-- Mathlib/RingTheory/Etale/Field.lean
lemma of_isSeparable [Algebra.IsSeparable K L] : FormallyEtale K L

Proof state:
case comp_bijective.refine_2.intro.intro
K L : Type u
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : Algebra.IsSeparable K L
B : Type u
inst✝¹ : CommRing B
inst✝ : Algebra K B
I : Ideal B
h : I ^ 2 = ⊥
f : L →ₐ[K] B ⧸ I
g : (k : L) → ↥K⟮k⟯ →ₐ[K] B
hg₁ : ∀ (k : L), (fun g => (Ideal.Quotient.mkₐ K I).comp g = f.comp (IsScalarTower.toAlgHom K (↥K⟮k⟯) L)) (g k)
hg₂ :
  ∀ (k : L) (y : ↥K⟮k⟯ →ₐ[K] B),
    (fun g => (Ideal.Quotient.mkₐ K I).comp g = f.comp (IsScalarTower.toAlgHom K (↥K⟮k⟯) L)) y → y = g k
hg₃ : ∀ (x y : L) (h : x ∈ K⟮y⟯), (g y) ⟨x, h⟩ = (g x) (IntermediateField.AdjoinSimple.gen K x)
H : ∀ (x y : L), ∃ α, x ∈ K⟮α⟯ ∧ y ∈ K⟮α⟯
x y α : L
hx : x ∈ K⟮α⟯
hy : y ∈ K⟮α⟯
⊢ (g α) ⟨x * y, ⋯⟩ = (g α) (⟨x, hx⟩ * ⟨y, hy⟩)

Tactic:
rfl

Result:
no goals

Proof state:
case intro.mk
K L : Type u
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : Algebra.IsSeparable K L
B : Type u
inst✝¹ : CommRing B
inst✝ : Algebra K B
I : Ideal B
h : I ^ 2 = ⊥
f : L →ₐ[K] B ⧸ I
g : (k : L) → ↥K⟮k⟯ →ₐ[K] B
hg₁ : ∀ (k : L), (fun g => (Ideal.Quotient.mkₐ K I).comp g = f.comp (IsScalarTower.toAlgHom K (↥K⟮k⟯) L)) (g k)
hg₂ :
  ∀ (k : L) (y : ↥K⟮k⟯ →ₐ[K] B),
    (fun g => (Ideal.Quotient.mkₐ K I).comp g = f.comp (IsScalarTower.toAlgHom K (↥K⟮k⟯) L)) y → y = g k
hg₃ : ∀ (x y : L) (h : x ∈ K⟮y⟯), (g y) ⟨x, h⟩ = (g x) (IntermediateField.AdjoinSimple.gen K x)
x y : L
this✝ : FiniteDimensional K ↥K⟮x, y⟯
this : Algebra.IsSeparable K ↥K⟮x, y⟯
α : L
hα : α ∈ K⟮x, y⟯
e : K⟮α⟯ = K⟮x, y⟯
⊢ ∃ α, x ∈ K⟮α⟯ ∧ y ∈ K⟮α⟯

Tactic:
have hx : x ∈ K⟮α⟯ := e ▸ IntermediateField.subset_adjoin K {x, y} (by simp)

Result:
case intro.mk
K L : Type u
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : Algebra.IsSeparable K L
B : Type u
inst✝¹ : CommRing B
inst✝ : Algebra K B
I : Ideal B
h : I ^ 2 = ⊥
f : L →ₐ[K] B ⧸ I
g : (k : L) → ↥K⟮k⟯ →ₐ[K] B
hg₁ : ∀ (k : L), (fun g => (Ideal.Quotient.mkₐ K I).comp g = f.comp (IsScalarTower.toAlgHom K (↥K⟮k⟯) L)) (g k)
hg₂ :
  ∀ (k : L) (y : ↥K⟮k⟯ →ₐ[K] B),
    (fun g => (Ideal.Quotient.mkₐ K I).comp g = f.comp (IsScalarTower.toAlgHom K (↥K⟮k⟯) L)) y → y = g k
hg₃ : ∀ (x y : L) (h : x ∈ K⟮y⟯), (g y) ⟨x, h⟩ = (g x) (IntermediateField.AdjoinSimple.gen K x)
x y : L
this✝ : FiniteDimensional K ↥K⟮x, y⟯
this : Algebra.IsSeparable K ↥K⟮x, y⟯
α : L
hα : α ∈ K⟮x, y⟯
e : K⟮α⟯ = K⟮x, y⟯
hx : x ∈ K⟮α⟯
⊢ ∃ α, x ∈ K⟮α⟯ ∧ y ∈ K⟮α⟯

Proof state:
case comp_bijective
K L : Type u
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : Algebra.IsSeparable K L
B : Type u
inst✝¹ : CommRing B
inst✝ : Algebra K B
I : Ideal B
h : I ^ 2 = ⊥
f : L →ₐ[K] B ⧸ I
this : ∀ (k : L), ∃! g, (Ideal.Quotient.mkₐ K I).comp g = f.comp (IsScalarTower.toAlgHom K (↥K⟮k⟯) L)
⊢ ∃ a, (Ideal.Quotient.mkₐ K I).comp a = f

Tactic:
choose g hg₁ hg₂ using this

Result:
case comp_bijective
K L : Type u
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : Algebra.IsSeparable K L
B : Type u
inst✝¹ : CommRing B
inst✝ : Algebra K B
I : Ideal B
h : I ^ 2 = ⊥
f : L →ₐ[K] B ⧸ I
g : (k : L) → ↥K⟮k⟯ →ₐ[K] B
hg₁ : ∀ (k : L), (fun g => (Ideal.Quotient.mkₐ K I).comp g = f.comp (IsScalarTower.toAlgHom K (↥K⟮k⟯) L)) (g k)
hg₂ :
  ∀ (k : L) (y : ↥K⟮k⟯ →ₐ[K] B),
    (fun g => (Ideal.Quotient.mkₐ K I).comp g = f.comp (IsScalarTower.toAlgHom K (↥K⟮k⟯) L)) y → y = g k
⊢ ∃ a, (Ideal.Quotient.mkₐ K I).comp a = f

Proof state:
case comp_bijective
K L : Type u
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : Algebra.IsSeparable K L
B : Type u
inst✝¹ : CommRing B
inst✝ : Algebra K B
I : Ideal B
h : I ^ 2 = ⊥
f : L →ₐ[K] B ⧸ I
⊢ ∃ a, (Ideal.Quotient.mkₐ K I).comp a = f

Tactic:
have : ∀ k : L, ∃! g : K⟮k⟯ →ₐ[K] B,
    (Ideal.Quotient.mkₐ K I).comp g = f.comp (IsScalarTower.toAlgHom K _ L) := by
  intro k
  have := IsSeparable.of_algHom _ _ (IsScalarTower.toAlgHom K (K⟮k⟯) L)
  have := IntermediateField.adjoin.finiteDimensional
    (Algebra.IsSeparable.isSeparable K k).isIntegral
  have := FormallyEtale.of_isSeparable_aux K (K⟮k⟯)
  have := FormallyEtale.comp_bijective (R := K) (A := K⟮k⟯) I h
  exact this.existsUnique _

Result:
case comp_bijective
K L : Type u
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : Algebra.IsSeparable K L
B : Type u
inst✝¹ : CommRing B
inst✝ : Algebra K B
I : Ideal B
h : I ^ 2 = ⊥
f : L →ₐ[K] B ⧸ I
this : ∀ (k : L), ∃! g, (Ideal.Quotient.mkₐ K I).comp g = f.comp (IsScalarTower.toAlgHom K (↥K⟮k⟯) L)
⊢ ∃ a, (Ideal.Quotient.mkₐ K I).comp a = f

Proof state:
case intro.mk
K L : Type u
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : Algebra.IsSeparable K L
B : Type u
inst✝¹ : CommRing B
inst✝ : Algebra K B
I : Ideal B
h : I ^ 2 = ⊥
f : L →ₐ[K] B ⧸ I
g : (k : L) → ↥K⟮k⟯ →ₐ[K] B
hg₁ : ∀ (k : L), (fun g => (Ideal.Quotient.mkₐ K I).comp g = f.comp (IsScalarTower.toAlgHom K (↥K⟮k⟯) L)) (g k)
hg₂ :
  ∀ (k : L) (y : ↥K⟮k⟯ →ₐ[K] B),
    (fun g => (Ideal.Quotient.mkₐ K I).comp g = f.comp (IsScalarTower.toAlgHom K (↥K⟮k⟯) L)) y → y = g k
hg₃ : ∀ (x y : L) (h : x ∈ K⟮y⟯), (g y) ⟨x, h⟩ = (g x) (IntermediateField.AdjoinSimple.gen K x)
x y : L
this✝ : FiniteDimensional K ↥K⟮x, y⟯
this : Algebra.IsSeparable K ↥K⟮x, y⟯
α : L
hα : α ∈ K⟮x, y⟯
e : K⟮⟨α, hα⟩⟯ = ⊤
⊢ ∃ α, x ∈ K⟮α⟯ ∧ y ∈ K⟮α⟯

Tactic:
apply_fun (IntermediateField.map (IntermediateField.val _)) at e

Result:
case intro.mk
K L : Type u
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : Algebra.IsSeparable K L
B : Type u
inst✝¹ : CommRing B
inst✝ : Algebra K B
I : Ideal B
h : I ^ 2 = ⊥
f : L →ₐ[K] B ⧸ I
g : (k : L) → ↥K⟮k⟯ →ₐ[K] B
hg₁ : ∀ (k : L), (fun g => (Ideal.Quotient.mkₐ K I).comp g = f.comp (IsScalarTower.toAlgHom K (↥K⟮k⟯) L)) (g k)
hg₂ :
  ∀ (k : L) (y : ↥K⟮k⟯ →ₐ[K] B),
    (fun g => (Ideal.Quotient.mkₐ K I).comp g = f.comp (IsScalarTower.toAlgHom K (↥K⟮k⟯) L)) y → y = g k
hg₃ : ∀ (x y : L) (h : x ∈ K⟮y⟯), (g y) ⟨x, h⟩ = (g x) (IntermediateField.AdjoinSimple.gen K x)
x y : L
this✝ : FiniteDimensional K ↥K⟮x, y⟯
this : Algebra.IsSeparable K ↥K⟮x, y⟯
α : L
hα : α ∈ K⟮x, y⟯
e : IntermediateField.map K⟮x, y⟯.val K⟮⟨α, hα⟩⟯ = IntermediateField.map K⟮x, y⟯.val ⊤
⊢ ∃ α, x ∈ K⟮α⟯ ∧ y ∈ K⟮α⟯

Proof state:
K L : Type u
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : Algebra.IsSeparable K L
B : Type u
inst✝¹ : CommRing B
inst✝ : Algebra K B
I : Ideal B
h : I ^ 2 = ⊥
f : L →ₐ[K] B ⧸ I
g : (k : L) → ↥K⟮k⟯ →ₐ[K] B
hg₁ : ∀ (k : L), (fun g => (Ideal.Quotient.mkₐ K I).comp g = f.comp (IsScalarTower.toAlgHom K (↥K⟮k⟯) L)) (g k)
hg₂ :
  ∀ (k : L) (y : ↥K⟮k⟯ →ₐ[K] B),
    (fun g => (Ideal.Quotient.mkₐ K I).comp g = f.comp (IsScalarTower.toAlgHom K (↥K⟮k⟯) L)) y → y = g k
hg₃ : ∀ (x y : L) (h : x ∈ K⟮y⟯), (g y) ⟨x, h⟩ = (g x) (IntermediateField.AdjoinSimple.gen K x)
x y : L
⊢ ∃ α, x ∈ K⟮α⟯ ∧ y ∈ K⟮α⟯

Tactic:
have : FiniteDimensional K K⟮x, y⟯ := by
  apply IntermediateField.finiteDimensional_adjoin
  intro x _; exact (Algebra.IsSeparable.isSeparable K x).isIntegral

Result:
K L : Type u
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : Algebra.IsSeparable K L
B : Type u
inst✝¹ : CommRing B
inst✝ : Algebra K B
I : Ideal B
h : I ^ 2 = ⊥
f : L →ₐ[K] B ⧸ I
g : (k : L) → ↥K⟮k⟯ →ₐ[K] B
hg₁ : ∀ (k : L), (fun g => (Ideal.Quotient.mkₐ K I).comp g = f.comp (IsScalarTower.toAlgHom K (↥K⟮k⟯) L)) (g k)
hg₂ :
  ∀ (k : L) (y : ↥K⟮k⟯ →ₐ[K] B),
    (fun g => (Ideal.Quotient.mkₐ K I).comp g = f.comp (IsScalarTower.toAlgHom K (↥K⟮k⟯) L)) y → y = g k
hg₃ : ∀ (x y : L) (h : x ∈ K⟮y⟯), (g y) ⟨x, h⟩ = (g x) (IntermediateField.AdjoinSimple.gen K x)
x y : L
this : FiniteDimensional K ↥K⟮x, y⟯
⊢ ∃ α, x ∈ K⟮α⟯ ∧ y ∈ K⟮α⟯