awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.FieldTheory.Galois.Infinite (goal)
Goal:
k : Type u_1
K : Type u_2
inst : Field k
inst_1 : Field K
inst_2 : Algebra k K
L : IntermediateField k K
inst_3 : IsGalois k K
σ : Gal(K/k)
h : σ ∈ (↑L.fixingSubgroup : Set Gal(K/k))ᶜ
y : K
yL : y ∈ (↑L : Set K)
ne : y ∉ fun (a : K) ↦ σ • a = a
f : Gal(K/k)
hf : f ∈ σ • (↑(IntermediateField.fixingSubgroup (toIntermediateField (adjoin k {y}))) : Set Gal(K/k))
⊢ f ∈ (↑L.fixingSubgroup : Set Gal(K/k))ᶜ
have h₁ : ∀ (g : Gal(K/k)) (hg : g ∈ (↑(IntermediateField.fixingSubgroup (toIntermediateField (adjoin k {y}))) : Set Gal(K/k)))
  (eq : σ • g = f), f ∈ (↑L.fixingSubgroup : Set Gal(K/k))ᶜ
Tactic: rcases (Set.mem_smul_set.mp hf) with ⟨g, hg, eq⟩
-- Lean 4 proof step from Mathlib.FieldTheory.Galois.Infinite (goal)
Goal:
k : Type u_1
K : Type u_2
inst : Field k
inst_1 : Field K
inst_2 : Algebra k K
L : IntermediateField k K
inst_3 : IsGalois k K
σ : Gal(K/k)
h : σ ∈ (↑L.fixingSubgroup : Set Gal(K/k))ᶜ
y : K
yL : y ∈ (↑L : Set K)
ne : y ∉ fun (a : K) ↦ σ • a = a
f : Gal(K/k)
hf : f ∈ σ • (↑(IntermediateField.fixingSubgroup (toIntermediateField (adjoin k {y}))) : Set Gal(K/k))
g : Gal(K/k)
hg : g ∈ (↑(IntermediateField.fixingSubgroup (toIntermediateField (adjoin k {y}))) : Set Gal(K/k))
eq : σ • g = f
⊢ f ∈ (↑L.fixingSubgroup : Set Gal(K/k))ᶜ
have h₁ : σ • g ∉ L.fixingSubgroup
Tactic: simp only [Set.mem_compl_iff, SetLike.mem_coe, ← eq]
-- Lean 4 proof step from Mathlib.FieldTheory.Galois.Infinite (goal)
Goal:
k : Type u_1
K : Type u_2
inst : Field k
inst_1 : Field K
inst_2 : Algebra k K
L : IntermediateField k K
inst_3 : IsGalois k K
σ : Gal(K/k)
h : σ ∈ (↑L.fixingSubgroup : Set Gal(K/k))ᶜ
y : K
yL : y ∈ (↑L : Set K)
ne : y ∉ fun (a : K) ↦ σ • a = a
f : Gal(K/k)
hf : f ∈ σ • (↑(IntermediateField.fixingSubgroup (toIntermediateField (adjoin k {y}))) : Set Gal(K/k))
g : Gal(K/k)
hg : g ∈ (↑(IntermediateField.fixingSubgroup (toIntermediateField (adjoin k {y}))) : Set Gal(K/k))
eq : σ • g = f
⊢ σ • g ∉ L.fixingSubgroup
have h₁ : ¬∀ y ∈ (↑L : Set K), (σ • g) • y = y
Tactic: apply (mem_fixingSubgroup_iff Gal(K/k)).not.mpr
-- Lean 4 proof step from Mathlib.FieldTheory.Galois.Infinite (goal)
Goal:
k : Type u_1
K : Type u_2
inst : Field k
inst_1 : Field K
inst_2 : Algebra k K
L : IntermediateField k K
inst_3 : IsGalois k K
σ : Gal(K/k)
h : σ ∈ (↑L.fixingSubgroup : Set Gal(K/k))ᶜ
y : K
yL : y ∈ (↑L : Set K)
ne : y ∉ fun (a : K) ↦ σ • a = a
f : Gal(K/k)
hf : f ∈ σ • (↑(IntermediateField.fixingSubgroup (toIntermediateField (adjoin k {y}))) : Set Gal(K/k))
g : Gal(K/k)
hg : g ∈ (↑(IntermediateField.fixingSubgroup (toIntermediateField (adjoin k {y}))) : Set Gal(K/k))
eq : σ • g = f
⊢ ¬∀ y ∈ (↑L : Set K), (σ • g) • y = y
have h₁ : ∃ y ∈ (↑L : Set K), (σ • g) • y ≠ y
Tactic: push Not
-- Lean 4 proof step from Mathlib.FieldTheory.Galois.Infinite (goal)
Goal:
k : Type u_1
K : Type u_2
inst : Field k
inst_1 : Field K
inst_2 : Algebra k K
L : IntermediateField k K
inst_3 : IsGalois k K
σ : Gal(K/k)
h : σ ∈ (↑L.fixingSubgroup : Set Gal(K/k))ᶜ
y : K
yL : y ∈ (↑L : Set K)
ne : y ∉ fun (a : K) ↦ σ • a = a
f : Gal(K/k)
hf : f ∈ σ • (↑(IntermediateField.fixingSubgroup (toIntermediateField (adjoin k {y}))) : Set Gal(K/k))
g : Gal(K/k)
hg : g ∈ (↑(IntermediateField.fixingSubgroup (toIntermediateField (adjoin k {y}))) : Set Gal(K/k))
eq : σ • g = f
⊢ ∃ y ∈ (↑L : Set K), (σ • g) • y ≠ y
have h₁ : y ∈ (↑L : Set K) ∧ (σ • g) • y ≠ y
Tactic: use y
-- Lean 4 proof step from Mathlib.FieldTheory.Galois.Infinite (goal)
Goal:
k : Type u_1
K : Type u_2
inst : Field k
inst_1 : Field K
inst_2 : Algebra k K
L : IntermediateField k K
inst_3 : IsGalois k K
σ : Gal(K/k)
h : σ ∈ (↑L.fixingSubgroup : Set Gal(K/k))ᶜ
y : K
yL : y ∈ (↑L : Set K)
ne : y ∉ fun (a : K) ↦ σ • a = a
f : Gal(K/k)
hf : f ∈ σ • (↑(IntermediateField.fixingSubgroup (toIntermediateField (adjoin k {y}))) : Set Gal(K/k))
g : Gal(K/k)
hg : g ∈ (↑(IntermediateField.fixingSubgroup (toIntermediateField (adjoin k {y}))) : Set Gal(K/k))
eq : σ • g = f
⊢ y ∈ (↑L : Set K) ∧ (σ • g) • y ≠ y
have h₁ : ¬(σ : (a : K) → K) ((g : (a : K) → K) y : K) = y
Tactic: simp only [yL, smul_eq_mul, AlgEquiv.smul_def, AlgEquiv.mul_apply, ne_eq, true_and]
-- Lean 4 proof step from Mathlib.FieldTheory.Galois.Infinite (hypothesis)
Goal:
k : Type u_1
K : Type u_2
inst : Field k
inst_1 : Field K
inst_2 : Algebra k K
L : IntermediateField k K
inst_3 : IsGalois k K
σ : Gal(K/k)
h : σ ∈ (↑L.fixingSubgroup : Set Gal(K/k))ᶜ
y : K
yL : y ∈ (↑L : Set K)
ne : y ∉ fun (a : K) ↦ σ • a = a
f : Gal(K/k)
hf : f ∈ σ • (↑(IntermediateField.fixingSubgroup (toIntermediateField (adjoin k {y}))) : Set Gal(K/k))
g : Gal(K/k)
hg : g ∈ (↑(IntermediateField.fixingSubgroup (toIntermediateField (adjoin k {y}))) : Set Gal(K/k))
eq : σ • g = f
⊢ ¬(σ : (a : K) → K) ((g : (a : K) → K) y : K) = y
have this : (g : (a : K) → K) y = y
Tactic: have : g y = y := (mem_fixingSubgroup_iff Gal(K/k)).mp hg y <|
        adjoin_simple_le_iff.mp le_rfl
-- Lean 4 proof step from Mathlib.FieldTheory.Galois.Infinite (goal)
Goal:
k : Type u_1
K : Type u_2
inst : Field k
inst_1 : Field K
inst_2 : Algebra k K
L : IntermediateField k K
inst_3 : IsGalois k K
σ : Gal(K/k)
h : σ ∈ (↑L.fixingSubgroup : Set Gal(K/k))ᶜ
y : K
yL : y ∈ (↑L : Set K)
ne : y ∉ fun (a : K) ↦ σ • a = a
⊢ IsOpen (σ • (↑(IntermediateField.fixingSubgroup (toIntermediateField (adjoin k {y}))) : Set Gal(K/k))) ∧
    σ ∈ σ • (↑(IntermediateField.fixingSubgroup (toIntermediateField (adjoin k {y}))) : Set Gal(K/k))
have h₁ : σ ∈ σ • (↑(IntermediateField.fixingSubgroup (toIntermediateField (adjoin k {y}))) : Set Gal(K/k))
Tactic: simp only [(IntermediateField.fixingSubgroup_isOpen (adjoin k {y}).1).smul σ, true_and]
-- Lean 4 proof step from Mathlib.FieldTheory.Galois.Infinite (goal)
Goal:
k : Type u_1
K : Type u_2
inst : Field k
inst_1 : Field K
inst_2 : Algebra k K
L : IntermediateField k K
inst_3 : IsGalois k K
σ : Gal(K/k)
h : σ ∈ (↑L.fixingSubgroup : Set Gal(K/k))ᶜ
y : K
yL : y ∈ (↑L : Set K)
ne : y ∉ fun (a : K) ↦ σ • a = a
⊢ σ ∈ σ • (↑(IntermediateField.fixingSubgroup (toIntermediateField (adjoin k {y}))) : Set Gal(K/k))
have h₁ : 1 ∈ (↑(IntermediateField.fixingSubgroup (toIntermediateField (adjoin k {y}))) : Set Gal(K/k)) ∧
  (fun (x : Gal(K/k)) ↦ σ • x) 1 = σ
Tactic: use 1
-- Lean 4 proof step from Mathlib.FieldTheory.Galois.Infinite (goal)
Goal:
k : Type u_1
K : Type u_2
inst : Field k
inst_1 : Field K
inst_2 : Algebra k K
L : IntermediateField k K
inst_3 : IsGalois k K
⊢ IntermediateField.fixedField L.fixingSubgroup = L
have a : IntermediateField.fixedField L.fixingSubgroup ≤ L
Tactic: apply le_antisymm
-- Lean 4 proof step from Mathlib.FieldTheory.Galois.Infinite (goal)
Goal:
k : Type u_1
K : Type u_2
inst : Field k
inst_1 : Field K
inst_2 : Algebra k K
L : IntermediateField k K
inst_3 : IsGalois k K
a : IntermediateField.fixedField L.fixingSubgroup ≤ L
⊢ IntermediateField.fixedField L.fixingSubgroup = L
have a₁ : L ≤ IntermediateField.fixedField L.fixingSubgroup
Tactic: apply le_antisymm
-- Lean 4 proof step from Mathlib.FieldTheory.Galois.Infinite (hypothesis)
Goal:
k : Type u_1
K : Type u_2
inst : Field k
inst_1 : Field K
inst_2 : Algebra k K
L : IntermediateField k K
inst_3 : IsGalois k K
x : K
hx : x ∈ IntermediateField.fixedField L.fixingSubgroup
⊢ x ∈ L
have hx₁ : ∀ f ∈ L.fixingSubgroup, (f : (a : K) → K) x = x
Tactic: rw [IntermediateField.mem_fixedField_iff] at hx
Cleared hypotheses: hx
-- Lean 4 proof step from Mathlib.FieldTheory.Galois.Infinite (hypothesis)
Goal:
k : Type u_1
K : Type u_2
inst : Field k
inst_1 : Field K
inst_2 : Algebra k K
L : IntermediateField k K
inst_3 : IsGalois k K
x : K
hx : ∀ f ∈ L.fixingSubgroup, (f : (a : K) → K) x = x
⊢ x ∈ L
have mem : Membership.mem (γ := IntermediateField (↥L) K) (toIntermediateField (adjoin ↥L {x})) x
Tactic: have mem : x ∈ (adjoin L {x}).1 := subset_adjoin _ _ rfl
-- Lean 4 proof step from Mathlib.FieldTheory.Galois.Infinite (hypothesis)
Goal:
k : Type u_1
K : Type u_2
inst : Field k
inst_1 : Field K
inst_2 : Algebra k K
L : IntermediateField k K
inst_3 : IsGalois k K
x : K
hx : ∀ f ∈ L.fixingSubgroup, (f : (a : K) → K) x = x
mem : Membership.mem (γ := IntermediateField (↥L) K) (toIntermediateField (adjoin ↥L {x})) x
⊢ x ∈ L
have this : Eq.{u_2 + 1} (α := IntermediateField ↥L ↥(toIntermediateField (adjoin ↥L {x}))) (IntermediateField.fixedField ⊤) ⊥
Tactic: have : IntermediateField.fixedField (⊤ : Subgroup ((adjoin L {x}) ≃ₐ[L] (adjoin L {x}))) = ⊥ :=
      (IsGalois.tfae.out 0 1).mp (by infer_instance)
-- Lean 4 proof step from Mathlib.FieldTheory.Galois.Infinite (goal)
Goal:
k : Type u_1
K : Type u_2
inst : Field k
inst_1 : Field K
inst_2 : Algebra k K
L : IntermediateField k K
inst_3 : IsGalois k K
x : K
hx : ∀ f ∈ L.fixingSubgroup, (f : (a : K) → K) x = x
mem : Membership.mem (γ := IntermediateField (↥L) K) (toIntermediateField (adjoin ↥L {x})) x
this :
  Eq.{u_2 + 1} (α := IntermediateField ↥L ↥(toIntermediateField (adjoin ↥L {x}))) (IntermediateField.fixedField ⊤) ⊥
⊢ Membership.mem (γ := IntermediateField ↥L ↥(toIntermediateField (adjoin ↥L {x}))) ⊥
    (Subtype.mk (p := fun (x_1 : K) ↦
      Membership.mem (γ := IntermediateField (↥L) K) (toIntermediateField (adjoin ↥L {x})) x_1) x mem)
have rw : ∀ (f : Gal(↥(toIntermediateField (adjoin ↥L {x}))/↥L)),
  Membership.mem (γ := Subgroup Gal(↥(toIntermediateField (adjoin ↥L {x}))/↥L)) ⊤ f →
    (f : (a : ↥(adjoin ↥L {x}).toIntermediateField) → ↥(adjoin ↥L {x}).toIntermediateField)
        (Subtype.mk (p := fun (x_1 : K) ↦
          Membership.mem (γ := IntermediateField (↥L) K) (toIntermediateField (adjoin ↥L {x})) x_1) x mem) =
      Subtype.mk (p := fun (x_1 : K) ↦
        Membership.mem (γ := IntermediateField (↥L) K) (toIntermediateField (adjoin ↥L {x})) x_1) x mem
Tactic: rw [← this, IntermediateField.mem_fixedField_iff]
-- Lean 4 proof step from Mathlib.FieldTheory.Galois.Infinite (goal)
Goal:
k : Type u_1
K : Type u_2
inst : Field k
inst_1 : Field K
inst_2 : Algebra k K
L : IntermediateField k K
inst_3 : IsGalois k K
x : K
hx : ∀ f ∈ L.fixingSubgroup, (f : (a : K) → K) x = x
mem : Membership.mem (γ := IntermediateField (↥L) K) (toIntermediateField (adjoin ↥L {x})) x
this :
  Eq.{u_2 + 1} (α := IntermediateField ↥L ↥(toIntermediateField (adjoin ↥L {x}))) (IntermediateField.fixedField ⊤) ⊥
f : Gal(↥(toIntermediateField (adjoin ↥L {x}))/↥L)
h : Membership.mem (γ := Subgroup Gal(↥(toIntermediateField (adjoin ↥L {x}))/↥L)) ⊤ f
⊢ (f : (a : ↥(adjoin ↥L {x}).toIntermediateField) → ↥(adjoin ↥L {x}).toIntermediateField)
      (Subtype.mk (p := fun (x_1 : K) ↦
        Membership.mem (γ := IntermediateField (↥L) K) (toIntermediateField (adjoin ↥L {x})) x_1) x mem) =
    Subtype.mk (p := fun (x_1 : K) ↦
      Membership.mem (γ := IntermediateField (↥L) K) (toIntermediateField (adjoin ↥L {x})) x_1) x mem
have rcases : ∀ (σ : Gal(K/↥L))
  (hσ :
    (restrictNormalHom ↥(toIntermediateField (adjoin ↥L {x})) :
          (a : Gal(K/↥L)) → Gal(↥(adjoin ↥L {x}).toIntermediateField/↥L))
        σ =
      f),
  (f : (a : ↥(adjoin ↥L {x}).toIntermediateField) → ↥(adjoin ↥L {x}).toIntermediateField)
      (Subtype.mk (p := fun (x_1 : K) ↦
        Membership.mem (γ := IntermediateField (↥L) K) (toIntermediateField (adjoin ↥L {x})) x_1) x mem) =
    Subtype.mk (p := fun (x_1 : K) ↦
      Membership.mem (γ := IntermediateField (↥L) K) (toIntermediateField (adjoin ↥L {x})) x_1) x mem
Tactic: rcases restrictNormalHom_surjective K f with ⟨σ, hσ⟩
-- Lean 4 proof step from Mathlib.FieldTheory.Galois.Infinite (goal)
Goal:
k : Type u_1
K : Type u_2
inst : Field k
inst_1 : Field K
inst_2 : Algebra k K
L : IntermediateField k K
inst_3 : IsGalois k K
x : K
hx : ∀ f ∈ L.fixingSubgroup, (f : (a : K) → K) x = x
mem : Membership.mem (γ := IntermediateField (↥L) K) (toIntermediateField (adjoin ↥L {x})) x
this :
  Eq.{u_2 + 1} (α := IntermediateField ↥L ↥(toIntermediateField (adjoin ↥L {x}))) (IntermediateField.fixedField ⊤) ⊥
f : Gal(↥(toIntermediateField (adjoin ↥L {x}))/↥L)
h : Membership.mem (γ := Subgroup Gal(↥(toIntermediateField (adjoin ↥L {x}))/↥L)) ⊤ f
σ : Gal(K/↥L)
hσ :
  (restrictNormalHom ↥(toIntermediateField (adjoin ↥L {x})) :
        (a : Gal(K/↥L)) → Gal(↥(adjoin ↥L {x}).toIntermediateField/↥L))
      σ =
    f
⊢ (f : (a : ↥(adjoin ↥L {x}).toIntermediateField) → ↥(adjoin ↥L {x}).toIntermediateField)
      (Subtype.mk (p := fun (x_1 : K) ↦
        Membership.mem (γ := IntermediateField (↥L) K) (toIntermediateField (adjoin ↥L {x})) x_1) x mem) =
    Subtype.mk (p := fun (x_1 : K) ↦
      Membership.mem (γ := IntermediateField (↥L) K) (toIntermediateField (adjoin ↥L {x})) x_1) x mem
have a : (↑((f : (a : ↥(adjoin ↥L {x}).toIntermediateField) → ↥(adjoin ↥L {x}).toIntermediateField)
        (Subtype.mk (p := fun (x_1 : K) ↦
          Membership.mem (γ := IntermediateField (↥L) K) (toIntermediateField (adjoin ↥L {x})) x_1) x mem)) :
    K) =
  (↑(Subtype.mk (p := fun (x_1 : K) ↦
        Membership.mem (γ := IntermediateField (↥L) K) (toIntermediateField (adjoin ↥L {x})) x_1) x mem) :
    K)
Tactic: apply Subtype.val_injective
-- Lean 4 proof step from Mathlib.FieldTheory.Galois.Infinite (goal)
Goal:
k : Type u_1
K : Type u_2
inst : Field k
inst_1 : Field K
inst_2 : Algebra k K
L : IntermediateField k K
inst_3 : IsGalois k K
x : K
hx : ∀ f ∈ L.fixingSubgroup, (f : (a : K) → K) x = x
mem : Membership.mem (γ := IntermediateField (↥L) K) (toIntermediateField (adjoin ↥L {x})) x
this :
  Eq.{u_2 + 1} (α := IntermediateField ↥L ↥(toIntermediateField (adjoin ↥L {x}))) (IntermediateField.fixedField ⊤) ⊥
f : Gal(↥(toIntermediateField (adjoin ↥L {x}))/↥L)
h : Membership.mem (γ := Subgroup Gal(↥(toIntermediateField (adjoin ↥L {x}))/↥L)) ⊤ f
σ : Gal(K/↥L)
hσ :
  (restrictNormalHom ↥(toIntermediateField (adjoin ↥L {x})) :
        (a : Gal(K/↥L)) → Gal(↥(adjoin ↥L {x}).toIntermediateField/↥L))
      σ =
    f
⊢ (↑((f : (a : ↥(adjoin ↥L {x}).toIntermediateField) → ↥(adjoin ↥L {x}).toIntermediateField)
          (Subtype.mk (p := fun (x_1 : K) ↦
            Membership.mem (γ := IntermediateField (↥L) K) (toIntermediateField (adjoin ↥L {x})) x_1) x mem)) :
      K) =
    (↑(Subtype.mk (p := fun (x_1 : K) ↦
          Membership.mem (γ := IntermediateField (↥L) K) (toIntermediateField (adjoin ↥L {x})) x_1) x mem) :
      K)
have a : (σ : (a : K) → K)
    (↑(Subtype.mk (p := fun (x_1 : K) ↦
          Membership.mem (γ := IntermediateField (↥L) K) (toIntermediateField (adjoin ↥L {x})) x_1) x mem) :
      K) =
  (↑(Subtype.mk (p := fun (x_1 : K) ↦
        Membership.mem (γ := IntermediateField (↥L) K) (toIntermediateField (adjoin ↥L {x})) x_1) x mem) :
    K)
Tactic: rw [← hσ, restrictNormalHom_apply (adjoin L {x}).1 σ ⟨x, mem⟩]
-- Lean 4 proof step from Mathlib.FieldTheory.Galois.Infinite (hypothesis)
Goal:
k : Type u_1
K : Type u_2
inst : Field k
inst_1 : Field K
inst_2 : Algebra k K
L : IntermediateField k K
inst_3 : IsGalois k K
x : K
hx : ∀ f ∈ L.fixingSubgroup, (f : (a : K) → K) x = x
mem : Membership.mem (γ := IntermediateField (↥L) K) (toIntermediateField (adjoin ↥L {x})) x
this :
  Eq.{u_2 + 1} (α := IntermediateField ↥L ↥(toIntermediateField (adjoin ↥L {x}))) (IntermediateField.fixedField ⊤) ⊥
f : Gal(↥(toIntermediateField (adjoin ↥L {x}))/↥L)
h : Membership.mem (γ := Subgroup Gal(↥(toIntermediateField (adjoin ↥L {x}))/↥L)) ⊤ f
σ : Gal(K/↥L)
hσ :
  (restrictNormalHom ↥(toIntermediateField (adjoin ↥L {x})) :
        (a : Gal(K/↥L)) → Gal(↥(adjoin ↥L {x}).toIntermediateField/↥L))
      σ =
    f
⊢ (σ : (a : K) → K)
      (↑(Subtype.mk (p := fun (x_1 : K) ↦
            Membership.mem (γ := IntermediateField (↥L) K) (toIntermediateField (adjoin ↥L {x})) x_1) x mem) :
        K) =
    (↑(Subtype.mk (p := fun (x_1 : K) ↦
          Membership.mem (γ := IntermediateField (↥L) K) (toIntermediateField (adjoin ↥L {x})) x_1) x mem) :
      K)
have this₁ : (↑((L.fixingSubgroupEquiv.symm : (a : Gal(K/↥L)) → ↥L.fixingSubgroup) σ) : Gal(K/k)) ∈ L.fixingSubgroup →
  ((↑((L.fixingSubgroupEquiv.symm : (a : Gal(K/↥L)) → ↥L.fixingSubgroup) σ) : Gal(K/k)) : (a : K) → K) x = x
Tactic: have := hx ((IntermediateField.fixingSubgroupEquiv L).symm σ)
-- Lean 4 proof step from Mathlib.FieldTheory.Galois.Infinite (hypothesis)
Goal:
k : Type u_1
K : Type u_2
inst : Field k
inst_1 : Field K
inst_2 : Algebra k K
L : IntermediateField k K
inst_3 : IsGalois k K
x : K
hx : ∀ f ∈ L.fixingSubgroup, (f : (a : K) → K) x = x
mem : Membership.mem (γ := IntermediateField (↥L) K) (toIntermediateField (adjoin ↥L {x})) x
this :
  Eq.{u_2 + 1} (α := IntermediateField ↥L ↥(toIntermediateField (adjoin ↥L {x}))) (IntermediateField.fixedField ⊤) ⊥
⊢ x ∈ L
have this₁ : Membership.mem (γ := IntermediateField ↥L ↥(toIntermediateField (adjoin ↥L {x}))) ⊥
  (Subtype.mk (p := fun (x_1 : K) ↦
    Membership.mem (γ := IntermediateField (↥L) K) (toIntermediateField (adjoin ↥L {x})) x_1) x mem)
Tactic: have : ⟨x, mem⟩ ∈ (⊥ : IntermediateField L (adjoin L {x})) := by
      rw [← this, IntermediateField.mem_fixedField_iff]
      intro f _
      rcases restrictNormalHom_surjective K f with ⟨σ, hσ⟩
      apply Subtype.val_injective
      rw [← hσ, restrictNormalHom_apply (adjoin L {x}).1 σ ⟨x, mem⟩]
      have := hx ((IntermediateField.fixingSubgroupEquiv L).symm σ)
      simpa only [SetLike.coe_mem, true_implies]
-- Lean 4 proof step from Mathlib.FieldTheory.Galois.Infinite (goal)
Goal:
k : Type u_1
K : Type u_2
inst : Field k
inst_1 : Field K
inst_2 : Algebra k K
L : IntermediateField k K
inst_3 : IsGalois k K
x : K
hx : ∀ f ∈ L.fixingSubgroup, (f : (a : K) → K) x = x
mem : Membership.mem (γ := IntermediateField (↥L) K) (toIntermediateField (adjoin ↥L {x})) x
this_1 :
  Eq.{u_2 + 1} (α := IntermediateField ↥L ↥(toIntermediateField (adjoin ↥L {x}))) (IntermediateField.fixedField ⊤) ⊥
this :
  Membership.mem (γ := IntermediateField ↥L ↥(toIntermediateField (adjoin ↥L {x}))) ⊥
    (Subtype.mk (p := fun (x_1 : K) ↦
      Membership.mem (γ := IntermediateField (↥L) K) (toIntermediateField (adjoin ↥L {x})) x_1) x mem)
⊢ x ∈ L
have a : ∀ (y : ↥L)
  (hy :
    (algebraMap ↥L ↥(toIntermediateField (adjoin ↥L {x})) : (a : ↥L) → ↥(adjoin ↥L {x}).toIntermediateField) y =
      Subtype.mk (p := fun (x_1 : K) ↦
        Membership.mem (γ := IntermediateField (↥L) K) (toIntermediateField (adjoin ↥L {x})) x_1) x mem),
  x ∈ L
Tactic: rcases IntermediateField.mem_bot.mp this with ⟨y, hy⟩
-- Lean 4 proof step from Mathlib.FieldTheory.Galois.Infinite (goal)
Goal:
k : Type u_1
K : Type u_2
inst : Field k
inst_1 : Field K
inst_2 : Algebra k K
L : IntermediateField k K
inst_3 : IsGalois k K
x : K
hx : ∀ f ∈ L.fixingSubgroup, (f : (a : K) → K) x = x
mem : Membership.mem (γ := IntermediateField (↥L) K) (toIntermediateField (adjoin ↥L {x})) x
this_1 :
  Eq.{u_2 + 1} (α := IntermediateField ↥L ↥(toIntermediateField (adjoin ↥L {x}))) (IntermediateField.fixedField ⊤) ⊥
this :
  Membership.mem (γ := IntermediateField ↥L ↥(toIntermediateField (adjoin ↥L {x}))) ⊥
    (Subtype.mk (p := fun (x_1 : K) ↦
      Membership.mem (γ := IntermediateField (↥L) K) (toIntermediateField (adjoin ↥L {x})) x_1) x mem)
y : ↥L
hy :
  (algebraMap ↥L ↥(toIntermediateField (adjoin ↥L {x})) : (a : ↥L) → ↥(adjoin ↥L {x}).toIntermediateField) y =
    Subtype.mk (p := fun (x_1 : K) ↦
      Membership.mem (γ := IntermediateField (↥L) K) (toIntermediateField (adjoin ↥L {x})) x_1) x mem
⊢ x ∈ L
have a : ∀ (hx : ∀ f ∈ L.fixingSubgroup, (f : (a : K) → K) (↑y : K) = (↑y : K))
  (mem : Membership.mem (γ := IntermediateField (↥L) K) (toIntermediateField (adjoin ↥L {(↑y : K)})) (↑y : K))
  (this :
    Eq.{u_2 + 1} (α := IntermediateField ↥L ↥(toIntermediateField (adjoin ↥L {(↑y : K)})))
      (IntermediateField.fixedField ⊤) ⊥)
  (this :
    Membership.mem (γ := IntermediateField ↥L ↥(toIntermediateField (adjoin ↥L {(↑y : K)}))) ⊥
      (Subtype.mk (p := fun (x : K) ↦
        Membership.mem (γ := IntermediateField (↥L) K) (toIntermediateField (adjoin ↥L {(↑y : K)})) x) (↑y : K) mem))
  (hy :
    (algebraMap ↥L ↥(toIntermediateField (adjoin ↥L {(↑y : K)})) :
          (a : ↥L) → ↥(adjoin ↥L {(↑y : K)}).toIntermediateField)
        y =
      Subtype.mk (p := fun (x : K) ↦
        Membership.mem (γ := IntermediateField (↥L) K) (toIntermediateField (adjoin ↥L {(↑y : K)})) x) (↑y : K) mem),
  (↑y : K) ∈ L
Tactic: obtain ⟨rfl⟩ : y = x := congrArg Subtype.val hy
Cleared hypotheses: x, hx, mem, this_1, this, hy
-- Lean 4 proof step from Mathlib.FieldTheory.Galois.Infinite (goal)
Goal:
k : Type u_1
K : Type u_2
inst : Field k
inst_1 : Field K
inst_2 : Algebra k K
inst_3 : IsGalois k K
⊢ Eq.{u_2 + 1} (α := IntermediateField k K) (IntermediateField.fixedField ⊤) ⊥
have rw : Eq.{u_2 + 1} (α := IntermediateField k K) (IntermediateField.fixedField (IntermediateField.fixingSubgroup ⊥)) ⊥
Tactic: rw [← IntermediateField.fixingSubgroup_bot, fixedField_fixingSubgroup]
-- Lean 4 proof step from Mathlib.FieldTheory.Galois.Infinite (goal)
Goal:
k : Type u_1
K : Type u_2
inst : Field k
inst_1 : Field K
inst_2 : Algebra k K
inst_3 : IsGalois k K
rw : Eq.{u_2 + 1} (α := IntermediateField k K) (IntermediateField.fixedField (IntermediateField.fixingSubgroup ⊥)) ⊥
⊢ Eq.{u_2 + 1} (α := IntermediateField k K) (IntermediateField.fixedField ⊤) ⊥
have rw₁ : Eq.{u_2 + 1} (α := IntermediateField k K) ⊥ ⊥
Tactic: rw [← IntermediateField.fixingSubgroup_bot, fixedField_fixingSubgroup]
-- Lean 4 proof step from Mathlib.FieldTheory.Galois.Infinite (goal)
Goal:
k : Type u_1
K : Type u_2
inst : Field k
inst_1 : Field K
inst_2 : Algebra k K
H : Subgroup Gal(K/k)
L : IntermediateField k K
inst_3 : Normal.{u_1, u_2} k ↥L
⊢ fixedField H ⊓ L = lift (F := L) (fixedField (Subgroup.map (restrictNormalHom ↥L) H))
have h : (↑(fixedField H ⊓ L) : Set K) = (↑(lift (F := L) (fixedField (Subgroup.map (restrictNormalHom ↥L) H))) : Set K)
Tactic: apply SetLike.ext'
-- Lean 4 proof step from Mathlib.FieldTheory.Galois.Infinite (goal)
Goal:
k : Type u_1
K : Type u_2
inst : Field k
inst_1 : Field K
inst_2 : Algebra k K
H : Subgroup Gal(K/k)
L : IntermediateField k K
inst_3 : Normal.{u_1, u_2} k ↥L
⊢ (↑(fixedField H ⊓ L) : Set K) = (↑(lift (F := L) (fixedField (Subgroup.map (restrictNormalHom ↥L) H))) : Set K)
have h : ∀ (x : K),
  x ∈ (↑(fixedField H ⊓ L) : Set K) ↔
    x ∈ (↑(lift (F := L) (fixedField (Subgroup.map (restrictNormalHom ↥L) H))) : Set K)
Tactic: ext x
-- Lean 4 proof step from Mathlib.FieldTheory.Galois.Infinite (goal)
Goal:
k : Type u_1
K : Type u_2
inst : Field k
inst_1 : Field K
inst_2 : Algebra k K
H : Subgroup Gal(K/k)
L : IntermediateField k K
inst_3 : Normal.{u_1, u_2} k ↥L
x : K
⊢ x ∈ (↑(fixedField H ⊓ L) : Set K) ↔
    x ∈ (↑(lift (F := L) (fixedField (Subgroup.map (restrictNormalHom ↥L) H))) : Set K)
have h : ∀ (h : x ∈ (↑(fixedField H ⊓ L) : Set K)),
  x ∈ (↑(lift (F := L) (fixedField (Subgroup.map (restrictNormalHom ↥L) H))) : Set K)
Tactic: refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩
-- Lean 4 proof step from Mathlib.FieldTheory.Galois.Infinite (goal)
Goal:
k : Type u_1
K : Type u_2
inst : Field k
inst_1 : Field K
inst_2 : Algebra k K
H : Subgroup Gal(K/k)
L : IntermediateField k K
inst_3 : Normal.{u_1, u_2} k ↥L
x : K
h :
  ∀ (h : x ∈ (↑(fixedField H ⊓ L) : Set K)),
    x ∈ (↑(lift (F := L) (fixedField (Subgroup.map (restrictNormalHom ↥L) H))) : Set K)
⊢ x ∈ (↑(fixedField H ⊓ L) : Set K) ↔
    x ∈ (↑(lift (F := L) (fixedField (Subgroup.map (restrictNormalHom ↥L) H))) : Set K)
have h₁ : ∀ (h : x ∈ (↑(lift (F := L) (fixedField (Subgroup.map (restrictNormalHom ↥L) H))) : Set K)),
  x ∈ (↑(fixedField H ⊓ L) : Set K)
Tactic: refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩
-- Lean 4 proof step from Mathlib.FieldTheory.Galois.Infinite (hypothesis)
Goal:
k : Type u_1
K : Type u_2
inst : Field k
inst_1 : Field K
inst_2 : Algebra k K
H : Subgroup Gal(K/k)
L : IntermediateField k K
inst_3 : Normal.{u_1, u_2} k ↥L
x : K
h : x ∈ (↑(fixedField H ⊓ L) : Set K)
⊢ x ∈ (↑(lift (F := L) (fixedField (Subgroup.map (restrictNormalHom ↥L) H))) : Set K)
have xL : Membership.mem (γ := Set K) ((fun (x : IntermediateField k K) ↦ (↑x : Set K)) L) x
Tactic: have xL := h.out.2