| {"state":"R : Type u\nS : Type v\nA : Type w\nB : Type u₁\nM : Type v₁\ninst✝¹⁰ : CommSemiring R\ninst✝⁹ : CommSemiring S\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R S\ninst✝⁵ : Algebra S A\ninst✝⁴ : Algebra R A\ninst✝³ : Algebra S B\ninst✝² : Algebra R B\ninst✝¹ : IsScalarTower R S A\ninst✝ : IsScalarTower R S B\nU : Subalgebra S A\nx : R\n⊢ (algebraMap R A) x ∈ U.carrier","srcUpToTactic":"/-\nCopyright (c) 2020 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Anne Baanen\n-/\nimport Mathlib.Algebra.Algebra.Subalgebra.Basic\nimport Mathlib.Algebra.Algebra.Tower\n\n#align_import algebra.algebra.subalgebra.tower from \"leanprover-community/mathlib\"@\"a35ddf20601f85f78cd57e7f5b09ed528d71b7af\"\n\n/-!\n# Subalgebras in towers of algebras\n\nIn this file we prove facts about subalgebras in towers of algebra.\n\nAn algebra tower A/S/R is expressed by having instances of `Algebra A S`,\n`Algebra R S`, `Algebra R A` and `IsScalarTower R S A`, the later asserting the\ncompatibility condition `(r • s) • a = r • (s • a)`.\n\n## Main results\n\n * `IsScalarTower.Subalgebra`: if `A/S/R` is a tower and `S₀` is a subalgebra\n between `S` and `R`, then `A/S/S₀` is a tower\n * `IsScalarTower.Subalgebra'`: if `A/S/R` is a tower and `S₀` is a subalgebra\n between `S` and `R`, then `A/S₀/R` is a tower\n * `Subalgebra.restrictScalars`: turn an `S`-subalgebra of `A` into an `R`-subalgebra of `A`,\n given that `A/S/R` is a tower\n\n-/\n\n\nopen Pointwise\n\nuniverse u v w u₁ v₁\n\nvariable (R : Type u) (S : Type v) (A : Type w) (B : Type u₁) (M : Type v₁)\n\nnamespace Algebra\n\nvariable [CommSemiring R] [Semiring A] [Algebra R A]\n\nvariable [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M]\n\nvariable {A}\n\ntheorem lmul_algebraMap (x : R) : Algebra.lmul R A (algebraMap R A x) = Algebra.lsmul R R A x :=\n Eq.symm <| LinearMap.ext <| smul_def x\n#align algebra.lmul_algebra_map Algebra.lmul_algebraMap\n\nend Algebra\n\nnamespace IsScalarTower\n\nsection Semiring\n\nvariable [CommSemiring R] [CommSemiring S] [Semiring A]\n\nvariable [Algebra R S] [Algebra S A]\n\ninstance subalgebra (S₀ : Subalgebra R S) : IsScalarTower S₀ S A :=\n of_algebraMap_eq fun _ ↦ rfl\n#align is_scalar_tower.subalgebra IsScalarTower.subalgebra\n\nvariable [Algebra R A] [IsScalarTower R S A]\n\ninstance subalgebra' (S₀ : Subalgebra R S) : IsScalarTower R S₀ A :=\n @IsScalarTower.of_algebraMap_eq R S₀ A _ _ _ _ _ _ fun _ ↦\n (IsScalarTower.algebraMap_apply R S A _ : _)\n#align is_scalar_tower.subalgebra' IsScalarTower.subalgebra'\n\nend Semiring\n\nend IsScalarTower\n\nnamespace Subalgebra\n\nopen IsScalarTower\n\nsection Semiring\n\nvariable {S A B} [CommSemiring R] [CommSemiring S] [Semiring A] [Semiring B]\n\nvariable [Algebra R S] [Algebra S A] [Algebra R A] [Algebra S B] [Algebra R B]\n\nvariable [IsScalarTower R S A] [IsScalarTower R S B]\n\n/-- Given a tower `A / ↥U / S / R` of algebras, where `U` is an `S`-subalgebra of `A`, reinterpret\n`U` as an `R`-subalgebra of `A`. -/\ndef restrictScalars (U : Subalgebra S A) : Subalgebra R A :=\n { U with\n algebraMap_mem' := fun x ↦ by\n ","nextTactic":"rw [algebraMap_apply R S A]","declUpToTactic":"/-- Given a tower `A / ↥U / S / R` of algebras, where `U` is an `S`-subalgebra of `A`, reinterpret\n`U` as an `R`-subalgebra of `A`. -/\ndef restrictScalars (U : Subalgebra S A) : Subalgebra R A :=\n { U with\n algebraMap_mem' := fun x ↦ by\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Subalgebra_Tower.87_0.Zq8PWcMlDFAlf8P","decl":"/-- Given a tower `A / ↥U / S / R` of algebras, where `U` is an `S`-subalgebra of `A`, reinterpret\n`U` as an `R`-subalgebra of `A`. -/\ndef restrictScalars (U : Subalgebra S A) : Subalgebra R A "} | |
| {"state":"R : Type u\nS : Type v\nA : Type w\nB : Type u₁\nM : Type v₁\ninst✝¹⁰ : CommSemiring R\ninst✝⁹ : CommSemiring S\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R S\ninst✝⁵ : Algebra S A\ninst✝⁴ : Algebra R A\ninst✝³ : Algebra S B\ninst✝² : Algebra R B\ninst✝¹ : IsScalarTower R S A\ninst✝ : IsScalarTower R S B\nU : Subalgebra S A\nx : R\n⊢ (algebraMap S A) ((algebraMap R S) x) ∈ U.carrier","srcUpToTactic":"/-\nCopyright (c) 2020 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Anne Baanen\n-/\nimport Mathlib.Algebra.Algebra.Subalgebra.Basic\nimport Mathlib.Algebra.Algebra.Tower\n\n#align_import algebra.algebra.subalgebra.tower from \"leanprover-community/mathlib\"@\"a35ddf20601f85f78cd57e7f5b09ed528d71b7af\"\n\n/-!\n# Subalgebras in towers of algebras\n\nIn this file we prove facts about subalgebras in towers of algebra.\n\nAn algebra tower A/S/R is expressed by having instances of `Algebra A S`,\n`Algebra R S`, `Algebra R A` and `IsScalarTower R S A`, the later asserting the\ncompatibility condition `(r • s) • a = r • (s • a)`.\n\n## Main results\n\n * `IsScalarTower.Subalgebra`: if `A/S/R` is a tower and `S₀` is a subalgebra\n between `S` and `R`, then `A/S/S₀` is a tower\n * `IsScalarTower.Subalgebra'`: if `A/S/R` is a tower and `S₀` is a subalgebra\n between `S` and `R`, then `A/S₀/R` is a tower\n * `Subalgebra.restrictScalars`: turn an `S`-subalgebra of `A` into an `R`-subalgebra of `A`,\n given that `A/S/R` is a tower\n\n-/\n\n\nopen Pointwise\n\nuniverse u v w u₁ v₁\n\nvariable (R : Type u) (S : Type v) (A : Type w) (B : Type u₁) (M : Type v₁)\n\nnamespace Algebra\n\nvariable [CommSemiring R] [Semiring A] [Algebra R A]\n\nvariable [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M]\n\nvariable {A}\n\ntheorem lmul_algebraMap (x : R) : Algebra.lmul R A (algebraMap R A x) = Algebra.lsmul R R A x :=\n Eq.symm <| LinearMap.ext <| smul_def x\n#align algebra.lmul_algebra_map Algebra.lmul_algebraMap\n\nend Algebra\n\nnamespace IsScalarTower\n\nsection Semiring\n\nvariable [CommSemiring R] [CommSemiring S] [Semiring A]\n\nvariable [Algebra R S] [Algebra S A]\n\ninstance subalgebra (S₀ : Subalgebra R S) : IsScalarTower S₀ S A :=\n of_algebraMap_eq fun _ ↦ rfl\n#align is_scalar_tower.subalgebra IsScalarTower.subalgebra\n\nvariable [Algebra R A] [IsScalarTower R S A]\n\ninstance subalgebra' (S₀ : Subalgebra R S) : IsScalarTower R S₀ A :=\n @IsScalarTower.of_algebraMap_eq R S₀ A _ _ _ _ _ _ fun _ ↦\n (IsScalarTower.algebraMap_apply R S A _ : _)\n#align is_scalar_tower.subalgebra' IsScalarTower.subalgebra'\n\nend Semiring\n\nend IsScalarTower\n\nnamespace Subalgebra\n\nopen IsScalarTower\n\nsection Semiring\n\nvariable {S A B} [CommSemiring R] [CommSemiring S] [Semiring A] [Semiring B]\n\nvariable [Algebra R S] [Algebra S A] [Algebra R A] [Algebra S B] [Algebra R B]\n\nvariable [IsScalarTower R S A] [IsScalarTower R S B]\n\n/-- Given a tower `A / ↥U / S / R` of algebras, where `U` is an `S`-subalgebra of `A`, reinterpret\n`U` as an `R`-subalgebra of `A`. -/\ndef restrictScalars (U : Subalgebra S A) : Subalgebra R A :=\n { U with\n algebraMap_mem' := fun x ↦ by\n rw [algebraMap_apply R S A]\n ","nextTactic":"exact U.algebraMap_mem _","declUpToTactic":"/-- Given a tower `A / ↥U / S / R` of algebras, where `U` is an `S`-subalgebra of `A`, reinterpret\n`U` as an `R`-subalgebra of `A`. -/\ndef restrictScalars (U : Subalgebra S A) : Subalgebra R A :=\n { U with\n algebraMap_mem' := fun x ↦ by\n rw [algebraMap_apply R S A]\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Subalgebra_Tower.87_0.Zq8PWcMlDFAlf8P","decl":"/-- Given a tower `A / ↥U / S / R` of algebras, where `U` is an `S`-subalgebra of `A`, reinterpret\n`U` as an `R`-subalgebra of `A`. -/\ndef restrictScalars (U : Subalgebra S A) : Subalgebra R A "} | |
| {"state":"R : Type u\nS : Type v\nA : Type w\nB : Type u₁\nM : Type v₁\ninst✝¹⁰ : CommSemiring R\ninst✝⁹ : CommSemiring S\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R S\ninst✝⁵ : Algebra S A\ninst✝⁴ : Algebra R A\ninst✝³ : Algebra S B\ninst✝² : Algebra R B\ninst✝¹ : IsScalarTower R S A\ninst✝ : IsScalarTower R S B\n⊢ ↑(restrictScalars R ⊤) = ↑⊤","srcUpToTactic":"/-\nCopyright (c) 2020 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Anne Baanen\n-/\nimport Mathlib.Algebra.Algebra.Subalgebra.Basic\nimport Mathlib.Algebra.Algebra.Tower\n\n#align_import algebra.algebra.subalgebra.tower from \"leanprover-community/mathlib\"@\"a35ddf20601f85f78cd57e7f5b09ed528d71b7af\"\n\n/-!\n# Subalgebras in towers of algebras\n\nIn this file we prove facts about subalgebras in towers of algebra.\n\nAn algebra tower A/S/R is expressed by having instances of `Algebra A S`,\n`Algebra R S`, `Algebra R A` and `IsScalarTower R S A`, the later asserting the\ncompatibility condition `(r • s) • a = r • (s • a)`.\n\n## Main results\n\n * `IsScalarTower.Subalgebra`: if `A/S/R` is a tower and `S₀` is a subalgebra\n between `S` and `R`, then `A/S/S₀` is a tower\n * `IsScalarTower.Subalgebra'`: if `A/S/R` is a tower and `S₀` is a subalgebra\n between `S` and `R`, then `A/S₀/R` is a tower\n * `Subalgebra.restrictScalars`: turn an `S`-subalgebra of `A` into an `R`-subalgebra of `A`,\n given that `A/S/R` is a tower\n\n-/\n\n\nopen Pointwise\n\nuniverse u v w u₁ v₁\n\nvariable (R : Type u) (S : Type v) (A : Type w) (B : Type u₁) (M : Type v₁)\n\nnamespace Algebra\n\nvariable [CommSemiring R] [Semiring A] [Algebra R A]\n\nvariable [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M]\n\nvariable {A}\n\ntheorem lmul_algebraMap (x : R) : Algebra.lmul R A (algebraMap R A x) = Algebra.lsmul R R A x :=\n Eq.symm <| LinearMap.ext <| smul_def x\n#align algebra.lmul_algebra_map Algebra.lmul_algebraMap\n\nend Algebra\n\nnamespace IsScalarTower\n\nsection Semiring\n\nvariable [CommSemiring R] [CommSemiring S] [Semiring A]\n\nvariable [Algebra R S] [Algebra S A]\n\ninstance subalgebra (S₀ : Subalgebra R S) : IsScalarTower S₀ S A :=\n of_algebraMap_eq fun _ ↦ rfl\n#align is_scalar_tower.subalgebra IsScalarTower.subalgebra\n\nvariable [Algebra R A] [IsScalarTower R S A]\n\ninstance subalgebra' (S₀ : Subalgebra R S) : IsScalarTower R S₀ A :=\n @IsScalarTower.of_algebraMap_eq R S₀ A _ _ _ _ _ _ fun _ ↦\n (IsScalarTower.algebraMap_apply R S A _ : _)\n#align is_scalar_tower.subalgebra' IsScalarTower.subalgebra'\n\nend Semiring\n\nend IsScalarTower\n\nnamespace Subalgebra\n\nopen IsScalarTower\n\nsection Semiring\n\nvariable {S A B} [CommSemiring R] [CommSemiring S] [Semiring A] [Semiring B]\n\nvariable [Algebra R S] [Algebra S A] [Algebra R A] [Algebra S B] [Algebra R B]\n\nvariable [IsScalarTower R S A] [IsScalarTower R S B]\n\n/-- Given a tower `A / ↥U / S / R` of algebras, where `U` is an `S`-subalgebra of `A`, reinterpret\n`U` as an `R`-subalgebra of `A`. -/\ndef restrictScalars (U : Subalgebra S A) : Subalgebra R A :=\n { U with\n algebraMap_mem' := fun x ↦ by\n rw [algebraMap_apply R S A]\n exact U.algebraMap_mem _ }\n#align subalgebra.restrict_scalars Subalgebra.restrictScalars\n\n@[simp]\ntheorem coe_restrictScalars {U : Subalgebra S A} : (restrictScalars R U : Set A) = (U : Set A) :=\n rfl\n#align subalgebra.coe_restrict_scalars Subalgebra.coe_restrictScalars\n\n@[simp]\ntheorem restrictScalars_top : restrictScalars R (⊤ : Subalgebra S A) = ⊤ :=\n SetLike.coe_injective $ by ","nextTactic":"dsimp","declUpToTactic":"@[simp]\ntheorem restrictScalars_top : restrictScalars R (⊤ : Subalgebra S A) = ⊤ :=\n SetLike.coe_injective $ by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Subalgebra_Tower.101_0.Zq8PWcMlDFAlf8P","decl":"@[simp]\ntheorem restrictScalars_top : restrictScalars R (⊤ : Subalgebra S A) = ⊤ "} | |
| {"state":"R : Type u\nS : Type v\nA : Type w\nB : Type u₁\nM : Type v₁\ninst✝¹⁰ : CommSemiring R\ninst✝⁹ : CommSemiring S\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R S\ninst✝⁵ : Algebra S A\ninst✝⁴ : Algebra R A\ninst✝³ : Algebra S B\ninst✝² : Algebra R B\ninst✝¹ : IsScalarTower R S A\ninst✝ : IsScalarTower R S B\nU V : Subalgebra S A\nH : restrictScalars R U = restrictScalars R V\nx : A\n⊢ x ∈ U ↔ x ∈ V","srcUpToTactic":"/-\nCopyright (c) 2020 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Anne Baanen\n-/\nimport Mathlib.Algebra.Algebra.Subalgebra.Basic\nimport Mathlib.Algebra.Algebra.Tower\n\n#align_import algebra.algebra.subalgebra.tower from \"leanprover-community/mathlib\"@\"a35ddf20601f85f78cd57e7f5b09ed528d71b7af\"\n\n/-!\n# Subalgebras in towers of algebras\n\nIn this file we prove facts about subalgebras in towers of algebra.\n\nAn algebra tower A/S/R is expressed by having instances of `Algebra A S`,\n`Algebra R S`, `Algebra R A` and `IsScalarTower R S A`, the later asserting the\ncompatibility condition `(r • s) • a = r • (s • a)`.\n\n## Main results\n\n * `IsScalarTower.Subalgebra`: if `A/S/R` is a tower and `S₀` is a subalgebra\n between `S` and `R`, then `A/S/S₀` is a tower\n * `IsScalarTower.Subalgebra'`: if `A/S/R` is a tower and `S₀` is a subalgebra\n between `S` and `R`, then `A/S₀/R` is a tower\n * `Subalgebra.restrictScalars`: turn an `S`-subalgebra of `A` into an `R`-subalgebra of `A`,\n given that `A/S/R` is a tower\n\n-/\n\n\nopen Pointwise\n\nuniverse u v w u₁ v₁\n\nvariable (R : Type u) (S : Type v) (A : Type w) (B : Type u₁) (M : Type v₁)\n\nnamespace Algebra\n\nvariable [CommSemiring R] [Semiring A] [Algebra R A]\n\nvariable [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M]\n\nvariable {A}\n\ntheorem lmul_algebraMap (x : R) : Algebra.lmul R A (algebraMap R A x) = Algebra.lsmul R R A x :=\n Eq.symm <| LinearMap.ext <| smul_def x\n#align algebra.lmul_algebra_map Algebra.lmul_algebraMap\n\nend Algebra\n\nnamespace IsScalarTower\n\nsection Semiring\n\nvariable [CommSemiring R] [CommSemiring S] [Semiring A]\n\nvariable [Algebra R S] [Algebra S A]\n\ninstance subalgebra (S₀ : Subalgebra R S) : IsScalarTower S₀ S A :=\n of_algebraMap_eq fun _ ↦ rfl\n#align is_scalar_tower.subalgebra IsScalarTower.subalgebra\n\nvariable [Algebra R A] [IsScalarTower R S A]\n\ninstance subalgebra' (S₀ : Subalgebra R S) : IsScalarTower R S₀ A :=\n @IsScalarTower.of_algebraMap_eq R S₀ A _ _ _ _ _ _ fun _ ↦\n (IsScalarTower.algebraMap_apply R S A _ : _)\n#align is_scalar_tower.subalgebra' IsScalarTower.subalgebra'\n\nend Semiring\n\nend IsScalarTower\n\nnamespace Subalgebra\n\nopen IsScalarTower\n\nsection Semiring\n\nvariable {S A B} [CommSemiring R] [CommSemiring S] [Semiring A] [Semiring B]\n\nvariable [Algebra R S] [Algebra S A] [Algebra R A] [Algebra S B] [Algebra R B]\n\nvariable [IsScalarTower R S A] [IsScalarTower R S B]\n\n/-- Given a tower `A / ↥U / S / R` of algebras, where `U` is an `S`-subalgebra of `A`, reinterpret\n`U` as an `R`-subalgebra of `A`. -/\ndef restrictScalars (U : Subalgebra S A) : Subalgebra R A :=\n { U with\n algebraMap_mem' := fun x ↦ by\n rw [algebraMap_apply R S A]\n exact U.algebraMap_mem _ }\n#align subalgebra.restrict_scalars Subalgebra.restrictScalars\n\n@[simp]\ntheorem coe_restrictScalars {U : Subalgebra S A} : (restrictScalars R U : Set A) = (U : Set A) :=\n rfl\n#align subalgebra.coe_restrict_scalars Subalgebra.coe_restrictScalars\n\n@[simp]\ntheorem restrictScalars_top : restrictScalars R (⊤ : Subalgebra S A) = ⊤ :=\n SetLike.coe_injective $ by dsimp -- porting note: why does `rfl` not work instead of `by dsimp`?\n#align subalgebra.restrict_scalars_top Subalgebra.restrictScalars_top\n\n@[simp]\ntheorem restrictScalars_toSubmodule {U : Subalgebra S A} :\n Subalgebra.toSubmodule (U.restrictScalars R) = U.toSubmodule.restrictScalars R :=\n SetLike.coe_injective rfl\n#align subalgebra.restrict_scalars_to_submodule Subalgebra.restrictScalars_toSubmodule\n\n@[simp]\ntheorem mem_restrictScalars {U : Subalgebra S A} {x : A} : x ∈ restrictScalars R U ↔ x ∈ U :=\n Iff.rfl\n#align subalgebra.mem_restrict_scalars Subalgebra.mem_restrictScalars\n\ntheorem restrictScalars_injective :\n Function.Injective (restrictScalars R : Subalgebra S A → Subalgebra R A) := fun U V H ↦\n ext fun x ↦ by ","nextTactic":"rw [← mem_restrictScalars R, H, mem_restrictScalars]","declUpToTactic":"theorem restrictScalars_injective :\n Function.Injective (restrictScalars R : Subalgebra S A → Subalgebra R A) := fun U V H ↦\n ext fun x ↦ by ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Subalgebra_Tower.117_0.Zq8PWcMlDFAlf8P","decl":"theorem restrictScalars_injective :\n Function.Injective (restrictScalars R : Subalgebra S A → Subalgebra R A) "} | |
| {"state":"R : Type u\nS✝ : Type v\nA : Type w\nB : Type u₁\nM : Type v₁\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nS : Subalgebra R A\n⊢ LinearMap.range (toAlgHom R (↥S) A) = toSubmodule S","srcUpToTactic":"/-\nCopyright (c) 2020 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Anne Baanen\n-/\nimport Mathlib.Algebra.Algebra.Subalgebra.Basic\nimport Mathlib.Algebra.Algebra.Tower\n\n#align_import algebra.algebra.subalgebra.tower from \"leanprover-community/mathlib\"@\"a35ddf20601f85f78cd57e7f5b09ed528d71b7af\"\n\n/-!\n# Subalgebras in towers of algebras\n\nIn this file we prove facts about subalgebras in towers of algebra.\n\nAn algebra tower A/S/R is expressed by having instances of `Algebra A S`,\n`Algebra R S`, `Algebra R A` and `IsScalarTower R S A`, the later asserting the\ncompatibility condition `(r • s) • a = r • (s • a)`.\n\n## Main results\n\n * `IsScalarTower.Subalgebra`: if `A/S/R` is a tower and `S₀` is a subalgebra\n between `S` and `R`, then `A/S/S₀` is a tower\n * `IsScalarTower.Subalgebra'`: if `A/S/R` is a tower and `S₀` is a subalgebra\n between `S` and `R`, then `A/S₀/R` is a tower\n * `Subalgebra.restrictScalars`: turn an `S`-subalgebra of `A` into an `R`-subalgebra of `A`,\n given that `A/S/R` is a tower\n\n-/\n\n\nopen Pointwise\n\nuniverse u v w u₁ v₁\n\nvariable (R : Type u) (S : Type v) (A : Type w) (B : Type u₁) (M : Type v₁)\n\nnamespace Algebra\n\nvariable [CommSemiring R] [Semiring A] [Algebra R A]\n\nvariable [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M]\n\nvariable {A}\n\ntheorem lmul_algebraMap (x : R) : Algebra.lmul R A (algebraMap R A x) = Algebra.lsmul R R A x :=\n Eq.symm <| LinearMap.ext <| smul_def x\n#align algebra.lmul_algebra_map Algebra.lmul_algebraMap\n\nend Algebra\n\nnamespace IsScalarTower\n\nsection Semiring\n\nvariable [CommSemiring R] [CommSemiring S] [Semiring A]\n\nvariable [Algebra R S] [Algebra S A]\n\ninstance subalgebra (S₀ : Subalgebra R S) : IsScalarTower S₀ S A :=\n of_algebraMap_eq fun _ ↦ rfl\n#align is_scalar_tower.subalgebra IsScalarTower.subalgebra\n\nvariable [Algebra R A] [IsScalarTower R S A]\n\ninstance subalgebra' (S₀ : Subalgebra R S) : IsScalarTower R S₀ A :=\n @IsScalarTower.of_algebraMap_eq R S₀ A _ _ _ _ _ _ fun _ ↦\n (IsScalarTower.algebraMap_apply R S A _ : _)\n#align is_scalar_tower.subalgebra' IsScalarTower.subalgebra'\n\nend Semiring\n\nend IsScalarTower\n\nnamespace Subalgebra\n\nopen IsScalarTower\n\nsection Semiring\n\nvariable {S A B} [CommSemiring R] [CommSemiring S] [Semiring A] [Semiring B]\n\nvariable [Algebra R S] [Algebra S A] [Algebra R A] [Algebra S B] [Algebra R B]\n\nvariable [IsScalarTower R S A] [IsScalarTower R S B]\n\n/-- Given a tower `A / ↥U / S / R` of algebras, where `U` is an `S`-subalgebra of `A`, reinterpret\n`U` as an `R`-subalgebra of `A`. -/\ndef restrictScalars (U : Subalgebra S A) : Subalgebra R A :=\n { U with\n algebraMap_mem' := fun x ↦ by\n rw [algebraMap_apply R S A]\n exact U.algebraMap_mem _ }\n#align subalgebra.restrict_scalars Subalgebra.restrictScalars\n\n@[simp]\ntheorem coe_restrictScalars {U : Subalgebra S A} : (restrictScalars R U : Set A) = (U : Set A) :=\n rfl\n#align subalgebra.coe_restrict_scalars Subalgebra.coe_restrictScalars\n\n@[simp]\ntheorem restrictScalars_top : restrictScalars R (⊤ : Subalgebra S A) = ⊤ :=\n SetLike.coe_injective $ by dsimp -- porting note: why does `rfl` not work instead of `by dsimp`?\n#align subalgebra.restrict_scalars_top Subalgebra.restrictScalars_top\n\n@[simp]\ntheorem restrictScalars_toSubmodule {U : Subalgebra S A} :\n Subalgebra.toSubmodule (U.restrictScalars R) = U.toSubmodule.restrictScalars R :=\n SetLike.coe_injective rfl\n#align subalgebra.restrict_scalars_to_submodule Subalgebra.restrictScalars_toSubmodule\n\n@[simp]\ntheorem mem_restrictScalars {U : Subalgebra S A} {x : A} : x ∈ restrictScalars R U ↔ x ∈ U :=\n Iff.rfl\n#align subalgebra.mem_restrict_scalars Subalgebra.mem_restrictScalars\n\ntheorem restrictScalars_injective :\n Function.Injective (restrictScalars R : Subalgebra S A → Subalgebra R A) := fun U V H ↦\n ext fun x ↦ by rw [← mem_restrictScalars R, H, mem_restrictScalars]\n#align subalgebra.restrict_scalars_injective Subalgebra.restrictScalars_injective\n\n/-- Produces an `R`-algebra map from `U.restrictScalars R` given an `S`-algebra map from `U`.\n\nThis is a special case of `AlgHom.restrictScalars` that can be helpful in elaboration. -/\n@[simp]\ndef ofRestrictScalars (U : Subalgebra S A) (f : U →ₐ[S] B) : U.restrictScalars R →ₐ[R] B :=\n f.restrictScalars R\n#align subalgebra.of_restrict_scalars Subalgebra.ofRestrictScalars\n\nend Semiring\n\nsection CommSemiring\n\n@[simp]\nlemma range_isScalarTower_toAlgHom [CommSemiring R] [CommSemiring A]\n [Algebra R A] (S : Subalgebra R A) :\n LinearMap.range (IsScalarTower.toAlgHom R S A) = Subalgebra.toSubmodule S := by\n ","nextTactic":"ext","declUpToTactic":"@[simp]\nlemma range_isScalarTower_toAlgHom [CommSemiring R] [CommSemiring A]\n [Algebra R A] (S : Subalgebra R A) :\n LinearMap.range (IsScalarTower.toAlgHom R S A) = Subalgebra.toSubmodule S := by\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Subalgebra_Tower.134_0.Zq8PWcMlDFAlf8P","decl":"@[simp]\nlemma range_isScalarTower_toAlgHom [CommSemiring R] [CommSemiring A]\n [Algebra R A] (S : Subalgebra R A) :\n LinearMap.range (IsScalarTower.toAlgHom R S A) = Subalgebra.toSubmodule S "} | |
| {"state":"case h\nR : Type u\nS✝ : Type v\nA : Type w\nB : Type u₁\nM : Type v₁\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nS : Subalgebra R A\nx✝ : A\n⊢ x✝ ∈ LinearMap.range (toAlgHom R (↥S) A) ↔ x✝ ∈ toSubmodule S","srcUpToTactic":"/-\nCopyright (c) 2020 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Anne Baanen\n-/\nimport Mathlib.Algebra.Algebra.Subalgebra.Basic\nimport Mathlib.Algebra.Algebra.Tower\n\n#align_import algebra.algebra.subalgebra.tower from \"leanprover-community/mathlib\"@\"a35ddf20601f85f78cd57e7f5b09ed528d71b7af\"\n\n/-!\n# Subalgebras in towers of algebras\n\nIn this file we prove facts about subalgebras in towers of algebra.\n\nAn algebra tower A/S/R is expressed by having instances of `Algebra A S`,\n`Algebra R S`, `Algebra R A` and `IsScalarTower R S A`, the later asserting the\ncompatibility condition `(r • s) • a = r • (s • a)`.\n\n## Main results\n\n * `IsScalarTower.Subalgebra`: if `A/S/R` is a tower and `S₀` is a subalgebra\n between `S` and `R`, then `A/S/S₀` is a tower\n * `IsScalarTower.Subalgebra'`: if `A/S/R` is a tower and `S₀` is a subalgebra\n between `S` and `R`, then `A/S₀/R` is a tower\n * `Subalgebra.restrictScalars`: turn an `S`-subalgebra of `A` into an `R`-subalgebra of `A`,\n given that `A/S/R` is a tower\n\n-/\n\n\nopen Pointwise\n\nuniverse u v w u₁ v₁\n\nvariable (R : Type u) (S : Type v) (A : Type w) (B : Type u₁) (M : Type v₁)\n\nnamespace Algebra\n\nvariable [CommSemiring R] [Semiring A] [Algebra R A]\n\nvariable [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M]\n\nvariable {A}\n\ntheorem lmul_algebraMap (x : R) : Algebra.lmul R A (algebraMap R A x) = Algebra.lsmul R R A x :=\n Eq.symm <| LinearMap.ext <| smul_def x\n#align algebra.lmul_algebra_map Algebra.lmul_algebraMap\n\nend Algebra\n\nnamespace IsScalarTower\n\nsection Semiring\n\nvariable [CommSemiring R] [CommSemiring S] [Semiring A]\n\nvariable [Algebra R S] [Algebra S A]\n\ninstance subalgebra (S₀ : Subalgebra R S) : IsScalarTower S₀ S A :=\n of_algebraMap_eq fun _ ↦ rfl\n#align is_scalar_tower.subalgebra IsScalarTower.subalgebra\n\nvariable [Algebra R A] [IsScalarTower R S A]\n\ninstance subalgebra' (S₀ : Subalgebra R S) : IsScalarTower R S₀ A :=\n @IsScalarTower.of_algebraMap_eq R S₀ A _ _ _ _ _ _ fun _ ↦\n (IsScalarTower.algebraMap_apply R S A _ : _)\n#align is_scalar_tower.subalgebra' IsScalarTower.subalgebra'\n\nend Semiring\n\nend IsScalarTower\n\nnamespace Subalgebra\n\nopen IsScalarTower\n\nsection Semiring\n\nvariable {S A B} [CommSemiring R] [CommSemiring S] [Semiring A] [Semiring B]\n\nvariable [Algebra R S] [Algebra S A] [Algebra R A] [Algebra S B] [Algebra R B]\n\nvariable [IsScalarTower R S A] [IsScalarTower R S B]\n\n/-- Given a tower `A / ↥U / S / R` of algebras, where `U` is an `S`-subalgebra of `A`, reinterpret\n`U` as an `R`-subalgebra of `A`. -/\ndef restrictScalars (U : Subalgebra S A) : Subalgebra R A :=\n { U with\n algebraMap_mem' := fun x ↦ by\n rw [algebraMap_apply R S A]\n exact U.algebraMap_mem _ }\n#align subalgebra.restrict_scalars Subalgebra.restrictScalars\n\n@[simp]\ntheorem coe_restrictScalars {U : Subalgebra S A} : (restrictScalars R U : Set A) = (U : Set A) :=\n rfl\n#align subalgebra.coe_restrict_scalars Subalgebra.coe_restrictScalars\n\n@[simp]\ntheorem restrictScalars_top : restrictScalars R (⊤ : Subalgebra S A) = ⊤ :=\n SetLike.coe_injective $ by dsimp -- porting note: why does `rfl` not work instead of `by dsimp`?\n#align subalgebra.restrict_scalars_top Subalgebra.restrictScalars_top\n\n@[simp]\ntheorem restrictScalars_toSubmodule {U : Subalgebra S A} :\n Subalgebra.toSubmodule (U.restrictScalars R) = U.toSubmodule.restrictScalars R :=\n SetLike.coe_injective rfl\n#align subalgebra.restrict_scalars_to_submodule Subalgebra.restrictScalars_toSubmodule\n\n@[simp]\ntheorem mem_restrictScalars {U : Subalgebra S A} {x : A} : x ∈ restrictScalars R U ↔ x ∈ U :=\n Iff.rfl\n#align subalgebra.mem_restrict_scalars Subalgebra.mem_restrictScalars\n\ntheorem restrictScalars_injective :\n Function.Injective (restrictScalars R : Subalgebra S A → Subalgebra R A) := fun U V H ↦\n ext fun x ↦ by rw [← mem_restrictScalars R, H, mem_restrictScalars]\n#align subalgebra.restrict_scalars_injective Subalgebra.restrictScalars_injective\n\n/-- Produces an `R`-algebra map from `U.restrictScalars R` given an `S`-algebra map from `U`.\n\nThis is a special case of `AlgHom.restrictScalars` that can be helpful in elaboration. -/\n@[simp]\ndef ofRestrictScalars (U : Subalgebra S A) (f : U →ₐ[S] B) : U.restrictScalars R →ₐ[R] B :=\n f.restrictScalars R\n#align subalgebra.of_restrict_scalars Subalgebra.ofRestrictScalars\n\nend Semiring\n\nsection CommSemiring\n\n@[simp]\nlemma range_isScalarTower_toAlgHom [CommSemiring R] [CommSemiring A]\n [Algebra R A] (S : Subalgebra R A) :\n LinearMap.range (IsScalarTower.toAlgHom R S A) = Subalgebra.toSubmodule S := by\n ext\n ","nextTactic":"simp only [← Submodule.range_subtype (Subalgebra.toSubmodule S), LinearMap.mem_range,\n IsScalarTower.coe_toAlgHom', Subalgebra.mem_toSubmodule]","declUpToTactic":"@[simp]\nlemma range_isScalarTower_toAlgHom [CommSemiring R] [CommSemiring A]\n [Algebra R A] (S : Subalgebra R A) :\n LinearMap.range (IsScalarTower.toAlgHom R S A) = Subalgebra.toSubmodule S := by\n ext\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Subalgebra_Tower.134_0.Zq8PWcMlDFAlf8P","decl":"@[simp]\nlemma range_isScalarTower_toAlgHom [CommSemiring R] [CommSemiring A]\n [Algebra R A] (S : Subalgebra R A) :\n LinearMap.range (IsScalarTower.toAlgHom R S A) = Subalgebra.toSubmodule S "} | |
| {"state":"case h\nR : Type u\nS✝ : Type v\nA : Type w\nB : Type u₁\nM : Type v₁\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nS : Subalgebra R A\nx✝ : A\n⊢ (∃ y, (algebraMap (↥S) A) y = x✝) ↔ ∃ y, (Submodule.subtype (toSubmodule S)) y = x✝","srcUpToTactic":"/-\nCopyright (c) 2020 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Anne Baanen\n-/\nimport Mathlib.Algebra.Algebra.Subalgebra.Basic\nimport Mathlib.Algebra.Algebra.Tower\n\n#align_import algebra.algebra.subalgebra.tower from \"leanprover-community/mathlib\"@\"a35ddf20601f85f78cd57e7f5b09ed528d71b7af\"\n\n/-!\n# Subalgebras in towers of algebras\n\nIn this file we prove facts about subalgebras in towers of algebra.\n\nAn algebra tower A/S/R is expressed by having instances of `Algebra A S`,\n`Algebra R S`, `Algebra R A` and `IsScalarTower R S A`, the later asserting the\ncompatibility condition `(r • s) • a = r • (s • a)`.\n\n## Main results\n\n * `IsScalarTower.Subalgebra`: if `A/S/R` is a tower and `S₀` is a subalgebra\n between `S` and `R`, then `A/S/S₀` is a tower\n * `IsScalarTower.Subalgebra'`: if `A/S/R` is a tower and `S₀` is a subalgebra\n between `S` and `R`, then `A/S₀/R` is a tower\n * `Subalgebra.restrictScalars`: turn an `S`-subalgebra of `A` into an `R`-subalgebra of `A`,\n given that `A/S/R` is a tower\n\n-/\n\n\nopen Pointwise\n\nuniverse u v w u₁ v₁\n\nvariable (R : Type u) (S : Type v) (A : Type w) (B : Type u₁) (M : Type v₁)\n\nnamespace Algebra\n\nvariable [CommSemiring R] [Semiring A] [Algebra R A]\n\nvariable [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M]\n\nvariable {A}\n\ntheorem lmul_algebraMap (x : R) : Algebra.lmul R A (algebraMap R A x) = Algebra.lsmul R R A x :=\n Eq.symm <| LinearMap.ext <| smul_def x\n#align algebra.lmul_algebra_map Algebra.lmul_algebraMap\n\nend Algebra\n\nnamespace IsScalarTower\n\nsection Semiring\n\nvariable [CommSemiring R] [CommSemiring S] [Semiring A]\n\nvariable [Algebra R S] [Algebra S A]\n\ninstance subalgebra (S₀ : Subalgebra R S) : IsScalarTower S₀ S A :=\n of_algebraMap_eq fun _ ↦ rfl\n#align is_scalar_tower.subalgebra IsScalarTower.subalgebra\n\nvariable [Algebra R A] [IsScalarTower R S A]\n\ninstance subalgebra' (S₀ : Subalgebra R S) : IsScalarTower R S₀ A :=\n @IsScalarTower.of_algebraMap_eq R S₀ A _ _ _ _ _ _ fun _ ↦\n (IsScalarTower.algebraMap_apply R S A _ : _)\n#align is_scalar_tower.subalgebra' IsScalarTower.subalgebra'\n\nend Semiring\n\nend IsScalarTower\n\nnamespace Subalgebra\n\nopen IsScalarTower\n\nsection Semiring\n\nvariable {S A B} [CommSemiring R] [CommSemiring S] [Semiring A] [Semiring B]\n\nvariable [Algebra R S] [Algebra S A] [Algebra R A] [Algebra S B] [Algebra R B]\n\nvariable [IsScalarTower R S A] [IsScalarTower R S B]\n\n/-- Given a tower `A / ↥U / S / R` of algebras, where `U` is an `S`-subalgebra of `A`, reinterpret\n`U` as an `R`-subalgebra of `A`. -/\ndef restrictScalars (U : Subalgebra S A) : Subalgebra R A :=\n { U with\n algebraMap_mem' := fun x ↦ by\n rw [algebraMap_apply R S A]\n exact U.algebraMap_mem _ }\n#align subalgebra.restrict_scalars Subalgebra.restrictScalars\n\n@[simp]\ntheorem coe_restrictScalars {U : Subalgebra S A} : (restrictScalars R U : Set A) = (U : Set A) :=\n rfl\n#align subalgebra.coe_restrict_scalars Subalgebra.coe_restrictScalars\n\n@[simp]\ntheorem restrictScalars_top : restrictScalars R (⊤ : Subalgebra S A) = ⊤ :=\n SetLike.coe_injective $ by dsimp -- porting note: why does `rfl` not work instead of `by dsimp`?\n#align subalgebra.restrict_scalars_top Subalgebra.restrictScalars_top\n\n@[simp]\ntheorem restrictScalars_toSubmodule {U : Subalgebra S A} :\n Subalgebra.toSubmodule (U.restrictScalars R) = U.toSubmodule.restrictScalars R :=\n SetLike.coe_injective rfl\n#align subalgebra.restrict_scalars_to_submodule Subalgebra.restrictScalars_toSubmodule\n\n@[simp]\ntheorem mem_restrictScalars {U : Subalgebra S A} {x : A} : x ∈ restrictScalars R U ↔ x ∈ U :=\n Iff.rfl\n#align subalgebra.mem_restrict_scalars Subalgebra.mem_restrictScalars\n\ntheorem restrictScalars_injective :\n Function.Injective (restrictScalars R : Subalgebra S A → Subalgebra R A) := fun U V H ↦\n ext fun x ↦ by rw [← mem_restrictScalars R, H, mem_restrictScalars]\n#align subalgebra.restrict_scalars_injective Subalgebra.restrictScalars_injective\n\n/-- Produces an `R`-algebra map from `U.restrictScalars R` given an `S`-algebra map from `U`.\n\nThis is a special case of `AlgHom.restrictScalars` that can be helpful in elaboration. -/\n@[simp]\ndef ofRestrictScalars (U : Subalgebra S A) (f : U →ₐ[S] B) : U.restrictScalars R →ₐ[R] B :=\n f.restrictScalars R\n#align subalgebra.of_restrict_scalars Subalgebra.ofRestrictScalars\n\nend Semiring\n\nsection CommSemiring\n\n@[simp]\nlemma range_isScalarTower_toAlgHom [CommSemiring R] [CommSemiring A]\n [Algebra R A] (S : Subalgebra R A) :\n LinearMap.range (IsScalarTower.toAlgHom R S A) = Subalgebra.toSubmodule S := by\n ext\n simp only [← Submodule.range_subtype (Subalgebra.toSubmodule S), LinearMap.mem_range,\n IsScalarTower.coe_toAlgHom', Subalgebra.mem_toSubmodule]\n ","nextTactic":"rfl","declUpToTactic":"@[simp]\nlemma range_isScalarTower_toAlgHom [CommSemiring R] [CommSemiring A]\n [Algebra R A] (S : Subalgebra R A) :\n LinearMap.range (IsScalarTower.toAlgHom R S A) = Subalgebra.toSubmodule S := by\n ext\n simp only [← Submodule.range_subtype (Subalgebra.toSubmodule S), LinearMap.mem_range,\n IsScalarTower.coe_toAlgHom', Subalgebra.mem_toSubmodule]\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Subalgebra_Tower.134_0.Zq8PWcMlDFAlf8P","decl":"@[simp]\nlemma range_isScalarTower_toAlgHom [CommSemiring R] [CommSemiring A]\n [Algebra R A] (S : Subalgebra R A) :\n LinearMap.range (IsScalarTower.toAlgHom R S A) = Subalgebra.toSubmodule S "} | |
| {"state":"R : Type u\nS : Type v\nA : Type w\nB : Type u₁\nM : Type v₁\ninst✝⁶ : CommSemiring R\ninst✝⁵ : CommSemiring S\ninst✝⁴ : CommSemiring A\ninst✝³ : Algebra R S\ninst✝² : Algebra S A\ninst✝¹ : Algebra R A\ninst✝ : IsScalarTower R S A\nt : Set A\nz : A\n⊢ z ∈ Subsemiring.closure (Set.range ⇑(algebraMap (↥(AlgHom.range (toAlgHom R S A))) A) ∪ t) ↔\n z ∈ Subsemiring.closure (Set.range ⇑(algebraMap S A) ∪ t)","srcUpToTactic":"/-\nCopyright (c) 2020 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Anne Baanen\n-/\nimport Mathlib.Algebra.Algebra.Subalgebra.Basic\nimport Mathlib.Algebra.Algebra.Tower\n\n#align_import algebra.algebra.subalgebra.tower from \"leanprover-community/mathlib\"@\"a35ddf20601f85f78cd57e7f5b09ed528d71b7af\"\n\n/-!\n# Subalgebras in towers of algebras\n\nIn this file we prove facts about subalgebras in towers of algebra.\n\nAn algebra tower A/S/R is expressed by having instances of `Algebra A S`,\n`Algebra R S`, `Algebra R A` and `IsScalarTower R S A`, the later asserting the\ncompatibility condition `(r • s) • a = r • (s • a)`.\n\n## Main results\n\n * `IsScalarTower.Subalgebra`: if `A/S/R` is a tower and `S₀` is a subalgebra\n between `S` and `R`, then `A/S/S₀` is a tower\n * `IsScalarTower.Subalgebra'`: if `A/S/R` is a tower and `S₀` is a subalgebra\n between `S` and `R`, then `A/S₀/R` is a tower\n * `Subalgebra.restrictScalars`: turn an `S`-subalgebra of `A` into an `R`-subalgebra of `A`,\n given that `A/S/R` is a tower\n\n-/\n\n\nopen Pointwise\n\nuniverse u v w u₁ v₁\n\nvariable (R : Type u) (S : Type v) (A : Type w) (B : Type u₁) (M : Type v₁)\n\nnamespace Algebra\n\nvariable [CommSemiring R] [Semiring A] [Algebra R A]\n\nvariable [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M]\n\nvariable {A}\n\ntheorem lmul_algebraMap (x : R) : Algebra.lmul R A (algebraMap R A x) = Algebra.lsmul R R A x :=\n Eq.symm <| LinearMap.ext <| smul_def x\n#align algebra.lmul_algebra_map Algebra.lmul_algebraMap\n\nend Algebra\n\nnamespace IsScalarTower\n\nsection Semiring\n\nvariable [CommSemiring R] [CommSemiring S] [Semiring A]\n\nvariable [Algebra R S] [Algebra S A]\n\ninstance subalgebra (S₀ : Subalgebra R S) : IsScalarTower S₀ S A :=\n of_algebraMap_eq fun _ ↦ rfl\n#align is_scalar_tower.subalgebra IsScalarTower.subalgebra\n\nvariable [Algebra R A] [IsScalarTower R S A]\n\ninstance subalgebra' (S₀ : Subalgebra R S) : IsScalarTower R S₀ A :=\n @IsScalarTower.of_algebraMap_eq R S₀ A _ _ _ _ _ _ fun _ ↦\n (IsScalarTower.algebraMap_apply R S A _ : _)\n#align is_scalar_tower.subalgebra' IsScalarTower.subalgebra'\n\nend Semiring\n\nend IsScalarTower\n\nnamespace Subalgebra\n\nopen IsScalarTower\n\nsection Semiring\n\nvariable {S A B} [CommSemiring R] [CommSemiring S] [Semiring A] [Semiring B]\n\nvariable [Algebra R S] [Algebra S A] [Algebra R A] [Algebra S B] [Algebra R B]\n\nvariable [IsScalarTower R S A] [IsScalarTower R S B]\n\n/-- Given a tower `A / ↥U / S / R` of algebras, where `U` is an `S`-subalgebra of `A`, reinterpret\n`U` as an `R`-subalgebra of `A`. -/\ndef restrictScalars (U : Subalgebra S A) : Subalgebra R A :=\n { U with\n algebraMap_mem' := fun x ↦ by\n rw [algebraMap_apply R S A]\n exact U.algebraMap_mem _ }\n#align subalgebra.restrict_scalars Subalgebra.restrictScalars\n\n@[simp]\ntheorem coe_restrictScalars {U : Subalgebra S A} : (restrictScalars R U : Set A) = (U : Set A) :=\n rfl\n#align subalgebra.coe_restrict_scalars Subalgebra.coe_restrictScalars\n\n@[simp]\ntheorem restrictScalars_top : restrictScalars R (⊤ : Subalgebra S A) = ⊤ :=\n SetLike.coe_injective $ by dsimp -- porting note: why does `rfl` not work instead of `by dsimp`?\n#align subalgebra.restrict_scalars_top Subalgebra.restrictScalars_top\n\n@[simp]\ntheorem restrictScalars_toSubmodule {U : Subalgebra S A} :\n Subalgebra.toSubmodule (U.restrictScalars R) = U.toSubmodule.restrictScalars R :=\n SetLike.coe_injective rfl\n#align subalgebra.restrict_scalars_to_submodule Subalgebra.restrictScalars_toSubmodule\n\n@[simp]\ntheorem mem_restrictScalars {U : Subalgebra S A} {x : A} : x ∈ restrictScalars R U ↔ x ∈ U :=\n Iff.rfl\n#align subalgebra.mem_restrict_scalars Subalgebra.mem_restrictScalars\n\ntheorem restrictScalars_injective :\n Function.Injective (restrictScalars R : Subalgebra S A → Subalgebra R A) := fun U V H ↦\n ext fun x ↦ by rw [← mem_restrictScalars R, H, mem_restrictScalars]\n#align subalgebra.restrict_scalars_injective Subalgebra.restrictScalars_injective\n\n/-- Produces an `R`-algebra map from `U.restrictScalars R` given an `S`-algebra map from `U`.\n\nThis is a special case of `AlgHom.restrictScalars` that can be helpful in elaboration. -/\n@[simp]\ndef ofRestrictScalars (U : Subalgebra S A) (f : U →ₐ[S] B) : U.restrictScalars R →ₐ[R] B :=\n f.restrictScalars R\n#align subalgebra.of_restrict_scalars Subalgebra.ofRestrictScalars\n\nend Semiring\n\nsection CommSemiring\n\n@[simp]\nlemma range_isScalarTower_toAlgHom [CommSemiring R] [CommSemiring A]\n [Algebra R A] (S : Subalgebra R A) :\n LinearMap.range (IsScalarTower.toAlgHom R S A) = Subalgebra.toSubmodule S := by\n ext\n simp only [← Submodule.range_subtype (Subalgebra.toSubmodule S), LinearMap.mem_range,\n IsScalarTower.coe_toAlgHom', Subalgebra.mem_toSubmodule]\n rfl\n\nend CommSemiring\n\nend Subalgebra\n\nnamespace IsScalarTower\n\nopen Subalgebra\n\nvariable [CommSemiring R] [CommSemiring S] [CommSemiring A]\n\nvariable [Algebra R S] [Algebra S A] [Algebra R A] [IsScalarTower R S A]\n\ntheorem adjoin_range_toAlgHom (t : Set A) :\n (Algebra.adjoin (toAlgHom R S A).range t).restrictScalars R =\n (Algebra.adjoin S t).restrictScalars R :=\n Subalgebra.ext fun z ↦\n show z ∈ Subsemiring.closure (Set.range (algebraMap (toAlgHom R S A).range A) ∪ t : Set A) ↔\n z ∈ Subsemiring.closure (Set.range (algebraMap S A) ∪ t : Set A) by\n ","nextTactic":"suffices Set.range (algebraMap (toAlgHom R S A).range A) = Set.range (algebraMap S A) by\n rw [this]","declUpToTactic":"theorem adjoin_range_toAlgHom (t : Set A) :\n (Algebra.adjoin (toAlgHom R S A).range t).restrictScalars R =\n (Algebra.adjoin S t).restrictScalars R :=\n Subalgebra.ext fun z ↦\n show z ∈ Subsemiring.closure (Set.range (algebraMap (toAlgHom R S A).range A) ∪ t : Set A) ↔\n z ∈ Subsemiring.closure (Set.range (algebraMap S A) ∪ t : Set A) by\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Subalgebra_Tower.155_0.Zq8PWcMlDFAlf8P","decl":"theorem adjoin_range_toAlgHom (t : Set A) :\n (Algebra.adjoin (toAlgHom R S A).range t).restrictScalars R =\n (Algebra.adjoin S t).restrictScalars R "} | |
| {"state":"R : Type u\nS : Type v\nA : Type w\nB : Type u₁\nM : Type v₁\ninst✝⁶ : CommSemiring R\ninst✝⁵ : CommSemiring S\ninst✝⁴ : CommSemiring A\ninst✝³ : Algebra R S\ninst✝² : Algebra S A\ninst✝¹ : Algebra R A\ninst✝ : IsScalarTower R S A\nt : Set A\nz : A\nthis : Set.range ⇑(algebraMap (↥(AlgHom.range (toAlgHom R S A))) A) = Set.range ⇑(algebraMap S A)\n⊢ z ∈ Subsemiring.closure (Set.range ⇑(algebraMap (↥(AlgHom.range (toAlgHom R S A))) A) ∪ t) ↔\n z ∈ Subsemiring.closure (Set.range ⇑(algebraMap S A) ∪ t)","srcUpToTactic":"/-\nCopyright (c) 2020 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Anne Baanen\n-/\nimport Mathlib.Algebra.Algebra.Subalgebra.Basic\nimport Mathlib.Algebra.Algebra.Tower\n\n#align_import algebra.algebra.subalgebra.tower from \"leanprover-community/mathlib\"@\"a35ddf20601f85f78cd57e7f5b09ed528d71b7af\"\n\n/-!\n# Subalgebras in towers of algebras\n\nIn this file we prove facts about subalgebras in towers of algebra.\n\nAn algebra tower A/S/R is expressed by having instances of `Algebra A S`,\n`Algebra R S`, `Algebra R A` and `IsScalarTower R S A`, the later asserting the\ncompatibility condition `(r • s) • a = r • (s • a)`.\n\n## Main results\n\n * `IsScalarTower.Subalgebra`: if `A/S/R` is a tower and `S₀` is a subalgebra\n between `S` and `R`, then `A/S/S₀` is a tower\n * `IsScalarTower.Subalgebra'`: if `A/S/R` is a tower and `S₀` is a subalgebra\n between `S` and `R`, then `A/S₀/R` is a tower\n * `Subalgebra.restrictScalars`: turn an `S`-subalgebra of `A` into an `R`-subalgebra of `A`,\n given that `A/S/R` is a tower\n\n-/\n\n\nopen Pointwise\n\nuniverse u v w u₁ v₁\n\nvariable (R : Type u) (S : Type v) (A : Type w) (B : Type u₁) (M : Type v₁)\n\nnamespace Algebra\n\nvariable [CommSemiring R] [Semiring A] [Algebra R A]\n\nvariable [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M]\n\nvariable {A}\n\ntheorem lmul_algebraMap (x : R) : Algebra.lmul R A (algebraMap R A x) = Algebra.lsmul R R A x :=\n Eq.symm <| LinearMap.ext <| smul_def x\n#align algebra.lmul_algebra_map Algebra.lmul_algebraMap\n\nend Algebra\n\nnamespace IsScalarTower\n\nsection Semiring\n\nvariable [CommSemiring R] [CommSemiring S] [Semiring A]\n\nvariable [Algebra R S] [Algebra S A]\n\ninstance subalgebra (S₀ : Subalgebra R S) : IsScalarTower S₀ S A :=\n of_algebraMap_eq fun _ ↦ rfl\n#align is_scalar_tower.subalgebra IsScalarTower.subalgebra\n\nvariable [Algebra R A] [IsScalarTower R S A]\n\ninstance subalgebra' (S₀ : Subalgebra R S) : IsScalarTower R S₀ A :=\n @IsScalarTower.of_algebraMap_eq R S₀ A _ _ _ _ _ _ fun _ ↦\n (IsScalarTower.algebraMap_apply R S A _ : _)\n#align is_scalar_tower.subalgebra' IsScalarTower.subalgebra'\n\nend Semiring\n\nend IsScalarTower\n\nnamespace Subalgebra\n\nopen IsScalarTower\n\nsection Semiring\n\nvariable {S A B} [CommSemiring R] [CommSemiring S] [Semiring A] [Semiring B]\n\nvariable [Algebra R S] [Algebra S A] [Algebra R A] [Algebra S B] [Algebra R B]\n\nvariable [IsScalarTower R S A] [IsScalarTower R S B]\n\n/-- Given a tower `A / ↥U / S / R` of algebras, where `U` is an `S`-subalgebra of `A`, reinterpret\n`U` as an `R`-subalgebra of `A`. -/\ndef restrictScalars (U : Subalgebra S A) : Subalgebra R A :=\n { U with\n algebraMap_mem' := fun x ↦ by\n rw [algebraMap_apply R S A]\n exact U.algebraMap_mem _ }\n#align subalgebra.restrict_scalars Subalgebra.restrictScalars\n\n@[simp]\ntheorem coe_restrictScalars {U : Subalgebra S A} : (restrictScalars R U : Set A) = (U : Set A) :=\n rfl\n#align subalgebra.coe_restrict_scalars Subalgebra.coe_restrictScalars\n\n@[simp]\ntheorem restrictScalars_top : restrictScalars R (⊤ : Subalgebra S A) = ⊤ :=\n SetLike.coe_injective $ by dsimp -- porting note: why does `rfl` not work instead of `by dsimp`?\n#align subalgebra.restrict_scalars_top Subalgebra.restrictScalars_top\n\n@[simp]\ntheorem restrictScalars_toSubmodule {U : Subalgebra S A} :\n Subalgebra.toSubmodule (U.restrictScalars R) = U.toSubmodule.restrictScalars R :=\n SetLike.coe_injective rfl\n#align subalgebra.restrict_scalars_to_submodule Subalgebra.restrictScalars_toSubmodule\n\n@[simp]\ntheorem mem_restrictScalars {U : Subalgebra S A} {x : A} : x ∈ restrictScalars R U ↔ x ∈ U :=\n Iff.rfl\n#align subalgebra.mem_restrict_scalars Subalgebra.mem_restrictScalars\n\ntheorem restrictScalars_injective :\n Function.Injective (restrictScalars R : Subalgebra S A → Subalgebra R A) := fun U V H ↦\n ext fun x ↦ by rw [← mem_restrictScalars R, H, mem_restrictScalars]\n#align subalgebra.restrict_scalars_injective Subalgebra.restrictScalars_injective\n\n/-- Produces an `R`-algebra map from `U.restrictScalars R` given an `S`-algebra map from `U`.\n\nThis is a special case of `AlgHom.restrictScalars` that can be helpful in elaboration. -/\n@[simp]\ndef ofRestrictScalars (U : Subalgebra S A) (f : U →ₐ[S] B) : U.restrictScalars R →ₐ[R] B :=\n f.restrictScalars R\n#align subalgebra.of_restrict_scalars Subalgebra.ofRestrictScalars\n\nend Semiring\n\nsection CommSemiring\n\n@[simp]\nlemma range_isScalarTower_toAlgHom [CommSemiring R] [CommSemiring A]\n [Algebra R A] (S : Subalgebra R A) :\n LinearMap.range (IsScalarTower.toAlgHom R S A) = Subalgebra.toSubmodule S := by\n ext\n simp only [← Submodule.range_subtype (Subalgebra.toSubmodule S), LinearMap.mem_range,\n IsScalarTower.coe_toAlgHom', Subalgebra.mem_toSubmodule]\n rfl\n\nend CommSemiring\n\nend Subalgebra\n\nnamespace IsScalarTower\n\nopen Subalgebra\n\nvariable [CommSemiring R] [CommSemiring S] [CommSemiring A]\n\nvariable [Algebra R S] [Algebra S A] [Algebra R A] [IsScalarTower R S A]\n\ntheorem adjoin_range_toAlgHom (t : Set A) :\n (Algebra.adjoin (toAlgHom R S A).range t).restrictScalars R =\n (Algebra.adjoin S t).restrictScalars R :=\n Subalgebra.ext fun z ↦\n show z ∈ Subsemiring.closure (Set.range (algebraMap (toAlgHom R S A).range A) ∪ t : Set A) ↔\n z ∈ Subsemiring.closure (Set.range (algebraMap S A) ∪ t : Set A) by\n suffices Set.range (algebraMap (toAlgHom R S A).range A) = Set.range (algebraMap S A) by\n ","nextTactic":"rw [this]","declUpToTactic":"theorem adjoin_range_toAlgHom (t : Set A) :\n (Algebra.adjoin (toAlgHom R S A).range t).restrictScalars R =\n (Algebra.adjoin S t).restrictScalars R :=\n Subalgebra.ext fun z ↦\n show z ∈ Subsemiring.closure (Set.range (algebraMap (toAlgHom R S A).range A) ∪ t : Set A) ↔\n z ∈ Subsemiring.closure (Set.range (algebraMap S A) ∪ t : Set A) by\n suffices Set.range (algebraMap (toAlgHom R S A).range A) = Set.range (algebraMap S A) by\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Subalgebra_Tower.155_0.Zq8PWcMlDFAlf8P","decl":"theorem adjoin_range_toAlgHom (t : Set A) :\n (Algebra.adjoin (toAlgHom R S A).range t).restrictScalars R =\n (Algebra.adjoin S t).restrictScalars R "} | |
| {"state":"R : Type u\nS : Type v\nA : Type w\nB : Type u₁\nM : Type v₁\ninst✝⁶ : CommSemiring R\ninst✝⁵ : CommSemiring S\ninst✝⁴ : CommSemiring A\ninst✝³ : Algebra R S\ninst✝² : Algebra S A\ninst✝¹ : Algebra R A\ninst✝ : IsScalarTower R S A\nt : Set A\nz : A\n⊢ Set.range ⇑(algebraMap (↥(AlgHom.range (toAlgHom R S A))) A) = Set.range ⇑(algebraMap S A)","srcUpToTactic":"/-\nCopyright (c) 2020 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Anne Baanen\n-/\nimport Mathlib.Algebra.Algebra.Subalgebra.Basic\nimport Mathlib.Algebra.Algebra.Tower\n\n#align_import algebra.algebra.subalgebra.tower from \"leanprover-community/mathlib\"@\"a35ddf20601f85f78cd57e7f5b09ed528d71b7af\"\n\n/-!\n# Subalgebras in towers of algebras\n\nIn this file we prove facts about subalgebras in towers of algebra.\n\nAn algebra tower A/S/R is expressed by having instances of `Algebra A S`,\n`Algebra R S`, `Algebra R A` and `IsScalarTower R S A`, the later asserting the\ncompatibility condition `(r • s) • a = r • (s • a)`.\n\n## Main results\n\n * `IsScalarTower.Subalgebra`: if `A/S/R` is a tower and `S₀` is a subalgebra\n between `S` and `R`, then `A/S/S₀` is a tower\n * `IsScalarTower.Subalgebra'`: if `A/S/R` is a tower and `S₀` is a subalgebra\n between `S` and `R`, then `A/S₀/R` is a tower\n * `Subalgebra.restrictScalars`: turn an `S`-subalgebra of `A` into an `R`-subalgebra of `A`,\n given that `A/S/R` is a tower\n\n-/\n\n\nopen Pointwise\n\nuniverse u v w u₁ v₁\n\nvariable (R : Type u) (S : Type v) (A : Type w) (B : Type u₁) (M : Type v₁)\n\nnamespace Algebra\n\nvariable [CommSemiring R] [Semiring A] [Algebra R A]\n\nvariable [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M]\n\nvariable {A}\n\ntheorem lmul_algebraMap (x : R) : Algebra.lmul R A (algebraMap R A x) = Algebra.lsmul R R A x :=\n Eq.symm <| LinearMap.ext <| smul_def x\n#align algebra.lmul_algebra_map Algebra.lmul_algebraMap\n\nend Algebra\n\nnamespace IsScalarTower\n\nsection Semiring\n\nvariable [CommSemiring R] [CommSemiring S] [Semiring A]\n\nvariable [Algebra R S] [Algebra S A]\n\ninstance subalgebra (S₀ : Subalgebra R S) : IsScalarTower S₀ S A :=\n of_algebraMap_eq fun _ ↦ rfl\n#align is_scalar_tower.subalgebra IsScalarTower.subalgebra\n\nvariable [Algebra R A] [IsScalarTower R S A]\n\ninstance subalgebra' (S₀ : Subalgebra R S) : IsScalarTower R S₀ A :=\n @IsScalarTower.of_algebraMap_eq R S₀ A _ _ _ _ _ _ fun _ ↦\n (IsScalarTower.algebraMap_apply R S A _ : _)\n#align is_scalar_tower.subalgebra' IsScalarTower.subalgebra'\n\nend Semiring\n\nend IsScalarTower\n\nnamespace Subalgebra\n\nopen IsScalarTower\n\nsection Semiring\n\nvariable {S A B} [CommSemiring R] [CommSemiring S] [Semiring A] [Semiring B]\n\nvariable [Algebra R S] [Algebra S A] [Algebra R A] [Algebra S B] [Algebra R B]\n\nvariable [IsScalarTower R S A] [IsScalarTower R S B]\n\n/-- Given a tower `A / ↥U / S / R` of algebras, where `U` is an `S`-subalgebra of `A`, reinterpret\n`U` as an `R`-subalgebra of `A`. -/\ndef restrictScalars (U : Subalgebra S A) : Subalgebra R A :=\n { U with\n algebraMap_mem' := fun x ↦ by\n rw [algebraMap_apply R S A]\n exact U.algebraMap_mem _ }\n#align subalgebra.restrict_scalars Subalgebra.restrictScalars\n\n@[simp]\ntheorem coe_restrictScalars {U : Subalgebra S A} : (restrictScalars R U : Set A) = (U : Set A) :=\n rfl\n#align subalgebra.coe_restrict_scalars Subalgebra.coe_restrictScalars\n\n@[simp]\ntheorem restrictScalars_top : restrictScalars R (⊤ : Subalgebra S A) = ⊤ :=\n SetLike.coe_injective $ by dsimp -- porting note: why does `rfl` not work instead of `by dsimp`?\n#align subalgebra.restrict_scalars_top Subalgebra.restrictScalars_top\n\n@[simp]\ntheorem restrictScalars_toSubmodule {U : Subalgebra S A} :\n Subalgebra.toSubmodule (U.restrictScalars R) = U.toSubmodule.restrictScalars R :=\n SetLike.coe_injective rfl\n#align subalgebra.restrict_scalars_to_submodule Subalgebra.restrictScalars_toSubmodule\n\n@[simp]\ntheorem mem_restrictScalars {U : Subalgebra S A} {x : A} : x ∈ restrictScalars R U ↔ x ∈ U :=\n Iff.rfl\n#align subalgebra.mem_restrict_scalars Subalgebra.mem_restrictScalars\n\ntheorem restrictScalars_injective :\n Function.Injective (restrictScalars R : Subalgebra S A → Subalgebra R A) := fun U V H ↦\n ext fun x ↦ by rw [← mem_restrictScalars R, H, mem_restrictScalars]\n#align subalgebra.restrict_scalars_injective Subalgebra.restrictScalars_injective\n\n/-- Produces an `R`-algebra map from `U.restrictScalars R` given an `S`-algebra map from `U`.\n\nThis is a special case of `AlgHom.restrictScalars` that can be helpful in elaboration. -/\n@[simp]\ndef ofRestrictScalars (U : Subalgebra S A) (f : U →ₐ[S] B) : U.restrictScalars R →ₐ[R] B :=\n f.restrictScalars R\n#align subalgebra.of_restrict_scalars Subalgebra.ofRestrictScalars\n\nend Semiring\n\nsection CommSemiring\n\n@[simp]\nlemma range_isScalarTower_toAlgHom [CommSemiring R] [CommSemiring A]\n [Algebra R A] (S : Subalgebra R A) :\n LinearMap.range (IsScalarTower.toAlgHom R S A) = Subalgebra.toSubmodule S := by\n ext\n simp only [← Submodule.range_subtype (Subalgebra.toSubmodule S), LinearMap.mem_range,\n IsScalarTower.coe_toAlgHom', Subalgebra.mem_toSubmodule]\n rfl\n\nend CommSemiring\n\nend Subalgebra\n\nnamespace IsScalarTower\n\nopen Subalgebra\n\nvariable [CommSemiring R] [CommSemiring S] [CommSemiring A]\n\nvariable [Algebra R S] [Algebra S A] [Algebra R A] [IsScalarTower R S A]\n\ntheorem adjoin_range_toAlgHom (t : Set A) :\n (Algebra.adjoin (toAlgHom R S A).range t).restrictScalars R =\n (Algebra.adjoin S t).restrictScalars R :=\n Subalgebra.ext fun z ↦\n show z ∈ Subsemiring.closure (Set.range (algebraMap (toAlgHom R S A).range A) ∪ t : Set A) ↔\n z ∈ Subsemiring.closure (Set.range (algebraMap S A) ∪ t : Set A) by\n suffices Set.range (algebraMap (toAlgHom R S A).range A) = Set.range (algebraMap S A) by\n rw [this]\n ","nextTactic":"ext z","declUpToTactic":"theorem adjoin_range_toAlgHom (t : Set A) :\n (Algebra.adjoin (toAlgHom R S A).range t).restrictScalars R =\n (Algebra.adjoin S t).restrictScalars R :=\n Subalgebra.ext fun z ↦\n show z ∈ Subsemiring.closure (Set.range (algebraMap (toAlgHom R S A).range A) ∪ t : Set A) ↔\n z ∈ Subsemiring.closure (Set.range (algebraMap S A) ∪ t : Set A) by\n suffices Set.range (algebraMap (toAlgHom R S A).range A) = Set.range (algebraMap S A) by\n rw [this]\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Subalgebra_Tower.155_0.Zq8PWcMlDFAlf8P","decl":"theorem adjoin_range_toAlgHom (t : Set A) :\n (Algebra.adjoin (toAlgHom R S A).range t).restrictScalars R =\n (Algebra.adjoin S t).restrictScalars R "} | |
| {"state":"case h\nR : Type u\nS : Type v\nA : Type w\nB : Type u₁\nM : Type v₁\ninst✝⁶ : CommSemiring R\ninst✝⁵ : CommSemiring S\ninst✝⁴ : CommSemiring A\ninst✝³ : Algebra R S\ninst✝² : Algebra S A\ninst✝¹ : Algebra R A\ninst✝ : IsScalarTower R S A\nt : Set A\nz✝ z : A\n⊢ z ∈ Set.range ⇑(algebraMap (↥(AlgHom.range (toAlgHom R S A))) A) ↔ z ∈ Set.range ⇑(algebraMap S A)","srcUpToTactic":"/-\nCopyright (c) 2020 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Anne Baanen\n-/\nimport Mathlib.Algebra.Algebra.Subalgebra.Basic\nimport Mathlib.Algebra.Algebra.Tower\n\n#align_import algebra.algebra.subalgebra.tower from \"leanprover-community/mathlib\"@\"a35ddf20601f85f78cd57e7f5b09ed528d71b7af\"\n\n/-!\n# Subalgebras in towers of algebras\n\nIn this file we prove facts about subalgebras in towers of algebra.\n\nAn algebra tower A/S/R is expressed by having instances of `Algebra A S`,\n`Algebra R S`, `Algebra R A` and `IsScalarTower R S A`, the later asserting the\ncompatibility condition `(r • s) • a = r • (s • a)`.\n\n## Main results\n\n * `IsScalarTower.Subalgebra`: if `A/S/R` is a tower and `S₀` is a subalgebra\n between `S` and `R`, then `A/S/S₀` is a tower\n * `IsScalarTower.Subalgebra'`: if `A/S/R` is a tower and `S₀` is a subalgebra\n between `S` and `R`, then `A/S₀/R` is a tower\n * `Subalgebra.restrictScalars`: turn an `S`-subalgebra of `A` into an `R`-subalgebra of `A`,\n given that `A/S/R` is a tower\n\n-/\n\n\nopen Pointwise\n\nuniverse u v w u₁ v₁\n\nvariable (R : Type u) (S : Type v) (A : Type w) (B : Type u₁) (M : Type v₁)\n\nnamespace Algebra\n\nvariable [CommSemiring R] [Semiring A] [Algebra R A]\n\nvariable [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M]\n\nvariable {A}\n\ntheorem lmul_algebraMap (x : R) : Algebra.lmul R A (algebraMap R A x) = Algebra.lsmul R R A x :=\n Eq.symm <| LinearMap.ext <| smul_def x\n#align algebra.lmul_algebra_map Algebra.lmul_algebraMap\n\nend Algebra\n\nnamespace IsScalarTower\n\nsection Semiring\n\nvariable [CommSemiring R] [CommSemiring S] [Semiring A]\n\nvariable [Algebra R S] [Algebra S A]\n\ninstance subalgebra (S₀ : Subalgebra R S) : IsScalarTower S₀ S A :=\n of_algebraMap_eq fun _ ↦ rfl\n#align is_scalar_tower.subalgebra IsScalarTower.subalgebra\n\nvariable [Algebra R A] [IsScalarTower R S A]\n\ninstance subalgebra' (S₀ : Subalgebra R S) : IsScalarTower R S₀ A :=\n @IsScalarTower.of_algebraMap_eq R S₀ A _ _ _ _ _ _ fun _ ↦\n (IsScalarTower.algebraMap_apply R S A _ : _)\n#align is_scalar_tower.subalgebra' IsScalarTower.subalgebra'\n\nend Semiring\n\nend IsScalarTower\n\nnamespace Subalgebra\n\nopen IsScalarTower\n\nsection Semiring\n\nvariable {S A B} [CommSemiring R] [CommSemiring S] [Semiring A] [Semiring B]\n\nvariable [Algebra R S] [Algebra S A] [Algebra R A] [Algebra S B] [Algebra R B]\n\nvariable [IsScalarTower R S A] [IsScalarTower R S B]\n\n/-- Given a tower `A / ↥U / S / R` of algebras, where `U` is an `S`-subalgebra of `A`, reinterpret\n`U` as an `R`-subalgebra of `A`. -/\ndef restrictScalars (U : Subalgebra S A) : Subalgebra R A :=\n { U with\n algebraMap_mem' := fun x ↦ by\n rw [algebraMap_apply R S A]\n exact U.algebraMap_mem _ }\n#align subalgebra.restrict_scalars Subalgebra.restrictScalars\n\n@[simp]\ntheorem coe_restrictScalars {U : Subalgebra S A} : (restrictScalars R U : Set A) = (U : Set A) :=\n rfl\n#align subalgebra.coe_restrict_scalars Subalgebra.coe_restrictScalars\n\n@[simp]\ntheorem restrictScalars_top : restrictScalars R (⊤ : Subalgebra S A) = ⊤ :=\n SetLike.coe_injective $ by dsimp -- porting note: why does `rfl` not work instead of `by dsimp`?\n#align subalgebra.restrict_scalars_top Subalgebra.restrictScalars_top\n\n@[simp]\ntheorem restrictScalars_toSubmodule {U : Subalgebra S A} :\n Subalgebra.toSubmodule (U.restrictScalars R) = U.toSubmodule.restrictScalars R :=\n SetLike.coe_injective rfl\n#align subalgebra.restrict_scalars_to_submodule Subalgebra.restrictScalars_toSubmodule\n\n@[simp]\ntheorem mem_restrictScalars {U : Subalgebra S A} {x : A} : x ∈ restrictScalars R U ↔ x ∈ U :=\n Iff.rfl\n#align subalgebra.mem_restrict_scalars Subalgebra.mem_restrictScalars\n\ntheorem restrictScalars_injective :\n Function.Injective (restrictScalars R : Subalgebra S A → Subalgebra R A) := fun U V H ↦\n ext fun x ↦ by rw [← mem_restrictScalars R, H, mem_restrictScalars]\n#align subalgebra.restrict_scalars_injective Subalgebra.restrictScalars_injective\n\n/-- Produces an `R`-algebra map from `U.restrictScalars R` given an `S`-algebra map from `U`.\n\nThis is a special case of `AlgHom.restrictScalars` that can be helpful in elaboration. -/\n@[simp]\ndef ofRestrictScalars (U : Subalgebra S A) (f : U →ₐ[S] B) : U.restrictScalars R →ₐ[R] B :=\n f.restrictScalars R\n#align subalgebra.of_restrict_scalars Subalgebra.ofRestrictScalars\n\nend Semiring\n\nsection CommSemiring\n\n@[simp]\nlemma range_isScalarTower_toAlgHom [CommSemiring R] [CommSemiring A]\n [Algebra R A] (S : Subalgebra R A) :\n LinearMap.range (IsScalarTower.toAlgHom R S A) = Subalgebra.toSubmodule S := by\n ext\n simp only [← Submodule.range_subtype (Subalgebra.toSubmodule S), LinearMap.mem_range,\n IsScalarTower.coe_toAlgHom', Subalgebra.mem_toSubmodule]\n rfl\n\nend CommSemiring\n\nend Subalgebra\n\nnamespace IsScalarTower\n\nopen Subalgebra\n\nvariable [CommSemiring R] [CommSemiring S] [CommSemiring A]\n\nvariable [Algebra R S] [Algebra S A] [Algebra R A] [IsScalarTower R S A]\n\ntheorem adjoin_range_toAlgHom (t : Set A) :\n (Algebra.adjoin (toAlgHom R S A).range t).restrictScalars R =\n (Algebra.adjoin S t).restrictScalars R :=\n Subalgebra.ext fun z ↦\n show z ∈ Subsemiring.closure (Set.range (algebraMap (toAlgHom R S A).range A) ∪ t : Set A) ↔\n z ∈ Subsemiring.closure (Set.range (algebraMap S A) ∪ t : Set A) by\n suffices Set.range (algebraMap (toAlgHom R S A).range A) = Set.range (algebraMap S A) by\n rw [this]\n ext z\n ","nextTactic":"exact ⟨fun ⟨⟨_, y, h1⟩, h2⟩ ↦ ⟨y, h2 ▸ h1⟩, fun ⟨y, hy⟩ ↦ ⟨⟨z, y, hy⟩, rfl⟩⟩","declUpToTactic":"theorem adjoin_range_toAlgHom (t : Set A) :\n (Algebra.adjoin (toAlgHom R S A).range t).restrictScalars R =\n (Algebra.adjoin S t).restrictScalars R :=\n Subalgebra.ext fun z ↦\n show z ∈ Subsemiring.closure (Set.range (algebraMap (toAlgHom R S A).range A) ∪ t : Set A) ↔\n z ∈ Subsemiring.closure (Set.range (algebraMap S A) ∪ t : Set A) by\n suffices Set.range (algebraMap (toAlgHom R S A).range A) = Set.range (algebraMap S A) by\n rw [this]\n ext z\n ","declId":"Examples.Mathlib.SplitRw.Mathlib_Algebra_Algebra_Subalgebra_Tower.155_0.Zq8PWcMlDFAlf8P","decl":"theorem adjoin_range_toAlgHom (t : Set A) :\n (Algebra.adjoin (toAlgHom R S A).range t).restrictScalars R =\n (Algebra.adjoin S t).restrictScalars R "} | |