Hugging Face's logo

Datasets:
l3lab
/
ntp-mathlib

Modalities:
Text
Formats:
json
Size:
100K - 1M
ArXiv:
Tags:
theorem-proving
math
lean
Libraries:
Datasets
pandas
Dataset card Data Studio Files Community
1
ntp-mathlib / Extracted /Mathlib /TacticPrediction /Mathlib_Algebra_Algebra_Subalgebra_Pointwise.jsonl
wellecks's picture
wellecks
Upload folder using huggingface_hub
01d1bb5 verified
raw
history blame
66.7 kB
 {"state":"R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\n⊢ toSubmodule S * toSubmodule T ≤ toSubmodule (S ⊔ T)","srcUpToTactic":"/-\nCopyright (c) 2021 Eric Weiser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\nimport Mathlib.Algebra.Algebra.Operations\nimport Mathlib.Algebra.Algebra.Subalgebra.Basic\nimport Mathlib.RingTheory.Subring.Pointwise\nimport Mathlib.RingTheory.Adjoin.Basic\n\n#align_import algebra.algebra.subalgebra.pointwise from \"leanprover-community/mathlib\"@\"b2c707cd190a58ea0565c86695a19e99ccecc215\"\n\n/-!\n# Pointwise actions on subalgebras.\n\nIf `R'` acts on an `R`-algebra `A` (so that `R'` and `R` actions commute)\nthen we get an `R'` action on the collection of `R`-subalgebras.\n-/\n\n\nnamespace Subalgebra\n\nsection Pointwise\n\nvariable {R : Type*} {A : Type*} [CommSemiring R] [Semiring A] [Algebra R A]\n\ntheorem mul_toSubmodule_le (S T : Subalgebra R A) :\n (Subalgebra.toSubmodule S)* (Subalgebra.toSubmodule T) ≤ Subalgebra.toSubmodule (S ⊔ T) := by\n ","nextTactic":"rw [Submodule.mul_le]","declUpToTactic":"theorem mul_toSubmodule_le (S T : Subalgebra R A) :\n (Subalgebra.toSubmodule S)* (Subalgebra.toSubmodule T) ≤ Subalgebra.toSubmodule (S ⊔ T) := by\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Subalgebra_Pointwise.27_0.A8LWG0bRLHJclKD","decl":"theorem mul_toSubmodule_le (S T : Subalgebra R A) :\n (Subalgebra.toSubmodule S)* (Subalgebra.toSubmodule T) ≤ Subalgebra.toSubmodule (S ⊔ T) "}
{"state":"R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\n⊢ ∀ m ∈ toSubmodule S, ∀ n ∈ toSubmodule T, m * n ∈ toSubmodule (S ⊔ T)","srcUpToTactic":"/-\nCopyright (c) 2021 Eric Weiser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\nimport Mathlib.Algebra.Algebra.Operations\nimport Mathlib.Algebra.Algebra.Subalgebra.Basic\nimport Mathlib.RingTheory.Subring.Pointwise\nimport Mathlib.RingTheory.Adjoin.Basic\n\n#align_import algebra.algebra.subalgebra.pointwise from \"leanprover-community/mathlib\"@\"b2c707cd190a58ea0565c86695a19e99ccecc215\"\n\n/-!\n# Pointwise actions on subalgebras.\n\nIf `R'` acts on an `R`-algebra `A` (so that `R'` and `R` actions commute)\nthen we get an `R'` action on the collection of `R`-subalgebras.\n-/\n\n\nnamespace Subalgebra\n\nsection Pointwise\n\nvariable {R : Type*} {A : Type*} [CommSemiring R] [Semiring A] [Algebra R A]\n\ntheorem mul_toSubmodule_le (S T : Subalgebra R A) :\n (Subalgebra.toSubmodule S)* (Subalgebra.toSubmodule T) ≤ Subalgebra.toSubmodule (S ⊔ T) := by\n rw [Submodule.mul_le]\n ","nextTactic":"intro y hy z hz","declUpToTactic":"theorem mul_toSubmodule_le (S T : Subalgebra R A) :\n (Subalgebra.toSubmodule S)* (Subalgebra.toSubmodule T) ≤ Subalgebra.toSubmodule (S ⊔ T) := by\n rw [Submodule.mul_le]\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Subalgebra_Pointwise.27_0.A8LWG0bRLHJclKD","decl":"theorem mul_toSubmodule_le (S T : Subalgebra R A) :\n (Subalgebra.toSubmodule S)* (Subalgebra.toSubmodule T) ≤ Subalgebra.toSubmodule (S ⊔ T) "}
{"state":"R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\ny : A\nhy : y ∈ toSubmodule S\nz : A\nhz : z ∈ toSubmodule T\n⊢ y * z ∈ toSubmodule (S ⊔ T)","srcUpToTactic":"/-\nCopyright (c) 2021 Eric Weiser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\nimport Mathlib.Algebra.Algebra.Operations\nimport Mathlib.Algebra.Algebra.Subalgebra.Basic\nimport Mathlib.RingTheory.Subring.Pointwise\nimport Mathlib.RingTheory.Adjoin.Basic\n\n#align_import algebra.algebra.subalgebra.pointwise from \"leanprover-community/mathlib\"@\"b2c707cd190a58ea0565c86695a19e99ccecc215\"\n\n/-!\n# Pointwise actions on subalgebras.\n\nIf `R'` acts on an `R`-algebra `A` (so that `R'` and `R` actions commute)\nthen we get an `R'` action on the collection of `R`-subalgebras.\n-/\n\n\nnamespace Subalgebra\n\nsection Pointwise\n\nvariable {R : Type*} {A : Type*} [CommSemiring R] [Semiring A] [Algebra R A]\n\ntheorem mul_toSubmodule_le (S T : Subalgebra R A) :\n (Subalgebra.toSubmodule S)* (Subalgebra.toSubmodule T) ≤ Subalgebra.toSubmodule (S ⊔ T) := by\n rw [Submodule.mul_le]\n intro y hy z hz\n ","nextTactic":"show y * z ∈ S ⊔ T","declUpToTactic":"theorem mul_toSubmodule_le (S T : Subalgebra R A) :\n (Subalgebra.toSubmodule S)* (Subalgebra.toSubmodule T) ≤ Subalgebra.toSubmodule (S ⊔ T) := by\n rw [Submodule.mul_le]\n intro y hy z hz\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Subalgebra_Pointwise.27_0.A8LWG0bRLHJclKD","decl":"theorem mul_toSubmodule_le (S T : Subalgebra R A) :\n (Subalgebra.toSubmodule S)* (Subalgebra.toSubmodule T) ≤ Subalgebra.toSubmodule (S ⊔ T) "}
{"state":"R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\ny : A\nhy : y ∈ toSubmodule S\nz : A\nhz : z ∈ toSubmodule T\n⊢ y * z ∈ S ⊔ T","srcUpToTactic":"/-\nCopyright (c) 2021 Eric Weiser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\nimport Mathlib.Algebra.Algebra.Operations\nimport Mathlib.Algebra.Algebra.Subalgebra.Basic\nimport Mathlib.RingTheory.Subring.Pointwise\nimport Mathlib.RingTheory.Adjoin.Basic\n\n#align_import algebra.algebra.subalgebra.pointwise from \"leanprover-community/mathlib\"@\"b2c707cd190a58ea0565c86695a19e99ccecc215\"\n\n/-!\n# Pointwise actions on subalgebras.\n\nIf `R'` acts on an `R`-algebra `A` (so that `R'` and `R` actions commute)\nthen we get an `R'` action on the collection of `R`-subalgebras.\n-/\n\n\nnamespace Subalgebra\n\nsection Pointwise\n\nvariable {R : Type*} {A : Type*} [CommSemiring R] [Semiring A] [Algebra R A]\n\ntheorem mul_toSubmodule_le (S T : Subalgebra R A) :\n (Subalgebra.toSubmodule S)* (Subalgebra.toSubmodule T) ≤ Subalgebra.toSubmodule (S ⊔ T) := by\n rw [Submodule.mul_le]\n intro y hy z hz\n show y * z ∈ S ⊔ T\n ","nextTactic":"exact mul_mem (Algebra.mem_sup_left hy) (Algebra.mem_sup_right hz)","declUpToTactic":"theorem mul_toSubmodule_le (S T : Subalgebra R A) :\n (Subalgebra.toSubmodule S)* (Subalgebra.toSubmodule T) ≤ Subalgebra.toSubmodule (S ⊔ T) := by\n rw [Submodule.mul_le]\n intro y hy z hz\n show y * z ∈ S ⊔ T\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Subalgebra_Pointwise.27_0.A8LWG0bRLHJclKD","decl":"theorem mul_toSubmodule_le (S T : Subalgebra R A) :\n (Subalgebra.toSubmodule S)* (Subalgebra.toSubmodule T) ≤ Subalgebra.toSubmodule (S ⊔ T) "}
{"state":"R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS : Subalgebra R A\n⊢ toSubmodule S * toSubmodule S = toSubmodule S","srcUpToTactic":"/-\nCopyright (c) 2021 Eric Weiser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\nimport Mathlib.Algebra.Algebra.Operations\nimport Mathlib.Algebra.Algebra.Subalgebra.Basic\nimport Mathlib.RingTheory.Subring.Pointwise\nimport Mathlib.RingTheory.Adjoin.Basic\n\n#align_import algebra.algebra.subalgebra.pointwise from \"leanprover-community/mathlib\"@\"b2c707cd190a58ea0565c86695a19e99ccecc215\"\n\n/-!\n# Pointwise actions on subalgebras.\n\nIf `R'` acts on an `R`-algebra `A` (so that `R'` and `R` actions commute)\nthen we get an `R'` action on the collection of `R`-subalgebras.\n-/\n\n\nnamespace Subalgebra\n\nsection Pointwise\n\nvariable {R : Type*} {A : Type*} [CommSemiring R] [Semiring A] [Algebra R A]\n\ntheorem mul_toSubmodule_le (S T : Subalgebra R A) :\n (Subalgebra.toSubmodule S)* (Subalgebra.toSubmodule T) ≤ Subalgebra.toSubmodule (S ⊔ T) := by\n rw [Submodule.mul_le]\n intro y hy z hz\n show y * z ∈ S ⊔ T\n exact mul_mem (Algebra.mem_sup_left hy) (Algebra.mem_sup_right hz)\n#align subalgebra.mul_to_submodule_le Subalgebra.mul_toSubmodule_le\n\n/-- As submodules, subalgebras are idempotent. -/\n@[simp]\ntheorem mul_self (S : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule S)\n = (Subalgebra.toSubmodule S) := by\n ","nextTactic":"apply le_antisymm","declUpToTactic":"/-- As submodules, subalgebras are idempotent. -/\n@[simp]\ntheorem mul_self (S : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule S)\n = (Subalgebra.toSubmodule S) := by\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Subalgebra_Pointwise.35_0.A8LWG0bRLHJclKD","decl":"/-- As submodules, subalgebras are idempotent. -/\n@[simp]\ntheorem mul_self (S : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule S)\n = (Subalgebra.toSubmodule S) "}
{"state":"case a\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS : Subalgebra R A\n⊢ toSubmodule S * toSubmodule S ≤ toSubmodule S","srcUpToTactic":"/-\nCopyright (c) 2021 Eric Weiser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\nimport Mathlib.Algebra.Algebra.Operations\nimport Mathlib.Algebra.Algebra.Subalgebra.Basic\nimport Mathlib.RingTheory.Subring.Pointwise\nimport Mathlib.RingTheory.Adjoin.Basic\n\n#align_import algebra.algebra.subalgebra.pointwise from \"leanprover-community/mathlib\"@\"b2c707cd190a58ea0565c86695a19e99ccecc215\"\n\n/-!\n# Pointwise actions on subalgebras.\n\nIf `R'` acts on an `R`-algebra `A` (so that `R'` and `R` actions commute)\nthen we get an `R'` action on the collection of `R`-subalgebras.\n-/\n\n\nnamespace Subalgebra\n\nsection Pointwise\n\nvariable {R : Type*} {A : Type*} [CommSemiring R] [Semiring A] [Algebra R A]\n\ntheorem mul_toSubmodule_le (S T : Subalgebra R A) :\n (Subalgebra.toSubmodule S)* (Subalgebra.toSubmodule T) ≤ Subalgebra.toSubmodule (S ⊔ T) := by\n rw [Submodule.mul_le]\n intro y hy z hz\n show y * z ∈ S ⊔ T\n exact mul_mem (Algebra.mem_sup_left hy) (Algebra.mem_sup_right hz)\n#align subalgebra.mul_to_submodule_le Subalgebra.mul_toSubmodule_le\n\n/-- As submodules, subalgebras are idempotent. -/\n@[simp]\ntheorem mul_self (S : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule S)\n = (Subalgebra.toSubmodule S) := by\n apply le_antisymm\n · ","nextTactic":"refine' (mul_toSubmodule_le _ _).trans_eq _","declUpToTactic":"/-- As submodules, subalgebras are idempotent. -/\n@[simp]\ntheorem mul_self (S : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule S)\n = (Subalgebra.toSubmodule S) := by\n apply le_antisymm\n · ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Subalgebra_Pointwise.35_0.A8LWG0bRLHJclKD","decl":"/-- As submodules, subalgebras are idempotent. -/\n@[simp]\ntheorem mul_self (S : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule S)\n = (Subalgebra.toSubmodule S) "}
{"state":"case a\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS : Subalgebra R A\n⊢ toSubmodule (S ⊔ S) = toSubmodule S","srcUpToTactic":"/-\nCopyright (c) 2021 Eric Weiser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\nimport Mathlib.Algebra.Algebra.Operations\nimport Mathlib.Algebra.Algebra.Subalgebra.Basic\nimport Mathlib.RingTheory.Subring.Pointwise\nimport Mathlib.RingTheory.Adjoin.Basic\n\n#align_import algebra.algebra.subalgebra.pointwise from \"leanprover-community/mathlib\"@\"b2c707cd190a58ea0565c86695a19e99ccecc215\"\n\n/-!\n# Pointwise actions on subalgebras.\n\nIf `R'` acts on an `R`-algebra `A` (so that `R'` and `R` actions commute)\nthen we get an `R'` action on the collection of `R`-subalgebras.\n-/\n\n\nnamespace Subalgebra\n\nsection Pointwise\n\nvariable {R : Type*} {A : Type*} [CommSemiring R] [Semiring A] [Algebra R A]\n\ntheorem mul_toSubmodule_le (S T : Subalgebra R A) :\n (Subalgebra.toSubmodule S)* (Subalgebra.toSubmodule T) ≤ Subalgebra.toSubmodule (S ⊔ T) := by\n rw [Submodule.mul_le]\n intro y hy z hz\n show y * z ∈ S ⊔ T\n exact mul_mem (Algebra.mem_sup_left hy) (Algebra.mem_sup_right hz)\n#align subalgebra.mul_to_submodule_le Subalgebra.mul_toSubmodule_le\n\n/-- As submodules, subalgebras are idempotent. -/\n@[simp]\ntheorem mul_self (S : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule S)\n = (Subalgebra.toSubmodule S) := by\n apply le_antisymm\n · refine' (mul_toSubmodule_le _ _).trans_eq _\n ","nextTactic":"rw [sup_idem]","declUpToTactic":"/-- As submodules, subalgebras are idempotent. -/\n@[simp]\ntheorem mul_self (S : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule S)\n = (Subalgebra.toSubmodule S) := by\n apply le_antisymm\n · refine' (mul_toSubmodule_le _ _).trans_eq _\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Subalgebra_Pointwise.35_0.A8LWG0bRLHJclKD","decl":"/-- As submodules, subalgebras are idempotent. -/\n@[simp]\ntheorem mul_self (S : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule S)\n = (Subalgebra.toSubmodule S) "}
{"state":"case a\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS : Subalgebra R A\n⊢ toSubmodule S ≤ toSubmodule S * toSubmodule S","srcUpToTactic":"/-\nCopyright (c) 2021 Eric Weiser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\nimport Mathlib.Algebra.Algebra.Operations\nimport Mathlib.Algebra.Algebra.Subalgebra.Basic\nimport Mathlib.RingTheory.Subring.Pointwise\nimport Mathlib.RingTheory.Adjoin.Basic\n\n#align_import algebra.algebra.subalgebra.pointwise from \"leanprover-community/mathlib\"@\"b2c707cd190a58ea0565c86695a19e99ccecc215\"\n\n/-!\n# Pointwise actions on subalgebras.\n\nIf `R'` acts on an `R`-algebra `A` (so that `R'` and `R` actions commute)\nthen we get an `R'` action on the collection of `R`-subalgebras.\n-/\n\n\nnamespace Subalgebra\n\nsection Pointwise\n\nvariable {R : Type*} {A : Type*} [CommSemiring R] [Semiring A] [Algebra R A]\n\ntheorem mul_toSubmodule_le (S T : Subalgebra R A) :\n (Subalgebra.toSubmodule S)* (Subalgebra.toSubmodule T) ≤ Subalgebra.toSubmodule (S ⊔ T) := by\n rw [Submodule.mul_le]\n intro y hy z hz\n show y * z ∈ S ⊔ T\n exact mul_mem (Algebra.mem_sup_left hy) (Algebra.mem_sup_right hz)\n#align subalgebra.mul_to_submodule_le Subalgebra.mul_toSubmodule_le\n\n/-- As submodules, subalgebras are idempotent. -/\n@[simp]\ntheorem mul_self (S : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule S)\n = (Subalgebra.toSubmodule S) := by\n apply le_antisymm\n · refine' (mul_toSubmodule_le _ _).trans_eq _\n rw [sup_idem]\n · ","nextTactic":"intro x hx1","declUpToTactic":"/-- As submodules, subalgebras are idempotent. -/\n@[simp]\ntheorem mul_self (S : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule S)\n = (Subalgebra.toSubmodule S) := by\n apply le_antisymm\n · refine' (mul_toSubmodule_le _ _).trans_eq _\n rw [sup_idem]\n · ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Subalgebra_Pointwise.35_0.A8LWG0bRLHJclKD","decl":"/-- As submodules, subalgebras are idempotent. -/\n@[simp]\ntheorem mul_self (S : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule S)\n = (Subalgebra.toSubmodule S) "}
{"state":"case a\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS : Subalgebra R A\nx : A\nhx1 : x ∈ toSubmodule S\n⊢ x ∈ toSubmodule S * toSubmodule S","srcUpToTactic":"/-\nCopyright (c) 2021 Eric Weiser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\nimport Mathlib.Algebra.Algebra.Operations\nimport Mathlib.Algebra.Algebra.Subalgebra.Basic\nimport Mathlib.RingTheory.Subring.Pointwise\nimport Mathlib.RingTheory.Adjoin.Basic\n\n#align_import algebra.algebra.subalgebra.pointwise from \"leanprover-community/mathlib\"@\"b2c707cd190a58ea0565c86695a19e99ccecc215\"\n\n/-!\n# Pointwise actions on subalgebras.\n\nIf `R'` acts on an `R`-algebra `A` (so that `R'` and `R` actions commute)\nthen we get an `R'` action on the collection of `R`-subalgebras.\n-/\n\n\nnamespace Subalgebra\n\nsection Pointwise\n\nvariable {R : Type*} {A : Type*} [CommSemiring R] [Semiring A] [Algebra R A]\n\ntheorem mul_toSubmodule_le (S T : Subalgebra R A) :\n (Subalgebra.toSubmodule S)* (Subalgebra.toSubmodule T) ≤ Subalgebra.toSubmodule (S ⊔ T) := by\n rw [Submodule.mul_le]\n intro y hy z hz\n show y * z ∈ S ⊔ T\n exact mul_mem (Algebra.mem_sup_left hy) (Algebra.mem_sup_right hz)\n#align subalgebra.mul_to_submodule_le Subalgebra.mul_toSubmodule_le\n\n/-- As submodules, subalgebras are idempotent. -/\n@[simp]\ntheorem mul_self (S : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule S)\n = (Subalgebra.toSubmodule S) := by\n apply le_antisymm\n · refine' (mul_toSubmodule_le _ _).trans_eq _\n rw [sup_idem]\n · intro x hx1\n ","nextTactic":"rw [← mul_one x]","declUpToTactic":"/-- As submodules, subalgebras are idempotent. -/\n@[simp]\ntheorem mul_self (S : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule S)\n = (Subalgebra.toSubmodule S) := by\n apply le_antisymm\n · refine' (mul_toSubmodule_le _ _).trans_eq _\n rw [sup_idem]\n · intro x hx1\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Subalgebra_Pointwise.35_0.A8LWG0bRLHJclKD","decl":"/-- As submodules, subalgebras are idempotent. -/\n@[simp]\ntheorem mul_self (S : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule S)\n = (Subalgebra.toSubmodule S) "}
{"state":"case a\nR : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS : Subalgebra R A\nx : A\nhx1 : x ∈ toSubmodule S\n⊢ x * 1 ∈ toSubmodule S * toSubmodule S","srcUpToTactic":"/-\nCopyright (c) 2021 Eric Weiser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\nimport Mathlib.Algebra.Algebra.Operations\nimport Mathlib.Algebra.Algebra.Subalgebra.Basic\nimport Mathlib.RingTheory.Subring.Pointwise\nimport Mathlib.RingTheory.Adjoin.Basic\n\n#align_import algebra.algebra.subalgebra.pointwise from \"leanprover-community/mathlib\"@\"b2c707cd190a58ea0565c86695a19e99ccecc215\"\n\n/-!\n# Pointwise actions on subalgebras.\n\nIf `R'` acts on an `R`-algebra `A` (so that `R'` and `R` actions commute)\nthen we get an `R'` action on the collection of `R`-subalgebras.\n-/\n\n\nnamespace Subalgebra\n\nsection Pointwise\n\nvariable {R : Type*} {A : Type*} [CommSemiring R] [Semiring A] [Algebra R A]\n\ntheorem mul_toSubmodule_le (S T : Subalgebra R A) :\n (Subalgebra.toSubmodule S)* (Subalgebra.toSubmodule T) ≤ Subalgebra.toSubmodule (S ⊔ T) := by\n rw [Submodule.mul_le]\n intro y hy z hz\n show y * z ∈ S ⊔ T\n exact mul_mem (Algebra.mem_sup_left hy) (Algebra.mem_sup_right hz)\n#align subalgebra.mul_to_submodule_le Subalgebra.mul_toSubmodule_le\n\n/-- As submodules, subalgebras are idempotent. -/\n@[simp]\ntheorem mul_self (S : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule S)\n = (Subalgebra.toSubmodule S) := by\n apply le_antisymm\n · refine' (mul_toSubmodule_le _ _).trans_eq _\n rw [sup_idem]\n · intro x hx1\n rw [← mul_one x]\n ","nextTactic":"exact Submodule.mul_mem_mul hx1 (show (1 : A) ∈ S from one_mem S)","declUpToTactic":"/-- As submodules, subalgebras are idempotent. -/\n@[simp]\ntheorem mul_self (S : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule S)\n = (Subalgebra.toSubmodule S) := by\n apply le_antisymm\n · refine' (mul_toSubmodule_le _ _).trans_eq _\n rw [sup_idem]\n · intro x hx1\n rw [← mul_one x]\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Subalgebra_Pointwise.35_0.A8LWG0bRLHJclKD","decl":"/-- As submodules, subalgebras are idempotent. -/\n@[simp]\ntheorem mul_self (S : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule S)\n = (Subalgebra.toSubmodule S) "}
{"state":"R✝ : Type u_1\nA✝ : Type u_2\ninst✝⁵ : CommSemiring R✝\ninst✝⁴ : Semiring A✝\ninst✝³ : Algebra R✝ A✝\nR : Type u_3\nA : Type u_4\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\n⊢ toSubmodule S * toSubmodule T = toSubmodule (S ⊔ T)","srcUpToTactic":"/-\nCopyright (c) 2021 Eric Weiser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\nimport Mathlib.Algebra.Algebra.Operations\nimport Mathlib.Algebra.Algebra.Subalgebra.Basic\nimport Mathlib.RingTheory.Subring.Pointwise\nimport Mathlib.RingTheory.Adjoin.Basic\n\n#align_import algebra.algebra.subalgebra.pointwise from \"leanprover-community/mathlib\"@\"b2c707cd190a58ea0565c86695a19e99ccecc215\"\n\n/-!\n# Pointwise actions on subalgebras.\n\nIf `R'` acts on an `R`-algebra `A` (so that `R'` and `R` actions commute)\nthen we get an `R'` action on the collection of `R`-subalgebras.\n-/\n\n\nnamespace Subalgebra\n\nsection Pointwise\n\nvariable {R : Type*} {A : Type*} [CommSemiring R] [Semiring A] [Algebra R A]\n\ntheorem mul_toSubmodule_le (S T : Subalgebra R A) :\n (Subalgebra.toSubmodule S)* (Subalgebra.toSubmodule T) ≤ Subalgebra.toSubmodule (S ⊔ T) := by\n rw [Submodule.mul_le]\n intro y hy z hz\n show y * z ∈ S ⊔ T\n exact mul_mem (Algebra.mem_sup_left hy) (Algebra.mem_sup_right hz)\n#align subalgebra.mul_to_submodule_le Subalgebra.mul_toSubmodule_le\n\n/-- As submodules, subalgebras are idempotent. -/\n@[simp]\ntheorem mul_self (S : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule S)\n = (Subalgebra.toSubmodule S) := by\n apply le_antisymm\n · refine' (mul_toSubmodule_le _ _).trans_eq _\n rw [sup_idem]\n · intro x hx1\n rw [← mul_one x]\n exact Submodule.mul_mem_mul hx1 (show (1 : A) ∈ S from one_mem S)\n#align subalgebra.mul_self Subalgebra.mul_self\n\n/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/\ntheorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]\n (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)\n = Subalgebra.toSubmodule (S ⊔ T) := by\n ","nextTactic":"refine' le_antisymm (mul_toSubmodule_le _ _) _","declUpToTactic":"/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/\ntheorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]\n (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)\n = Subalgebra.toSubmodule (S ⊔ T) := by\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Subalgebra_Pointwise.47_0.A8LWG0bRLHJclKD","decl":"/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/\ntheorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]\n (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)\n = Subalgebra.toSubmodule (S ⊔ T) "}
{"state":"R✝ : Type u_1\nA✝ : Type u_2\ninst✝⁵ : CommSemiring R✝\ninst✝⁴ : Semiring A✝\ninst✝³ : Algebra R✝ A✝\nR : Type u_3\nA : Type u_4\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\n⊢ toSubmodule (S ⊔ T) ≤ toSubmodule S * toSubmodule T","srcUpToTactic":"/-\nCopyright (c) 2021 Eric Weiser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\nimport Mathlib.Algebra.Algebra.Operations\nimport Mathlib.Algebra.Algebra.Subalgebra.Basic\nimport Mathlib.RingTheory.Subring.Pointwise\nimport Mathlib.RingTheory.Adjoin.Basic\n\n#align_import algebra.algebra.subalgebra.pointwise from \"leanprover-community/mathlib\"@\"b2c707cd190a58ea0565c86695a19e99ccecc215\"\n\n/-!\n# Pointwise actions on subalgebras.\n\nIf `R'` acts on an `R`-algebra `A` (so that `R'` and `R` actions commute)\nthen we get an `R'` action on the collection of `R`-subalgebras.\n-/\n\n\nnamespace Subalgebra\n\nsection Pointwise\n\nvariable {R : Type*} {A : Type*} [CommSemiring R] [Semiring A] [Algebra R A]\n\ntheorem mul_toSubmodule_le (S T : Subalgebra R A) :\n (Subalgebra.toSubmodule S)* (Subalgebra.toSubmodule T) ≤ Subalgebra.toSubmodule (S ⊔ T) := by\n rw [Submodule.mul_le]\n intro y hy z hz\n show y * z ∈ S ⊔ T\n exact mul_mem (Algebra.mem_sup_left hy) (Algebra.mem_sup_right hz)\n#align subalgebra.mul_to_submodule_le Subalgebra.mul_toSubmodule_le\n\n/-- As submodules, subalgebras are idempotent. -/\n@[simp]\ntheorem mul_self (S : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule S)\n = (Subalgebra.toSubmodule S) := by\n apply le_antisymm\n · refine' (mul_toSubmodule_le _ _).trans_eq _\n rw [sup_idem]\n · intro x hx1\n rw [← mul_one x]\n exact Submodule.mul_mem_mul hx1 (show (1 : A) ∈ S from one_mem S)\n#align subalgebra.mul_self Subalgebra.mul_self\n\n/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/\ntheorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]\n (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)\n = Subalgebra.toSubmodule (S ⊔ T) := by\n refine' le_antisymm (mul_toSubmodule_le _ _) _\n ","nextTactic":"rintro x (hx : x ∈ Algebra.adjoin R (S ∪ T : Set A))","declUpToTactic":"/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/\ntheorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]\n (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)\n = Subalgebra.toSubmodule (S ⊔ T) := by\n refine' le_antisymm (mul_toSubmodule_le _ _) _\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Subalgebra_Pointwise.47_0.A8LWG0bRLHJclKD","decl":"/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/\ntheorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]\n (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)\n = Subalgebra.toSubmodule (S ⊔ T) "}
{"state":"R✝ : Type u_1\nA✝ : Type u_2\ninst✝⁵ : CommSemiring R✝\ninst✝⁴ : Semiring A✝\ninst✝³ : Algebra R✝ A✝\nR : Type u_3\nA : Type u_4\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\nx : A\nhx : x ∈ Algebra.adjoin R (↑S ∪ ↑T)\n⊢ x ∈ toSubmodule S * toSubmodule T","srcUpToTactic":"/-\nCopyright (c) 2021 Eric Weiser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\nimport Mathlib.Algebra.Algebra.Operations\nimport Mathlib.Algebra.Algebra.Subalgebra.Basic\nimport Mathlib.RingTheory.Subring.Pointwise\nimport Mathlib.RingTheory.Adjoin.Basic\n\n#align_import algebra.algebra.subalgebra.pointwise from \"leanprover-community/mathlib\"@\"b2c707cd190a58ea0565c86695a19e99ccecc215\"\n\n/-!\n# Pointwise actions on subalgebras.\n\nIf `R'` acts on an `R`-algebra `A` (so that `R'` and `R` actions commute)\nthen we get an `R'` action on the collection of `R`-subalgebras.\n-/\n\n\nnamespace Subalgebra\n\nsection Pointwise\n\nvariable {R : Type*} {A : Type*} [CommSemiring R] [Semiring A] [Algebra R A]\n\ntheorem mul_toSubmodule_le (S T : Subalgebra R A) :\n (Subalgebra.toSubmodule S)* (Subalgebra.toSubmodule T) ≤ Subalgebra.toSubmodule (S ⊔ T) := by\n rw [Submodule.mul_le]\n intro y hy z hz\n show y * z ∈ S ⊔ T\n exact mul_mem (Algebra.mem_sup_left hy) (Algebra.mem_sup_right hz)\n#align subalgebra.mul_to_submodule_le Subalgebra.mul_toSubmodule_le\n\n/-- As submodules, subalgebras are idempotent. -/\n@[simp]\ntheorem mul_self (S : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule S)\n = (Subalgebra.toSubmodule S) := by\n apply le_antisymm\n · refine' (mul_toSubmodule_le _ _).trans_eq _\n rw [sup_idem]\n · intro x hx1\n rw [← mul_one x]\n exact Submodule.mul_mem_mul hx1 (show (1 : A) ∈ S from one_mem S)\n#align subalgebra.mul_self Subalgebra.mul_self\n\n/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/\ntheorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]\n (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)\n = Subalgebra.toSubmodule (S ⊔ T) := by\n refine' le_antisymm (mul_toSubmodule_le _ _) _\n rintro x (hx : x ∈ Algebra.adjoin R (S ∪ T : Set A))\n ","nextTactic":"refine'\n Algebra.adjoin_induction hx (fun x hx => _) (fun r => _) (fun _ _ => Submodule.add_mem _)\n fun x y hx hy => _","declUpToTactic":"/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/\ntheorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]\n (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)\n = Subalgebra.toSubmodule (S ⊔ T) := by\n refine' le_antisymm (mul_toSubmodule_le _ _) _\n rintro x (hx : x ∈ Algebra.adjoin R (S ∪ T : Set A))\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Subalgebra_Pointwise.47_0.A8LWG0bRLHJclKD","decl":"/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/\ntheorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]\n (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)\n = Subalgebra.toSubmodule (S ⊔ T) "}
{"state":"case refine'_1\nR✝ : Type u_1\nA✝ : Type u_2\ninst✝⁵ : CommSemiring R✝\ninst✝⁴ : Semiring A✝\ninst✝³ : Algebra R✝ A✝\nR : Type u_3\nA : Type u_4\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\nx✝ : A\nhx✝ : x✝ ∈ Algebra.adjoin R (↑S ∪ ↑T)\nx : A\nhx : x ∈ ↑S ∪ ↑T\n⊢ x ∈ toSubmodule S * toSubmodule T","srcUpToTactic":"/-\nCopyright (c) 2021 Eric Weiser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\nimport Mathlib.Algebra.Algebra.Operations\nimport Mathlib.Algebra.Algebra.Subalgebra.Basic\nimport Mathlib.RingTheory.Subring.Pointwise\nimport Mathlib.RingTheory.Adjoin.Basic\n\n#align_import algebra.algebra.subalgebra.pointwise from \"leanprover-community/mathlib\"@\"b2c707cd190a58ea0565c86695a19e99ccecc215\"\n\n/-!\n# Pointwise actions on subalgebras.\n\nIf `R'` acts on an `R`-algebra `A` (so that `R'` and `R` actions commute)\nthen we get an `R'` action on the collection of `R`-subalgebras.\n-/\n\n\nnamespace Subalgebra\n\nsection Pointwise\n\nvariable {R : Type*} {A : Type*} [CommSemiring R] [Semiring A] [Algebra R A]\n\ntheorem mul_toSubmodule_le (S T : Subalgebra R A) :\n (Subalgebra.toSubmodule S)* (Subalgebra.toSubmodule T) ≤ Subalgebra.toSubmodule (S ⊔ T) := by\n rw [Submodule.mul_le]\n intro y hy z hz\n show y * z ∈ S ⊔ T\n exact mul_mem (Algebra.mem_sup_left hy) (Algebra.mem_sup_right hz)\n#align subalgebra.mul_to_submodule_le Subalgebra.mul_toSubmodule_le\n\n/-- As submodules, subalgebras are idempotent. -/\n@[simp]\ntheorem mul_self (S : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule S)\n = (Subalgebra.toSubmodule S) := by\n apply le_antisymm\n · refine' (mul_toSubmodule_le _ _).trans_eq _\n rw [sup_idem]\n · intro x hx1\n rw [← mul_one x]\n exact Submodule.mul_mem_mul hx1 (show (1 : A) ∈ S from one_mem S)\n#align subalgebra.mul_self Subalgebra.mul_self\n\n/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/\ntheorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]\n (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)\n = Subalgebra.toSubmodule (S ⊔ T) := by\n refine' le_antisymm (mul_toSubmodule_le _ _) _\n rintro x (hx : x ∈ Algebra.adjoin R (S ∪ T : Set A))\n refine'\n Algebra.adjoin_induction hx (fun x hx => _) (fun r => _) (fun _ _ => Submodule.add_mem _)\n fun x y hx hy => _\n · ","nextTactic":"cases' hx with hxS hxT","declUpToTactic":"/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/\ntheorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]\n (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)\n = Subalgebra.toSubmodule (S ⊔ T) := by\n refine' le_antisymm (mul_toSubmodule_le _ _) _\n rintro x (hx : x ∈ Algebra.adjoin R (S ∪ T : Set A))\n refine'\n Algebra.adjoin_induction hx (fun x hx => _) (fun r => _) (fun _ _ => Submodule.add_mem _)\n fun x y hx hy => _\n · ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Subalgebra_Pointwise.47_0.A8LWG0bRLHJclKD","decl":"/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/\ntheorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]\n (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)\n = Subalgebra.toSubmodule (S ⊔ T) "}
{"state":"case refine'_1.inl\nR✝ : Type u_1\nA✝ : Type u_2\ninst✝⁵ : CommSemiring R✝\ninst✝⁴ : Semiring A✝\ninst✝³ : Algebra R✝ A✝\nR : Type u_3\nA : Type u_4\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\nx✝ : A\nhx : x✝ ∈ Algebra.adjoin R (↑S ∪ ↑T)\nx : A\nhxS : x ∈ ↑S\n⊢ x ∈ toSubmodule S * toSubmodule T","srcUpToTactic":"/-\nCopyright (c) 2021 Eric Weiser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\nimport Mathlib.Algebra.Algebra.Operations\nimport Mathlib.Algebra.Algebra.Subalgebra.Basic\nimport Mathlib.RingTheory.Subring.Pointwise\nimport Mathlib.RingTheory.Adjoin.Basic\n\n#align_import algebra.algebra.subalgebra.pointwise from \"leanprover-community/mathlib\"@\"b2c707cd190a58ea0565c86695a19e99ccecc215\"\n\n/-!\n# Pointwise actions on subalgebras.\n\nIf `R'` acts on an `R`-algebra `A` (so that `R'` and `R` actions commute)\nthen we get an `R'` action on the collection of `R`-subalgebras.\n-/\n\n\nnamespace Subalgebra\n\nsection Pointwise\n\nvariable {R : Type*} {A : Type*} [CommSemiring R] [Semiring A] [Algebra R A]\n\ntheorem mul_toSubmodule_le (S T : Subalgebra R A) :\n (Subalgebra.toSubmodule S)* (Subalgebra.toSubmodule T) ≤ Subalgebra.toSubmodule (S ⊔ T) := by\n rw [Submodule.mul_le]\n intro y hy z hz\n show y * z ∈ S ⊔ T\n exact mul_mem (Algebra.mem_sup_left hy) (Algebra.mem_sup_right hz)\n#align subalgebra.mul_to_submodule_le Subalgebra.mul_toSubmodule_le\n\n/-- As submodules, subalgebras are idempotent. -/\n@[simp]\ntheorem mul_self (S : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule S)\n = (Subalgebra.toSubmodule S) := by\n apply le_antisymm\n · refine' (mul_toSubmodule_le _ _).trans_eq _\n rw [sup_idem]\n · intro x hx1\n rw [← mul_one x]\n exact Submodule.mul_mem_mul hx1 (show (1 : A) ∈ S from one_mem S)\n#align subalgebra.mul_self Subalgebra.mul_self\n\n/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/\ntheorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]\n (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)\n = Subalgebra.toSubmodule (S ⊔ T) := by\n refine' le_antisymm (mul_toSubmodule_le _ _) _\n rintro x (hx : x ∈ Algebra.adjoin R (S ∪ T : Set A))\n refine'\n Algebra.adjoin_induction hx (fun x hx => _) (fun r => _) (fun _ _ => Submodule.add_mem _)\n fun x y hx hy => _\n · cases' hx with hxS hxT\n · ","nextTactic":"rw [← mul_one x]","declUpToTactic":"/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/\ntheorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]\n (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)\n = Subalgebra.toSubmodule (S ⊔ T) := by\n refine' le_antisymm (mul_toSubmodule_le _ _) _\n rintro x (hx : x ∈ Algebra.adjoin R (S ∪ T : Set A))\n refine'\n Algebra.adjoin_induction hx (fun x hx => _) (fun r => _) (fun _ _ => Submodule.add_mem _)\n fun x y hx hy => _\n · cases' hx with hxS hxT\n · ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Subalgebra_Pointwise.47_0.A8LWG0bRLHJclKD","decl":"/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/\ntheorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]\n (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)\n = Subalgebra.toSubmodule (S ⊔ T) "}
{"state":"case refine'_1.inl\nR✝ : Type u_1\nA✝ : Type u_2\ninst✝⁵ : CommSemiring R✝\ninst✝⁴ : Semiring A✝\ninst✝³ : Algebra R✝ A✝\nR : Type u_3\nA : Type u_4\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\nx✝ : A\nhx : x✝ ∈ Algebra.adjoin R (↑S ∪ ↑T)\nx : A\nhxS : x ∈ ↑S\n⊢ x * 1 ∈ toSubmodule S * toSubmodule T","srcUpToTactic":"/-\nCopyright (c) 2021 Eric Weiser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\nimport Mathlib.Algebra.Algebra.Operations\nimport Mathlib.Algebra.Algebra.Subalgebra.Basic\nimport Mathlib.RingTheory.Subring.Pointwise\nimport Mathlib.RingTheory.Adjoin.Basic\n\n#align_import algebra.algebra.subalgebra.pointwise from \"leanprover-community/mathlib\"@\"b2c707cd190a58ea0565c86695a19e99ccecc215\"\n\n/-!\n# Pointwise actions on subalgebras.\n\nIf `R'` acts on an `R`-algebra `A` (so that `R'` and `R` actions commute)\nthen we get an `R'` action on the collection of `R`-subalgebras.\n-/\n\n\nnamespace Subalgebra\n\nsection Pointwise\n\nvariable {R : Type*} {A : Type*} [CommSemiring R] [Semiring A] [Algebra R A]\n\ntheorem mul_toSubmodule_le (S T : Subalgebra R A) :\n (Subalgebra.toSubmodule S)* (Subalgebra.toSubmodule T) ≤ Subalgebra.toSubmodule (S ⊔ T) := by\n rw [Submodule.mul_le]\n intro y hy z hz\n show y * z ∈ S ⊔ T\n exact mul_mem (Algebra.mem_sup_left hy) (Algebra.mem_sup_right hz)\n#align subalgebra.mul_to_submodule_le Subalgebra.mul_toSubmodule_le\n\n/-- As submodules, subalgebras are idempotent. -/\n@[simp]\ntheorem mul_self (S : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule S)\n = (Subalgebra.toSubmodule S) := by\n apply le_antisymm\n · refine' (mul_toSubmodule_le _ _).trans_eq _\n rw [sup_idem]\n · intro x hx1\n rw [← mul_one x]\n exact Submodule.mul_mem_mul hx1 (show (1 : A) ∈ S from one_mem S)\n#align subalgebra.mul_self Subalgebra.mul_self\n\n/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/\ntheorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]\n (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)\n = Subalgebra.toSubmodule (S ⊔ T) := by\n refine' le_antisymm (mul_toSubmodule_le _ _) _\n rintro x (hx : x ∈ Algebra.adjoin R (S ∪ T : Set A))\n refine'\n Algebra.adjoin_induction hx (fun x hx => _) (fun r => _) (fun _ _ => Submodule.add_mem _)\n fun x y hx hy => _\n · cases' hx with hxS hxT\n · rw [← mul_one x]\n ","nextTactic":"exact Submodule.mul_mem_mul hxS (show (1 : A) ∈ T from one_mem T)","declUpToTactic":"/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/\ntheorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]\n (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)\n = Subalgebra.toSubmodule (S ⊔ T) := by\n refine' le_antisymm (mul_toSubmodule_le _ _) _\n rintro x (hx : x ∈ Algebra.adjoin R (S ∪ T : Set A))\n refine'\n Algebra.adjoin_induction hx (fun x hx => _) (fun r => _) (fun _ _ => Submodule.add_mem _)\n fun x y hx hy => _\n · cases' hx with hxS hxT\n · rw [← mul_one x]\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Subalgebra_Pointwise.47_0.A8LWG0bRLHJclKD","decl":"/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/\ntheorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]\n (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)\n = Subalgebra.toSubmodule (S ⊔ T) "}
{"state":"case refine'_1.inr\nR✝ : Type u_1\nA✝ : Type u_2\ninst✝⁵ : CommSemiring R✝\ninst✝⁴ : Semiring A✝\ninst✝³ : Algebra R✝ A✝\nR : Type u_3\nA : Type u_4\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\nx✝ : A\nhx : x✝ ∈ Algebra.adjoin R (↑S ∪ ↑T)\nx : A\nhxT : x ∈ ↑T\n⊢ x ∈ toSubmodule S * toSubmodule T","srcUpToTactic":"/-\nCopyright (c) 2021 Eric Weiser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\nimport Mathlib.Algebra.Algebra.Operations\nimport Mathlib.Algebra.Algebra.Subalgebra.Basic\nimport Mathlib.RingTheory.Subring.Pointwise\nimport Mathlib.RingTheory.Adjoin.Basic\n\n#align_import algebra.algebra.subalgebra.pointwise from \"leanprover-community/mathlib\"@\"b2c707cd190a58ea0565c86695a19e99ccecc215\"\n\n/-!\n# Pointwise actions on subalgebras.\n\nIf `R'` acts on an `R`-algebra `A` (so that `R'` and `R` actions commute)\nthen we get an `R'` action on the collection of `R`-subalgebras.\n-/\n\n\nnamespace Subalgebra\n\nsection Pointwise\n\nvariable {R : Type*} {A : Type*} [CommSemiring R] [Semiring A] [Algebra R A]\n\ntheorem mul_toSubmodule_le (S T : Subalgebra R A) :\n (Subalgebra.toSubmodule S)* (Subalgebra.toSubmodule T) ≤ Subalgebra.toSubmodule (S ⊔ T) := by\n rw [Submodule.mul_le]\n intro y hy z hz\n show y * z ∈ S ⊔ T\n exact mul_mem (Algebra.mem_sup_left hy) (Algebra.mem_sup_right hz)\n#align subalgebra.mul_to_submodule_le Subalgebra.mul_toSubmodule_le\n\n/-- As submodules, subalgebras are idempotent. -/\n@[simp]\ntheorem mul_self (S : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule S)\n = (Subalgebra.toSubmodule S) := by\n apply le_antisymm\n · refine' (mul_toSubmodule_le _ _).trans_eq _\n rw [sup_idem]\n · intro x hx1\n rw [← mul_one x]\n exact Submodule.mul_mem_mul hx1 (show (1 : A) ∈ S from one_mem S)\n#align subalgebra.mul_self Subalgebra.mul_self\n\n/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/\ntheorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]\n (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)\n = Subalgebra.toSubmodule (S ⊔ T) := by\n refine' le_antisymm (mul_toSubmodule_le _ _) _\n rintro x (hx : x ∈ Algebra.adjoin R (S ∪ T : Set A))\n refine'\n Algebra.adjoin_induction hx (fun x hx => _) (fun r => _) (fun _ _ => Submodule.add_mem _)\n fun x y hx hy => _\n · cases' hx with hxS hxT\n · rw [← mul_one x]\n exact Submodule.mul_mem_mul hxS (show (1 : A) ∈ T from one_mem T)\n · ","nextTactic":"rw [← one_mul x]","declUpToTactic":"/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/\ntheorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]\n (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)\n = Subalgebra.toSubmodule (S ⊔ T) := by\n refine' le_antisymm (mul_toSubmodule_le _ _) _\n rintro x (hx : x ∈ Algebra.adjoin R (S ∪ T : Set A))\n refine'\n Algebra.adjoin_induction hx (fun x hx => _) (fun r => _) (fun _ _ => Submodule.add_mem _)\n fun x y hx hy => _\n · cases' hx with hxS hxT\n · rw [← mul_one x]\n exact Submodule.mul_mem_mul hxS (show (1 : A) ∈ T from one_mem T)\n · ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Subalgebra_Pointwise.47_0.A8LWG0bRLHJclKD","decl":"/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/\ntheorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]\n (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)\n = Subalgebra.toSubmodule (S ⊔ T) "}
{"state":"case refine'_1.inr\nR✝ : Type u_1\nA✝ : Type u_2\ninst✝⁵ : CommSemiring R✝\ninst✝⁴ : Semiring A✝\ninst✝³ : Algebra R✝ A✝\nR : Type u_3\nA : Type u_4\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\nx✝ : A\nhx : x✝ ∈ Algebra.adjoin R (↑S ∪ ↑T)\nx : A\nhxT : x ∈ ↑T\n⊢ 1 * x ∈ toSubmodule S * toSubmodule T","srcUpToTactic":"/-\nCopyright (c) 2021 Eric Weiser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\nimport Mathlib.Algebra.Algebra.Operations\nimport Mathlib.Algebra.Algebra.Subalgebra.Basic\nimport Mathlib.RingTheory.Subring.Pointwise\nimport Mathlib.RingTheory.Adjoin.Basic\n\n#align_import algebra.algebra.subalgebra.pointwise from \"leanprover-community/mathlib\"@\"b2c707cd190a58ea0565c86695a19e99ccecc215\"\n\n/-!\n# Pointwise actions on subalgebras.\n\nIf `R'` acts on an `R`-algebra `A` (so that `R'` and `R` actions commute)\nthen we get an `R'` action on the collection of `R`-subalgebras.\n-/\n\n\nnamespace Subalgebra\n\nsection Pointwise\n\nvariable {R : Type*} {A : Type*} [CommSemiring R] [Semiring A] [Algebra R A]\n\ntheorem mul_toSubmodule_le (S T : Subalgebra R A) :\n (Subalgebra.toSubmodule S)* (Subalgebra.toSubmodule T) ≤ Subalgebra.toSubmodule (S ⊔ T) := by\n rw [Submodule.mul_le]\n intro y hy z hz\n show y * z ∈ S ⊔ T\n exact mul_mem (Algebra.mem_sup_left hy) (Algebra.mem_sup_right hz)\n#align subalgebra.mul_to_submodule_le Subalgebra.mul_toSubmodule_le\n\n/-- As submodules, subalgebras are idempotent. -/\n@[simp]\ntheorem mul_self (S : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule S)\n = (Subalgebra.toSubmodule S) := by\n apply le_antisymm\n · refine' (mul_toSubmodule_le _ _).trans_eq _\n rw [sup_idem]\n · intro x hx1\n rw [← mul_one x]\n exact Submodule.mul_mem_mul hx1 (show (1 : A) ∈ S from one_mem S)\n#align subalgebra.mul_self Subalgebra.mul_self\n\n/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/\ntheorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]\n (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)\n = Subalgebra.toSubmodule (S ⊔ T) := by\n refine' le_antisymm (mul_toSubmodule_le _ _) _\n rintro x (hx : x ∈ Algebra.adjoin R (S ∪ T : Set A))\n refine'\n Algebra.adjoin_induction hx (fun x hx => _) (fun r => _) (fun _ _ => Submodule.add_mem _)\n fun x y hx hy => _\n · cases' hx with hxS hxT\n · rw [← mul_one x]\n exact Submodule.mul_mem_mul hxS (show (1 : A) ∈ T from one_mem T)\n · rw [← one_mul x]\n ","nextTactic":"exact Submodule.mul_mem_mul (show (1 : A) ∈ S from one_mem S) hxT","declUpToTactic":"/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/\ntheorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]\n (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)\n = Subalgebra.toSubmodule (S ⊔ T) := by\n refine' le_antisymm (mul_toSubmodule_le _ _) _\n rintro x (hx : x ∈ Algebra.adjoin R (S ∪ T : Set A))\n refine'\n Algebra.adjoin_induction hx (fun x hx => _) (fun r => _) (fun _ _ => Submodule.add_mem _)\n fun x y hx hy => _\n · cases' hx with hxS hxT\n · rw [← mul_one x]\n exact Submodule.mul_mem_mul hxS (show (1 : A) ∈ T from one_mem T)\n · rw [← one_mul x]\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Subalgebra_Pointwise.47_0.A8LWG0bRLHJclKD","decl":"/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/\ntheorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]\n (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)\n = Subalgebra.toSubmodule (S ⊔ T) "}
{"state":"case refine'_2\nR✝ : Type u_1\nA✝ : Type u_2\ninst✝⁵ : CommSemiring R✝\ninst✝⁴ : Semiring A✝\ninst✝³ : Algebra R✝ A✝\nR : Type u_3\nA : Type u_4\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\nx : A\nhx : x ∈ Algebra.adjoin R (↑S ∪ ↑T)\nr : R\n⊢ (algebraMap R A) r ∈ toSubmodule S * toSubmodule T","srcUpToTactic":"/-\nCopyright (c) 2021 Eric Weiser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\nimport Mathlib.Algebra.Algebra.Operations\nimport Mathlib.Algebra.Algebra.Subalgebra.Basic\nimport Mathlib.RingTheory.Subring.Pointwise\nimport Mathlib.RingTheory.Adjoin.Basic\n\n#align_import algebra.algebra.subalgebra.pointwise from \"leanprover-community/mathlib\"@\"b2c707cd190a58ea0565c86695a19e99ccecc215\"\n\n/-!\n# Pointwise actions on subalgebras.\n\nIf `R'` acts on an `R`-algebra `A` (so that `R'` and `R` actions commute)\nthen we get an `R'` action on the collection of `R`-subalgebras.\n-/\n\n\nnamespace Subalgebra\n\nsection Pointwise\n\nvariable {R : Type*} {A : Type*} [CommSemiring R] [Semiring A] [Algebra R A]\n\ntheorem mul_toSubmodule_le (S T : Subalgebra R A) :\n (Subalgebra.toSubmodule S)* (Subalgebra.toSubmodule T) ≤ Subalgebra.toSubmodule (S ⊔ T) := by\n rw [Submodule.mul_le]\n intro y hy z hz\n show y * z ∈ S ⊔ T\n exact mul_mem (Algebra.mem_sup_left hy) (Algebra.mem_sup_right hz)\n#align subalgebra.mul_to_submodule_le Subalgebra.mul_toSubmodule_le\n\n/-- As submodules, subalgebras are idempotent. -/\n@[simp]\ntheorem mul_self (S : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule S)\n = (Subalgebra.toSubmodule S) := by\n apply le_antisymm\n · refine' (mul_toSubmodule_le _ _).trans_eq _\n rw [sup_idem]\n · intro x hx1\n rw [← mul_one x]\n exact Submodule.mul_mem_mul hx1 (show (1 : A) ∈ S from one_mem S)\n#align subalgebra.mul_self Subalgebra.mul_self\n\n/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/\ntheorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]\n (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)\n = Subalgebra.toSubmodule (S ⊔ T) := by\n refine' le_antisymm (mul_toSubmodule_le _ _) _\n rintro x (hx : x ∈ Algebra.adjoin R (S ∪ T : Set A))\n refine'\n Algebra.adjoin_induction hx (fun x hx => _) (fun r => _) (fun _ _ => Submodule.add_mem _)\n fun x y hx hy => _\n · cases' hx with hxS hxT\n · rw [← mul_one x]\n exact Submodule.mul_mem_mul hxS (show (1 : A) ∈ T from one_mem T)\n · rw [← one_mul x]\n exact Submodule.mul_mem_mul (show (1 : A) ∈ S from one_mem S) hxT\n · ","nextTactic":"rw [← one_mul (algebraMap _ _ _)]","declUpToTactic":"/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/\ntheorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]\n (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)\n = Subalgebra.toSubmodule (S ⊔ T) := by\n refine' le_antisymm (mul_toSubmodule_le _ _) _\n rintro x (hx : x ∈ Algebra.adjoin R (S ∪ T : Set A))\n refine'\n Algebra.adjoin_induction hx (fun x hx => _) (fun r => _) (fun _ _ => Submodule.add_mem _)\n fun x y hx hy => _\n · cases' hx with hxS hxT\n · rw [← mul_one x]\n exact Submodule.mul_mem_mul hxS (show (1 : A) ∈ T from one_mem T)\n · rw [← one_mul x]\n exact Submodule.mul_mem_mul (show (1 : A) ∈ S from one_mem S) hxT\n · ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Subalgebra_Pointwise.47_0.A8LWG0bRLHJclKD","decl":"/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/\ntheorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]\n (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)\n = Subalgebra.toSubmodule (S ⊔ T) "}
{"state":"case refine'_2\nR✝ : Type u_1\nA✝ : Type u_2\ninst✝⁵ : CommSemiring R✝\ninst✝⁴ : Semiring A✝\ninst✝³ : Algebra R✝ A✝\nR : Type u_3\nA : Type u_4\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\nx : A\nhx : x ∈ Algebra.adjoin R (↑S ∪ ↑T)\nr : R\n⊢ 1 * (algebraMap R A) r ∈ toSubmodule S * toSubmodule T","srcUpToTactic":"/-\nCopyright (c) 2021 Eric Weiser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\nimport Mathlib.Algebra.Algebra.Operations\nimport Mathlib.Algebra.Algebra.Subalgebra.Basic\nimport Mathlib.RingTheory.Subring.Pointwise\nimport Mathlib.RingTheory.Adjoin.Basic\n\n#align_import algebra.algebra.subalgebra.pointwise from \"leanprover-community/mathlib\"@\"b2c707cd190a58ea0565c86695a19e99ccecc215\"\n\n/-!\n# Pointwise actions on subalgebras.\n\nIf `R'` acts on an `R`-algebra `A` (so that `R'` and `R` actions commute)\nthen we get an `R'` action on the collection of `R`-subalgebras.\n-/\n\n\nnamespace Subalgebra\n\nsection Pointwise\n\nvariable {R : Type*} {A : Type*} [CommSemiring R] [Semiring A] [Algebra R A]\n\ntheorem mul_toSubmodule_le (S T : Subalgebra R A) :\n (Subalgebra.toSubmodule S)* (Subalgebra.toSubmodule T) ≤ Subalgebra.toSubmodule (S ⊔ T) := by\n rw [Submodule.mul_le]\n intro y hy z hz\n show y * z ∈ S ⊔ T\n exact mul_mem (Algebra.mem_sup_left hy) (Algebra.mem_sup_right hz)\n#align subalgebra.mul_to_submodule_le Subalgebra.mul_toSubmodule_le\n\n/-- As submodules, subalgebras are idempotent. -/\n@[simp]\ntheorem mul_self (S : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule S)\n = (Subalgebra.toSubmodule S) := by\n apply le_antisymm\n · refine' (mul_toSubmodule_le _ _).trans_eq _\n rw [sup_idem]\n · intro x hx1\n rw [← mul_one x]\n exact Submodule.mul_mem_mul hx1 (show (1 : A) ∈ S from one_mem S)\n#align subalgebra.mul_self Subalgebra.mul_self\n\n/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/\ntheorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]\n (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)\n = Subalgebra.toSubmodule (S ⊔ T) := by\n refine' le_antisymm (mul_toSubmodule_le _ _) _\n rintro x (hx : x ∈ Algebra.adjoin R (S ∪ T : Set A))\n refine'\n Algebra.adjoin_induction hx (fun x hx => _) (fun r => _) (fun _ _ => Submodule.add_mem _)\n fun x y hx hy => _\n · cases' hx with hxS hxT\n · rw [← mul_one x]\n exact Submodule.mul_mem_mul hxS (show (1 : A) ∈ T from one_mem T)\n · rw [← one_mul x]\n exact Submodule.mul_mem_mul (show (1 : A) ∈ S from one_mem S) hxT\n · rw [← one_mul (algebraMap _ _ _)]\n ","nextTactic":"exact Submodule.mul_mem_mul (show (1 : A) ∈ S from one_mem S) (algebraMap_mem T _)","declUpToTactic":"/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/\ntheorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]\n (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)\n = Subalgebra.toSubmodule (S ⊔ T) := by\n refine' le_antisymm (mul_toSubmodule_le _ _) _\n rintro x (hx : x ∈ Algebra.adjoin R (S ∪ T : Set A))\n refine'\n Algebra.adjoin_induction hx (fun x hx => _) (fun r => _) (fun _ _ => Submodule.add_mem _)\n fun x y hx hy => _\n · cases' hx with hxS hxT\n · rw [← mul_one x]\n exact Submodule.mul_mem_mul hxS (show (1 : A) ∈ T from one_mem T)\n · rw [← one_mul x]\n exact Submodule.mul_mem_mul (show (1 : A) ∈ S from one_mem S) hxT\n · rw [← one_mul (algebraMap _ _ _)]\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Subalgebra_Pointwise.47_0.A8LWG0bRLHJclKD","decl":"/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/\ntheorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]\n (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)\n = Subalgebra.toSubmodule (S ⊔ T) "}
{"state":"case refine'_3\nR✝ : Type u_1\nA✝ : Type u_2\ninst✝⁵ : CommSemiring R✝\ninst✝⁴ : Semiring A✝\ninst✝³ : Algebra R✝ A✝\nR : Type u_3\nA : Type u_4\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\nx✝ : A\nhx✝ : x✝ ∈ Algebra.adjoin R (↑S ∪ ↑T)\nx y : A\nhx : x ∈ toSubmodule S * toSubmodule T\nhy : y ∈ toSubmodule S * toSubmodule T\n⊢ x * y ∈ toSubmodule S * toSubmodule T","srcUpToTactic":"/-\nCopyright (c) 2021 Eric Weiser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\nimport Mathlib.Algebra.Algebra.Operations\nimport Mathlib.Algebra.Algebra.Subalgebra.Basic\nimport Mathlib.RingTheory.Subring.Pointwise\nimport Mathlib.RingTheory.Adjoin.Basic\n\n#align_import algebra.algebra.subalgebra.pointwise from \"leanprover-community/mathlib\"@\"b2c707cd190a58ea0565c86695a19e99ccecc215\"\n\n/-!\n# Pointwise actions on subalgebras.\n\nIf `R'` acts on an `R`-algebra `A` (so that `R'` and `R` actions commute)\nthen we get an `R'` action on the collection of `R`-subalgebras.\n-/\n\n\nnamespace Subalgebra\n\nsection Pointwise\n\nvariable {R : Type*} {A : Type*} [CommSemiring R] [Semiring A] [Algebra R A]\n\ntheorem mul_toSubmodule_le (S T : Subalgebra R A) :\n (Subalgebra.toSubmodule S)* (Subalgebra.toSubmodule T) ≤ Subalgebra.toSubmodule (S ⊔ T) := by\n rw [Submodule.mul_le]\n intro y hy z hz\n show y * z ∈ S ⊔ T\n exact mul_mem (Algebra.mem_sup_left hy) (Algebra.mem_sup_right hz)\n#align subalgebra.mul_to_submodule_le Subalgebra.mul_toSubmodule_le\n\n/-- As submodules, subalgebras are idempotent. -/\n@[simp]\ntheorem mul_self (S : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule S)\n = (Subalgebra.toSubmodule S) := by\n apply le_antisymm\n · refine' (mul_toSubmodule_le _ _).trans_eq _\n rw [sup_idem]\n · intro x hx1\n rw [← mul_one x]\n exact Submodule.mul_mem_mul hx1 (show (1 : A) ∈ S from one_mem S)\n#align subalgebra.mul_self Subalgebra.mul_self\n\n/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/\ntheorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]\n (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)\n = Subalgebra.toSubmodule (S ⊔ T) := by\n refine' le_antisymm (mul_toSubmodule_le _ _) _\n rintro x (hx : x ∈ Algebra.adjoin R (S ∪ T : Set A))\n refine'\n Algebra.adjoin_induction hx (fun x hx => _) (fun r => _) (fun _ _ => Submodule.add_mem _)\n fun x y hx hy => _\n · cases' hx with hxS hxT\n · rw [← mul_one x]\n exact Submodule.mul_mem_mul hxS (show (1 : A) ∈ T from one_mem T)\n · rw [← one_mul x]\n exact Submodule.mul_mem_mul (show (1 : A) ∈ S from one_mem S) hxT\n · rw [← one_mul (algebraMap _ _ _)]\n exact Submodule.mul_mem_mul (show (1 : A) ∈ S from one_mem S) (algebraMap_mem T _)\n ","nextTactic":"have := Submodule.mul_mem_mul hx hy","declUpToTactic":"/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/\ntheorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]\n (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)\n = Subalgebra.toSubmodule (S ⊔ T) := by\n refine' le_antisymm (mul_toSubmodule_le _ _) _\n rintro x (hx : x ∈ Algebra.adjoin R (S ∪ T : Set A))\n refine'\n Algebra.adjoin_induction hx (fun x hx => _) (fun r => _) (fun _ _ => Submodule.add_mem _)\n fun x y hx hy => _\n · cases' hx with hxS hxT\n · rw [← mul_one x]\n exact Submodule.mul_mem_mul hxS (show (1 : A) ∈ T from one_mem T)\n · rw [← one_mul x]\n exact Submodule.mul_mem_mul (show (1 : A) ∈ S from one_mem S) hxT\n · rw [← one_mul (algebraMap _ _ _)]\n exact Submodule.mul_mem_mul (show (1 : A) ∈ S from one_mem S) (algebraMap_mem T _)\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Subalgebra_Pointwise.47_0.A8LWG0bRLHJclKD","decl":"/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/\ntheorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]\n (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)\n = Subalgebra.toSubmodule (S ⊔ T) "}
{"state":"case refine'_3\nR✝ : Type u_1\nA✝ : Type u_2\ninst✝⁵ : CommSemiring R✝\ninst✝⁴ : Semiring A✝\ninst✝³ : Algebra R✝ A✝\nR : Type u_3\nA : Type u_4\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\nx✝ : A\nhx✝ : x✝ ∈ Algebra.adjoin R (↑S ∪ ↑T)\nx y : A\nhx : x ∈ toSubmodule S * toSubmodule T\nhy : y ∈ toSubmodule S * toSubmodule T\nthis : x * y ∈ toSubmodule S * toSubmodule T * (toSubmodule S * toSubmodule T)\n⊢ x * y ∈ toSubmodule S * toSubmodule T","srcUpToTactic":"/-\nCopyright (c) 2021 Eric Weiser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\nimport Mathlib.Algebra.Algebra.Operations\nimport Mathlib.Algebra.Algebra.Subalgebra.Basic\nimport Mathlib.RingTheory.Subring.Pointwise\nimport Mathlib.RingTheory.Adjoin.Basic\n\n#align_import algebra.algebra.subalgebra.pointwise from \"leanprover-community/mathlib\"@\"b2c707cd190a58ea0565c86695a19e99ccecc215\"\n\n/-!\n# Pointwise actions on subalgebras.\n\nIf `R'` acts on an `R`-algebra `A` (so that `R'` and `R` actions commute)\nthen we get an `R'` action on the collection of `R`-subalgebras.\n-/\n\n\nnamespace Subalgebra\n\nsection Pointwise\n\nvariable {R : Type*} {A : Type*} [CommSemiring R] [Semiring A] [Algebra R A]\n\ntheorem mul_toSubmodule_le (S T : Subalgebra R A) :\n (Subalgebra.toSubmodule S)* (Subalgebra.toSubmodule T) ≤ Subalgebra.toSubmodule (S ⊔ T) := by\n rw [Submodule.mul_le]\n intro y hy z hz\n show y * z ∈ S ⊔ T\n exact mul_mem (Algebra.mem_sup_left hy) (Algebra.mem_sup_right hz)\n#align subalgebra.mul_to_submodule_le Subalgebra.mul_toSubmodule_le\n\n/-- As submodules, subalgebras are idempotent. -/\n@[simp]\ntheorem mul_self (S : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule S)\n = (Subalgebra.toSubmodule S) := by\n apply le_antisymm\n · refine' (mul_toSubmodule_le _ _).trans_eq _\n rw [sup_idem]\n · intro x hx1\n rw [← mul_one x]\n exact Submodule.mul_mem_mul hx1 (show (1 : A) ∈ S from one_mem S)\n#align subalgebra.mul_self Subalgebra.mul_self\n\n/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/\ntheorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]\n (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)\n = Subalgebra.toSubmodule (S ⊔ T) := by\n refine' le_antisymm (mul_toSubmodule_le _ _) _\n rintro x (hx : x ∈ Algebra.adjoin R (S ∪ T : Set A))\n refine'\n Algebra.adjoin_induction hx (fun x hx => _) (fun r => _) (fun _ _ => Submodule.add_mem _)\n fun x y hx hy => _\n · cases' hx with hxS hxT\n · rw [← mul_one x]\n exact Submodule.mul_mem_mul hxS (show (1 : A) ∈ T from one_mem T)\n · rw [← one_mul x]\n exact Submodule.mul_mem_mul (show (1 : A) ∈ S from one_mem S) hxT\n · rw [← one_mul (algebraMap _ _ _)]\n exact Submodule.mul_mem_mul (show (1 : A) ∈ S from one_mem S) (algebraMap_mem T _)\n have := Submodule.mul_mem_mul hx hy\n ","nextTactic":"rwa [mul_assoc, mul_comm _ (Subalgebra.toSubmodule T), ← mul_assoc _ _ (Subalgebra.toSubmodule S),\n mul_self, mul_comm (Subalgebra.toSubmodule T), ← mul_assoc, mul_self] at this","declUpToTactic":"/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/\ntheorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]\n (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)\n = Subalgebra.toSubmodule (S ⊔ T) := by\n refine' le_antisymm (mul_toSubmodule_le _ _) _\n rintro x (hx : x ∈ Algebra.adjoin R (S ∪ T : Set A))\n refine'\n Algebra.adjoin_induction hx (fun x hx => _) (fun r => _) (fun _ _ => Submodule.add_mem _)\n fun x y hx hy => _\n · cases' hx with hxS hxT\n · rw [← mul_one x]\n exact Submodule.mul_mem_mul hxS (show (1 : A) ∈ T from one_mem T)\n · rw [← one_mul x]\n exact Submodule.mul_mem_mul (show (1 : A) ∈ S from one_mem S) hxT\n · rw [← one_mul (algebraMap _ _ _)]\n exact Submodule.mul_mem_mul (show (1 : A) ∈ S from one_mem S) (algebraMap_mem T _)\n have := Submodule.mul_mem_mul hx hy\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Subalgebra_Pointwise.47_0.A8LWG0bRLHJclKD","decl":"/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/\ntheorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]\n (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)\n = Subalgebra.toSubmodule (S ⊔ T) "}