hom.coe_monoid_hom_injective AlgHom.coe_monoidHom_injective\n\ntheorem coe_addMonoidHom_injective : Function.Injective ((↑) : (A →ₐ[R] B) → A →+ B) :=\n RingHom.coe_addMonoidHom_injective.comp coe_ringHom_injective\n#align alg_hom.coe_add_monoid_hom_injective AlgHom.coe_addMonoidHom_injective\n\nprotected theorem congr_fun {φ₁ φ₂ : A →ₐ[R] B} (H : φ₁ = φ₂) (x : A) : φ₁ x = φ₂ x :=\n FunLike.congr_fun H x\n#align alg_hom.congr_fun AlgHom.congr_fun\n\nprotected theorem congr_arg (φ : A →ₐ[R] B) {x y : A} (h : x = y) : φ x = φ y :=\n FunLike.congr_arg φ h\n#align alg_hom.congr_arg AlgHom.congr_arg\n\n@[ext]\ntheorem ext {φ₁ φ₂ : A →ₐ[R] B} (H : ∀ x, φ₁ x = φ₂ x) : φ₁ = φ₂ :=\n FunLike.ext _ _ H\n#align alg_hom.ext AlgHom.ext\n\ntheorem ext_iff {φ₁ φ₂ : A →ₐ[R] B} : φ₁ = φ₂ ↔ ∀ x, φ₁ x = φ₂ x :=\n FunLike.ext_iff\n#align alg_hom.ext_iff AlgHom.ext_iff\n\n@[simp]\ntheorem mk_coe {f : A →ₐ[R] B} (h₁ h₂ h₃ h₄ h₅) : (⟨⟨⟨⟨f, h₁⟩, h₂⟩, h₃, h₄⟩, h₅⟩ : A →ₐ[R] B) = f :=\n ext fun _ => rfl\n#align alg_hom.mk_coe AlgHom.mk_coe\n\n@[simp]\ntheorem commutes (r : R) : φ (algebraMap R A r) = algebraMap R B r :=\n φ.commutes' r\n#align alg_hom.commutes AlgHom.commutes\n\ntheorem comp_algebraMap : (φ : A →+* B).comp (algebraMap R A) = algebraMap R B :=\n RingHom.ext <| φ.commutes\n#align alg_hom.comp_algebra_map AlgHom.comp_algebraMap\n\nprotected theorem map_add (r s : A) : φ (r + s) = φ r + φ s :=\n map_add _ _ _\n#align alg_hom.map_add AlgHom.map_add\n\nprotected theorem map_zero : φ 0 = 0 :=\n map_zero _\n#align alg_hom.map_zero AlgHom.map_zero\n\nprotected theorem map_mul (x y) : φ (x * y) = φ x * φ y :=\n map_mul _ _ _\n#align alg_hom.map_mul AlgHom.map_mul\n\nprotected theorem map_one : φ 1 = 1 :=\n map_one _\n#align alg_hom.map_one AlgHom.map_one\n\nprotected theorem map_pow (x : A) (n : ℕ) : φ (x ^ n) = φ x ^ n :=\n map_pow _ _ _\n#align alg_hom.map_pow AlgHom.map_pow\n\n-- @[simp] -- Porting note: simp can prove this\nprotected theorem map_smul (r : R) (x : A) : φ (r • x) = r • φ x :=\n map_smul _ _ _\n#align alg_hom.map_smul AlgHom.map_smul\n\nprotected theorem map_sum {ι : Type*} (f : ι → A) (s : Finset ι) :\n φ (∑ x in s, f x) = ∑ x in s, φ (f x) :=\n map_sum _ _ _\n#align alg_hom.map_sum AlgHom.map_sum\n\nprotected theorem map_finsupp_sum {α : Type*} [Zero α] {ι : Type*} (f : ι →₀ α) (g : ι → α → A) :\n φ (f.sum g) = f.sum fun i a => φ (g i a) :=\n map_finsupp_sum _ _ _\n#align alg_hom.map_finsupp_sum AlgHom.map_finsupp_sum\n\nset_option linter.deprecated false in\nprotected theorem map_bit0 (x) : φ (bit0 x) = bit0 (φ x) :=\n map_bit0 _ _\n#align alg_hom.map_bit0 AlgHom.map_bit0\n\nset_option linter.deprecated false in\nprotected theorem map_bit1 (x) : φ (bit1 x) = bit1 (φ x) :=\n map_bit1 _ _\n#align alg_hom.map_bit1 AlgHom.map_bit1\n\n/-- If a `RingHom` is `R`-linear, then it is an `AlgHom`. -/\ndef mk' (f : A →+* B) (h : ∀ (c : R) (x), f (c • x) = c • f x) : A →ₐ[R] B :=\n { f with\n toFun := f\n commutes' := fun c => by simp only [Algebra.algebraMap_eq_smul_one, h, f.map_one] }\n#align alg_hom.mk' AlgHom.mk'\n\n@[simp]\ntheorem coe_mk' (f : A →+* B) (h : ∀ (c : R) (x), f (c • x) = c • f x) : ⇑(mk' f h) = f :=\n rfl\n#align alg_hom.coe_mk' AlgHom.coe_mk'\n\nsection\n\nvariable (R A)\n\n/-- Identity map as an `AlgHom`. -/\nprotected def id : A →ₐ[R] A :=\n { RingHom.id A with commutes' := fun _ => rfl }\n#align alg_hom.id AlgHom.id\n\n@[simp]\ntheorem coe_id : ⇑(AlgHom.id R A) = id :=\n rfl\n#align alg_hom.coe_id AlgHom.coe_id\n\n@[simp]\ntheorem id_toRingHom : (AlgHom.id R A : A →+* A) = RingHom.id _ :=\n rfl\n#align alg_hom.id_to_ring_hom AlgHom.id_toRingHom\n\nend\n\ntheorem id_apply (p : A) : AlgHom.id R A p = p :=\n rfl\n#align alg_hom.id_apply AlgHom.id_apply\n\n/-- Composition of algebra homeomorphisms. -/\ndef comp (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) : A →ₐ[R] C :=\n { φ₁.toRingHom.comp ↑φ₂ with\n commutes' := fun r : R => by rw [← φ₁.commutes, ← φ₂.commutes]; rfl }\n#align alg_hom.comp AlgHom.comp\n\n@[simp]\ntheorem coe_comp (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) : ⇑(φ₁.comp φ₂) = φ₁ ∘ φ₂ :=\n rfl\n#align alg_hom.coe_comp AlgHom.coe_comp\n\ntheorem comp_apply (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) (p : A) : φ₁.comp φ₂ p = φ₁ (φ₂ p) :=\n rfl\n#align alg_hom.comp_apply AlgHom.comp_apply\n\ntheorem comp_toRingHom (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) :\n (φ₁.comp φ₂ : A →+* C) = (φ₁ : B →+* C).comp ↑φ₂ :=\n rfl\n#align alg_hom.comp_to_ring_hom AlgHom.comp_toRingHom\n\n@[simp]\ntheorem comp_id : φ.comp (AlgHom.id R A) = φ :=\n ext fun _x => rfl\n#align alg_hom.comp_id AlgHom.comp_id\n\n@[simp]\ntheorem id_comp : (AlgHom.id R B).comp φ = φ :=\n ext fun _x => rfl\n#align alg_hom.id_comp AlgHom.id_comp\n\ntheorem comp_assoc (φ₁ : C →ₐ[R] D) (φ₂ : B →ₐ[R] C) (φ₃ : A →ₐ[R] B) :\n (φ₁.comp φ₂).comp φ₃ = φ₁.comp (φ₂.comp φ₃) :=\n ext fun _x => rfl\n#align alg_hom.comp_assoc AlgHom.comp_assoc\n\n/-- R-Alg ⥤ R-Mod -/\ndef toLinearMap : A →ₗ[R] B where\n toFun := φ\n map_add' := map_add _\n map_smul' := map_smul _\n#align alg_hom.to_linear_map AlgHom.toLinearMap\n\n@[simp]\ntheorem toLinearMap_apply (p : A) : φ.toLinearMap p = φ p :=\n rfl\n#align alg_hom.to_linear_map_apply AlgHom.toLinearMap_apply\n\ntheorem toLinearMap_injective :\n Function.Injective (toLinearMap : _ → A →ₗ[R] B) := fun _φ₁ _φ₂ h =>\n ext <| LinearMap.congr_fun h\n#align alg_hom.to_linear_map_injective AlgHom.toLinearMap_injective\n\n@[simp]\ntheorem comp_toLinearMap (f : A →ₐ[R] B) (g : B →ₐ[R] C) :\n (g.comp f).toLinearMap = g.toLinearMap.comp f.toLinearMap :=\n rfl\n#align alg_hom.comp_to_linear_map AlgHom.comp_toLinearMap\n\n@[simp]\ntheorem toLinearMap_id : toLinearMap (AlgHom.id R A) = LinearMap.id :=\n LinearMap.ext fun _ => rfl\n#align alg_hom.to_linear_map_id AlgHom.toLinearMap_id\n\n/-- Promote a `LinearMap` to an `AlgHom` by supplying proofs about the behavior on `1` and `*`. -/\n@[simps]\ndef ofLinearMap (f : A →ₗ[R] B) (map_one : f 1 = 1) (map_mul : ∀ x y, f (x * y) = f x * f y) :\n A →ₐ[R] B :=\n { f.toAddMonoidHom with\n toFun := f\n map_one' := map_one\n map_mul' := map_mul\n commutes' := fun c => by simp only [Algebra.algebraMap_eq_smul_one, f.map_smul, map_one] }\n#align alg_hom.of_linear_map AlgHom.ofLinearMap\n\n@[simp]\ntheorem ofLinearMap_toLinearMap (map_one) (map_mul) :\n ofLinearMap φ.toLinearMap map_one map_mul = φ := by\n ext\n rfl\n#align alg_hom.of_linear_map_to_linear_map AlgHom.ofLinearMap_toLinearMap\n\n@[simp]\ntheorem toLinearMap_ofLinearMap (f : A →ₗ[R] B) (map_one) (map_mul) :\n toLinearMap (ofLinearMap f map_one map_mul) = f := by\n ext\n rfl\n#align alg_hom.to_linear_map_of_linear_map AlgHom.toLinearMap_ofLinearMap\n\n@[simp]\ntheorem ofLinearMap_id (map_one) (map_mul) :\n ofLinearMap LinearMap.id map_one map_mul = AlgHom.id R A :=\n ext fun _ => rfl\n#align alg_hom.of_linear_map_id AlgHom.ofLinearMap_id\n\ntheorem map_smul_of_tower {R'} [SMul R' A] [SMul R' B] [LinearMap.CompatibleSMul A B R' R] (r : R')\n (x : A) : φ (r • x) = r • φ x :=\n φ.toLinearMap.map_smul_of_tower r x\n#align alg_hom.map_smul_of_tower AlgHom.map_smul_of_tower\n\ntheorem map_list_prod (s : List A) : φ s.prod = (s.map φ).prod :=\n φ.toRingHom.map_list_prod s\n#align alg_hom.map_list_prod AlgHom.map_list_prod\n\n@[simps (config := .lemmasOnly) toSemigroup_toMul_mul toOne_one]\ninstance End : Monoid (A →ₐ[R] A) where\n mul := comp\n mul_assoc ϕ ψ χ := rfl\n one := AlgHom.id R A\n one_mul ϕ := ext fun x => rfl\n mul_one ϕ := ext fun x => rfl\n#align alg_hom.End AlgHom.End\n\n@[simp]\ntheorem one_apply (x : A) : (1 : A →ₐ[R] A) x = x :=\n rfl\n#align alg_hom.one_apply AlgHom.one_apply\n\n@[simp]\ntheorem mul_apply (φ ψ : A →ₐ[R] A) (x : A) : (φ * ψ) x = φ (ψ x) :=\n rfl\n#align alg_hom.mul_apply AlgHom.mul_apply\n\ntheorem algebraMap_eq_apply (f : A →ₐ[R] B) {y : R} {x : A} (h : algebraMap R A y = x) :\n algebraMap R B y = f x :=\n h ▸ (f.commutes _).symm\n#align alg_hom.algebra_map_eq_apply AlgHom.algebraMap_eq_apply\n\nend Semiring\n\nsection CommSemiring\n\nvariable [CommSemiring R] [CommSemiring A] [CommSemiring B]\n\nvariable [Algebra R A] [Algebra R B] (φ : A →ₐ[R] B)\n\nprotected theorem map_multiset_prod (s : Multiset A) : φ s.prod = (s.map φ).prod :=\n map_multiset_prod _ _\n#align alg_hom.map_multiset_prod AlgHom.map_multiset_prod\n\nprotected theorem map_prod {ι : Type*} (f : ι → A) (s : Finset ι) :\n φ (∏ x in s, f x) = ∏ x in s, φ (f x) :=\n map_prod _ _ _\n#align alg_hom.map_prod AlgHom.map_prod\n\nprotected theorem map_finsupp_prod {α : Type*} [Zero α] {ι : Type*} (f : ι →₀ α) (g : ι → α → A) :\n φ (f.prod g) = f.prod fun i a => φ (g i a) :=\n map_finsupp_prod _ _ _\n#align alg_hom.map_finsupp_prod AlgHom.map_finsupp_prod\n\nend CommSemiring\n\nsection Ring\n\nvariable [CommSemiring R] [Ring A] [Ring B]\n\nvariable [Algebra R A] [Algebra R B] (φ : A →ₐ[R] B)\n\nprotected theorem map_neg (x) : φ (-x) = -φ x :=\n map_neg _ _\n#align alg_hom.map_neg AlgHom.map_neg\n\nprotected theorem map_sub (x y) : φ (x - y) = φ x - φ y :=\n map_sub _ _ _\n#align alg_hom.map_sub AlgHom.map_sub\n\nend Ring\n\nend AlgHom\n\nnamespace RingHom\n\nvariable {R S : Type*}\n\n/-- Reinterpret a `RingHom` as an `ℕ`-algebra homomorphism. -/\ndef toNatAlgHom [Semiring R] [Semiring S] (f : R →+* S) : R →ₐ[ℕ] S :=\n { f with\n toFun := f\n commutes' := fun n => by simp }\n#align ring_hom.to_nat_alg_hom RingHom.toNatAlgHom\n\n/-- Reinterpret a `RingHom` as a `ℤ`-algebra homomorphism. -/\ndef toIntAlgHom [Ring R] [Ring S] [Algebra ℤ R] [Algebra ℤ S] (f : R →+* S) : R →ₐ[ℤ] S :=\n { f with commutes' := fun n => by simp }\n#align ring_hom.to_int_alg_hom RingHom.toIntAlgHom\n\nlemma toIntAlgHom_injective [Ring R] [Ring S] [Algebra ℤ R] [Algebra ℤ S] :\n Function.Injective (RingHom.toIntAlgHom : (R →+* S) → _) :=\n fun _ _ e ↦ FunLike.ext _ _ (fun x ↦ FunLike.congr_fun e x)\n\n/-- Reinterpret a `RingHom` as a `ℚ`-algebra homomorphism. This actually yields an equivalence,\nsee `RingHom.equivRatAlgHom`. -/\ndef toRatAlgHom [Ring R] [Ring S] [Algebra ℚ R] [Algebra ℚ S] (f : R →+* S) : R →ₐ[ℚ] S :=\n { f with commutes' := f.map_rat_algebraMap }\n#align ring_hom.to_rat_alg_hom RingHom.toRatAlgHom\n\n@[simp]\ntheorem toRatAlgHom_toRingHom [Ring R] [Ring S] [Algebra ℚ R] [Algebra ℚ S] (f : R →+* S) :\n ↑f.toRatAlgHom = f :=\n RingHom.ext fun _x => rfl\n#align ring_hom.to_rat_alg_hom_to_ring_hom RingHom.toRatAlgHom_toRingHom\n\nend RingHom\n\nsection\n\nvariable {R S : Type*}\n\n@[simp]\ntheorem AlgHom.toRingHom_toRatAlgHom [Ring R] [Ring S] [Algebra ℚ R] [Algebra ℚ S]\n (f : R →ₐ[ℚ] S) : (f : R →+* S).toRatAlgHom = f :=\n AlgHom.ext fun _x => rfl\n#align alg_hom.to_ring_hom_to_rat_alg_hom AlgHom.toRingHom_toRatAlgHom\n\n/-- The equivalence between `RingHom` and `ℚ`-algebra homomorphisms. -/\n@[simps]\ndef RingHom.equivRatAlgHom [Ring R] [Ring S] [Algebra ℚ R] [Algebra ℚ S] : (R →+* S) ≃ (R →ₐ[ℚ] S)\n where\n toFun := RingHom.toRatAlgHom\n invFun := AlgHom.toRingHom\n left_inv f := RingHom.toRatAlgHom_toRingHom f\n right_inv f := AlgHom.toRingHom_toRatAlgHom f\n#align ring_hom.equiv_rat_alg_hom RingHom.equivRatAlgHom\n\nend\n\nnamespace Algebra\n\nvariable (R : Type u) (A : Type v)\n\nvariable [CommSemiring R] [Semiring A] [Algebra R A]\n\n/-- `AlgebraMap` as an `AlgHom`. -/\ndef ofId : R →ₐ[R] A :=\n { algebraMap R A with commutes' := fun _ => rfl }\n#align algebra.of_id Algebra.ofId\n\nvariable {R}\n\ntheorem ofId_apply (r) : ofId R A r = algebraMap R A r :=\n rfl\n#align algebra.of_id_apply Algebra.ofId_apply\n\n/-- This is a special case of a more general instance that we define in a later file. -/\ninstance subsingleton_id : Subsingleton (R →ₐ[R] A) :=\n ⟨fun f g => AlgHom.ext fun _ => (f.commutes _).trans (g.commutes _).symm⟩\n\n/-- This ext lemma closes trivial subgoals create when chaining heterobasic ext lemmas. -/\n@[ext high]\ntheorem ext_id (f g : R →ₐ[R] A) : f = g := Subsingleton.elim _ _\n\nsection MulDistribMulAction\n\ninstance : MulDistribMulAction (A →ₐ[R] A) Aˣ where\n smul := fun f => Units.map f\n one_smul := fun x => by ext; rfl\n mul_smul := fun x y z => by ext; rfl\n smul_mul := fun x y z => by ext; ","nextTactic":"exact x.map_mul _ _","declUpToTactic":"instance : MulDistribMulAction (A →ₐ[R] A) Aˣ where\n smul := fun f => Units.map f\n one_smul := fun x => by ext; rfl\n mul_smul := fun x y z => by ext; rfl\n smul_mul := fun x y z => by ext; ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Hom.579_0.FxqJeqR3qcIVlrq","decl":"instance : MulDistribMulAction (A →ₐ[R] A) Aˣ where\n smul "} +{"state":"R : Type u\nA : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nx : A →ₐ[R] A\n⊢ x • 1 = 1","srcUpToTactic":"/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Yury Kudryashov\n-/\nimport Mathlib.Algebra.Algebra.Basic\nimport Mathlib.Algebra.BigOperators.Finsupp\n\n#align_import algebra.algebra.hom from \"leanprover-community/mathlib\"@\"e97cf15cd1aec9bd5c193b2ffac5a6dc9118912b\"\n\n/-!\n# Homomorphisms of `R`-algebras\n\nThis file defines bundled homomorphisms of `R`-algebras.\n\n## Main definitions\n\n* `AlgHom R A B`: the type of `R`-algebra morphisms from `A` to `B`.\n* `Algebra.ofId R A : R →ₐ[R] A`: the canonical map from `R` to `A`, as an `AlgHom`.\n\n## Notations\n\n* `A →ₐ[R] B` : `R`-algebra homomorphism from `A` to `B`.\n-/\n\nset_option autoImplicit true\n\n\nopen BigOperators\n\nuniverse u v w u₁ v₁\n\n/-- Defining the homomorphism in the category R-Alg. -/\n-- @[nolint has_nonempty_instance] -- Porting note: This linter does not exist yet.\nstructure AlgHom (R : Type u) (A : Type v) (B : Type w) [CommSemiring R] [Semiring A] [Semiring B]\n [Algebra R A] [Algebra R B] extends RingHom A B where\n commutes' : ∀ r : R, toFun (algebraMap R A r) = algebraMap R B r\n#align alg_hom AlgHom\n\n/-- Reinterpret an `AlgHom` as a `RingHom` -/\nadd_decl_doc AlgHom.toRingHom\n\n@[inherit_doc AlgHom]\ninfixr:25 \" →ₐ \" => AlgHom _\n\n@[inherit_doc]\nnotation:25 A \" →ₐ[\" R \"] \" B => AlgHom R A B\n\n/-- `AlgHomClass F R A B` asserts `F` is a type of bundled algebra homomorphisms\nfrom `A` to `B`. -/\nclass AlgHomClass (F : Type*) (R : outParam (Type*)) (A : outParam (Type*))\n (B : outParam (Type*)) [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A]\n [Algebra R B] extends RingHomClass F A B where\n commutes : ∀ (f : F) (r : R), f (algebraMap R A r) = algebraMap R B r\n#align alg_hom_class AlgHomClass\n\n-- Porting note: `dangerousInstance` linter has become smarter about `outParam`s\n-- attribute [nolint dangerousInstance] AlgHomClass.toRingHomClass\n\n-- Porting note: simp can prove this\n-- attribute [simp] AlgHomClass.commutes\n\nnamespace AlgHomClass\n\nvariable {R : Type*} {A : Type*} {B : Type*} [CommSemiring R] [Semiring A] [Semiring B]\n [Algebra R A] [Algebra R B]\n\n-- see Note [lower instance priority]\ninstance (priority := 100) linearMapClass [AlgHomClass F R A B] : LinearMapClass F R A B :=\n { ‹AlgHomClass F R A B› with\n map_smulₛₗ := fun f r x => by\n simp only [Algebra.smul_def, map_mul, commutes, RingHom.id_apply] }\n#align alg_hom_class.linear_map_class AlgHomClass.linearMapClass\n\n-- Porting note: A new definition underlying a coercion `↑`.\n/-- Turn an element of a type `F` satisfying `AlgHomClass F α β` into an actual\n`AlgHom`. This is declared as the default coercion from `F` to `α →+* β`. -/\n@[coe]\ndef toAlgHom {F : Type*} [AlgHomClass F R A B] (f : F) : A →ₐ[R] B :=\n { (f : A →+* B) with\n toFun := f\n commutes' := AlgHomClass.commutes f }\n\ninstance coeTC {F : Type*} [AlgHomClass F R A B] : CoeTC F (A →ₐ[R] B) :=\n ⟨AlgHomClass.toAlgHom⟩\n#align alg_hom_class.alg_hom.has_coe_t AlgHomClass.coeTC\n\nend AlgHomClass\n\nnamespace AlgHom\n\nvariable {R : Type u} {A : Type v} {B : Type w} {C : Type u₁} {D : Type v₁}\n\nsection Semiring\n\nvariable [CommSemiring R] [Semiring A] [Semiring B] [Semiring C] [Semiring D]\n\nvariable [Algebra R A] [Algebra R B] [Algebra R C] [Algebra R D]\n\n-- Porting note: we don't port specialized `CoeFun` instances if there is `FunLike` instead\n#noalign alg_hom.has_coe_to_fun\n\n-- Porting note: This instance is moved.\ninstance algHomClass : AlgHomClass (A →ₐ[R] B) R A B where\n coe f := f.toFun\n coe_injective' f g h := by\n rcases f with ⟨⟨⟨⟨_, _⟩, _⟩, _, _⟩, _⟩\n rcases g with ⟨⟨⟨⟨_, _⟩, _⟩, _, _⟩, _⟩\n congr\n map_add f := f.map_add'\n map_zero f := f.map_zero'\n map_mul f := f.map_mul'\n map_one f := f.map_one'\n commutes f := f.commutes'\n#align alg_hom.alg_hom_class AlgHom.algHomClass\n\n/-- See Note [custom simps projection] -/\ndef Simps.apply {R : Type u} {α : Type v} {β : Type w} [CommSemiring R]\n [Semiring α] [Semiring β] [Algebra R α] [Algebra R β] (f : α →ₐ[R] β) : α → β := f\n\ninitialize_simps_projections AlgHom (toFun → apply)\n\n@[simp]\nprotected theorem coe_coe {F : Type*} [AlgHomClass F R A B] (f : F) : ⇑(f : A →ₐ[R] B) = f :=\n rfl\n#align alg_hom.coe_coe AlgHom.coe_coe\n\n@[simp]\ntheorem toFun_eq_coe (f : A →ₐ[R] B) : f.toFun = f :=\n rfl\n#align alg_hom.to_fun_eq_coe AlgHom.toFun_eq_coe\n\nattribute [coe] AlgHom.toRingHom\n\ninstance coeOutRingHom : CoeOut (A →ₐ[R] B) (A →+* B) :=\n ⟨AlgHom.toRingHom⟩\n#align alg_hom.coe_ring_hom AlgHom.coeOutRingHom\n\n-- Porting note: A new definition underlying a coercion `↑`.\n@[coe]\ndef toMonoidHom' (f : A →ₐ[R] B) : A →* B := (f : A →+* B)\n\ninstance coeOutMonoidHom : CoeOut (A →ₐ[R] B) (A →* B) :=\n ⟨AlgHom.toMonoidHom'⟩\n#align alg_hom.coe_monoid_hom AlgHom.coeOutMonoidHom\n\n-- Porting note: A new definition underlying a coercion `↑`.\n@[coe]\ndef toAddMonoidHom' (f : A →ₐ[R] B) : A →+ B := (f : A →+* B)\n\ninstance coeOutAddMonoidHom : CoeOut (A →ₐ[R] B) (A →+ B) :=\n ⟨AlgHom.toAddMonoidHom'⟩\n#align alg_hom.coe_add_monoid_hom AlgHom.coeOutAddMonoidHom\n\n-- Porting note: Lean 3: `@[simp, norm_cast] coe_mk`\n-- Lean 4: `@[simp] coe_mk` & `@[norm_cast] coe_mks`\n@[simp]\ntheorem coe_mk {f : A →+* B} (h) : ((⟨f, h⟩ : A →ₐ[R] B) : A → B) = f :=\n rfl\n\n@[norm_cast]\ntheorem coe_mks {f : A → B} (h₁ h₂ h₃ h₄ h₅) : ⇑(⟨⟨⟨⟨f, h₁⟩, h₂⟩, h₃, h₄⟩, h₅⟩ : A →ₐ[R] B) = f :=\n rfl\n#align alg_hom.coe_mk AlgHom.coe_mks\n\n-- Porting note: This theorem is new.\n@[simp, norm_cast]\ntheorem coe_ringHom_mk {f : A →+* B} (h) : ((⟨f, h⟩ : A →ₐ[R] B) : A →+* B) = f :=\n rfl\n\n-- make the coercion the simp-normal form\n@[simp]\ntheorem toRingHom_eq_coe (f : A →ₐ[R] B) : f.toRingHom = f :=\n rfl\n#align alg_hom.to_ring_hom_eq_coe AlgHom.toRingHom_eq_coe\n\n@[simp, norm_cast]\ntheorem coe_toRingHom (f : A →ₐ[R] B) : ⇑(f : A →+* B) = f :=\n rfl\n#align alg_hom.coe_to_ring_hom AlgHom.coe_toRingHom\n\n@[simp, norm_cast]\ntheorem coe_toMonoidHom (f : A →ₐ[R] B) : ⇑(f : A →* B) = f :=\n rfl\n#align alg_hom.coe_to_monoid_hom AlgHom.coe_toMonoidHom\n\n@[simp, norm_cast]\ntheorem coe_toAddMonoidHom (f : A →ₐ[R] B) : ⇑(f : A →+ B) = f :=\n rfl\n#align alg_hom.coe_to_add_monoid_hom AlgHom.coe_toAddMonoidHom\n\nvariable (φ : A →ₐ[R] B)\n\ntheorem coe_fn_injective : @Function.Injective (A →ₐ[R] B) (A → B) (↑) :=\n FunLike.coe_injective\n#align alg_hom.coe_fn_injective AlgHom.coe_fn_injective\n\ntheorem coe_fn_inj {φ₁ φ₂ : A →ₐ[R] B} : (φ₁ : A → B) = φ₂ ↔ φ₁ = φ₂ :=\n FunLike.coe_fn_eq\n#align alg_hom.coe_fn_inj AlgHom.coe_fn_inj\n\ntheorem coe_ringHom_injective : Function.Injective ((↑) : (A →ₐ[R] B) → A →+* B) := fun φ₁ φ₂ H =>\n coe_fn_injective <| show ((φ₁ : A →+* B) : A → B) = ((φ₂ : A →+* B) : A → B) from congr_arg _ H\n#align alg_hom.coe_ring_hom_injective AlgHom.coe_ringHom_injective\n\ntheorem coe_monoidHom_injective : Function.Injective ((↑) : (A →ₐ[R] B) → A →* B) :=\n RingHom.coe_monoidHom_injective.comp coe_ringHom_injective\n#align alg_hom.coe_monoid_hom_injective AlgHom.coe_monoidHom_injective\n\ntheorem coe_addMonoidHom_injective : Function.Injective ((↑) : (A →ₐ[R] B) → A →+ B) :=\n RingHom.coe_addMonoidHom_injective.comp coe_ringHom_injective\n#align alg_hom.coe_add_monoid_hom_injective AlgHom.coe_addMonoidHom_injective\n\nprotected theorem congr_fun {φ₁ φ₂ : A →ₐ[R] B} (H : φ₁ = φ₂) (x : A) : φ₁ x = φ₂ x :=\n FunLike.congr_fun H x\n#align alg_hom.congr_fun AlgHom.congr_fun\n\nprotected theorem congr_arg (φ : A →ₐ[R] B) {x y : A} (h : x = y) : φ x = φ y :=\n FunLike.congr_arg φ h\n#align alg_hom.congr_arg AlgHom.congr_arg\n\n@[ext]\ntheorem ext {φ₁ φ₂ : A →ₐ[R] B} (H : ∀ x, φ₁ x = φ₂ x) : φ₁ = φ₂ :=\n FunLike.ext _ _ H\n#align alg_hom.ext AlgHom.ext\n\ntheorem ext_iff {φ₁ φ₂ : A →ₐ[R] B} : φ₁ = φ₂ ↔ ∀ x, φ₁ x = φ₂ x :=\n FunLike.ext_iff\n#align alg_hom.ext_iff AlgHom.ext_iff\n\n@[simp]\ntheorem mk_coe {f : A →ₐ[R] B} (h₁ h₂ h₃ h₄ h₅) : (⟨⟨⟨⟨f, h₁⟩, h₂⟩, h₃, h₄⟩, h₅⟩ : A →ₐ[R] B) = f :=\n ext fun _ => rfl\n#align alg_hom.mk_coe AlgHom.mk_coe\n\n@[simp]\ntheorem commutes (r : R) : φ (algebraMap R A r) = algebraMap R B r :=\n φ.commutes' r\n#align alg_hom.commutes AlgHom.commutes\n\ntheorem comp_algebraMap : (φ : A →+* B).comp (algebraMap R A) = algebraMap R B :=\n RingHom.ext <| φ.commutes\n#align alg_hom.comp_algebra_map AlgHom.comp_algebraMap\n\nprotected theorem map_add (r s : A) : φ (r + s) = φ r + φ s :=\n map_add _ _ _\n#align alg_hom.map_add AlgHom.map_add\n\nprotected theorem map_zero : φ 0 = 0 :=\n map_zero _\n#align alg_hom.map_zero AlgHom.map_zero\n\nprotected theorem map_mul (x y) : φ (x * y) = φ x * φ y :=\n map_mul _ _ _\n#align alg_hom.map_mul AlgHom.map_mul\n\nprotected theorem map_one : φ 1 = 1 :=\n map_one _\n#align alg_hom.map_one AlgHom.map_one\n\nprotected theorem map_pow (x : A) (n : ℕ) : φ (x ^ n) = φ x ^ n :=\n map_pow _ _ _\n#align alg_hom.map_pow AlgHom.map_pow\n\n-- @[simp] -- Porting note: simp can prove this\nprotected theorem map_smul (r : R) (x : A) : φ (r • x) = r • φ x :=\n map_smul _ _ _\n#align alg_hom.map_smul AlgHom.map_smul\n\nprotected theorem map_sum {ι : Type*} (f : ι → A) (s : Finset ι) :\n φ (∑ x in s, f x) = ∑ x in s, φ (f x) :=\n map_sum _ _ _\n#align alg_hom.map_sum AlgHom.map_sum\n\nprotected theorem map_finsupp_sum {α : Type*} [Zero α] {ι : Type*} (f : ι →₀ α) (g : ι → α → A) :\n φ (f.sum g) = f.sum fun i a => φ (g i a) :=\n map_finsupp_sum _ _ _\n#align alg_hom.map_finsupp_sum AlgHom.map_finsupp_sum\n\nset_option linter.deprecated false in\nprotected theorem map_bit0 (x) : φ (bit0 x) = bit0 (φ x) :=\n map_bit0 _ _\n#align alg_hom.map_bit0 AlgHom.map_bit0\n\nset_option linter.deprecated false in\nprotected theorem map_bit1 (x) : φ (bit1 x) = bit1 (φ x) :=\n map_bit1 _ _\n#align alg_hom.map_bit1 AlgHom.map_bit1\n\n/-- If a `RingHom` is `R`-linear, then it is an `AlgHom`. -/\ndef mk' (f : A →+* B) (h : ∀ (c : R) (x), f (c • x) = c • f x) : A →ₐ[R] B :=\n { f with\n toFun := f\n commutes' := fun c => by simp only [Algebra.algebraMap_eq_smul_one, h, f.map_one] }\n#align alg_hom.mk' AlgHom.mk'\n\n@[simp]\ntheorem coe_mk' (f : A →+* B) (h : ∀ (c : R) (x), f (c • x) = c • f x) : ⇑(mk' f h) = f :=\n rfl\n#align alg_hom.coe_mk' AlgHom.coe_mk'\n\nsection\n\nvariable (R A)\n\n/-- Identity map as an `AlgHom`. -/\nprotected def id : A →ₐ[R] A :=\n { RingHom.id A with commutes' := fun _ => rfl }\n#align alg_hom.id AlgHom.id\n\n@[simp]\ntheorem coe_id : ⇑(AlgHom.id R A) = id :=\n rfl\n#align alg_hom.coe_id AlgHom.coe_id\n\n@[simp]\ntheorem id_toRingHom : (AlgHom.id R A : A →+* A) = RingHom.id _ :=\n rfl\n#align alg_hom.id_to_ring_hom AlgHom.id_toRingHom\n\nend\n\ntheorem id_apply (p : A) : AlgHom.id R A p = p :=\n rfl\n#align alg_hom.id_apply AlgHom.id_apply\n\n/-- Composition of algebra homeomorphisms. -/\ndef comp (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) : A →ₐ[R] C :=\n { φ₁.toRingHom.comp ↑φ₂ with\n commutes' := fun r : R => by rw [← φ₁.commutes, ← φ₂.commutes]; rfl }\n#align alg_hom.comp AlgHom.comp\n\n@[simp]\ntheorem coe_comp (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) : ⇑(φ₁.comp φ₂) = φ₁ ∘ φ₂ :=\n rfl\n#align alg_hom.coe_comp AlgHom.coe_comp\n\ntheorem comp_apply (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) (p : A) : φ₁.comp φ₂ p = φ₁ (φ₂ p) :=\n rfl\n#align alg_hom.comp_apply AlgHom.comp_apply\n\ntheorem comp_toRingHom (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) :\n (φ₁.comp φ₂ : A →+* C) = (φ₁ : B →+* C).comp ↑φ₂ :=\n rfl\n#align alg_hom.comp_to_ring_hom AlgHom.comp_toRingHom\n\n@[simp]\ntheorem comp_id : φ.comp (AlgHom.id R A) = φ :=\n ext fun _x => rfl\n#align alg_hom.comp_id AlgHom.comp_id\n\n@[simp]\ntheorem id_comp : (AlgHom.id R B).comp φ = φ :=\n ext fun _x => rfl\n#align alg_hom.id_comp AlgHom.id_comp\n\ntheorem comp_assoc (φ₁ : C →ₐ[R] D) (φ₂ : B →ₐ[R] C) (φ₃ : A →ₐ[R] B) :\n (φ₁.comp φ₂).comp φ₃ = φ₁.comp (φ₂.comp φ₃) :=\n ext fun _x => rfl\n#align alg_hom.comp_assoc AlgHom.comp_assoc\n\n/-- R-Alg ⥤ R-Mod -/\ndef toLinearMap : A →ₗ[R] B where\n toFun := φ\n map_add' := map_add _\n map_smul' := map_smul _\n#align alg_hom.to_linear_map AlgHom.toLinearMap\n\n@[simp]\ntheorem toLinearMap_apply (p : A) : φ.toLinearMap p = φ p :=\n rfl\n#align alg_hom.to_linear_map_apply AlgHom.toLinearMap_apply\n\ntheorem toLinearMap_injective :\n Function.Injective (toLinearMap : _ → A →ₗ[R] B) := fun _φ₁ _φ₂ h =>\n ext <| LinearMap.congr_fun h\n#align alg_hom.to_linear_map_injective AlgHom.toLinearMap_injective\n\n@[simp]\ntheorem comp_toLinearMap (f : A →ₐ[R] B) (g : B →ₐ[R] C) :\n (g.comp f).toLinearMap = g.toLinearMap.comp f.toLinearMap :=\n rfl\n#align alg_hom.comp_to_linear_map AlgHom.comp_toLinearMap\n\n@[simp]\ntheorem toLinearMap_id : toLinearMap (AlgHom.id R A) = LinearMap.id :=\n LinearMap.ext fun _ => rfl\n#align alg_hom.to_linear_map_id AlgHom.toLinearMap_id\n\n/-- Promote a `LinearMap` to an `AlgHom` by supplying proofs about the behavior on `1` and `*`. -/\n@[simps]\ndef ofLinearMap (f : A →ₗ[R] B) (map_one : f 1 = 1) (map_mul : ∀ x y, f (x * y) = f x * f y) :\n A →ₐ[R] B :=\n { f.toAddMonoidHom with\n toFun := f\n map_one' := map_one\n map_mul' := map_mul\n commutes' := fun c => by simp only [Algebra.algebraMap_eq_smul_one, f.map_smul, map_one] }\n#align alg_hom.of_linear_map AlgHom.ofLinearMap\n\n@[simp]\ntheorem ofLinearMap_toLinearMap (map_one) (map_mul) :\n ofLinearMap φ.toLinearMap map_one map_mul = φ := by\n ext\n rfl\n#align alg_hom.of_linear_map_to_linear_map AlgHom.ofLinearMap_toLinearMap\n\n@[simp]\ntheorem toLinearMap_ofLinearMap (f : A →ₗ[R] B) (map_one) (map_mul) :\n toLinearMap (ofLinearMap f map_one map_mul) = f := by\n ext\n rfl\n#align alg_hom.to_linear_map_of_linear_map AlgHom.toLinearMap_ofLinearMap\n\n@[simp]\ntheorem ofLinearMap_id (map_one) (map_mul) :\n ofLinearMap LinearMap.id map_one map_mul = AlgHom.id R A :=\n ext fun _ => rfl\n#align alg_hom.of_linear_map_id AlgHom.ofLinearMap_id\n\ntheorem map_smul_of_tower {R'} [SMul R' A] [SMul R' B] [LinearMap.CompatibleSMul A B R' R] (r : R')\n (x : A) : φ (r • x) = r • φ x :=\n φ.toLinearMap.map_smul_of_tower r x\n#align alg_hom.map_smul_of_tower AlgHom.map_smul_of_tower\n\ntheorem map_list_prod (s : List A) : φ s.prod = (s.map φ).prod :=\n φ.toRingHom.map_list_prod s\n#align alg_hom.map_list_prod AlgHom.map_list_prod\n\n@[simps (config := .lemmasOnly) toSemigroup_toMul_mul toOne_one]\ninstance End : Monoid (A →ₐ[R] A) where\n mul := comp\n mul_assoc ϕ ψ χ := rfl\n one := AlgHom.id R A\n one_mul ϕ := ext fun x => rfl\n mul_one ϕ := ext fun x => rfl\n#align alg_hom.End AlgHom.End\n\n@[simp]\ntheorem one_apply (x : A) : (1 : A →ₐ[R] A) x = x :=\n rfl\n#align alg_hom.one_apply AlgHom.one_apply\n\n@[simp]\ntheorem mul_apply (φ ψ : A →ₐ[R] A) (x : A) : (φ * ψ) x = φ (ψ x) :=\n rfl\n#align alg_hom.mul_apply AlgHom.mul_apply\n\ntheorem algebraMap_eq_apply (f : A →ₐ[R] B) {y : R} {x : A} (h : algebraMap R A y = x) :\n algebraMap R B y = f x :=\n h ▸ (f.commutes _).symm\n#align alg_hom.algebra_map_eq_apply AlgHom.algebraMap_eq_apply\n\nend Semiring\n\nsection CommSemiring\n\nvariable [CommSemiring R] [CommSemiring A] [CommSemiring B]\n\nvariable [Algebra R A] [Algebra R B] (φ : A →ₐ[R] B)\n\nprotected theorem map_multiset_prod (s : Multiset A) : φ s.prod = (s.map φ).prod :=\n map_multiset_prod _ _\n#align alg_hom.map_multiset_prod AlgHom.map_multiset_prod\n\nprotected theorem map_prod {ι : Type*} (f : ι → A) (s : Finset ι) :\n φ (∏ x in s, f x) = ∏ x in s, φ (f x) :=\n map_prod _ _ _\n#align alg_hom.map_prod AlgHom.map_prod\n\nprotected theorem map_finsupp_prod {α : Type*} [Zero α] {ι : Type*} (f : ι →₀ α) (g : ι → α → A) :\n φ (f.prod g) = f.prod fun i a => φ (g i a) :=\n map_finsupp_prod _ _ _\n#align alg_hom.map_finsupp_prod AlgHom.map_finsupp_prod\n\nend CommSemiring\n\nsection Ring\n\nvariable [CommSemiring R] [Ring A] [Ring B]\n\nvariable [Algebra R A] [Algebra R B] (φ : A →ₐ[R] B)\n\nprotected theorem map_neg (x) : φ (-x) = -φ x :=\n map_neg _ _\n#align alg_hom.map_neg AlgHom.map_neg\n\nprotected theorem map_sub (x y) : φ (x - y) = φ x - φ y :=\n map_sub _ _ _\n#align alg_hom.map_sub AlgHom.map_sub\n\nend Ring\n\nend AlgHom\n\nnamespace RingHom\n\nvariable {R S : Type*}\n\n/-- Reinterpret a `RingHom` as an `ℕ`-algebra homomorphism. -/\ndef toNatAlgHom [Semiring R] [Semiring S] (f : R →+* S) : R →ₐ[ℕ] S :=\n { f with\n toFun := f\n commutes' := fun n => by simp }\n#align ring_hom.to_nat_alg_hom RingHom.toNatAlgHom\n\n/-- Reinterpret a `RingHom` as a `ℤ`-algebra homomorphism. -/\ndef toIntAlgHom [Ring R] [Ring S] [Algebra ℤ R] [Algebra ℤ S] (f : R →+* S) : R →ₐ[ℤ] S :=\n { f with commutes' := fun n => by simp }\n#align ring_hom.to_int_alg_hom RingHom.toIntAlgHom\n\nlemma toIntAlgHom_injective [Ring R] [Ring S] [Algebra ℤ R] [Algebra ℤ S] :\n Function.Injective (RingHom.toIntAlgHom : (R →+* S) → _) :=\n fun _ _ e ↦ FunLike.ext _ _ (fun x ↦ FunLike.congr_fun e x)\n\n/-- Reinterpret a `RingHom` as a `ℚ`-algebra homomorphism. This actually yields an equivalence,\nsee `RingHom.equivRatAlgHom`. -/\ndef toRatAlgHom [Ring R] [Ring S] [Algebra ℚ R] [Algebra ℚ S] (f : R →+* S) : R →ₐ[ℚ] S :=\n { f with commutes' := f.map_rat_algebraMap }\n#align ring_hom.to_rat_alg_hom RingHom.toRatAlgHom\n\n@[simp]\ntheorem toRatAlgHom_toRingHom [Ring R] [Ring S] [Algebra ℚ R] [Algebra ℚ S] (f : R →+* S) :\n ↑f.toRatAlgHom = f :=\n RingHom.ext fun _x => rfl\n#align ring_hom.to_rat_alg_hom_to_ring_hom RingHom.toRatAlgHom_toRingHom\n\nend RingHom\n\nsection\n\nvariable {R S : Type*}\n\n@[simp]\ntheorem AlgHom.toRingHom_toRatAlgHom [Ring R] [Ring S] [Algebra ℚ R] [Algebra ℚ S]\n (f : R →ₐ[ℚ] S) : (f : R →+* S).toRatAlgHom = f :=\n AlgHom.ext fun _x => rfl\n#align alg_hom.to_ring_hom_to_rat_alg_hom AlgHom.toRingHom_toRatAlgHom\n\n/-- The equivalence between `RingHom` and `ℚ`-algebra homomorphisms. -/\n@[simps]\ndef RingHom.equivRatAlgHom [Ring R] [Ring S] [Algebra ℚ R] [Algebra ℚ S] : (R →+* S) ≃ (R →ₐ[ℚ] S)\n where\n toFun := RingHom.toRatAlgHom\n invFun := AlgHom.toRingHom\n left_inv f := RingHom.toRatAlgHom_toRingHom f\n right_inv f := AlgHom.toRingHom_toRatAlgHom f\n#align ring_hom.equiv_rat_alg_hom RingHom.equivRatAlgHom\n\nend\n\nnamespace Algebra\n\nvariable (R : Type u) (A : Type v)\n\nvariable [CommSemiring R] [Semiring A] [Algebra R A]\n\n/-- `AlgebraMap` as an `AlgHom`. -/\ndef ofId : R →ₐ[R] A :=\n { algebraMap R A with commutes' := fun _ => rfl }\n#align algebra.of_id Algebra.ofId\n\nvariable {R}\n\ntheorem ofId_apply (r) : ofId R A r = algebraMap R A r :=\n rfl\n#align algebra.of_id_apply Algebra.ofId_apply\n\n/-- This is a special case of a more general instance that we define in a later file. -/\ninstance subsingleton_id : Subsingleton (R →ₐ[R] A) :=\n ⟨fun f g => AlgHom.ext fun _ => (f.commutes _).trans (g.commutes _).symm⟩\n\n/-- This ext lemma closes trivial subgoals create when chaining heterobasic ext lemmas. -/\n@[ext high]\ntheorem ext_id (f g : R →ₐ[R] A) : f = g := Subsingleton.elim _ _\n\nsection MulDistribMulAction\n\ninstance : MulDistribMulAction (A →ₐ[R] A) Aˣ where\n smul := fun f => Units.map f\n one_smul := fun x => by ext; rfl\n mul_smul := fun x y z => by ext; rfl\n smul_mul := fun x y z => by ext; exact x.map_mul _ _\n smul_one := fun x => by ","nextTactic":"ext","declUpToTactic":"instance : MulDistribMulAction (A →ₐ[R] A) Aˣ where\n smul := fun f => Units.map f\n one_smul := fun x => by ext; rfl\n mul_smul := fun x y z => by ext; rfl\n smul_mul := fun x y z => by ext; exact x.map_mul _ _\n smul_one := fun x => by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Hom.579_0.FxqJeqR3qcIVlrq","decl":"instance : MulDistribMulAction (A →ₐ[R] A) Aˣ where\n smul "} +{"state":"case a\nR : Type u\nA : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nx : A →ₐ[R] A\n⊢ ↑(x • 1) = ↑1","srcUpToTactic":"/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Yury Kudryashov\n-/\nimport Mathlib.Algebra.Algebra.Basic\nimport Mathlib.Algebra.BigOperators.Finsupp\n\n#align_import algebra.algebra.hom from \"leanprover-community/mathlib\"@\"e97cf15cd1aec9bd5c193b2ffac5a6dc9118912b\"\n\n/-!\n# Homomorphisms of `R`-algebras\n\nThis file defines bundled homomorphisms of `R`-algebras.\n\n## Main definitions\n\n* `AlgHom R A B`: the type of `R`-algebra morphisms from `A` to `B`.\n* `Algebra.ofId R A : R →ₐ[R] A`: the canonical map from `R` to `A`, as an `AlgHom`.\n\n## Notations\n\n* `A →ₐ[R] B` : `R`-algebra homomorphism from `A` to `B`.\n-/\n\nset_option autoImplicit true\n\n\nopen BigOperators\n\nuniverse u v w u₁ v₁\n\n/-- Defining the homomorphism in the category R-Alg. -/\n-- @[nolint has_nonempty_instance] -- Porting note: This linter does not exist yet.\nstructure AlgHom (R : Type u) (A : Type v) (B : Type w) [CommSemiring R] [Semiring A] [Semiring B]\n [Algebra R A] [Algebra R B] extends RingHom A B where\n commutes' : ∀ r : R, toFun (algebraMap R A r) = algebraMap R B r\n#align alg_hom AlgHom\n\n/-- Reinterpret an `AlgHom` as a `RingHom` -/\nadd_decl_doc AlgHom.toRingHom\n\n@[inherit_doc AlgHom]\ninfixr:25 \" →ₐ \" => AlgHom _\n\n@[inherit_doc]\nnotation:25 A \" →ₐ[\" R \"] \" B => AlgHom R A B\n\n/-- `AlgHomClass F R A B` asserts `F` is a type of bundled algebra homomorphisms\nfrom `A` to `B`. -/\nclass AlgHomClass (F : Type*) (R : outParam (Type*)) (A : outParam (Type*))\n (B : outParam (Type*)) [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A]\n [Algebra R B] extends RingHomClass F A B where\n commutes : ∀ (f : F) (r : R), f (algebraMap R A r) = algebraMap R B r\n#align alg_hom_class AlgHomClass\n\n-- Porting note: `dangerousInstance` linter has become smarter about `outParam`s\n-- attribute [nolint dangerousInstance] AlgHomClass.toRingHomClass\n\n-- Porting note: simp can prove this\n-- attribute [simp] AlgHomClass.commutes\n\nnamespace AlgHomClass\n\nvariable {R : Type*} {A : Type*} {B : Type*} [CommSemiring R] [Semiring A] [Semiring B]\n [Algebra R A] [Algebra R B]\n\n-- see Note [lower instance priority]\ninstance (priority := 100) linearMapClass [AlgHomClass F R A B] : LinearMapClass F R A B :=\n { ‹AlgHomClass F R A B› with\n map_smulₛₗ := fun f r x => by\n simp only [Algebra.smul_def, map_mul, commutes, RingHom.id_apply] }\n#align alg_hom_class.linear_map_class AlgHomClass.linearMapClass\n\n-- Porting note: A new definition underlying a coercion `↑`.\n/-- Turn an element of a type `F` satisfying `AlgHomClass F α β` into an actual\n`AlgHom`. This is declared as the default coercion from `F` to `α →+* β`. -/\n@[coe]\ndef toAlgHom {F : Type*} [AlgHomClass F R A B] (f : F) : A →ₐ[R] B :=\n { (f : A →+* B) with\n toFun := f\n commutes' := AlgHomClass.commutes f }\n\ninstance coeTC {F : Type*} [AlgHomClass F R A B] : CoeTC F (A →ₐ[R] B) :=\n ⟨AlgHomClass.toAlgHom⟩\n#align alg_hom_class.alg_hom.has_coe_t AlgHomClass.coeTC\n\nend AlgHomClass\n\nnamespace AlgHom\n\nvariable {R : Type u} {A : Type v} {B : Type w} {C : Type u₁} {D : Type v₁}\n\nsection Semiring\n\nvariable [CommSemiring R] [Semiring A] [Semiring B] [Semiring C] [Semiring D]\n\nvariable [Algebra R A] [Algebra R B] [Algebra R C] [Algebra R D]\n\n-- Porting note: we don't port specialized `CoeFun` instances if there is `FunLike` instead\n#noalign alg_hom.has_coe_to_fun\n\n-- Porting note: This instance is moved.\ninstance algHomClass : AlgHomClass (A →ₐ[R] B) R A B where\n coe f := f.toFun\n coe_injective' f g h := by\n rcases f with ⟨⟨⟨⟨_, _⟩, _⟩, _, _⟩, _⟩\n rcases g with ⟨⟨⟨⟨_, _⟩, _⟩, _, _⟩, _⟩\n congr\n map_add f := f.map_add'\n map_zero f := f.map_zero'\n map_mul f := f.map_mul'\n map_one f := f.map_one'\n commutes f := f.commutes'\n#align alg_hom.alg_hom_class AlgHom.algHomClass\n\n/-- See Note [custom simps projection] -/\ndef Simps.apply {R : Type u} {α : Type v} {β : Type w} [CommSemiring R]\n [Semiring α] [Semiring β] [Algebra R α] [Algebra R β] (f : α →ₐ[R] β) : α → β := f\n\ninitialize_simps_projections AlgHom (toFun → apply)\n\n@[simp]\nprotected theorem coe_coe {F : Type*} [AlgHomClass F R A B] (f : F) : ⇑(f : A →ₐ[R] B) = f :=\n rfl\n#align alg_hom.coe_coe AlgHom.coe_coe\n\n@[simp]\ntheorem toFun_eq_coe (f : A →ₐ[R] B) : f.toFun = f :=\n rfl\n#align alg_hom.to_fun_eq_coe AlgHom.toFun_eq_coe\n\nattribute [coe] AlgHom.toRingHom\n\ninstance coeOutRingHom : CoeOut (A →ₐ[R] B) (A →+* B) :=\n ⟨AlgHom.toRingHom⟩\n#align alg_hom.coe_ring_hom AlgHom.coeOutRingHom\n\n-- Porting note: A new definition underlying a coercion `↑`.\n@[coe]\ndef toMonoidHom' (f : A →ₐ[R] B) : A →* B := (f : A →+* B)\n\ninstance coeOutMonoidHom : CoeOut (A →ₐ[R] B) (A →* B) :=\n ⟨AlgHom.toMonoidHom'⟩\n#align alg_hom.coe_monoid_hom AlgHom.coeOutMonoidHom\n\n-- Porting note: A new definition underlying a coercion `↑`.\n@[coe]\ndef toAddMonoidHom' (f : A →ₐ[R] B) : A →+ B := (f : A →+* B)\n\ninstance coeOutAddMonoidHom : CoeOut (A →ₐ[R] B) (A →+ B) :=\n ⟨AlgHom.toAddMonoidHom'⟩\n#align alg_hom.coe_add_monoid_hom AlgHom.coeOutAddMonoidHom\n\n-- Porting note: Lean 3: `@[simp, norm_cast] coe_mk`\n-- Lean 4: `@[simp] coe_mk` & `@[norm_cast] coe_mks`\n@[simp]\ntheorem coe_mk {f : A →+* B} (h) : ((⟨f, h⟩ : A →ₐ[R] B) : A → B) = f :=\n rfl\n\n@[norm_cast]\ntheorem coe_mks {f : A → B} (h₁ h₂ h₃ h₄ h₅) : ⇑(⟨⟨⟨⟨f, h₁⟩, h₂⟩, h₃, h₄⟩, h₅⟩ : A →ₐ[R] B) = f :=\n rfl\n#align alg_hom.coe_mk AlgHom.coe_mks\n\n-- Porting note: This theorem is new.\n@[simp, norm_cast]\ntheorem coe_ringHom_mk {f : A →+* B} (h) : ((⟨f, h⟩ : A →ₐ[R] B) : A →+* B) = f :=\n rfl\n\n-- make the coercion the simp-normal form\n@[simp]\ntheorem toRingHom_eq_coe (f : A →ₐ[R] B) : f.toRingHom = f :=\n rfl\n#align alg_hom.to_ring_hom_eq_coe AlgHom.toRingHom_eq_coe\n\n@[simp, norm_cast]\ntheorem coe_toRingHom (f : A →ₐ[R] B) : ⇑(f : A →+* B) = f :=\n rfl\n#align alg_hom.coe_to_ring_hom AlgHom.coe_toRingHom\n\n@[simp, norm_cast]\ntheorem coe_toMonoidHom (f : A →ₐ[R] B) : ⇑(f : A →* B) = f :=\n rfl\n#align alg_hom.coe_to_monoid_hom AlgHom.coe_toMonoidHom\n\n@[simp, norm_cast]\ntheorem coe_toAddMonoidHom (f : A →ₐ[R] B) : ⇑(f : A →+ B) = f :=\n rfl\n#align alg_hom.coe_to_add_monoid_hom AlgHom.coe_toAddMonoidHom\n\nvariable (φ : A →ₐ[R] B)\n\ntheorem coe_fn_injective : @Function.Injective (A →ₐ[R] B) (A → B) (↑) :=\n FunLike.coe_injective\n#align alg_hom.coe_fn_injective AlgHom.coe_fn_injective\n\ntheorem coe_fn_inj {φ₁ φ₂ : A →ₐ[R] B} : (φ₁ : A → B) = φ₂ ↔ φ₁ = φ₂ :=\n FunLike.coe_fn_eq\n#align alg_hom.coe_fn_inj AlgHom.coe_fn_inj\n\ntheorem coe_ringHom_injective : Function.Injective ((↑) : (A →ₐ[R] B) → A →+* B) := fun φ₁ φ₂ H =>\n coe_fn_injective <| show ((φ₁ : A →+* B) : A → B) = ((φ₂ : A →+* B) : A → B) from congr_arg _ H\n#align alg_hom.coe_ring_hom_injective AlgHom.coe_ringHom_injective\n\ntheorem coe_monoidHom_injective : Function.Injective ((↑) : (A →ₐ[R] B) → A →* B) :=\n RingHom.coe_monoidHom_injective.comp coe_ringHom_injective\n#align alg_hom.coe_monoid_hom_injective AlgHom.coe_monoidHom_injective\n\ntheorem coe_addMonoidHom_injective : Function.Injective ((↑) : (A →ₐ[R] B) → A →+ B) :=\n RingHom.coe_addMonoidHom_injective.comp coe_ringHom_injective\n#align alg_hom.coe_add_monoid_hom_injective AlgHom.coe_addMonoidHom_injective\n\nprotected theorem congr_fun {φ₁ φ₂ : A →ₐ[R] B} (H : φ₁ = φ₂) (x : A) : φ₁ x = φ₂ x :=\n FunLike.congr_fun H x\n#align alg_hom.congr_fun AlgHom.congr_fun\n\nprotected theorem congr_arg (φ : A →ₐ[R] B) {x y : A} (h : x = y) : φ x = φ y :=\n FunLike.congr_arg φ h\n#align alg_hom.congr_arg AlgHom.congr_arg\n\n@[ext]\ntheorem ext {φ₁ φ₂ : A →ₐ[R] B} (H : ∀ x, φ₁ x = φ₂ x) : φ₁ = φ₂ :=\n FunLike.ext _ _ H\n#align alg_hom.ext AlgHom.ext\n\ntheorem ext_iff {φ₁ φ₂ : A →ₐ[R] B} : φ₁ = φ₂ ↔ ∀ x, φ₁ x = φ₂ x :=\n FunLike.ext_iff\n#align alg_hom.ext_iff AlgHom.ext_iff\n\n@[simp]\ntheorem mk_coe {f : A →ₐ[R] B} (h₁ h₂ h₃ h₄ h₅) : (⟨⟨⟨⟨f, h₁⟩, h₂⟩, h₃, h₄⟩, h₅⟩ : A →ₐ[R] B) = f :=\n ext fun _ => rfl\n#align alg_hom.mk_coe AlgHom.mk_coe\n\n@[simp]\ntheorem commutes (r : R) : φ (algebraMap R A r) = algebraMap R B r :=\n φ.commutes' r\n#align alg_hom.commutes AlgHom.commutes\n\ntheorem comp_algebraMap : (φ : A →+* B).comp (algebraMap R A) = algebraMap R B :=\n RingHom.ext <| φ.commutes\n#align alg_hom.comp_algebra_map AlgHom.comp_algebraMap\n\nprotected theorem map_add (r s : A) : φ (r + s) = φ r + φ s :=\n map_add _ _ _\n#align alg_hom.map_add AlgHom.map_add\n\nprotected theorem map_zero : φ 0 = 0 :=\n map_zero _\n#align alg_hom.map_zero AlgHom.map_zero\n\nprotected theorem map_mul (x y) : φ (x * y) = φ x * φ y :=\n map_mul _ _ _\n#align alg_hom.map_mul AlgHom.map_mul\n\nprotected theorem map_one : φ 1 = 1 :=\n map_one _\n#align alg_hom.map_one AlgHom.map_one\n\nprotected theorem map_pow (x : A) (n : ℕ) : φ (x ^ n) = φ x ^ n :=\n map_pow _ _ _\n#align alg_hom.map_pow AlgHom.map_pow\n\n-- @[simp] -- Porting note: simp can prove this\nprotected theorem map_smul (r : R) (x : A) : φ (r • x) = r • φ x :=\n map_smul _ _ _\n#align alg_hom.map_smul AlgHom.map_smul\n\nprotected theorem map_sum {ι : Type*} (f : ι → A) (s : Finset ι) :\n φ (∑ x in s, f x) = ∑ x in s, φ (f x) :=\n map_sum _ _ _\n#align alg_hom.map_sum AlgHom.map_sum\n\nprotected theorem map_finsupp_sum {α : Type*} [Zero α] {ι : Type*} (f : ι →₀ α) (g : ι → α → A) :\n φ (f.sum g) = f.sum fun i a => φ (g i a) :=\n map_finsupp_sum _ _ _\n#align alg_hom.map_finsupp_sum AlgHom.map_finsupp_sum\n\nset_option linter.deprecated false in\nprotected theorem map_bit0 (x) : φ (bit0 x) = bit0 (φ x) :=\n map_bit0 _ _\n#align alg_hom.map_bit0 AlgHom.map_bit0\n\nset_option linter.deprecated false in\nprotected theorem map_bit1 (x) : φ (bit1 x) = bit1 (φ x) :=\n map_bit1 _ _\n#align alg_hom.map_bit1 AlgHom.map_bit1\n\n/-- If a `RingHom` is `R`-linear, then it is an `AlgHom`. -/\ndef mk' (f : A →+* B) (h : ∀ (c : R) (x), f (c • x) = c • f x) : A →ₐ[R] B :=\n { f with\n toFun := f\n commutes' := fun c => by simp only [Algebra.algebraMap_eq_smul_one, h, f.map_one] }\n#align alg_hom.mk' AlgHom.mk'\n\n@[simp]\ntheorem coe_mk' (f : A →+* B) (h : ∀ (c : R) (x), f (c • x) = c • f x) : ⇑(mk' f h) = f :=\n rfl\n#align alg_hom.coe_mk' AlgHom.coe_mk'\n\nsection\n\nvariable (R A)\n\n/-- Identity map as an `AlgHom`. -/\nprotected def id : A →ₐ[R] A :=\n { RingHom.id A with commutes' := fun _ => rfl }\n#align alg_hom.id AlgHom.id\n\n@[simp]\ntheorem coe_id : ⇑(AlgHom.id R A) = id :=\n rfl\n#align alg_hom.coe_id AlgHom.coe_id\n\n@[simp]\ntheorem id_toRingHom : (AlgHom.id R A : A →+* A) = RingHom.id _ :=\n rfl\n#align alg_hom.id_to_ring_hom AlgHom.id_toRingHom\n\nend\n\ntheorem id_apply (p : A) : AlgHom.id R A p = p :=\n rfl\n#align alg_hom.id_apply AlgHom.id_apply\n\n/-- Composition of algebra homeomorphisms. -/\ndef comp (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) : A →ₐ[R] C :=\n { φ₁.toRingHom.comp ↑φ₂ with\n commutes' := fun r : R => by rw [← φ₁.commutes, ← φ₂.commutes]; rfl }\n#align alg_hom.comp AlgHom.comp\n\n@[simp]\ntheorem coe_comp (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) : ⇑(φ₁.comp φ₂) = φ₁ ∘ φ₂ :=\n rfl\n#align alg_hom.coe_comp AlgHom.coe_comp\n\ntheorem comp_apply (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) (p : A) : φ₁.comp φ₂ p = φ₁ (φ₂ p) :=\n rfl\n#align alg_hom.comp_apply AlgHom.comp_apply\n\ntheorem comp_toRingHom (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) :\n (φ₁.comp φ₂ : A →+* C) = (φ₁ : B →+* C).comp ↑φ₂ :=\n rfl\n#align alg_hom.comp_to_ring_hom AlgHom.comp_toRingHom\n\n@[simp]\ntheorem comp_id : φ.comp (AlgHom.id R A) = φ :=\n ext fun _x => rfl\n#align alg_hom.comp_id AlgHom.comp_id\n\n@[simp]\ntheorem id_comp : (AlgHom.id R B).comp φ = φ :=\n ext fun _x => rfl\n#align alg_hom.id_comp AlgHom.id_comp\n\ntheorem comp_assoc (φ₁ : C →ₐ[R] D) (φ₂ : B →ₐ[R] C) (φ₃ : A →ₐ[R] B) :\n (φ₁.comp φ₂).comp φ₃ = φ₁.comp (φ₂.comp φ₃) :=\n ext fun _x => rfl\n#align alg_hom.comp_assoc AlgHom.comp_assoc\n\n/-- R-Alg ⥤ R-Mod -/\ndef toLinearMap : A →ₗ[R] B where\n toFun := φ\n map_add' := map_add _\n map_smul' := map_smul _\n#align alg_hom.to_linear_map AlgHom.toLinearMap\n\n@[simp]\ntheorem toLinearMap_apply (p : A) : φ.toLinearMap p = φ p :=\n rfl\n#align alg_hom.to_linear_map_apply AlgHom.toLinearMap_apply\n\ntheorem toLinearMap_injective :\n Function.Injective (toLinearMap : _ → A →ₗ[R] B) := fun _φ₁ _φ₂ h =>\n ext <| LinearMap.congr_fun h\n#align alg_hom.to_linear_map_injective AlgHom.toLinearMap_injective\n\n@[simp]\ntheorem comp_toLinearMap (f : A →ₐ[R] B) (g : B →ₐ[R] C) :\n (g.comp f).toLinearMap = g.toLinearMap.comp f.toLinearMap :=\n rfl\n#align alg_hom.comp_to_linear_map AlgHom.comp_toLinearMap\n\n@[simp]\ntheorem toLinearMap_id : toLinearMap (AlgHom.id R A) = LinearMap.id :=\n LinearMap.ext fun _ => rfl\n#align alg_hom.to_linear_map_id AlgHom.toLinearMap_id\n\n/-- Promote a `LinearMap` to an `AlgHom` by supplying proofs about the behavior on `1` and `*`. -/\n@[simps]\ndef ofLinearMap (f : A →ₗ[R] B) (map_one : f 1 = 1) (map_mul : ∀ x y, f (x * y) = f x * f y) :\n A →ₐ[R] B :=\n { f.toAddMonoidHom with\n toFun := f\n map_one' := map_one\n map_mul' := map_mul\n commutes' := fun c => by simp only [Algebra.algebraMap_eq_smul_one, f.map_smul, map_one] }\n#align alg_hom.of_linear_map AlgHom.ofLinearMap\n\n@[simp]\ntheorem ofLinearMap_toLinearMap (map_one) (map_mul) :\n ofLinearMap φ.toLinearMap map_one map_mul = φ := by\n ext\n rfl\n#align alg_hom.of_linear_map_to_linear_map AlgHom.ofLinearMap_toLinearMap\n\n@[simp]\ntheorem toLinearMap_ofLinearMap (f : A →ₗ[R] B) (map_one) (map_mul) :\n toLinearMap (ofLinearMap f map_one map_mul) = f := by\n ext\n rfl\n#align alg_hom.to_linear_map_of_linear_map AlgHom.toLinearMap_ofLinearMap\n\n@[simp]\ntheorem ofLinearMap_id (map_one) (map_mul) :\n ofLinearMap LinearMap.id map_one map_mul = AlgHom.id R A :=\n ext fun _ => rfl\n#align alg_hom.of_linear_map_id AlgHom.ofLinearMap_id\n\ntheorem map_smul_of_tower {R'} [SMul R' A] [SMul R' B] [LinearMap.CompatibleSMul A B R' R] (r : R')\n (x : A) : φ (r • x) = r • φ x :=\n φ.toLinearMap.map_smul_of_tower r x\n#align alg_hom.map_smul_of_tower AlgHom.map_smul_of_tower\n\ntheorem map_list_prod (s : List A) : φ s.prod = (s.map φ).prod :=\n φ.toRingHom.map_list_prod s\n#align alg_hom.map_list_prod AlgHom.map_list_prod\n\n@[simps (config := .lemmasOnly) toSemigroup_toMul_mul toOne_one]\ninstance End : Monoid (A →ₐ[R] A) where\n mul := comp\n mul_assoc ϕ ψ χ := rfl\n one := AlgHom.id R A\n one_mul ϕ := ext fun x => rfl\n mul_one ϕ := ext fun x => rfl\n#align alg_hom.End AlgHom.End\n\n@[simp]\ntheorem one_apply (x : A) : (1 : A →ₐ[R] A) x = x :=\n rfl\n#align alg_hom.one_apply AlgHom.one_apply\n\n@[simp]\ntheorem mul_apply (φ ψ : A →ₐ[R] A) (x : A) : (φ * ψ) x = φ (ψ x) :=\n rfl\n#align alg_hom.mul_apply AlgHom.mul_apply\n\ntheorem algebraMap_eq_apply (f : A →ₐ[R] B) {y : R} {x : A} (h : algebraMap R A y = x) :\n algebraMap R B y = f x :=\n h ▸ (f.commutes _).symm\n#align alg_hom.algebra_map_eq_apply AlgHom.algebraMap_eq_apply\n\nend Semiring\n\nsection CommSemiring\n\nvariable [CommSemiring R] [CommSemiring A] [CommSemiring B]\n\nvariable [Algebra R A] [Algebra R B] (φ : A →ₐ[R] B)\n\nprotected theorem map_multiset_prod (s : Multiset A) : φ s.prod = (s.map φ).prod :=\n map_multiset_prod _ _\n#align alg_hom.map_multiset_prod AlgHom.map_multiset_prod\n\nprotected theorem map_prod {ι : Type*} (f : ι → A) (s : Finset ι) :\n φ (∏ x in s, f x) = ∏ x in s, φ (f x) :=\n map_prod _ _ _\n#align alg_hom.map_prod AlgHom.map_prod\n\nprotected theorem map_finsupp_prod {α : Type*} [Zero α] {ι : Type*} (f : ι →₀ α) (g : ι → α → A) :\n φ (f.prod g) = f.prod fun i a => φ (g i a) :=\n map_finsupp_prod _ _ _\n#align alg_hom.map_finsupp_prod AlgHom.map_finsupp_prod\n\nend CommSemiring\n\nsection Ring\n\nvariable [CommSemiring R] [Ring A] [Ring B]\n\nvariable [Algebra R A] [Algebra R B] (φ : A →ₐ[R] B)\n\nprotected theorem map_neg (x) : φ (-x) = -φ x :=\n map_neg _ _\n#align alg_hom.map_neg AlgHom.map_neg\n\nprotected theorem map_sub (x y) : φ (x - y) = φ x - φ y :=\n map_sub _ _ _\n#align alg_hom.map_sub AlgHom.map_sub\n\nend Ring\n\nend AlgHom\n\nnamespace RingHom\n\nvariable {R S : Type*}\n\n/-- Reinterpret a `RingHom` as an `ℕ`-algebra homomorphism. -/\ndef toNatAlgHom [Semiring R] [Semiring S] (f : R →+* S) : R →ₐ[ℕ] S :=\n { f with\n toFun := f\n commutes' := fun n => by simp }\n#align ring_hom.to_nat_alg_hom RingHom.toNatAlgHom\n\n/-- Reinterpret a `RingHom` as a `ℤ`-algebra homomorphism. -/\ndef toIntAlgHom [Ring R] [Ring S] [Algebra ℤ R] [Algebra ℤ S] (f : R →+* S) : R →ₐ[ℤ] S :=\n { f with commutes' := fun n => by simp }\n#align ring_hom.to_int_alg_hom RingHom.toIntAlgHom\n\nlemma toIntAlgHom_injective [Ring R] [Ring S] [Algebra ℤ R] [Algebra ℤ S] :\n Function.Injective (RingHom.toIntAlgHom : (R →+* S) → _) :=\n fun _ _ e ↦ FunLike.ext _ _ (fun x ↦ FunLike.congr_fun e x)\n\n/-- Reinterpret a `RingHom` as a `ℚ`-algebra homomorphism. This actually yields an equivalence,\nsee `RingHom.equivRatAlgHom`. -/\ndef toRatAlgHom [Ring R] [Ring S] [Algebra ℚ R] [Algebra ℚ S] (f : R →+* S) : R →ₐ[ℚ] S :=\n { f with commutes' := f.map_rat_algebraMap }\n#align ring_hom.to_rat_alg_hom RingHom.toRatAlgHom\n\n@[simp]\ntheorem toRatAlgHom_toRingHom [Ring R] [Ring S] [Algebra ℚ R] [Algebra ℚ S] (f : R →+* S) :\n ↑f.toRatAlgHom = f :=\n RingHom.ext fun _x => rfl\n#align ring_hom.to_rat_alg_hom_to_ring_hom RingHom.toRatAlgHom_toRingHom\n\nend RingHom\n\nsection\n\nvariable {R S : Type*}\n\n@[simp]\ntheorem AlgHom.toRingHom_toRatAlgHom [Ring R] [Ring S] [Algebra ℚ R] [Algebra ℚ S]\n (f : R →ₐ[ℚ] S) : (f : R →+* S).toRatAlgHom = f :=\n AlgHom.ext fun _x => rfl\n#align alg_hom.to_ring_hom_to_rat_alg_hom AlgHom.toRingHom_toRatAlgHom\n\n/-- The equivalence between `RingHom` and `ℚ`-algebra homomorphisms. -/\n@[simps]\ndef RingHom.equivRatAlgHom [Ring R] [Ring S] [Algebra ℚ R] [Algebra ℚ S] : (R →+* S) ≃ (R →ₐ[ℚ] S)\n where\n toFun := RingHom.toRatAlgHom\n invFun := AlgHom.toRingHom\n left_inv f := RingHom.toRatAlgHom_toRingHom f\n right_inv f := AlgHom.toRingHom_toRatAlgHom f\n#align ring_hom.equiv_rat_alg_hom RingHom.equivRatAlgHom\n\nend\n\nnamespace Algebra\n\nvariable (R : Type u) (A : Type v)\n\nvariable [CommSemiring R] [Semiring A] [Algebra R A]\n\n/-- `AlgebraMap` as an `AlgHom`. -/\ndef ofId : R →ₐ[R] A :=\n { algebraMap R A with commutes' := fun _ => rfl }\n#align algebra.of_id Algebra.ofId\n\nvariable {R}\n\ntheorem ofId_apply (r) : ofId R A r = algebraMap R A r :=\n rfl\n#align algebra.of_id_apply Algebra.ofId_apply\n\n/-- This is a special case of a more general instance that we define in a later file. -/\ninstance subsingleton_id : Subsingleton (R →ₐ[R] A) :=\n ⟨fun f g => AlgHom.ext fun _ => (f.commutes _).trans (g.commutes _).symm⟩\n\n/-- This ext lemma closes trivial subgoals create when chaining heterobasic ext lemmas. -/\n@[ext high]\ntheorem ext_id (f g : R →ₐ[R] A) : f = g := Subsingleton.elim _ _\n\nsection MulDistribMulAction\n\ninstance : MulDistribMulAction (A →ₐ[R] A) Aˣ where\n smul := fun f => Units.map f\n one_smul := fun x => by ext; rfl\n mul_smul := fun x y z => by ext; rfl\n smul_mul := fun x y z => by ext; exact x.map_mul _ _\n smul_one := fun x => by ext; ","nextTactic":"exact x.map_one","declUpToTactic":"instance : MulDistribMulAction (A →ₐ[R] A) Aˣ where\n smul := fun f => Units.map f\n one_smul := fun x => by ext; rfl\n mul_smul := fun x y z => by ext; rfl\n smul_mul := fun x y z => by ext; exact x.map_mul _ _\n smul_one := fun x => by ext; ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Hom.579_0.FxqJeqR3qcIVlrq","decl":"instance : MulDistribMulAction (A →ₐ[R] A) Aˣ where\n smul "}