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
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)
xL : Membership.mem (γ := Set K) ((fun (x : IntermediateField k K) ↦ (↑x : Set K)) L) x
⊢ x ∈ (↑(lift (F := L) (fixedField (Subgroup.map (restrictNormalHom ↥L) H))) : Set K)
have h₁ : Membership.mem (γ := IntermediateField k ↥L) (fixedField (Subgroup.map (restrictNormalHom ↥L) H))
  (Subtype.mk (p := fun (x : K) ↦ x ∈ L) x xL)
Tactic: apply (mem_lift (⟨x, xL⟩ : L)).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 : 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)
xL : Membership.mem (γ := Set K) ((fun (x : IntermediateField k K) ↦ (↑x : Set K)) L) x
⊢ Membership.mem (γ := IntermediateField k ↥L) (fixedField (Subgroup.map (restrictNormalHom ↥L) H))
    (Subtype.mk (p := fun (x : K) ↦ x ∈ L) x xL)
have h₁ : ∀ a ∈ H,
  (((restrictNormalHom ↥L : (a : Gal(K/k)) → Gal(↥L/k)) a : Gal(↥L/k)) : (a : ↥L) → ↥L)
      (Subtype.mk (p := fun (x : K) ↦ x ∈ L) x xL) =
    Subtype.mk (p := fun (x : K) ↦ x ∈ L) x xL
Tactic: simp only [mem_fixedField_iff, Subgroup.mem_map, forall_exists_index, and_imp,
      forall_apply_eq_imp_iff₂]
-- 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 : x ∈ (↑(fixedField H ⊓ L) : Set K)
xL : Membership.mem (γ := Set K) ((fun (x : IntermediateField k K) ↦ (↑x : Set K)) L) x
σ : Gal(K/k)
hσ : σ ∈ H
⊢ (((restrictNormalHom ↥L : (a : Gal(K/k)) → Gal(↥L/k)) σ : Gal(↥L/k)) : (a : ↥L) → ↥L)
      (Subtype.mk (p := fun (x : K) ↦ x ∈ L) x xL) =
    Subtype.mk (p := fun (x : K) ↦ x ∈ L) x xL
have h₁ : (↑((((restrictNormalHom ↥L : (a : Gal(K/k)) → Gal(↥L/k)) σ : Gal(↥L/k)) : (a : ↥L) → ↥L)
        (Subtype.mk (p := fun (x : K) ↦ x ∈ L) x xL)) :
    K) =
  (↑(Subtype.mk (p := fun (x : K) ↦ x ∈ L) x xL) : 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
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)
xL : Membership.mem (γ := Set K) ((fun (x : IntermediateField k K) ↦ (↑x : Set K)) L) x
σ : Gal(K/k)
hσ : σ ∈ H
⊢ (↑((((restrictNormalHom ↥L : (a : Gal(K/k)) → Gal(↥L/k)) σ : Gal(↥L/k)) : (a : ↥L) → ↥L)
          (Subtype.mk (p := fun (x : K) ↦ x ∈ L) x xL)) :
      K) =
    (↑(Subtype.mk (p := fun (x : K) ↦ x ∈ L) x xL) : K)
have h₁ : (↑((((restrictNormalHom ↥L : (a : Gal(K/k)) → Gal(↥L/k)) σ : Gal(↥L/k)) : (a : ↥L) → ↥L)
        (Subtype.mk (p := fun (x : K) ↦ x ∈ L) x xL)) :
    K) =
  x
Tactic: dsimp only
-- 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 : x ∈ (↑(fixedField H ⊓ L) : Set K)
xL : Membership.mem (γ := Set K) ((fun (x : IntermediateField k K) ↦ (↑x : Set K)) L) x
σ : Gal(K/k)
hσ : σ ∈ H
⊢ (↑((((restrictNormalHom ↥L : (a : Gal(K/k)) → Gal(↥L/k)) σ : Gal(↥L/k)) : (a : ↥L) → ↥L)
          (Subtype.mk (p := fun (x : K) ↦ x ∈ L) x xL)) :
      K) =
    x
have h₁ : (↑((((restrictNormalHom ↥L : (a : Gal(K/k)) → Gal(↥L/k)) σ : Gal(↥L/k)) : (a : ↥L) → ↥L)
        (Subtype.mk (p := fun (x : K) ↦ x ∈ L) x xL)) :
    K) =
  Subtype.mk (p := fun (x : Gal(K/k)) ↦ x ∈ H) σ hσ • x
Tactic: nth_rw 2 [← (h.out.1 ⟨σ, hσ⟩)]
-- 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 ∈ (↑(lift (F := L) (fixedField (Subgroup.map (restrictNormalHom ↥L) H))) : Set K)
⊢ x ∈ (↑(fixedField H ⊓ L) : Set K)
have xL : x ∈ L
Tactic: have xL := lift_le _ 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 ∈ (↑(lift (F := L) (fixedField (Subgroup.map (restrictNormalHom ↥L) H))) : Set K)
xL : x ∈ L
⊢ x ∈ (↑(fixedField H ⊓ L) : Set K)
have h₁ : Membership.mem (γ := IntermediateField k ↥L) (fixedField (Subgroup.map (restrictNormalHom ↥L) H))
  (Subtype.mk (p := fun (x : K) ↦ x ∈ L) x xL)
Tactic: apply (mem_lift (⟨x, xL⟩ : L)).mp at h
Cleared hypotheses: 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
xL : x ∈ L
h :
  Membership.mem (γ := IntermediateField k ↥L) (fixedField (Subgroup.map (restrictNormalHom ↥L) H))
    (Subtype.mk (p := fun (x : K) ↦ x ∈ L) x xL)
⊢ x ∈ (↑(fixedField H ⊓ L) : Set K)
have h₁ : ∀ a ∈ H,
  (((restrictNormalHom ↥L : (a : Gal(K/k)) → Gal(↥L/k)) a : Gal(↥L/k)) : (a : ↥L) → ↥L)
      (Subtype.mk (p := fun (x : K) ↦ x ∈ L) x xL) =
    Subtype.mk (p := fun (x : K) ↦ x ∈ L) x xL
Tactic: simp only [mem_fixedField_iff, Subgroup.mem_map, forall_exists_index, and_imp,
      forall_apply_eq_imp_iff₂] at h
Cleared hypotheses: 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
xL : x ∈ L
h :
  ∀ a ∈ H,
    (((restrictNormalHom ↥L : (a : Gal(K/k)) → Gal(↥L/k)) a : Gal(↥L/k)) : (a : ↥L) → ↥L)
        (Subtype.mk (p := fun (x : K) ↦ x ∈ L) x xL) =
      Subtype.mk (p := fun (x : K) ↦ x ∈ L) x xL
⊢ x ∈ (↑(fixedField H ⊓ L) : Set K)
have h₁ : ∀ f ∈ H, (f : (a : K) → K) x = x
Tactic: simp only [coe_inf, Set.mem_inter_iff, SetLike.mem_coe, mem_fixedField_iff, xL, and_true]
-- 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
xL : x ∈ L
h :
  ∀ a ∈ H,
    (((restrictNormalHom ↥L : (a : Gal(K/k)) → Gal(↥L/k)) a : Gal(↥L/k)) : (a : ↥L) → ↥L)
        (Subtype.mk (p := fun (x : K) ↦ x ∈ L) x xL) =
      Subtype.mk (p := fun (x : K) ↦ x ∈ L) x xL
σ : Gal(K/k)
hσ : σ ∈ H
⊢ (↑((((restrictNormalHom ↥L : (a : Gal(K/k)) → Gal(↥L/k)) σ : Gal(↥L/k)) : (a : ↥L) → ↥L)
          (Subtype.mk (p := fun (x : K) ↦ x ∈ L) x xL)) :
      K) =
    x
have rw : (↑(Subtype.mk (p := fun (x : K) ↦ x ∈ L) x xL) : K) = x
Tactic: rw [h σ hσ]
-- 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
xL : x ∈ L
h :
  ∀ a ∈ H,
    (((restrictNormalHom ↥L : (a : Gal(K/k)) → Gal(↥L/k)) a : Gal(↥L/k)) : (a : ↥L) → ↥L)
        (Subtype.mk (p := fun (x : K) ↦ x ∈ L) x xL) =
      Subtype.mk (p := fun (x : K) ↦ x ∈ L) x xL
σ : Gal(K/k)
hσ : σ ∈ H
⊢ (σ : (a : K) → K) x = x
have this : (↑((((restrictNormalHom ↥L : (a : Gal(K/k)) → Gal(↥L/k)) σ : Gal(↥L/k)) : (a : ↥L) → ↥L)
        (Subtype.mk (p := fun (x : K) ↦ x ∈ L) x xL)) :
    K) =
  x
Tactic: have : ((restrictNormalHom L σ) ⟨x, xL⟩).1 = x := by rw [h 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
H : Subgroup Gal(K/k)
L : IntermediateField k K
inst_3 : Normal.{u_1, u_2} k ↥L
x : K
xL : x ∈ L
h :
  ∀ a ∈ H,
    (((restrictNormalHom ↥L : (a : Gal(K/k)) → Gal(↥L/k)) a : Gal(↥L/k)) : (a : ↥L) → ↥L)
        (Subtype.mk (p := fun (x : K) ↦ x ∈ L) x xL) =
      Subtype.mk (p := fun (x : K) ↦ x ∈ L) x xL
σ : Gal(K/k)
hσ : σ ∈ H
this :
  (↑((((restrictNormalHom ↥L : (a : Gal(K/k)) → Gal(↥L/k)) σ : Gal(↥L/k)) : (a : ↥L) → ↥L)
          (Subtype.mk (p := fun (x : K) ↦ x ∈ L) x xL)) :
      K) =
    x
⊢ (σ : (a : K) → K) x = x
have h₁ : Eq (α := K) ((σ : (a : K) → K) x : K)
  ((↑((((restrictNormalHom ↥L : (a : Gal(K/k)) → Gal(↥L/k)) σ : Gal(↥L/k)) : (a : ↥L) → ↥L)
          (Subtype.mk (p := fun (x : K) ↦ x ∈ L) x xL)) :
      K) :
    K)
Tactic: nth_rw 2 [← 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
H : ClosedSubgroup Gal(K/k)
inst_3 : IsGalois k K
⊢ (fixedField (↑H : Subgroup Gal(K/k))).fixingSubgroup = (↑H : Subgroup Gal(K/k))
have apply : (fixedField (↑H : Subgroup Gal(K/k))).fixingSubgroup ≤ (↑H : Subgroup Gal(K/k))
Tactic: apply le_antisymm _ ((IntermediateField.le_iff_le H.toSubgroup
    (IntermediateField.fixedField H.toSubgroup)).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
H : ClosedSubgroup Gal(K/k)
inst_3 : IsGalois k K
σ : Gal(K/k)
hσ : σ ∈ (fixedField (↑H : Subgroup Gal(K/k))).fixingSubgroup
⊢ σ ∈ (↑H : Subgroup Gal(K/k))
have by_contra : ∀ (h : σ ∉ (↑H : Subgroup Gal(K/k))), False
Tactic: by_contra 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 : ClosedSubgroup Gal(K/k)
inst_3 : IsGalois k K
σ : Gal(K/k)
hσ : σ ∈ (fixedField (↑H : Subgroup Gal(K/k))).fixingSubgroup
h : σ ∉ (↑H : Subgroup Gal(K/k))
⊢ False
have nhds : (↑H : Subgroup Gal(K/k)).carrierᶜ ∈ nhds σ
Tactic: have nhds : H.carrierᶜ ∈ nhds σ := H.isClosed'.isOpen_compl.mem_nhds 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 : ClosedSubgroup Gal(K/k)
inst_3 : IsGalois k K
σ : Gal(K/k)
hσ : σ ∈ (fixedField (↑H : Subgroup Gal(K/k))).fixingSubgroup
h : σ ∉ (↑H : Subgroup Gal(K/k))
nhds : (↑H : Subgroup Gal(K/k)).carrierᶜ ∈ _root_.nhds σ
⊢ False
have nhds₁ : (↑H : Subgroup Gal(K/k)).carrierᶜ ∈ GroupFilterBasis.N (galGroupBasis k K) σ
Tactic: rw [GroupFilterBasis.nhds_eq (x₀ := σ) (galGroupBasis k K)] at nhds
Cleared hypotheses: nhds
-- 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 : σ ∉ (↑H : Subgroup Gal(K/k))
nhds : (↑H : Subgroup Gal(K/k)).carrierᶜ ∈ GroupFilterBasis.N (galGroupBasis k K) σ
⊢ False
have rcases : ∀ (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)) (eq : (fun (g : Subgroup Gal(K/k)) ↦ g.carrier) gp = b) (L : IntermediateField k K)
  (hL : Membership.mem (γ := Set (IntermediateField k K)) (finiteExts k K) L) (eq' : L.fixingSubgroup = gp), False
Tactic: rcases nhds with ⟨b, ⟨gp, ⟨L, hL, eq'⟩, eq⟩, sub⟩
Cleared hypotheses: nhds
-- 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 : 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)
eq : (fun (g : Subgroup Gal(K/k)) ↦ g.carrier) gp = b
L : IntermediateField k K
hL : Membership.mem (γ := Set (IntermediateField k K)) (finiteExts k K) L
eq' : L.fixingSubgroup = gp
⊢ False
have eq₁ : (fun (g : Subgroup Gal(K/k)) ↦ g.carrier) L.fixingSubgroup = b
Tactic: rw [← eq'] at eq
Cleared hypotheses: eq, hL, 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
H : ClosedSubgroup Gal(K/k)
inst_3 : IsGalois k K
σ : Gal(K/k)
hσ : σ ∈ (fixedField (↑H : Subgroup Gal(K/k))).fixingSubgroup
h : 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 : (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
⊢ False
have this : FiniteDimensional.{u_1, u_2} k ↥L
Tactic: have := hL.out
-- 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 : 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 : (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 := ⋯ }
φ τ : Gal(K/k)
hτ : τ ∈ L'.fixingSubgroup.carrier
muleq : (fun (x : Gal(K/k)) ↦ σ • x) τ = φ
⊢ φ ∈ (↑H : Subgroup Gal(K/k)).carrierᶜ
have sub' : σ • b ⊆ (↑H : Subgroup Gal(K/k)).carrierᶜ
Tactic: have sub' : σ • b ⊆ H.carrierᶜ := Set.smul_set_subset_iff.mpr 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
H : ClosedSubgroup Gal(K/k)
inst_3 : IsGalois k K
σ : Gal(K/k)
hσ : σ ∈ (fixedField (↑H : Subgroup Gal(K/k))).fixingSubgroup
h : 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 : (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 := ⋯ }
φ τ : Gal(K/k)
hτ : τ ∈ L'.fixingSubgroup.carrier
muleq : (fun (x : Gal(K/k)) ↦ σ • x) τ = φ
sub' : σ • b ⊆ (↑H : Subgroup Gal(K/k)).carrierᶜ
⊢ φ ∈ (↑H : Subgroup Gal(K/k)).carrierᶜ
have a : φ ∈ σ • b
Tactic: apply 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
H : ClosedSubgroup Gal(K/k)
inst_3 : IsGalois k K
σ : Gal(K/k)
hσ : σ ∈ (fixedField (↑H : Subgroup Gal(K/k))).fixingSubgroup
h : 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 : (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 := ⋯ }
φ τ : Gal(K/k)
hτ : τ ∈ L'.fixingSubgroup.carrier
muleq : (fun (x : Gal(K/k)) ↦ σ • x) τ = φ
sub' : σ • b ⊆ (↑H : Subgroup Gal(K/k)).carrierᶜ
⊢ φ ∈ σ • b
have a : σ • τ ∈ σ • L.fixingSubgroup.carrier
Tactic: simp only [← muleq, ← 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
H : ClosedSubgroup Gal(K/k)
inst_3 : IsGalois k K
σ : Gal(K/k)
hσ : σ ∈ (fixedField (↑H : Subgroup Gal(K/k))).fixingSubgroup
h : 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 : (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 := ⋯ }
φ τ : Gal(K/k)
hτ : τ ∈ L'.fixingSubgroup.carrier
muleq : (fun (x : Gal(K/k)) ↦ σ • x) τ = φ
sub' : σ • b ⊆ (↑H : Subgroup Gal(K/k)).carrierᶜ
⊢ σ • τ ∈ σ • L.fixingSubgroup.carrier
have a : τ ∈ L.fixingSubgroup.carrier
Tactic: apply Set.smul_mem_smul_set
-- 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 : 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 : (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 := ⋯ }
⊢ False
have compl : σ • L'.fixingSubgroup.carrier ⊆ (↑H : Subgroup Gal(K/k)).carrierᶜ
Tactic: have compl : σ • L'.1.fixingSubgroup.carrier ⊆ H.carrierᶜ := by
    rintro φ ⟨τ, hτ, muleq⟩
    have sub' : σ • b ⊆ H.carrierᶜ := Set.smul_set_subset_iff.mpr sub
    apply sub'
    simp only [← muleq, ← eq]
    apply Set.smul_mem_smul_set
    exact (L.fixingSubgroup_le (IntermediateField.le_normalClosure L) hτ)
-- 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 : 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 : (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
⊢ False
have fix : ∀ x ∈ fixedField (↑H : Subgroup Gal(K/k)) ⊓ L'.toIntermediateField, (σ : (a : K) → K) x = x
Tactic: have fix : ∀ x ∈ IntermediateField.fixedField H.toSubgroup ⊓ ↑L', σ x = x :=
    fun x hx ↦ ((mem_fixingSubgroup_iff Gal(K/k)).mp hσ) x hx.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
H : ClosedSubgroup Gal(K/k)
inst_3 : IsGalois k K
σ : Gal(K/k)
hσ : σ ∈ (fixedField (↑H : Subgroup Gal(K/k))).fixingSubgroup
h : 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 : (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 ∈ fixedField (↑H : Subgroup Gal(K/k)) ⊓ L'.toIntermediateField, (σ : (a : K) → K) x = x
⊢ False
have fix₁ : ∀ (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
Tactic: rw [restrict_fixedField H.1 L'.1] at fix
Cleared hypotheses: fix
-- 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 : 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 : (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
⊢ Membership.mem (α := Gal(↥L'.toIntermediateField/k)) (γ := Subgroup Gal(↥L'.toIntermediateField/k))
    (Subgroup.map (restrictNormalHom ↥L'.toIntermediateField) (↑H : Subgroup Gal(K/k)))
    ((restrictNormalHom ↥L'.toIntermediateField : (a : Gal(K/k)) → Gal(↥L'.toIntermediateField/k)) σ :
      Gal(↥L'.toIntermediateField/k))
have rw : Membership.mem (α := Gal(↥L'.toIntermediateField/k)) (γ := Subgroup Gal(↥L'.toIntermediateField/k))
  (IntermediateField.fixingSubgroup
    (fixedField (Subgroup.map (restrictNormalHom ↥L'.toIntermediateField) (↑H : Subgroup Gal(K/k)))))
  ((restrictNormalHom ↥L'.toIntermediateField : (a : Gal(K/k)) → Gal(↥L'.toIntermediateField/k)) σ :
    Gal(↥L'.toIntermediateField/k))
Tactic: rw [← IntermediateField.fixingSubgroup_fixedField (Subgroup.map (restrictNormalHom L') H.1)]