-- Mathlib/FieldTheory/LinearDisjoint.lean
theorem isField_of_forall (A : Type v) [Field A] (B : Type w) [Field B]
    [Algebra F A] [Algebra F B]
    (H : ∀ (K : Type (max v w)) [Field K] [Algebra F K],
      ∀ (fa : A →ₐ[F] K) (fb : B →ₐ[F] K), fa.fieldRange.LinearDisjoint fb.fieldRange) :
    IsField (A ⊗[F] B)

Proof state:
case intro
F : Type u
inst✝⁴ : Field F
A : Type v
inst✝³ : Field A
B : Type w
inst✝² : Field B
inst✝¹ : Algebra F A
inst✝ : Algebra F B
M : Ideal (A ⊗[F] B)
hM : M.IsMaximal
K : Type (max v w) := A ⊗[F] B ⧸ M
this : Field K := Ideal.Quotient.field M
i : A ⊗[F] B →ₐ[F] K := IsScalarTower.toAlgHom F (A ⊗[F] B) K
fa : A →ₐ[F] K := i.comp Algebra.TensorProduct.includeLeft
fb : B →ₐ[F] K := i.comp Algebra.TensorProduct.includeRight
H : Function.Injective ⇑(fa.range.mulMap fb.range)
⊢ M = ⊥

Tactic:
have hi : i = (fa.range.mulMap fb.range).comp (Algebra.TensorProduct.congr
    (AlgEquiv.ofInjective fa fa.injective) (AlgEquiv.ofInjective fb fb.injective)) := by
  ext <;> simp [fa, fb]

Result:
case intro
F : Type u
inst✝⁴ : Field F
A : Type v
inst✝³ : Field A
B : Type w
inst✝² : Field B
inst✝¹ : Algebra F A
inst✝ : Algebra F B
M : Ideal (A ⊗[F] B)
hM : M.IsMaximal
K : Type (max v w) := A ⊗[F] B ⧸ M
this : Field K := Ideal.Quotient.field M
i : A ⊗[F] B →ₐ[F] K := IsScalarTower.toAlgHom F (A ⊗[F] B) K
fa : A →ₐ[F] K := i.comp Algebra.TensorProduct.includeLeft
fb : B →ₐ[F] K := i.comp Algebra.TensorProduct.includeRight
H : Function.Injective ⇑(fa.range.mulMap fb.range)
hi :
  i =
    (fa.range.mulMap fb.range).comp
      ↑(Algebra.TensorProduct.congr (AlgEquiv.ofInjective fa ⋯) (AlgEquiv.ofInjective fb ⋯))
⊢ M = ⊥

Proof state:
case intro
F : Type u
inst✝⁴ : Field F
A : Type v
inst✝³ : Field A
B : Type w
inst✝² : Field B
inst✝¹ : Algebra F A
inst✝ : Algebra F B
M : Ideal (A ⊗[F] B)
hM : M.IsMaximal
K : Type (max v w) := A ⊗[F] B ⧸ M
this : Field K := Ideal.Quotient.field M
i : A ⊗[F] B →ₐ[F] K := IsScalarTower.toAlgHom F (A ⊗[F] B) K
fa : A →ₐ[F] K := i.comp Algebra.TensorProduct.includeLeft
fb : B →ₐ[F] K := i.comp Algebra.TensorProduct.includeRight
hi :
  i =
    (fa.range.mulMap fb.range).comp
      ↑(Algebra.TensorProduct.congr (AlgEquiv.ofInjective fa ⋯) (AlgEquiv.ofInjective fb ⋯))
H : Function.Injective ⇑(Ideal.Quotient.mk M)
⊢ M = ⊥

Tactic:
rwa [RingHom.injective_iff_ker_eq_bot, Ideal.mk_ker] at H

Result:
no goals

Proof state:
case intro
F : Type u
inst✝⁴ : Field F
A : Type v
inst✝³ : Field A
B : Type w
inst✝² : Field B
inst✝¹ : Algebra F A
inst✝ : Algebra F B
H :
  ∀ (K : Type (max v w)) [inst : Field K] [inst_1 : Algebra F K] (fa : A →ₐ[F] K) (fb : B →ₐ[F] K),
    fa.fieldRange.LinearDisjoint ↥fb.fieldRange
M : Ideal (A ⊗[F] B)
hM : M.IsMaximal
⊢ IsField (A ⊗[F] B)

Tactic:
apply not_imp_not.1 (Ring.ne_bot_of_isMaximal_of_not_isField hM)

Result:
case intro
F : Type u
inst✝⁴ : Field F
A : Type v
inst✝³ : Field A
B : Type w
inst✝² : Field B
inst✝¹ : Algebra F A
inst✝ : Algebra F B
H :
  ∀ (K : Type (max v w)) [inst : Field K] [inst_1 : Algebra F K] (fa : A →ₐ[F] K) (fb : B →ₐ[F] K),
    fa.fieldRange.LinearDisjoint ↥fb.fieldRange
M : Ideal (A ⊗[F] B)
hM : M.IsMaximal
⊢ M = ⊥

Proof state:
case intro
F : Type u
inst✝⁴ : Field F
A : Type v
inst✝³ : Field A
B : Type w
inst✝² : Field B
inst✝¹ : Algebra F A
inst✝ : Algebra F B
H :
  ∀ (K : Type (max v w)) [inst : Field K] [inst_1 : Algebra F K] (fa : A →ₐ[F] K) (fb : B →ₐ[F] K),
    fa.fieldRange.LinearDisjoint ↥fb.fieldRange
M : Ideal (A ⊗[F] B)
hM : M.IsMaximal
K : Type (max v w) := A ⊗[F] B ⧸ M
⊢ M = ⊥

Tactic:
letI : Field K := Ideal.Quotient.field _

Result:
case intro
F : Type u
inst✝⁴ : Field F
A : Type v
inst✝³ : Field A
B : Type w
inst✝² : Field B
inst✝¹ : Algebra F A
inst✝ : Algebra F B
H :
  ∀ (K : Type (max v w)) [inst : Field K] [inst_1 : Algebra F K] (fa : A →ₐ[F] K) (fb : B →ₐ[F] K),
    fa.fieldRange.LinearDisjoint ↥fb.fieldRange
M : Ideal (A ⊗[F] B)
hM : M.IsMaximal
K : Type (max v w) := A ⊗[F] B ⧸ M
this : Field K := Ideal.Quotient.field M
⊢ M = ⊥

Proof state:
case intro
F : Type u
inst✝⁴ : Field F
A : Type v
inst✝³ : Field A
B : Type w
inst✝² : Field B
inst✝¹ : Algebra F A
inst✝ : Algebra F B
M : Ideal (A ⊗[F] B)
hM : M.IsMaximal
K : Type (max v w) := A ⊗[F] B ⧸ M
this : Field K := Ideal.Quotient.field M
i : A ⊗[F] B →ₐ[F] K := IsScalarTower.toAlgHom F (A ⊗[F] B) K
fa : A →ₐ[F] K := i.comp Algebra.TensorProduct.includeLeft
fb : B →ₐ[F] K := i.comp Algebra.TensorProduct.includeRight
hi :
  i =
    (fa.range.mulMap fb.range).comp
      ↑(Algebra.TensorProduct.congr (AlgEquiv.ofInjective fa ⋯) (AlgEquiv.ofInjective fb ⋯))
H : Function.Injective ⇑i
⊢ M = ⊥

Tactic:
change Function.Injective (Ideal.Quotient.mk M) at H

Result:
case intro
F : Type u
inst✝⁴ : Field F
A : Type v
inst✝³ : Field A
B : Type w
inst✝² : Field B
inst✝¹ : Algebra F A
inst✝ : Algebra F B
M : Ideal (A ⊗[F] B)
hM : M.IsMaximal
K : Type (max v w) := A ⊗[F] B ⧸ M
this : Field K := Ideal.Quotient.field M
i : A ⊗[F] B →ₐ[F] K := IsScalarTower.toAlgHom F (A ⊗[F] B) K
fa : A →ₐ[F] K := i.comp Algebra.TensorProduct.includeLeft
fb : B →ₐ[F] K := i.comp Algebra.TensorProduct.includeRight
hi :
  i =
    (fa.range.mulMap fb.range).comp
      ↑(Algebra.TensorProduct.congr (AlgEquiv.ofInjective fa ⋯) (AlgEquiv.ofInjective fb ⋯))
H : Function.Injective ⇑(Ideal.Quotient.mk M)
⊢ M = ⊥

Proof state:
case intro
F : Type u
inst✝⁴ : Field F
A : Type v
inst✝³ : Field A
B : Type w
inst✝² : Field B
inst✝¹ : Algebra F A
inst✝ : Algebra F B
H :
  ∀ (K : Type (max v w)) [inst : Field K] [inst_1 : Algebra F K] (fa : A →ₐ[F] K) (fb : B →ₐ[F] K),
    fa.fieldRange.LinearDisjoint ↥fb.fieldRange
M : Ideal (A ⊗[F] B)
hM : M.IsMaximal
K : Type (max v w) := A ⊗[F] B ⧸ M
this : Field K := Ideal.Quotient.field M
i : A ⊗[F] B →ₐ[F] K := IsScalarTower.toAlgHom F (A ⊗[F] B) K
fa : A →ₐ[F] K := i.comp Algebra.TensorProduct.includeLeft
⊢ M = ⊥

Tactic:
let fb := i.comp (Algebra.TensorProduct.includeRight : B →ₐ[F] _)

Result:
case intro
F : Type u
inst✝⁴ : Field F
A : Type v
inst✝³ : Field A
B : Type w
inst✝² : Field B
inst✝¹ : Algebra F A
inst✝ : Algebra F B
H :
  ∀ (K : Type (max v w)) [inst : Field K] [inst_1 : Algebra F K] (fa : A →ₐ[F] K) (fb : B →ₐ[F] K),
    fa.fieldRange.LinearDisjoint ↥fb.fieldRange
M : Ideal (A ⊗[F] B)
hM : M.IsMaximal
K : Type (max v w) := A ⊗[F] B ⧸ M
this : Field K := Ideal.Quotient.field M
i : A ⊗[F] B →ₐ[F] K := IsScalarTower.toAlgHom F (A ⊗[F] B) K
fa : A →ₐ[F] K := i.comp Algebra.TensorProduct.includeLeft
fb : B →ₐ[F] K := i.comp Algebra.TensorProduct.includeRight
⊢ M = ⊥

Proof state:
case intro
F : Type u
inst✝⁴ : Field F
A : Type v
inst✝³ : Field A
B : Type w
inst✝² : Field B
inst✝¹ : Algebra F A
inst✝ : Algebra F B
H :
  ∀ (K : Type (max v w)) [inst : Field K] [inst_1 : Algebra F K] (fa : A →ₐ[F] K) (fb : B →ₐ[F] K),
    fa.fieldRange.LinearDisjoint ↥fb.fieldRange
M : Ideal (A ⊗[F] B)
hM : M.IsMaximal
K : Type (max v w) := A ⊗[F] B ⧸ M
this : Field K := Ideal.Quotient.field M
i : A ⊗[F] B →ₐ[F] K := IsScalarTower.toAlgHom F (A ⊗[F] B) K
fa : A →ₐ[F] K := i.comp Algebra.TensorProduct.includeLeft
fb : B →ₐ[F] K := i.comp Algebra.TensorProduct.includeRight
⊢ M = ⊥

Tactic:
replace H := H K fa fb

Result:
case intro
F : Type u
inst✝⁴ : Field F
A : Type v
inst✝³ : Field A
B : Type w
inst✝² : Field B
inst✝¹ : Algebra F A
inst✝ : Algebra F B
M : Ideal (A ⊗[F] B)
hM : M.IsMaximal
K : Type (max v w) := A ⊗[F] B ⧸ M
this : Field K := Ideal.Quotient.field M
i : A ⊗[F] B →ₐ[F] K := IsScalarTower.toAlgHom F (A ⊗[F] B) K
fa : A →ₐ[F] K := i.comp Algebra.TensorProduct.includeLeft
fb : B →ₐ[F] K := i.comp Algebra.TensorProduct.includeRight
H : fa.fieldRange.LinearDisjoint ↥fb.fieldRange
⊢ M = ⊥