internlm/Lean-Github huggingface dataset:
Theorem: TensorProduct.Algebra.eq_of_fg_of_subtype_eq
Proof state before:
R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
A : Subalgebra R S
hA : A.FG
t t' : ↥A ⊗[R] N
h : (LinearMap.rTensor N A.val.toLinearMap) t = (LinearMap.rTensor N A.val.toLinearMap) t'
P : Submodule R ↥A
hP : P.FG
u : ↥P ⊗[R] N
hu : (LinearMap.rTensor N P.subtype) u = t
P' : Submodule R ↥A
hP' : P'.FG
u' : ↥P' ⊗[R] N
hu' : (LinearMap.rTensor N P'.subtype) u' = t'
P₁ : Submodule R S := Submodule.map (Subalgebra.toSubmodule A).subtype (P ⊔ P')
hP₁ : P₁.FG
j : ↥P →ₗ[R] ↥P₁ := (Subalgebra.toSubmodule A).subtype.restrict ⋯
j' : ↥P' →ₗ[R] ↥P₁ := (Subalgebra.toSubmodule A).subtype.restrict ⋯
u₁ : ↥P₁ ⊗[R] N := (LinearMap.rTensor N j) u
hu₁ : u₁ = (LinearMap.rTensor N j) u
u'₁ : ↥P₁ ⊗[R] N := (LinearMap.rTensor N j') u'
hu'₁ : u'₁ = (LinearMap.rTensor N j') u'
hj₁ : P₁.subtype ∘ₗ j = A.val.toLinearMap ∘ₗ P.subtype
hj'₁ : P₁.subtype ∘ₗ j' = A.val.toLinearMap ∘ₗ P'.subtype
⊢ (LinearMap.rTensor N (P₁.subtype ∘ₗ j)) u = (LinearMap.rTensor N (P₁.subtype ∘ₗ j')) u'
Tactic:
rw [hj₁, hj'₁]
Proof state after:
R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
A : Subalgebra R S
hA : A.FG
t t' : ↥A ⊗[R] N
h : (LinearMap.rTensor N A.val.toLinearMap) t = (LinearMap.rTensor N A.val.toLinearMap) t'
P : Submodule R ↥A
hP : P.FG
u : ↥P ⊗[R] N
hu : (LinearMap.rTensor N P.subtype) u = t
P' : Submodule R ↥A
hP' : P'.FG
u' : ↥P' ⊗[R] N
hu' : (LinearMap.rTensor N P'.subtype) u' = t'
P₁ : Submodule R S := Submodule.map (Subalgebra.toSubmodule A).subtype (P ⊔ P')
hP₁ : P₁.FG
j : ↥P →ₗ[R] ↥P₁ := (Subalgebra.toSubmodule A).subtype.restrict ⋯
j' : ↥P' →ₗ[R] ↥P₁ := (Subalgebra.toSubmodule A).subtype.restrict ⋯
u₁ : ↥P₁ ⊗[R] N := (LinearMap.rTensor N j) u
hu₁ : u₁ = (LinearMap.rTensor N j) u
u'₁ : ↥P₁ ⊗[R] N := (LinearMap.rTensor N j') u'
hu'₁ : u'₁ = (LinearMap.rTensor N j') u'
hj₁ : P₁.subtype ∘ₗ j = A.val.toLinearMap ∘ₗ P.subtype
hj'₁ : P₁.subtype ∘ₗ j' = A.val.toLinearMap ∘ₗ P'.subtype
⊢ (LinearMap.rTensor N (A.val.toLinearMap ∘ₗ P.subtype)) u = (LinearMap.rTensor N (A.val.toLinearMap ∘ₗ P'.subtype)) u'
Theorem: TensorProduct.Algebra.eq_of_fg_of_subtype_eq
Proof state before:
R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
A : Subalgebra R S
hA : A.FG
t t' : ↥A ⊗[R] N
h : (LinearMap.rTensor N A.val.toLinearMap) t = (LinearMap.rTensor N A.val.toLinearMap) t'
P : Submodule R ↥A
hP : P.FG
u : ↥P ⊗[R] N
hu : (LinearMap.rTensor N P.subtype) u = t
P' : Submodule R ↥A
hP' : P'.FG
u' : ↥P' ⊗[R] N
hu' : (LinearMap.rTensor N P'.subtype) u' = t'
P₁ : Submodule R S := Submodule.map (Subalgebra.toSubmodule A).subtype (P ⊔ P')
hP₁ : P₁.FG
j : ↥P →ₗ[R] ↥P₁ := (Subalgebra.toSubmodule A).subtype.restrict ⋯
j' : ↥P' →ₗ[R] ↥P₁ := (Subalgebra.toSubmodule A).subtype.restrict ⋯
u₁ : ↥P₁ ⊗[R] N := (LinearMap.rTensor N j) u
hu₁ : u₁ = (LinearMap.rTensor N j) u
u'₁ : ↥P₁ ⊗[R] N := (LinearMap.rTensor N j') u'
hu'₁ : u'₁ = (LinearMap.rTensor N j') u'
hj₁ : P₁.subtype ∘ₗ j = A.val.toLinearMap ∘ₗ P.subtype
hj'₁ : P₁.subtype ∘ₗ j' = A.val.toLinearMap ∘ₗ P'.subtype
⊢ (LinearMap.rTensor N (A.val.toLinearMap ∘ₗ P.subtype)) u = (LinearMap.rTensor N (A.val.toLinearMap ∘ₗ P'.subtype)) u'
Tactic:
simp only [LinearMap.rTensor_comp, LinearMap.comp_apply]
Proof state after:
R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
A : Subalgebra R S
hA : A.FG
t t' : ↥A ⊗[R] N
h : (LinearMap.rTensor N A.val.toLinearMap) t = (LinearMap.rTensor N A.val.toLinearMap) t'
P : Submodule R ↥A
hP : P.FG
u : ↥P ⊗[R] N
hu : (LinearMap.rTensor N P.subtype) u = t
P' : Submodule R ↥A
hP' : P'.FG
u' : ↥P' ⊗[R] N
hu' : (LinearMap.rTensor N P'.subtype) u' = t'
P₁ : Submodule R S := Submodule.map (Subalgebra.toSubmodule A).subtype (P ⊔ P')
hP₁ : P₁.FG
j : ↥P →ₗ[R] ↥P₁ := (Subalgebra.toSubmodule A).subtype.restrict ⋯
j' : ↥P' →ₗ[R] ↥P₁ := (Subalgebra.toSubmodule A).subtype.restrict ⋯
u₁ : ↥P₁ ⊗[R] N := (LinearMap.rTensor N j) u
hu₁ : u₁ = (LinearMap.rTensor N j) u
u'₁ : ↥P₁ ⊗[R] N := (LinearMap.rTensor N j') u'
hu'₁ : u'₁ = (LinearMap.rTensor N j') u'
hj₁ : P₁.subtype ∘ₗ j = A.val.toLinearMap ∘ₗ P.subtype
hj'₁ : P₁.subtype ∘ₗ j' = A.val.toLinearMap ∘ₗ P'.subtype
⊢ (LinearMap.rTensor N A.val.toLinearMap) ((LinearMap.rTensor N P.subtype) u) =
    (LinearMap.rTensor N A.val.toLinearMap) ((LinearMap.rTensor N P'.subtype) u')
Theorem: TensorProduct.Algebra.eq_of_fg_of_subtype_eq
Proof state before:
R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
A : Subalgebra R S
hA : A.FG
t t' : ↥A ⊗[R] N
h : (LinearMap.rTensor N A.val.toLinearMap) t = (LinearMap.rTensor N A.val.toLinearMap) t'
P : Submodule R ↥A
hP : P.FG
u : ↥P ⊗[R] N
hu : (LinearMap.rTensor N P.subtype) u = t
P' : Submodule R ↥A
hP' : P'.FG
u' : ↥P' ⊗[R] N
hu' : (LinearMap.rTensor N P'.subtype) u' = t'
P₁ : Submodule R S := Submodule.map (Subalgebra.toSubmodule A).subtype (P ⊔ P')
hP₁ : P₁.FG
j : ↥P →ₗ[R] ↥P₁ := (Subalgebra.toSubmodule A).subtype.restrict ⋯
j' : ↥P' →ₗ[R] ↥P₁ := (Subalgebra.toSubmodule A).subtype.restrict ⋯
u₁ : ↥P₁ ⊗[R] N := (LinearMap.rTensor N j) u
hu₁ : u₁ = (LinearMap.rTensor N j) u
u'₁ : ↥P₁ ⊗[R] N := (LinearMap.rTensor N j') u'
hu'₁ : u'₁ = (LinearMap.rTensor N j') u'
hj₁ : P₁.subtype ∘ₗ j = A.val.toLinearMap ∘ₗ P.subtype
hj'₁ : P₁.subtype ∘ₗ j' = A.val.toLinearMap ∘ₗ P'.subtype
⊢ (LinearMap.rTensor N A.val.toLinearMap) ((LinearMap.rTensor N P.subtype) u) =
    (LinearMap.rTensor N A.val.toLinearMap) ((LinearMap.rTensor N P'.subtype) u')
Tactic:
rw [hu, hu', h]
Proof state after:
no goals
Theorem: TensorProduct.Algebra.eq_of_fg_of_subtype_eq
Proof state before:
R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
A : Subalgebra R S
hA : A.FG
t t' : ↥A ⊗[R] N
h : (LinearMap.rTensor N A.val.toLinearMap) t = (LinearMap.rTensor N A.val.toLinearMap) t'
P : Submodule R ↥A
hP : P.FG
u : ↥P ⊗[R] N
hu : (LinearMap.rTensor N P.subtype) u = t
P' : Submodule R ↥A
hP' : P'.FG
u' : ↥P' ⊗[R] N
hu' : (LinearMap.rTensor N P'.subtype) u' = t'
P₁ : Submodule R S := Submodule.map (Subalgebra.toSubmodule A).subtype (P ⊔ P')
hP₁ : P₁.FG
j : ↥P →ₗ[R] ↥P₁ := (Subalgebra.toSubmodule A).subtype.restrict ⋯
j' : ↥P' →ₗ[R] ↥P₁ := (Subalgebra.toSubmodule A).subtype.restrict ⋯
u₁ : ↥P₁ ⊗[R] N := (LinearMap.rTensor N j) u
hu₁ : u₁ = (LinearMap.rTensor N j) u
u'₁ : ↥P₁ ⊗[R] N := (LinearMap.rTensor N j') u'
hu'₁ : u'₁ = (LinearMap.rTensor N j') u'
⊢ P₁.subtype ∘ₗ j = A.val.toLinearMap ∘ₗ P.subtype
Tactic:
ext
Proof state after:
case h
R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
A : Subalgebra R S
hA : A.FG
t t' : ↥A ⊗[R] N
h : (LinearMap.rTensor N A.val.toLinearMap) t = (LinearMap.rTensor N A.val.toLinearMap) t'
P : Submodule R ↥A
hP : P.FG
u : ↥P ⊗[R] N
hu : (LinearMap.rTensor N P.subtype) u = t
P' : Submodule R ↥A
hP' : P'.FG
u' : ↥P' ⊗[R] N
hu' : (LinearMap.rTensor N P'.subtype) u' = t'
P₁ : Submodule R S := Submodule.map (Subalgebra.toSubmodule A).subtype (P ⊔ P')
hP₁ : P₁.FG
j : ↥P →ₗ[R] ↥P₁ := (Subalgebra.toSubmodule A).subtype.restrict ⋯
j' : ↥P' →ₗ[R] ↥P₁ := (Subalgebra.toSubmodule A).subtype.restrict ⋯
u₁ : ↥P₁ ⊗[R] N := (LinearMap.rTensor N j) u
hu₁ : u₁ = (LinearMap.rTensor N j) u
u'₁ : ↥P₁ ⊗[R] N := (LinearMap.rTensor N j') u'
hu'₁ : u'₁ = (LinearMap.rTensor N j') u'
x✝ : ↥P
⊢ (P₁.subtype ∘ₗ j) x✝ = (A.val.toLinearMap ∘ₗ P.subtype) x✝
Theorem: TensorProduct.Algebra.eq_of_fg_of_subtype_eq
Proof state before:
case h
R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
A : Subalgebra R S
hA : A.FG
t t' : ↥A ⊗[R] N
h : (LinearMap.rTensor N A.val.toLinearMap) t = (LinearMap.rTensor N A.val.toLinearMap) t'
P : Submodule R ↥A
hP : P.FG
u : ↥P ⊗[R] N
hu : (LinearMap.rTensor N P.subtype) u = t
P' : Submodule R ↥A
hP' : P'.FG
u' : ↥P' ⊗[R] N
hu' : (LinearMap.rTensor N P'.subtype) u' = t'
P₁ : Submodule R S := Submodule.map (Subalgebra.toSubmodule A).subtype (P ⊔ P')
hP₁ : P₁.FG
j : ↥P →ₗ[R] ↥P₁ := (Subalgebra.toSubmodule A).subtype.restrict ⋯
j' : ↥P' →ₗ[R] ↥P₁ := (Subalgebra.toSubmodule A).subtype.restrict ⋯
u₁ : ↥P₁ ⊗[R] N := (LinearMap.rTensor N j) u
hu₁ : u₁ = (LinearMap.rTensor N j) u
u'₁ : ↥P₁ ⊗[R] N := (LinearMap.rTensor N j') u'
hu'₁ : u'₁ = (LinearMap.rTensor N j') u'
x✝ : ↥P
⊢ (P₁.subtype ∘ₗ j) x✝ = (A.val.toLinearMap ∘ₗ P.subtype) x✝
Tactic:
rfl
Proof state after:
no goals
Theorem: TensorProduct.Algebra.eq_of_fg_of_subtype_eq
Proof state before:
R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
A : Subalgebra R S
hA : A.FG
t t' : ↥A ⊗[R] N
h : (LinearMap.rTensor N A.val.toLinearMap) t = (LinearMap.rTensor N A.val.toLinearMap) t'
P : Submodule R ↥A
hP : P.FG
u : ↥P ⊗[R] N
hu : (LinearMap.rTensor N P.subtype) u = t
P' : Submodule R ↥A
hP' : P'.FG
u' : ↥P' ⊗[R] N
hu' : (LinearMap.rTensor N P'.subtype) u' = t'
P₁ : Submodule R S := Submodule.map (Subalgebra.toSubmodule A).subtype (P ⊔ P')
hP₁ : P₁.FG
j : ↥P →ₗ[R] ↥P₁ := (Subalgebra.toSubmodule A).subtype.restrict ⋯
j' : ↥P' →ₗ[R] ↥P₁ := (Subalgebra.toSubmodule A).subtype.restrict ⋯
u₁ : ↥P₁ ⊗[R] N := (LinearMap.rTensor N j) u
hu₁ : u₁ = (LinearMap.rTensor N j) u
u'₁ : ↥P₁ ⊗[R] N := (LinearMap.rTensor N j') u'
hu'₁ : u'₁ = (LinearMap.rTensor N j') u'
hj₁ : P₁.subtype ∘ₗ j = A.val.toLinearMap ∘ₗ P.subtype
⊢ P₁.subtype ∘ₗ j' = A.val.toLinearMap ∘ₗ P'.subtype
Tactic:
ext
Proof state after:
case h
R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
A : Subalgebra R S
hA : A.FG
t t' : ↥A ⊗[R] N
h : (LinearMap.rTensor N A.val.toLinearMap) t = (LinearMap.rTensor N A.val.toLinearMap) t'
P : Submodule R ↥A
hP : P.FG
u : ↥P ⊗[R] N
hu : (LinearMap.rTensor N P.subtype) u = t
P' : Submodule R ↥A
hP' : P'.FG
u' : ↥P' ⊗[R] N
hu' : (LinearMap.rTensor N P'.subtype) u' = t'
P₁ : Submodule R S := Submodule.map (Subalgebra.toSubmodule A).subtype (P ⊔ P')
hP₁ : P₁.FG
j : ↥P →ₗ[R] ↥P₁ := (Subalgebra.toSubmodule A).subtype.restrict ⋯
j' : ↥P' →ₗ[R] ↥P₁ := (Subalgebra.toSubmodule A).subtype.restrict ⋯
u₁ : ↥P₁ ⊗[R] N := (LinearMap.rTensor N j) u
hu₁ : u₁ = (LinearMap.rTensor N j) u
u'₁ : ↥P₁ ⊗[R] N := (LinearMap.rTensor N j') u'
hu'₁ : u'₁ = (LinearMap.rTensor N j') u'
hj₁ : P₁.subtype ∘ₗ j = A.val.toLinearMap ∘ₗ P.subtype
x✝ : ↥P'
⊢ (P₁.subtype ∘ₗ j') x✝ = (A.val.toLinearMap ∘ₗ P'.subtype) x✝
Theorem: TensorProduct.Algebra.eq_of_fg_of_subtype_eq
Proof state before:
case h
R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
A : Subalgebra R S
hA : A.FG
t t' : ↥A ⊗[R] N
h : (LinearMap.rTensor N A.val.toLinearMap) t = (LinearMap.rTensor N A.val.toLinearMap) t'
P : Submodule R ↥A
hP : P.FG
u : ↥P ⊗[R] N
hu : (LinearMap.rTensor N P.subtype) u = t
P' : Submodule R ↥A
hP' : P'.FG
u' : ↥P' ⊗[R] N
hu' : (LinearMap.rTensor N P'.subtype) u' = t'
P₁ : Submodule R S := Submodule.map (Subalgebra.toSubmodule A).subtype (P ⊔ P')
hP₁ : P₁.FG
j : ↥P →ₗ[R] ↥P₁ := (Subalgebra.toSubmodule A).subtype.restrict ⋯
j' : ↥P' →ₗ[R] ↥P₁ := (Subalgebra.toSubmodule A).subtype.restrict ⋯
u₁ : ↥P₁ ⊗[R] N := (LinearMap.rTensor N j) u
hu₁ : u₁ = (LinearMap.rTensor N j) u
u'₁ : ↥P₁ ⊗[R] N := (LinearMap.rTensor N j') u'
hu'₁ : u'₁ = (LinearMap.rTensor N j') u'
hj₁ : P₁.subtype ∘ₗ j = A.val.toLinearMap ∘ₗ P.subtype
x✝ : ↥P'
⊢ (P₁.subtype ∘ₗ j') x✝ = (A.val.toLinearMap ∘ₗ P'.subtype) x✝
Tactic:
rfl
Proof state after:
no goals
Theorem: TensorProduct.Algebra.eq_of_fg_of_subtype_eq
Proof state before:
R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
A : Subalgebra R S
hA : A.FG
t t' : ↥A ⊗[R] N
h✝ : (LinearMap.rTensor N A.val.toLinearMap) t = (LinearMap.rTensor N A.val.toLinearMap) t'
P : Submodule R ↥A
hP : P.FG
u : ↥P ⊗[R] N
hu : (LinearMap.rTensor N P.subtype) u = t
P' : Submodule R ↥A
hP' : P'.FG
u' : ↥P' ⊗[R] N
hu' : (LinearMap.rTensor N P'.subtype) u' = t'
P₁ : Submodule R S := Submodule.map (Subalgebra.toSubmodule A).subtype (P ⊔ P')
hP₁ : P₁.FG
j : ↥P →ₗ[R] ↥P₁ := (Subalgebra.toSubmodule A).subtype.restrict ⋯
j' : ↥P' →ₗ[R] ↥P₁ := (Subalgebra.toSubmodule A).subtype.restrict ⋯
u₁ : ↥P₁ ⊗[R] N := (LinearMap.rTensor N j) u
hu₁ : u₁ = (LinearMap.rTensor N j) u
u'₁ : ↥P₁ ⊗[R] N := (LinearMap.rTensor N j') u'
hu'₁ : u'₁ = (LinearMap.rTensor N j') u'
this : (LinearMap.rTensor N P₁.subtype) u₁ = (LinearMap.rTensor N P₁.subtype) u'₁
P'₁ : Submodule R S
hP₁_le : P₁ ≤ P'₁
hP'₁ : P'₁.FG
h : (LinearMap.rTensor N (Submodule.inclusion hP₁_le)) u₁ = (LinearMap.rTensor N (Submodule.inclusion hP₁_le)) u'₁
s : Finset S
hs : Submodule.span R ↑s = P'₁
w : Finset S
hw : Algebra.adjoin R ↑w = A
B : Subalgebra R S := Algebra.adjoin R ↑(s ∪ w)
⊢ A ≤ B
Tactic:
simp only [B, ← hw]
Proof state after:
R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
A : Subalgebra R S
hA : A.FG
t t' : ↥A ⊗[R] N
h✝ : (LinearMap.rTensor N A.val.toLinearMap) t = (LinearMap.rTensor N A.val.toLinearMap) t'
P : Submodule R ↥A
hP : P.FG
u : ↥P ⊗[R] N
hu : (LinearMap.rTensor N P.subtype) u = t
P' : Submodule R ↥A
hP' : P'.FG
u' : ↥P' ⊗[R] N
hu' : (LinearMap.rTensor N P'.subtype) u' = t'
P₁ : Submodule R S := Submodule.map (Subalgebra.toSubmodule A).subtype (P ⊔ P')
hP₁ : P₁.FG
j : ↥P →ₗ[R] ↥P₁ := (Subalgebra.toSubmodule A).subtype.restrict ⋯
j' : ↥P' →ₗ[R] ↥P₁ := (Subalgebra.toSubmodule A).subtype.restrict ⋯
u₁ : ↥P₁ ⊗[R] N := (LinearMap.rTensor N j) u
hu₁ : u₁ = (LinearMap.rTensor N j) u
u'₁ : ↥P₁ ⊗[R] N := (LinearMap.rTensor N j') u'
hu'₁ : u'₁ = (LinearMap.rTensor N j') u'
this : (LinearMap.rTensor N P₁.subtype) u₁ = (LinearMap.rTensor N P₁.subtype) u'₁
P'₁ : Submodule R S
hP₁_le : P₁ ≤ P'₁
hP'₁ : P'₁.FG
h : (LinearMap.rTensor N (Submodule.inclusion hP₁_le)) u₁ = (LinearMap.rTensor N (Submodule.inclusion hP₁_le)) u'₁
s : Finset S
hs : Submodule.span R ↑s = P'₁
w : Finset S
hw : Algebra.adjoin R ↑w = A
B : Subalgebra R S := Algebra.adjoin R ↑(s ∪ w)
⊢ Algebra.adjoin R ↑w ≤ Algebra.adjoin R ↑(s ∪ w)
Theorem: TensorProduct.Algebra.eq_of_fg_of_subtype_eq
Proof state before:
R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
A : Subalgebra R S
hA : A.FG
t t' : ↥A ⊗[R] N
h✝ : (LinearMap.rTensor N A.val.toLinearMap) t = (LinearMap.rTensor N A.val.toLinearMap) t'
P : Submodule R ↥A
hP : P.FG
u : ↥P ⊗[R] N
hu : (LinearMap.rTensor N P.subtype) u = t
P' : Submodule R ↥A
hP' : P'.FG
u' : ↥P' ⊗[R] N
hu' : (LinearMap.rTensor N P'.subtype) u' = t'
P₁ : Submodule R S := Submodule.map (Subalgebra.toSubmodule A).subtype (P ⊔ P')
hP₁ : P₁.FG
j : ↥P →ₗ[R] ↥P₁ := (Subalgebra.toSubmodule A).subtype.restrict ⋯
j' : ↥P' →ₗ[R] ↥P₁ := (Subalgebra.toSubmodule A).subtype.restrict ⋯
u₁ : ↥P₁ ⊗[R] N := (LinearMap.rTensor N j) u
hu₁ : u₁ = (LinearMap.rTensor N j) u
u'₁ : ↥P₁ ⊗[R] N := (LinearMap.rTensor N j') u'
hu'₁ : u'₁ = (LinearMap.rTensor N j') u'
this : (LinearMap.rTensor N P₁.subtype) u₁ = (LinearMap.rTensor N P₁.subtype) u'₁
P'₁ : Submodule R S
hP₁_le : P₁ ≤ P'₁
hP'₁ : P'₁.FG
h : (LinearMap.rTensor N (Submodule.inclusion hP₁_le)) u₁ = (LinearMap.rTensor N (Submodule.inclusion hP₁_le)) u'₁
s : Finset S
hs : Submodule.span R ↑s = P'₁
w : Finset S
hw : Algebra.adjoin R ↑w = A
B : Subalgebra R S := Algebra.adjoin R ↑(s ∪ w)
⊢ Algebra.adjoin R ↑w ≤ Algebra.adjoin R ↑(s ∪ w)
Tactic:
apply Algebra.adjoin_mono
Proof state after:
case H
R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
A : Subalgebra R S
hA : A.FG
t t' : ↥A ⊗[R] N
h✝ : (LinearMap.rTensor N A.val.toLinearMap) t = (LinearMap.rTensor N A.val.toLinearMap) t'
P : Submodule R ↥A
hP : P.FG
u : ↥P ⊗[R] N
hu : (LinearMap.rTensor N P.subtype) u = t
P' : Submodule R ↥A
hP' : P'.FG
u' : ↥P' ⊗[R] N
hu' : (LinearMap.rTensor N P'.subtype) u' = t'
P₁ : Submodule R S := Submodule.map (Subalgebra.toSubmodule A).subtype (P ⊔ P')
hP₁ : P₁.FG
j : ↥P →ₗ[R] ↥P₁ := (Subalgebra.toSubmodule A).subtype.restrict ⋯
j' : ↥P' →ₗ[R] ↥P₁ := (Subalgebra.toSubmodule A).subtype.restrict ⋯
u₁ : ↥P₁ ⊗[R] N := (LinearMap.rTensor N j) u
hu₁ : u₁ = (LinearMap.rTensor N j) u
u'₁ : ↥P₁ ⊗[R] N := (LinearMap.rTensor N j') u'
hu'₁ : u'₁ = (LinearMap.rTensor N j') u'
this : (LinearMap.rTensor N P₁.subtype) u₁ = (LinearMap.rTensor N P₁.subtype) u'₁
P'₁ : Submodule R S
hP₁_le : P₁ ≤ P'₁
hP'₁ : P'₁.FG
h : (LinearMap.rTensor N (Submodule.inclusion hP₁_le)) u₁ = (LinearMap.rTensor N (Submodule.inclusion hP₁_le)) u'₁
s : Finset S
hs : Submodule.span R ↑s = P'₁
w : Finset S
hw : Algebra.adjoin R ↑w = A
B : Subalgebra R S := Algebra.adjoin R ↑(s ∪ w)
⊢ ↑w ⊆ ↑(s ∪ w)
Theorem: TensorProduct.Algebra.eq_of_fg_of_subtype_eq
Proof state before:
case H
R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
A : Subalgebra R S
hA : A.FG
t t' : ↥A ⊗[R] N
h✝ : (LinearMap.rTensor N A.val.toLinearMap) t = (LinearMap.rTensor N A.val.toLinearMap) t'
P : Submodule R ↥A
hP : P.FG
u : ↥P ⊗[R] N
hu : (LinearMap.rTensor N P.subtype) u = t
P' : Submodule R ↥A
hP' : P'.FG
u' : ↥P' ⊗[R] N
hu' : (LinearMap.rTensor N P'.subtype) u' = t'
P₁ : Submodule R S := Submodule.map (Subalgebra.toSubmodule A).subtype (P ⊔ P')
hP₁ : P₁.FG
j : ↥P →ₗ[R] ↥P₁ := (Subalgebra.toSubmodule A).subtype.restrict ⋯
j' : ↥P' →ₗ[R] ↥P₁ := (Subalgebra.toSubmodule A).subtype.restrict ⋯
u₁ : ↥P₁ ⊗[R] N := (LinearMap.rTensor N j) u
hu₁ : u₁ = (LinearMap.rTensor N j) u
u'₁ : ↥P₁ ⊗[R] N := (LinearMap.rTensor N j') u'
hu'₁ : u'₁ = (LinearMap.rTensor N j') u'
this : (LinearMap.rTensor N P₁.subtype) u₁ = (LinearMap.rTensor N P₁.subtype) u'₁
P'₁ : Submodule R S
hP₁_le : P₁ ≤ P'₁
hP'₁ : P'₁.FG
h : (LinearMap.rTensor N (Submodule.inclusion hP₁_le)) u₁ = (LinearMap.rTensor N (Submodule.inclusion hP₁_le)) u'₁
s : Finset S
hs : Submodule.span R ↑s = P'₁
w : Finset S
hw : Algebra.adjoin R ↑w = A
B : Subalgebra R S := Algebra.adjoin R ↑(s ∪ w)
⊢ ↑w ⊆ ↑(s ∪ w)
Tactic:
simp only [Finset.coe_union, Set.subset_union_right]
Proof state after:
no goals
Theorem: TensorProduct.Algebra.eq_of_fg_of_subtype_eq
Proof state before:
R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
A : Subalgebra R S
hA : A.FG
t t' : ↥A ⊗[R] N
h✝ : (LinearMap.rTensor N A.val.toLinearMap) t = (LinearMap.rTensor N A.val.toLinearMap) t'
P : Submodule R ↥A
hP : P.FG
u : ↥P ⊗[R] N
hu : (LinearMap.rTensor N P.subtype) u = t
P' : Submodule R ↥A
hP' : P'.FG
u' : ↥P' ⊗[R] N
hu' : (LinearMap.rTensor N P'.subtype) u' = t'
P₁ : Submodule R S := Submodule.map (Subalgebra.toSubmodule A).subtype (P ⊔ P')
hP₁ : P₁.FG
j : ↥P →ₗ[R] ↥P₁ := (Subalgebra.toSubmodule A).subtype.restrict ⋯
j' : ↥P' →ₗ[R] ↥P₁ := (Subalgebra.toSubmodule A).subtype.restrict ⋯
u₁ : ↥P₁ ⊗[R] N := (LinearMap.rTensor N j) u
hu₁ : u₁ = (LinearMap.rTensor N j) u
u'₁ : ↥P₁ ⊗[R] N := (LinearMap.rTensor N j') u'
hu'₁ : u'₁ = (LinearMap.rTensor N j') u'
this : (LinearMap.rTensor N P₁.subtype) u₁ = (LinearMap.rTensor N P₁.subtype) u'₁
P'₁ : Submodule R S
hP₁_le : P₁ ≤ P'₁
hP'₁ : P'₁.FG
h : (LinearMap.rTensor N (Submodule.inclusion hP₁_le)) u₁ = (LinearMap.rTensor N (Submodule.inclusion hP₁_le)) u'₁
s : Finset S
hs : Submodule.span R ↑s = P'₁
w : Finset S
hw : Algebra.adjoin R ↑w = A
B : Subalgebra R S := Algebra.adjoin R ↑(s ∪ w)
hBA : A ≤ B
⊢ P'₁ ≤ Subalgebra.toSubmodule B
Tactic:
simp only [← hs, Finset.coe_union, Submodule.span_le, Subalgebra.coe_toSubmodule, B]
Proof state after:
R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
A : Subalgebra R S
hA : A.FG
t t' : ↥A ⊗[R] N
h✝ : (LinearMap.rTensor N A.val.toLinearMap) t = (LinearMap.rTensor N A.val.toLinearMap) t'
P : Submodule R ↥A
hP : P.FG
u : ↥P ⊗[R] N
hu : (LinearMap.rTensor N P.subtype) u = t
P' : Submodule R ↥A
hP' : P'.FG
u' : ↥P' ⊗[R] N
hu' : (LinearMap.rTensor N P'.subtype) u' = t'
P₁ : Submodule R S := Submodule.map (Subalgebra.toSubmodule A).subtype (P ⊔ P')
hP₁ : P₁.FG
j : ↥P →ₗ[R] ↥P₁ := (Subalgebra.toSubmodule A).subtype.restrict ⋯
j' : ↥P' →ₗ[R] ↥P₁ := (Subalgebra.toSubmodule A).subtype.restrict ⋯
u₁ : ↥P₁ ⊗[R] N := (LinearMap.rTensor N j) u
hu₁ : u₁ = (LinearMap.rTensor N j) u
u'₁ : ↥P₁ ⊗[R] N := (LinearMap.rTensor N j') u'
hu'₁ : u'₁ = (LinearMap.rTensor N j') u'
this : (LinearMap.rTensor N P₁.subtype) u₁ = (LinearMap.rTensor N P₁.subtype) u'₁
P'₁ : Submodule R S
hP₁_le : P₁ ≤ P'₁
hP'₁ : P'₁.FG
h : (LinearMap.rTensor N (Submodule.inclusion hP₁_le)) u₁ = (LinearMap.rTensor N (Submodule.inclusion hP₁_le)) u'₁
s : Finset S
hs : Submodule.span R ↑s = P'₁
w : Finset S
hw : Algebra.adjoin R ↑w = A
B : Subalgebra R S := Algebra.adjoin R ↑(s ∪ w)
hBA : A ≤ B
⊢ ↑s ⊆ ↑(Algebra.adjoin R (↑s ∪ ↑w))
Theorem: TensorProduct.Algebra.eq_of_fg_of_subtype_eq
Proof state before:
R : Type u_1
S : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : AddCommGroup N
inst✝ : Module R N
A : Subalgebra R S
hA : A.FG
t t' : ↥A ⊗[R] N
h✝ : (LinearMap.rTensor N A.val.toLinearMap) t = (LinearMap.rTensor N A.val.toLinearMap) t'
P : Submodule R ↥A
hP : P.FG
u : ↥P ⊗[R] N
hu : (LinearMap.rTensor N P.subtype) u = t
P' : Submodule R ↥A
hP' : P'.FG
u' : ↥P' ⊗[R] N
hu' : (LinearMap.rTensor N P'.subtype) u' = t'
P₁ : Submodule R S := Submodule.map (Subalgebra.toSubmodule A).subtype (P ⊔ P')
hP₁ : P₁.FG
j : ↥P →ₗ[R] ↥P₁ := (Subalgebra.toSubmodule A).subtype.restrict ⋯
j' : ↥P' →ₗ[R] ↥P₁ := (Subalgebra.toSubmodule A).subtype.restrict ⋯
u₁ : ↥P₁ ⊗[R] N := (LinearMap.rTensor N j) u
hu₁ : u₁ = (LinearMap.rTensor N j) u
u'₁ : ↥P₁ ⊗[R] N := (LinearMap.rTensor N j') u'
hu'₁ : u'₁ = (LinearMap.rTensor N j') u'
this : (LinearMap.rTensor N P₁.subtype) u₁ = (LinearMap.rTensor N P₁.subtype) u'₁
P'₁ : Submodule R S
hP₁_le : P₁ ≤ P'₁
hP'₁ : P'₁.FG
h : (LinearMap.rTensor N (Submodule.inclusion hP₁_le)) u₁ = (LinearMap.rTensor N (Submodule.inclusion hP₁_le)) u'₁
s : Finset S
hs : Submodule.span R ↑s = P'₁
w : Finset S
hw : Algebra.adjoin R ↑w = A
B : Subalgebra R S := Algebra.adjoin R ↑(s ∪ w)
hBA : A ≤ B
⊢ ↑s ⊆ ↑(Algebra.adjoin R (↑s ∪ ↑w))
Tactic:
exact subset_trans (Set.subset_union_left _ _) Algebra.subset_adjoin
Proof state after:
no goals