awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- 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 : ClosedSubgroup Gal(K/k)
inst_3 : IsGalois k K
σ : Gal(K/k)
hσ : σ ∈ (fixedField (↑H : Subgroup Gal(K/k))).fixingSubgroup
h_1 : σ ∉ (↑H : Subgroup Gal(K/k))
b : Set Gal(K/k)
sub : b ⊆ (fun (y : Gal(K/k)) ↦ σ * y) ⁻¹' (↑H : Subgroup Gal(K/k)).carrierᶜ
gp : Subgroup Gal(K/k)
L : IntermediateField k K
eq_1 : (fun (g : Subgroup Gal(K/k)) ↦ g.carrier) L.fixingSubgroup = b
hL : Membership.mem (γ := Set (IntermediateField k K)) (finiteExts k K) L
eq' : L.fixingSubgroup = gp
this : FiniteDimensional.{u_1, u_2} k ↥L
L' : FiniteGaloisIntermediateField k K :=
  let __src : IntermediateField k K := normalClosure k (↥L) K;
  { toIntermediateField := __src, finiteDimensional := ⋯, isGalois := ⋯ }
compl : σ • L'.fixingSubgroup.carrier ⊆ (↑H : Subgroup Gal(K/k)).carrierᶜ
fix :
  ∀ (x : K),
    Membership.mem (γ := IntermediateField k K)
        (lift (F := L'.toIntermediateField)
          (fixedField (Subgroup.map (restrictNormalHom ↥L'.toIntermediateField) (↑H : Subgroup Gal(K/k)))))
        x →
      (σ : (a : K) → K) x = x
h : Gal(K/k)
mem : h ∈ (↑(↑H : Subgroup Gal(K/k)) : Set Gal(K/k))
eq :
  Eq (α := Gal(↥L'.toIntermediateField/k))
    ((restrictNormalHom ↥L'.toIntermediateField : (a : Gal(K/k)) → Gal(↥L'.toIntermediateField/k)) h :
      Gal(↥L'.toIntermediateField/k))
    ((restrictNormalHom ↥L'.toIntermediateField : (a : Gal(K/k)) → Gal(↥L'.toIntermediateField/k)) σ :
      Gal(↥L'.toIntermediateField/k))
⊢ False
have this₁ : h ∈ σ • L'.fixingSubgroup.carrier
Tactic: have : h ∈ σ • L'.1.fixingSubgroup.carrier := by
    use σ⁻¹ * h
    simp only [Subsemigroup.mem_carrier, Submonoid.mem_toSubsemigroup, Subgroup.mem_toSubmonoid,
      smul_eq_mul, mul_inv_cancel_left, and_true]
    apply (mem_fixingSubgroup_iff Gal(K/k)).mpr
    intro y hy
    simp only [AlgEquiv.smul_def, AlgEquiv.mul_apply]
    have : ((restrictNormalHom L') h ⟨y,hy⟩).1 = ((restrictNormalHom L') σ ⟨y,hy⟩).1 := by rw [eq]
    rw [restrictNormalHom_apply L'.1 h ⟨y, hy⟩, restrictNormalHom_apply L'.1 σ ⟨y, hy⟩] at this
    simp only [this, ← AlgEquiv.mul_apply, inv_mul_cancel, one_apply]
-- 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 : ClosedSubgroup Gal(K/k)
inst_3 : IsGalois k K
σ : Gal(K/k)
hσ : σ ∈ (fixedField (↑H : Subgroup Gal(K/k))).fixingSubgroup
h_1 : σ ∉ (↑H : Subgroup Gal(K/k))
b : Set Gal(K/k)
sub : b ⊆ (fun (y : Gal(K/k)) ↦ σ * y) ⁻¹' (↑H : Subgroup Gal(K/k)).carrierᶜ
gp : Subgroup Gal(K/k)
L : IntermediateField k K
eq_1 : (fun (g : Subgroup Gal(K/k)) ↦ g.carrier) L.fixingSubgroup = b
hL : Membership.mem (γ := Set (IntermediateField k K)) (finiteExts k K) L
eq' : L.fixingSubgroup = gp
this_1 : FiniteDimensional.{u_1, u_2} k ↥L
L' : FiniteGaloisIntermediateField k K :=
  let __src : IntermediateField k K := normalClosure k (↥L) K;
  { toIntermediateField := __src, finiteDimensional := ⋯, isGalois := ⋯ }
compl : σ • L'.fixingSubgroup.carrier ⊆ (↑H : Subgroup Gal(K/k)).carrierᶜ
fix :
  ∀ (x : K),
    Membership.mem (γ := IntermediateField k K)
        (lift (F := L'.toIntermediateField)
          (fixedField (Subgroup.map (restrictNormalHom ↥L'.toIntermediateField) (↑H : Subgroup Gal(K/k)))))
        x →
      (σ : (a : K) → K) x = x
h : Gal(K/k)
mem : h ∈ (↑(↑H : Subgroup Gal(K/k)) : Set Gal(K/k))
eq :
  Eq (α := Gal(↥L'.toIntermediateField/k))
    ((restrictNormalHom ↥L'.toIntermediateField : (a : Gal(K/k)) → Gal(↥L'.toIntermediateField/k)) h :
      Gal(↥L'.toIntermediateField/k))
    ((restrictNormalHom ↥L'.toIntermediateField : (a : Gal(K/k)) → Gal(↥L'.toIntermediateField/k)) σ :
      Gal(↥L'.toIntermediateField/k))
this : h ∈ σ • L'.fixingSubgroup.carrier
⊢ False
have absurd : ¬σ • L'.fixingSubgroup.carrier ⊆ (↑H : Subgroup Gal(K/k)).carrierᶜ
Tactic: absurd compl
-- 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 : ClosedSubgroup Gal(K/k)
inst_3 : IsGalois k K
σ : Gal(K/k)
hσ : σ ∈ (fixedField (↑H : Subgroup Gal(K/k))).fixingSubgroup
h_1 : σ ∉ (↑H : Subgroup Gal(K/k))
b : Set Gal(K/k)
sub : b ⊆ (fun (y : Gal(K/k)) ↦ σ * y) ⁻¹' (↑H : Subgroup Gal(K/k)).carrierᶜ
gp : Subgroup Gal(K/k)
L : IntermediateField k K
eq_1 : (fun (g : Subgroup Gal(K/k)) ↦ g.carrier) L.fixingSubgroup = b
hL : Membership.mem (γ := Set (IntermediateField k K)) (finiteExts k K) L
eq' : L.fixingSubgroup = gp
this_1 : FiniteDimensional.{u_1, u_2} k ↥L
L' : FiniteGaloisIntermediateField k K :=
  let __src : IntermediateField k K := normalClosure k (↥L) K;
  { toIntermediateField := __src, finiteDimensional := ⋯, isGalois := ⋯ }
compl : σ • L'.fixingSubgroup.carrier ⊆ (↑H : Subgroup Gal(K/k)).carrierᶜ
fix :
  ∀ (x : K),
    Membership.mem (γ := IntermediateField k K)
        (lift (F := L'.toIntermediateField)
          (fixedField (Subgroup.map (restrictNormalHom ↥L'.toIntermediateField) (↑H : Subgroup Gal(K/k)))))
        x →
      (σ : (a : K) → K) x = x
h : Gal(K/k)
mem : h ∈ (↑(↑H : Subgroup Gal(K/k)) : Set Gal(K/k))
eq :
  Eq (α := Gal(↥L'.toIntermediateField/k))
    ((restrictNormalHom ↥L'.toIntermediateField : (a : Gal(K/k)) → Gal(↥L'.toIntermediateField/k)) h :
      Gal(↥L'.toIntermediateField/k))
    ((restrictNormalHom ↥L'.toIntermediateField : (a : Gal(K/k)) → Gal(↥L'.toIntermediateField/k)) σ :
      Gal(↥L'.toIntermediateField/k))
this : h ∈ σ • L'.fixingSubgroup.carrier
⊢ ¬σ • L'.fixingSubgroup.carrier ⊆ (↑H : Subgroup Gal(K/k)).carrierᶜ
have apply : ∃ a ∈ σ • L'.fixingSubgroup.carrier, a ∉ (↑H : Subgroup Gal(K/k)).carrierᶜ
Tactic: apply Set.not_subset.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
H : ClosedSubgroup Gal(K/k)
inst_3 : IsGalois k K
σ : Gal(K/k)
hσ : σ ∈ (fixedField (↑H : Subgroup Gal(K/k))).fixingSubgroup
h_1 : σ ∉ (↑H : Subgroup Gal(K/k))
b : Set Gal(K/k)
sub : b ⊆ (fun (y : Gal(K/k)) ↦ σ * y) ⁻¹' (↑H : Subgroup Gal(K/k)).carrierᶜ
gp : Subgroup Gal(K/k)
L : IntermediateField k K
eq_1 : (fun (g : Subgroup Gal(K/k)) ↦ g.carrier) L.fixingSubgroup = b
hL : Membership.mem (γ := Set (IntermediateField k K)) (finiteExts k K) L
eq' : L.fixingSubgroup = gp
this_1 : FiniteDimensional.{u_1, u_2} k ↥L
L' : FiniteGaloisIntermediateField k K :=
  let __src : IntermediateField k K := normalClosure k (↥L) K;
  { toIntermediateField := __src, finiteDimensional := ⋯, isGalois := ⋯ }
compl : σ • L'.fixingSubgroup.carrier ⊆ (↑H : Subgroup Gal(K/k)).carrierᶜ
fix :
  ∀ (x : K),
    Membership.mem (γ := IntermediateField k K)
        (lift (F := L'.toIntermediateField)
          (fixedField (Subgroup.map (restrictNormalHom ↥L'.toIntermediateField) (↑H : Subgroup Gal(K/k)))))
        x →
      (σ : (a : K) → K) x = x
h : Gal(K/k)
mem : h ∈ (↑(↑H : Subgroup Gal(K/k)) : Set Gal(K/k))
eq :
  Eq (α := Gal(↥L'.toIntermediateField/k))
    ((restrictNormalHom ↥L'.toIntermediateField : (a : Gal(K/k)) → Gal(↥L'.toIntermediateField/k)) h :
      Gal(↥L'.toIntermediateField/k))
    ((restrictNormalHom ↥L'.toIntermediateField : (a : Gal(K/k)) → Gal(↥L'.toIntermediateField/k)) σ :
      Gal(↥L'.toIntermediateField/k))
this : h ∈ σ • L'.fixingSubgroup.carrier
⊢ ∃ a ∈ σ • L'.fixingSubgroup.carrier, a ∉ (↑H : Subgroup Gal(K/k)).carrierᶜ
have h₁ : h ∈ σ • L'.fixingSubgroup.carrier ∧ h ∉ (↑H : Subgroup Gal(K/k)).carrierᶜ
Tactic: use 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
inst_3 : IsGalois k K
H : (ClosedSubgroup Gal(K/k))ᵒᵈ
⊢ (fun (L : IntermediateField k K) ↦ { toSubgroup := L.fixingSubgroup, isClosed' := ⋯ : ClosedSubgroup Gal(K/k) })
      ((fun (H : (ClosedSubgroup Gal(K/k))ᵒᵈ) ↦ IntermediateField.fixedField (↑H : Subgroup Gal(K/k))) H) =
    H
have simp_rw : { toSubgroup := (↑H : Subgroup Gal(K/k)), isClosed' := sorry : ClosedSubgroup Gal(K/k) } = H
Tactic: simp_rw [fixingSubgroup_fixedField H]
-- Lean 4 proof step from Mathlib.FieldTheory.Galois.Infinite (goal)
Goal:
k : Type u_1
K_1 : Type u_2
inst : Field k
inst_1 : Field K_1
inst_2 : Algebra k K_1
inst_3 : IsGalois k K_1
K L : IntermediateField k K_1
⊢ LE.le (α := (ClosedSubgroup Gal(K_1/k))ᵒᵈ)
      (({ toFun := fun (L : IntermediateField k K_1) ↦ { toSubgroup := L.fixingSubgroup, isClosed' := ⋯ },
              invFun := fun (H : (ClosedSubgroup Gal(K_1/k))ᵒᵈ) ↦
                IntermediateField.fixedField (↑H : Subgroup Gal(K_1/k)),
              left_inv := ⋯, right_inv := ⋯ } :
            (a : IntermediateField k K_1) → (ClosedSubgroup Gal(K_1/k))ᵒᵈ)
          K :
        (ClosedSubgroup Gal(K_1/k))ᵒᵈ)
      (({ toFun := fun (L : IntermediateField k K_1) ↦ { toSubgroup := L.fixingSubgroup, isClosed' := ⋯ },
              invFun := fun (H : (ClosedSubgroup Gal(K_1/k))ᵒᵈ) ↦
                IntermediateField.fixedField (↑H : Subgroup Gal(K_1/k)),
              left_inv := ⋯, right_inv := ⋯ } :
            (a : IntermediateField k K_1) → (ClosedSubgroup Gal(K_1/k))ᵒᵈ)
          L :
        (ClosedSubgroup Gal(K_1/k))ᵒᵈ) ↔
    K ≤ L
have rw : LE.le (α := (ClosedSubgroup Gal(K_1/k))ᵒᵈ)
    (({ toFun := fun (L : IntermediateField k K_1) ↦ { toSubgroup := L.fixingSubgroup, isClosed' := sorry },
            invFun := fun (H : (ClosedSubgroup Gal(K_1/k))ᵒᵈ) ↦ IntermediateField.fixedField (↑H : Subgroup Gal(K_1/k)),
            left_inv := sorry, right_inv := sorry } :
          (a : IntermediateField k K_1) → (ClosedSubgroup Gal(K_1/k))ᵒᵈ)
        K :
      (ClosedSubgroup Gal(K_1/k))ᵒᵈ)
    (({ toFun := fun (L : IntermediateField k K_1) ↦ { toSubgroup := L.fixingSubgroup, isClosed' := sorry },
            invFun := fun (H : (ClosedSubgroup Gal(K_1/k))ᵒᵈ) ↦ IntermediateField.fixedField (↑H : Subgroup Gal(K_1/k)),
            left_inv := sorry, right_inv := sorry } :
          (a : IntermediateField k K_1) → (ClosedSubgroup Gal(K_1/k))ᵒᵈ)
        L :
      (ClosedSubgroup Gal(K_1/k))ᵒᵈ) ↔
  L.fixingSubgroup ≤ K.fixingSubgroup
Tactic: rw [← fixedField_fixingSubgroup L, IntermediateField.le_iff_le, fixedField_fixingSubgroup L]
-- 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
⊢ IsOpen L.fixingSubgroup.carrier ↔ FiniteDimensional.{u_1, u_2} k ↥L
have refine : ∀ (h : IsOpen L.fixingSubgroup.carrier), FiniteDimensional.{u_1, u_2} k ↥L
Tactic: refine ⟨fun h ↦ ?_, fun h ↦ IntermediateField.fixingSubgroup_isOpen L⟩
-- 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
h : IsOpen L.fixingSubgroup.carrier
⊢ FiniteDimensional.{u_1, u_2} k ↥L
have this : Membership.mem (γ := Filter Gal(K/k)) (nhds 1)
  (↑(Equiv.toFun
          (RelIso.toEquiv (r := fun (x1 x2 : IntermediateField k K) ↦ x1 ≤ x2) (s :=
            fun (x1 x2 : (ClosedSubgroup Gal(K/k))ᵒᵈ) ↦ x1 ≤ x2) IntermediateFieldEquivClosedSubgroup)
          L) :
      Subgroup Gal(K/k)).carrier
Tactic: have : (IntermediateFieldEquivClosedSubgroup.toFun L).carrier ∈ nhds 1 :=
    IsOpen.mem_nhds h (congrFun 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
h : IsOpen L.fixingSubgroup.carrier
this :
  Membership.mem (γ := Filter Gal(K/k)) (nhds 1)
    (↑(Equiv.toFun
            (RelIso.toEquiv (r := fun (x1 x2 : IntermediateField k K) ↦ x1 ≤ x2) (s :=
              fun (x1 x2 : (ClosedSubgroup Gal(K/k))ᵒᵈ) ↦ x1 ≤ x2) IntermediateFieldEquivClosedSubgroup)
            L) :
        Subgroup Gal(K/k)).carrier
⊢ FiniteDimensional.{u_1, u_2} k ↥L
have this₁ : Membership.mem (γ := Filter Gal(K/k)) (FilterBasis.filter (@GroupFilterBasis.toFilterBasis _ aut _))
  (↑(Equiv.toFun
          (RelIso.toEquiv (r := fun (x1 x2 : IntermediateField k K) ↦ x1 ≤ x2) (s :=
            fun (x1 x2 : (ClosedSubgroup Gal(K/k))ᵒᵈ) ↦ x1 ≤ x2) IntermediateFieldEquivClosedSubgroup)
          L) :
      Subgroup Gal(K/k)).carrier
Tactic: rw [GroupFilterBasis.nhds_one_eq] at this
Cleared hypotheses: this
-- 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
h : IsOpen L.fixingSubgroup.carrier
this :
  Membership.mem (γ := Filter Gal(K/k)) (FilterBasis.filter (@GroupFilterBasis.toFilterBasis _ aut _))
    (↑(Equiv.toFun
            (RelIso.toEquiv (r := fun (x1 x2 : IntermediateField k K) ↦ x1 ≤ x2) (s :=
              fun (x1 x2 : (ClosedSubgroup Gal(K/k))ᵒᵈ) ↦ x1 ≤ x2) IntermediateFieldEquivClosedSubgroup)
            L) :
        Subgroup Gal(K/k)).carrier
⊢ FiniteDimensional.{u_1, u_2} k ↥L
have rcases : ∀ (S : Set Gal(K/k))
  (sub :
    S ⊆
      (↑(Equiv.toFun
              (RelIso.toEquiv (r := fun (x1 x2 : IntermediateField k K) ↦ x1 ≤ x2) (s :=
                fun (x1 x2 : (ClosedSubgroup Gal(K/k))ᵒᵈ) ↦ x1 ≤ x2) IntermediateFieldEquivClosedSubgroup)
              L) :
          Subgroup Gal(K/k)).carrier)
  (gp : Subgroup Gal(K/k)) (eq : (fun (g : Subgroup Gal(K/k)) ↦ g.carrier) gp = S) (M : IntermediateField k K)
  (hM : Membership.mem (γ := Set (IntermediateField k K)) (finiteExts k K) M) (eq' : M.fixingSubgroup = gp),
  FiniteDimensional.{u_1, u_2} k ↥L
Tactic: rcases this with ⟨S, ⟨gp, ⟨M, hM, eq'⟩, eq⟩, sub⟩
Cleared hypotheses: this
-- 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
h : IsOpen L.fixingSubgroup.carrier
S : Set Gal(K/k)
sub :
  S ⊆
    (↑(Equiv.toFun
            (RelIso.toEquiv (r := fun (x1 x2 : IntermediateField k K) ↦ x1 ≤ x2) (s :=
              fun (x1 x2 : (ClosedSubgroup Gal(K/k))ᵒᵈ) ↦ x1 ≤ x2) IntermediateFieldEquivClosedSubgroup)
            L) :
        Subgroup Gal(K/k)).carrier
gp : Subgroup Gal(K/k)
eq : (fun (g : Subgroup Gal(K/k)) ↦ g.carrier) gp = S
M : IntermediateField k K
hM : Membership.mem (γ := Set (IntermediateField k K)) (finiteExts k K) M
eq' : M.fixingSubgroup = gp
⊢ FiniteDimensional.{u_1, u_2} k ↥L
have rw : ∀ (eq : (fun (g : Subgroup Gal(K/k)) ↦ g.carrier) gp = S) (M : IntermediateField k K)
  (sub :
    (fun (g : Subgroup Gal(K/k)) ↦ g.carrier) M.fixingSubgroup ⊆
      (↑(Equiv.toFun
              (RelIso.toEquiv (r := fun (x1 x2 : IntermediateField k K) ↦ x1 ≤ x2) (s :=
                fun (x1 x2 : (ClosedSubgroup Gal(K/k))ᵒᵈ) ↦ x1 ≤ x2) IntermediateFieldEquivClosedSubgroup)
              L) :
          Subgroup Gal(K/k)).carrier)
  (hM : Membership.mem (γ := Set (IntermediateField k K)) (finiteExts k K) M) (eq' : M.fixingSubgroup = gp),
  FiniteDimensional.{u_1, u_2} k ↥L
Tactic: rw [← eq, ← eq'] at sub
Cleared hypotheses: sub, eq, M, hM, eq'
-- 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
h : IsOpen L.fixingSubgroup.carrier
S : Set Gal(K/k)
gp : Subgroup Gal(K/k)
eq : (fun (g : Subgroup Gal(K/k)) ↦ g.carrier) gp = S
M : IntermediateField k K
sub :
  (fun (g : Subgroup Gal(K/k)) ↦ g.carrier) M.fixingSubgroup ⊆
    (↑(Equiv.toFun
            (RelIso.toEquiv (r := fun (x1 x2 : IntermediateField k K) ↦ x1 ≤ x2) (s :=
              fun (x1 x2 : (ClosedSubgroup Gal(K/k))ᵒᵈ) ↦ x1 ≤ x2) IntermediateFieldEquivClosedSubgroup)
            L) :
        Subgroup Gal(K/k)).carrier
hM : Membership.mem (γ := Set (IntermediateField k K)) (finiteExts k K) M
eq' : M.fixingSubgroup = gp
⊢ FiniteDimensional.{u_1, u_2} k ↥L
have this : FiniteDimensional.{u_1, u_2} k ↥M
Tactic: have := hM.out
-- 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
h : IsOpen L.fixingSubgroup.carrier
S : Set Gal(K/k)
gp : Subgroup Gal(K/k)
eq : (fun (g : Subgroup Gal(K/k)) ↦ g.carrier) gp = S
M : IntermediateField k K
sub :
  (fun (g : Subgroup Gal(K/k)) ↦ g.carrier) M.fixingSubgroup ⊆
    (↑(Equiv.toFun
            (RelIso.toEquiv (r := fun (x1 x2 : IntermediateField k K) ↦ x1 ≤ x2) (s :=
              fun (x1 x2 : (ClosedSubgroup Gal(K/k))ᵒᵈ) ↦ x1 ≤ x2) IntermediateFieldEquivClosedSubgroup)
            L) :
        Subgroup Gal(K/k)).carrier
hM : Membership.mem (γ := Set (IntermediateField k K)) (finiteExts k K) M
eq' : M.fixingSubgroup = gp
this : FiniteDimensional.{u_1, u_2} k ↥M
L' : FiniteGaloisIntermediateField k K :=
  let __src : IntermediateField k K := normalClosure k (↥M) K;
  { toIntermediateField := __src, finiteDimensional := ⋯, isGalois := ⋯ }
⊢ L ≤ L'.toIntermediateField
have apply : L ≤ M
Tactic: apply le_trans _ (IntermediateField.le_normalClosure M)
-- 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
h : IsOpen L.fixingSubgroup.carrier
S : Set Gal(K/k)
gp : Subgroup Gal(K/k)
eq : (fun (g : Subgroup Gal(K/k)) ↦ g.carrier) gp = S
M : IntermediateField k K
sub :
  (fun (g : Subgroup Gal(K/k)) ↦ g.carrier) M.fixingSubgroup ⊆
    (↑(Equiv.toFun
            (RelIso.toEquiv (r := fun (x1 x2 : IntermediateField k K) ↦ x1 ≤ x2) (s :=
              fun (x1 x2 : (ClosedSubgroup Gal(K/k))ᵒᵈ) ↦ x1 ≤ x2) IntermediateFieldEquivClosedSubgroup)
            L) :
        Subgroup Gal(K/k)).carrier
hM : Membership.mem (γ := Set (IntermediateField k K)) (finiteExts k K) M
eq' : M.fixingSubgroup = gp
this : FiniteDimensional.{u_1, u_2} k ↥M
L' : FiniteGaloisIntermediateField k K :=
  let __src : IntermediateField k K := normalClosure k (↥M) K;
  { toIntermediateField := __src, finiteDimensional := ⋯, isGalois := ⋯ }
⊢ L ≤ M
have rw : M.fixingSubgroup ≤ L.fixingSubgroup
Tactic: rw [← fixedField_fixingSubgroup M, IntermediateField.le_iff_le]
-- 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
h : IsOpen L.fixingSubgroup.carrier
S : Set Gal(K/k)
gp : Subgroup Gal(K/k)
eq : (fun (g : Subgroup Gal(K/k)) ↦ g.carrier) gp = S
M : IntermediateField k K
sub :
  (fun (g : Subgroup Gal(K/k)) ↦ g.carrier) M.fixingSubgroup ⊆
    (↑(Equiv.toFun
            (RelIso.toEquiv (r := fun (x1 x2 : IntermediateField k K) ↦ x1 ≤ x2) (s :=
              fun (x1 x2 : (ClosedSubgroup Gal(K/k))ᵒᵈ) ↦ x1 ≤ x2) IntermediateFieldEquivClosedSubgroup)
            L) :
        Subgroup Gal(K/k)).carrier
hM : Membership.mem (γ := Set (IntermediateField k K)) (finiteExts k K) M
eq' : M.fixingSubgroup = gp
this : FiniteDimensional.{u_1, u_2} k ↥M
L' : FiniteGaloisIntermediateField k K :=
  let __src : IntermediateField k K := normalClosure k (↥M) K;
  { toIntermediateField := __src, finiteDimensional := ⋯, isGalois := ⋯ }
⊢ FiniteDimensional.{u_1, u_2} k ↥L
have this₁ : L ≤ L'.toIntermediateField
Tactic: have : L ≤ L'.1 := by
    apply le_trans _ (IntermediateField.le_normalClosure M)
    rw [← fixedField_fixingSubgroup M, IntermediateField.le_iff_le]
    exact sub
-- 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
⊢ L.fixingSubgroup.Normal ↔ IsGalois.{u_1, u_2} k ↥L
have refine_1 : ∀ (h : L.fixingSubgroup.Normal), IsGalois.{u_1, u_2} k ↥L
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
L : IntermediateField k K
inst_3 : IsGalois k K
refine_1 : ∀ (h : L.fixingSubgroup.Normal), IsGalois.{u_1, u_2} k ↥L
⊢ L.fixingSubgroup.Normal ↔ IsGalois.{u_1, u_2} k ↥L
have refine_2 : ∀ (h : IsGalois.{u_1, u_2} k ↥L), L.fixingSubgroup.Normal
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
L : IntermediateField k K
inst_3 : IsGalois k K
h : L.fixingSubgroup.Normal
g : (x : K) → Subgroup Gal(↥(toIntermediateField (adjoin k {x}))/k) :=
  fun (x : K) ↦ Subgroup.map (restrictNormalHom ↥(toIntermediateField (adjoin k {x}))) L.fixingSubgroup
f : (x : ↥L) → IntermediateField k K :=
  fun (x : ↥L) ↦
    IntermediateField.lift (F := toIntermediateField (adjoin k {(↑x : K)})) (IntermediateField.fixedField (g (↑x : K)))
⊢ IsGalois.{u_1, u_2} k ↥L
have this : ∀ (x : K), (g x).Normal
Tactic: have (x : K) : (g x).Normal :=
      Subgroup.Normal.map h (restrictNormalHom (adjoin k {x})) (restrictNormalHom_surjective K)
-- 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
h : L.fixingSubgroup.Normal
g : (x : K) → Subgroup Gal(↥(toIntermediateField (adjoin k {x}))/k) :=
  fun (x : K) ↦ Subgroup.map (restrictNormalHom ↥(toIntermediateField (adjoin k {x}))) L.fixingSubgroup
f : (x : ↥L) → IntermediateField k K :=
  fun (x : ↥L) ↦
    IntermediateField.lift (F := toIntermediateField (adjoin k {(↑x : K)})) (IntermediateField.fixedField (g (↑x : K)))
this : ∀ (x : K), (g x).Normal
⊢ IsGalois.{u_1, u_2} k ↥L
have this₁ : ∀ (l : ↥L), Normal.{u_1, u_2} k ↥(f l)
Tactic: have (l : L) : Normal k (f l) :=
      Normal.of_algEquiv <| IntermediateField.liftAlgEquiv <| IntermediateField.fixedField (g l.1)
-- 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
h : L.fixingSubgroup.Normal
g : (x : K) → Subgroup Gal(↥(toIntermediateField (adjoin k {x}))/k) :=
  fun (x : K) ↦ Subgroup.map (restrictNormalHom ↥(toIntermediateField (adjoin k {x}))) L.fixingSubgroup
f : (x : ↥L) → IntermediateField k K :=
  fun (x : ↥L) ↦
    IntermediateField.lift (F := toIntermediateField (adjoin k {(↑x : K)})) (IntermediateField.fixedField (g (↑x : K)))
this_1 : ∀ (x : K), (g x).Normal
this : ∀ (l : ↥L), Normal.{u_1, u_2} k ↥(f l)
⊢ IsGalois.{u_1, u_2} k ↥L
have n : Normal.{u_1, u_2} k ↥(⨆ (l : ↥L), f l)
Tactic: have n : Normal k ↥(⨆ l : L, f l) := IntermediateField.normal_iSup k K f
-- 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
h : L.fixingSubgroup.Normal
g : (x : K) → Subgroup Gal(↥(toIntermediateField (adjoin k {x}))/k) :=
  fun (x : K) ↦ Subgroup.map (restrictNormalHom ↥(toIntermediateField (adjoin k {x}))) L.fixingSubgroup
f : (x : ↥L) → IntermediateField k K :=
  fun (x : ↥L) ↦
    IntermediateField.lift (F := toIntermediateField (adjoin k {(↑x : K)})) (IntermediateField.fixedField (g (↑x : K)))
this_1 : ∀ (x : K), (g x).Normal
this : ∀ (l : ↥L), Normal.{u_1, u_2} k ↥(f l)
n : Normal.{u_1, u_2} k ↥(⨆ (l : ↥L), f l)
⊢ ⨆ (l : ↥L), f l = L
have a : ⨆ (l : ↥L), f l ≤ L
Tactic: apply le_antisymm