awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.FieldTheory.Galois.Profinite (goal)
Goal:
k : Type u_1
K : Type u_2
inst : Field k
inst_1 : Field K
inst_2 : Algebra k K
L₁ : IntermediateField k K
finiteDimensional : FiniteDimensional.{u_1, u_2} k ↥L₁
isGalois : IsGalois.{u_1, u_2} k ↥L₁
L₂ : IntermediateField k K
finiteDimensional_1 : FiniteDimensional.{u_1, u_2} k ↥L₂
isGalois_1 : IsGalois.{u_1, u_2} k ↥L₂
f :
  op
      { toIntermediateField := L₁, finiteDimensional := finiteDimensional, isGalois := isGalois :
        FiniteGaloisIntermediateField k K } ⟶
    op { toIntermediateField := L₂, finiteDimensional := finiteDimensional_1, isGalois := isGalois_1 }
L₃ : IntermediateField k K
finiteDimensional_2 : FiniteDimensional.{u_1, u_2} k ↥L₃
isGalois_2 : IsGalois.{u_1, u_2} k ↥L₃
g :
  op
      { toIntermediateField := L₂, finiteDimensional := finiteDimensional_1, isGalois := isGalois_1 :
        FiniteGaloisIntermediateField k K } ⟶
    op { toIntermediateField := L₃, finiteDimensional := finiteDimensional_2, isGalois := isGalois_2 }
algInst : Algebra
  ↥(unop
        (op
          { toIntermediateField := L₃, finiteDimensional := finiteDimensional_2, isGalois := isGalois_2 :
            FiniteGaloisIntermediateField k K })).toSubsemiring
  ↥(unop
        (op
          { toIntermediateField := L₂, finiteDimensional := finiteDimensional_1, isGalois := isGalois_1 :
            FiniteGaloisIntermediateField k K })).toSubsemiring :=
  (Subsemiring.inclusion ⋯).toAlgebra
algInst_1 : Algebra
  ↥(unop
        (op
          { toIntermediateField := L₂, finiteDimensional := finiteDimensional_1, isGalois := isGalois_1 :
            FiniteGaloisIntermediateField k K })).toSubsemiring
  ↥(unop
        (op
          { toIntermediateField := L₁, finiteDimensional := finiteDimensional, isGalois := isGalois :
            FiniteGaloisIntermediateField k K })).toSubsemiring :=
  (Subsemiring.inclusion ⋯).toAlgebra
algInst_2 : Algebra
  ↥(unop
        (op
          { toIntermediateField := L₃, finiteDimensional := finiteDimensional_2, isGalois := isGalois_2 :
            FiniteGaloisIntermediateField k K })).toSubsemiring
  ↥(unop
        (op
          { toIntermediateField := L₁, finiteDimensional := finiteDimensional, isGalois := isGalois :
            FiniteGaloisIntermediateField k K })).toSubsemiring :=
  (Subsemiring.inclusion ⋯).toAlgebra
this_1 : IsScalarTower.{u_1, u_2, u_2} k ↥L₂ ↥L₁
this_2 : IsScalarTower.{u_1, u_2, u_2} k ↥L₃ ↥L₁
this_3 : IsScalarTower.{u_1, u_2, u_2} k ↥L₃ ↥L₂
this : IsScalarTower.{u_2, u_2, u_2} ↥L₃ ↥L₂ ↥L₁
⊢ finGaloisGroupMap (f ≫ g) = finGaloisGroupMap f ≫ finGaloisGroupMap g
have op : ConcreteCategory.hom (FC := fun (x1 x2 : FiniteGrp.{u_2}) ↦ (↑x1.toGrp : Type u_2) →* (↑x2.toGrp : Type u_2)) (CC :=
    fun (X : FiniteGrp.{u_2}) ↦ (↑X.toGrp : Type u_2)) (finGaloisGroupMap (f ≫ g)) =
  ConcreteCategory.hom (FC := fun (x1 x2 : FiniteGrp.{u_2}) ↦ (↑x1.toGrp : Type u_2) →* (↑x2.toGrp : Type u_2)) (CC :=
    fun (X : FiniteGrp.{u_2}) ↦ (↑X.toGrp : Type u_2)) (finGaloisGroupMap f ≫ finGaloisGroupMap g)
Tactic: ext : 1
-- Lean 4 proof step from Mathlib.FieldTheory.Galois.Profinite (hypothesis)
Goal:
k_1 : Type u_1
K_1 : Type u_2
inst : Field k_1
inst_1 : Field K_1
inst_2 : Algebra k_1 K_1
k : Type u_3
K : Type u_4
inst_3 : Field k
inst_4 : Field K
inst_5 : Algebra k K
σ : Gal(K/k)
L₁ L₂ : (FiniteGaloisIntermediateField k K)ᵒᵖ
π : L₁ ⟶ L₂
algInst : Algebra ↥(unop L₂).toSubsemiring ↥(unop L₁).toSubsemiring := (Subsemiring.inclusion ⋯).toAlgebra
⊢ Eq (α :=
    (↑((Functor.obj (asProfiniteGaloisGroupFunctor k K) L₂).toProfinite.toTop (P := fun (X : TopCat.{u_4}) ↦
          TotallyDisconnectedSpace.{u_4} (↑X : Type u_4))) :
      Type u_4))
    ((Hom.hom (CategoryTheory.Functor.map (asProfiniteGaloisGroupFunctor k K) π) :
          (a : (↑((asProfiniteGaloisGroupFunctor k K).obj L₁).toProfinite.toTop : Type u_4)) →
            (↑((asProfiniteGaloisGroupFunctor k K).obj L₂).toProfinite.toTop : Type u_4))
        ((fun (L : (FiniteGaloisIntermediateField k K)ᵒᵖ) ↦
            ((AlgEquiv.restrictNormalHom ↥(unop L).toIntermediateField :
                  (a : Gal(K/k)) → Gal(↥(unop L).toIntermediateField/k))
                σ :
              Gal(↥(unop L).toIntermediateField/k)))
          L₁) :
      (↑((Functor.obj (asProfiniteGaloisGroupFunctor k K) L₂).toProfinite.toTop (P := fun (X : TopCat.{u_4}) ↦
            TotallyDisconnectedSpace.{u_4} (↑X : Type u_4))) :
        Type u_4))
    ((fun (L : (FiniteGaloisIntermediateField k K)ᵒᵖ) ↦
          ((AlgEquiv.restrictNormalHom ↥(unop L).toIntermediateField :
                (a : Gal(K/k)) → Gal(↥(unop L).toIntermediateField/k))
              σ :
            Gal(↥(unop L).toIntermediateField/k)))
        L₂ :
      Gal(↥(unop L₂).toIntermediateField/k))
have this : IsScalarTower.{u_3, u_4, u_4} k ↥(unop L₂).toIntermediateField ↥(unop L₁).toIntermediateField
Tactic: have : IsScalarTower k L₂.unop L₁.unop := IsScalarTower.of_algebraMap_eq (congrFun rfl)
-- Lean 4 proof step from Mathlib.FieldTheory.Galois.Profinite (hypothesis)
Goal:
k_1 : Type u_1
K_1 : Type u_2
inst : Field k_1
inst_1 : Field K_1
inst_2 : Algebra k_1 K_1
k : Type u_3
K : Type u_4
inst_3 : Field k
inst_4 : Field K
inst_5 : Algebra k K
σ : Gal(K/k)
L₁ L₂ : (FiniteGaloisIntermediateField k K)ᵒᵖ
π : L₁ ⟶ L₂
algInst : Algebra ↥(unop L₂).toSubsemiring ↥(unop L₁).toSubsemiring := (Subsemiring.inclusion ⋯).toAlgebra
this : IsScalarTower.{u_3, u_4, u_4} k ↥(unop L₂).toIntermediateField ↥(unop L₁).toIntermediateField
⊢ Eq (α :=
    (↑((Functor.obj (asProfiniteGaloisGroupFunctor k K) L₂).toProfinite.toTop (P := fun (X : TopCat.{u_4}) ↦
          TotallyDisconnectedSpace.{u_4} (↑X : Type u_4))) :
      Type u_4))
    ((Hom.hom (CategoryTheory.Functor.map (asProfiniteGaloisGroupFunctor k K) π) :
          (a : (↑((asProfiniteGaloisGroupFunctor k K).obj L₁).toProfinite.toTop : Type u_4)) →
            (↑((asProfiniteGaloisGroupFunctor k K).obj L₂).toProfinite.toTop : Type u_4))
        ((fun (L : (FiniteGaloisIntermediateField k K)ᵒᵖ) ↦
            ((AlgEquiv.restrictNormalHom ↥(unop L).toIntermediateField :
                  (a : Gal(K/k)) → Gal(↥(unop L).toIntermediateField/k))
                σ :
              Gal(↥(unop L).toIntermediateField/k)))
          L₁) :
      (↑((Functor.obj (asProfiniteGaloisGroupFunctor k K) L₂).toProfinite.toTop (P := fun (X : TopCat.{u_4}) ↦
            TotallyDisconnectedSpace.{u_4} (↑X : Type u_4))) :
        Type u_4))
    ((fun (L : (FiniteGaloisIntermediateField k K)ᵒᵖ) ↦
          ((AlgEquiv.restrictNormalHom ↥(unop L).toIntermediateField :
                (a : Gal(K/k)) → Gal(↥(unop L).toIntermediateField/k))
              σ :
            Gal(↥(unop L).toIntermediateField/k)))
        L₂ :
      Gal(↥(unop L₂).toIntermediateField/k))
have this₁ : IsScalarTower (↥(unop L₂).toIntermediateField) (↥(unop L₁).toIntermediateField) K
Tactic: have : IsScalarTower L₂.unop L₁.unop K := IsScalarTower.of_algebraMap_eq (congrFun rfl)
-- Lean 4 proof step from Mathlib.FieldTheory.Galois.Profinite (goal)
Goal:
k_1 : Type u_1
K_1 : Type u_2
inst : Field k_1
inst_1 : Field K_1
inst_2 : Algebra k_1 K_1
k : Type u_3
K : Type u_4
inst_3 : Field k
inst_4 : Field K
inst_5 : Algebra k K
⊢ Eq.{u_4 + 1} (α :=
    (↑((limit (J := (FiniteGaloisIntermediateField k K)ᵒᵖ) (asProfiniteGaloisGroupFunctor k K)).toProfinite.toTop (P :=
          fun (X : TopCat.{u_4}) ↦ TotallyDisconnectedSpace.{u_4} (↑X : Type u_4))) :
      Type u_4))
    (Subtype.mk (p :=
      fun
        (x :
          (j : (FiniteGaloisIntermediateField k K)ᵒᵖ) →
            (↑((Functor.obj (asProfiniteGaloisGroupFunctor k K) j).toProfinite.toTop (P := fun (X : TopCat.{u_4}) ↦
                  TotallyDisconnectedSpace.{u_4} (↑X : Type u_4))) :
              Type u_4)) ↦
      Membership.mem (γ :=
        Subgroup
          ((j : (FiniteGaloisIntermediateField k K)ᵒᵖ) →
            (↑((Functor.obj (asProfiniteGaloisGroupFunctor k K) j).toProfinite.toTop (P := fun (X : TopCat.{u_4}) ↦
                  TotallyDisconnectedSpace.{u_4} (↑X : Type u_4))) :
              Type u_4)))
        (limitConePtAux (asProfiniteGaloisGroupFunctor k K)) x)
      (fun (L : (FiniteGaloisIntermediateField k K)ᵒᵖ) ↦
        (AlgEquiv.restrictNormalHom ↥(unop L).toIntermediateField :
            (a : Gal(K/k)) → Gal(↥(unop L).toIntermediateField/k))
          1)
      ⋯)
    1
have simp : Eq.{u_4 + 1} (α := ↥(limitConePtAux (asProfiniteGaloisGroupFunctor k K)))
  (Subtype.mk (p :=
    fun
      (x :
        (j : (FiniteGaloisIntermediateField k K)ᵒᵖ) →
          (↑((Functor.obj (asProfiniteGaloisGroupFunctor k K) j).toProfinite.toTop (P := fun (X : TopCat.{u_4}) ↦
                TotallyDisconnectedSpace.{u_4} (↑X : Type u_4))) :
            Type u_4)) ↦
    Membership.mem (γ :=
      Subgroup
        ((j : (FiniteGaloisIntermediateField k K)ᵒᵖ) →
          (↑((Functor.obj (asProfiniteGaloisGroupFunctor k K) j).toProfinite.toTop (P := fun (X : TopCat.{u_4}) ↦
                TotallyDisconnectedSpace.{u_4} (↑X : Type u_4))) :
            Type u_4)))
      (limitConePtAux (asProfiniteGaloisGroupFunctor k K)) x)
    (fun (L : (FiniteGaloisIntermediateField k K)ᵒᵖ) ↦ 1) sorry)
  1
Tactic: simp only [map_one]
-- Lean 4 proof step from Mathlib.FieldTheory.Galois.Profinite (goal)
Goal:
k : Type u_3
K : Type u_4
inst : Field k
inst_1 : Field K
inst_2 : Algebra k K
L : IntermediateField k K
inst_3 : Normal.{u_3, u_4} k ↥L
⊢ Continuous.{u_4, u_4} (X := Gal(K/k)) (Y := Gal(↥L/k)) (⇑(AlgEquiv.restrictNormalHom ↥L) : (a : Gal(K/k)) → Gal(↥L/k))
have apply : ∀ (A : Set Gal(↥L/k)),
  Membership.mem (γ := Filter Gal(↥L/k))
      (nhds ((AlgEquiv.restrictNormalHom ↥L : (a : Gal(K/k)) → Gal(↥L/k)) 1) : Filter Gal(↥L/k)) A →
    Membership.mem (γ := Filter Gal(K/k)) (nhds 1)
      ((⇑(AlgEquiv.restrictNormalHom ↥L) : (a : Gal(K/k)) → Gal(↥L/k)) ⁻¹' A)
Tactic: apply continuous_of_continuousAt_one _ (continuousAt_def.mpr _)
-- Lean 4 proof step from Mathlib.FieldTheory.Galois.Profinite (hypothesis)
Goal:
k : Type u_3
K : Type u_4
inst : Field k
inst_1 : Field K
inst_2 : Algebra k K
L : IntermediateField k K
inst_3 : Normal.{u_3, u_4} k ↥L
N : Set Gal(↥L/k)
hN :
  Membership.mem (γ := Filter Gal(↥L/k))
    (nhds ((AlgEquiv.restrictNormalHom ↥L : (a : Gal(K/k)) → Gal(↥L/k)) 1) : Filter Gal(↥L/k)) N
⊢ Membership.mem (γ := Filter Gal(K/k)) (nhds 1) ((⇑(AlgEquiv.restrictNormalHom ↥L) : (a : Gal(K/k)) → Gal(↥L/k)) ⁻¹' N)
have hN₁ : ∃ (E : IntermediateField k ↥L), FiniteDimensional.{u_3, u_4} k ↥E ∧ (↑E.fixingSubgroup : Set Gal(↥L/k)) ⊆ N
Tactic: rw [map_one, krullTopology_mem_nhds_one_iff] at hN
Cleared hypotheses: hN
-- Lean 4 proof step from Mathlib.FieldTheory.Galois.Profinite (goal)
Goal:
k : Type u_3
K : Type u_4
inst : Field k
inst_1 : Field K
inst_2 : Algebra k K
L : IntermediateField k K
inst_3 : Normal.{u_3, u_4} k ↥L
N : Set Gal(↥L/k)
hN : ∃ (E : IntermediateField k ↥L), FiniteDimensional.{u_3, u_4} k ↥E ∧ (↑E.fixingSubgroup : Set Gal(↥L/k)) ⊆ N
⊢ Membership.mem (γ := Filter Gal(K/k)) (nhds 1) ((⇑(AlgEquiv.restrictNormalHom ↥L) : (a : Gal(K/k)) → Gal(↥L/k)) ⁻¹' N)
have obtain : ∀ (L' : IntermediateField k ↥L),
  FiniteDimensional.{u_3, u_4} k ↥L' →
    ∀ (hO : (↑L'.fixingSubgroup : Set Gal(↥L/k)) ⊆ N),
      Membership.mem (γ := Filter Gal(K/k)) (nhds 1)
        ((⇑(AlgEquiv.restrictNormalHom ↥L) : (a : Gal(K/k)) → Gal(↥L/k)) ⁻¹' N)
Tactic: obtain ⟨L', _, hO⟩ := hN
Cleared hypotheses: hN
-- Lean 4 proof step from Mathlib.FieldTheory.Galois.Profinite (hypothesis)
Goal:
k : Type u_3
K : Type u_4
inst : Field k
inst_1 : Field K
inst_2 : Algebra k K
L : IntermediateField k K
inst_3 : Normal.{u_3, u_4} k ↥L
N : Set Gal(↥L/k)
L' : IntermediateField k ↥L
left : FiniteDimensional.{u_3, u_4} k ↥L'
hO : (↑L'.fixingSubgroup : Set Gal(↥L/k)) ⊆ N
⊢ Membership.mem (γ := Filter Gal(K/k)) (nhds 1) ((⇑(AlgEquiv.restrictNormalHom ↥L) : (a : Gal(K/k)) → Gal(↥L/k)) ⁻¹' N)
have this : Module.Finite.{u_3, u_4} k ↥(IntermediateField.lift L')
Tactic: have := Module.Finite.equiv <| AlgEquiv.toLinearEquiv <| IntermediateField.liftAlgEquiv L'
-- Lean 4 proof step from Mathlib.FieldTheory.Galois.Profinite (goal)
Goal:
k : Type u_3
K : Type u_4
inst : Field k
inst_1 : Field K
inst_2 : Algebra k K
L : IntermediateField k K
inst_3 : Normal.{u_3, u_4} k ↥L
N : Set Gal(↥L/k)
L' : IntermediateField k ↥L
left : FiniteDimensional.{u_3, u_4} k ↥L'
hO : (↑L'.fixingSubgroup : Set Gal(↥L/k)) ⊆ N
this : Module.Finite.{u_3, u_4} k ↥(IntermediateField.lift L')
⊢ Membership.mem (γ := Filter Gal(K/k)) (nhds 1) ((⇑(AlgEquiv.restrictNormalHom ↥L) : (a : Gal(K/k)) → Gal(↥L/k)) ⁻¹' N)
have apply : ∃ t ⊆ (⇑(AlgEquiv.restrictNormalHom ↥L) : (a : Gal(K/k)) → Gal(↥L/k)) ⁻¹' N, IsOpen t ∧ 1 ∈ t
Tactic: apply mem_nhds_iff.mpr
-- Lean 4 proof step from Mathlib.FieldTheory.Galois.Profinite (goal)
Goal:
k : Type u_3
K : Type u_4
inst : Field k
inst_1 : Field K
inst_2 : Algebra k K
L : IntermediateField k K
inst_3 : Normal.{u_3, u_4} k ↥L
N : Set Gal(↥L/k)
L' : IntermediateField k ↥L
left : FiniteDimensional.{u_3, u_4} k ↥L'
hO : (↑L'.fixingSubgroup : Set Gal(↥L/k)) ⊆ N
this : Module.Finite.{u_3, u_4} k ↥(IntermediateField.lift L')
⊢ ∃ t ⊆ (⇑(AlgEquiv.restrictNormalHom ↥L) : (a : Gal(K/k)) → Gal(↥L/k)) ⁻¹' N, IsOpen t ∧ 1 ∈ t
have h : (↑(IntermediateField.lift L').fixingSubgroup : Set Gal(K/k)) ⊆
    (⇑(AlgEquiv.restrictNormalHom ↥L) : (a : Gal(K/k)) → Gal(↥L/k)) ⁻¹' N ∧
  IsOpen (↑(IntermediateField.lift L').fixingSubgroup : Set Gal(K/k)) ∧
    1 ∈ (↑(IntermediateField.lift L').fixingSubgroup : Set Gal(K/k))
Tactic: use (IntermediateField.lift L').fixingSubgroup
-- Lean 4 proof step from Mathlib.FieldTheory.Galois.Profinite (goal)
Goal:
k : Type u_3
K : Type u_4
inst : Field k
inst_1 : Field K
inst_2 : Algebra k K
L : IntermediateField k K
inst_3 : Normal.{u_3, u_4} k ↥L
N : Set Gal(↥L/k)
L' : IntermediateField k ↥L
left : FiniteDimensional.{u_3, u_4} k ↥L'
hO : (↑L'.fixingSubgroup : Set Gal(↥L/k)) ⊆ N
this : Module.Finite.{u_3, u_4} k ↥(IntermediateField.lift L')
⊢ (↑(IntermediateField.lift L').fixingSubgroup : Set Gal(K/k)) ⊆
      (⇑(AlgEquiv.restrictNormalHom ↥L) : (a : Gal(K/k)) → Gal(↥L/k)) ⁻¹' N ∧
    IsOpen (↑(IntermediateField.lift L').fixingSubgroup : Set Gal(K/k)) ∧
      1 ∈ (↑(IntermediateField.lift L').fixingSubgroup : Set Gal(K/k))
have h : (↑(IntermediateField.lift L').fixingSubgroup : Set Gal(K/k)) ⊆
  (⇑(AlgEquiv.restrictNormalHom ↥L) : (a : Gal(K/k)) → Gal(↥L/k)) ⁻¹' N
Tactic: constructor
-- Lean 4 proof step from Mathlib.FieldTheory.Galois.Profinite (goal)
Goal:
k : Type u_3
K : Type u_4
inst : Field k
inst_1 : Field K
inst_2 : Algebra k K
L : IntermediateField k K
inst_3 : Normal.{u_3, u_4} k ↥L
N : Set Gal(↥L/k)
L' : IntermediateField k ↥L
left : FiniteDimensional.{u_3, u_4} k ↥L'
hO : (↑L'.fixingSubgroup : Set Gal(↥L/k)) ⊆ N
this : Module.Finite.{u_3, u_4} k ↥(IntermediateField.lift L')
h :
  (↑(IntermediateField.lift L').fixingSubgroup : Set Gal(K/k)) ⊆
    (⇑(AlgEquiv.restrictNormalHom ↥L) : (a : Gal(K/k)) → Gal(↥L/k)) ⁻¹' N
⊢ (↑(IntermediateField.lift L').fixingSubgroup : Set Gal(K/k)) ⊆
      (⇑(AlgEquiv.restrictNormalHom ↥L) : (a : Gal(K/k)) → Gal(↥L/k)) ⁻¹' N ∧
    IsOpen (↑(IntermediateField.lift L').fixingSubgroup : Set Gal(K/k)) ∧
      1 ∈ (↑(IntermediateField.lift L').fixingSubgroup : Set Gal(K/k))
have h₁ : IsOpen (↑(IntermediateField.lift L').fixingSubgroup : Set Gal(K/k)) ∧
  1 ∈ (↑(IntermediateField.lift L').fixingSubgroup : Set Gal(K/k))
Tactic: constructor
-- Lean 4 proof step from Mathlib.FieldTheory.Galois.Profinite (goal)
Goal:
k : Type u_3
K : Type u_4
inst : Field k
inst_1 : Field K
inst_2 : Algebra k K
L : IntermediateField k K
inst_3 : Normal.{u_3, u_4} k ↥L
N : Set Gal(↥L/k)
L' : IntermediateField k ↥L
left : FiniteDimensional.{u_3, u_4} k ↥L'
hO : (↑L'.fixingSubgroup : Set Gal(↥L/k)) ⊆ N
this : Module.Finite.{u_3, u_4} k ↥(IntermediateField.lift L')
x : Gal(K/k)
hx : x ∈ (↑(IntermediateField.lift L').fixingSubgroup : Set Gal(K/k))
⊢ Membership.mem (γ := Set Gal(K/k)) ((⇑(AlgEquiv.restrictNormalHom ↥L) : (a : Gal(K/k)) → Gal(↥L/k)) ⁻¹' N) x
have h : Membership.mem (α := Gal(↥L/k)) (↑L'.fixingSubgroup : Set Gal(↥L/k))
  ((AlgEquiv.restrictNormalHom ↥L : (a : Gal(K/k)) → Gal(↥L/k)) x : Gal(↥L/k))
Tactic: apply hO
-- Lean 4 proof step from Mathlib.FieldTheory.Galois.Profinite (goal)
Goal:
k : Type u_3
K : Type u_4
inst : Field k
inst_1 : Field K
inst_2 : Algebra k K
L : IntermediateField k K
inst_3 : Normal.{u_3, u_4} k ↥L
N : Set Gal(↥L/k)
L' : IntermediateField k ↥L
left : FiniteDimensional.{u_3, u_4} k ↥L'
hO : (↑L'.fixingSubgroup : Set Gal(↥L/k)) ⊆ N
this : Module.Finite.{u_3, u_4} k ↥(IntermediateField.lift L')
x : Gal(K/k)
hx : x ∈ (↑(IntermediateField.lift L').fixingSubgroup : Set Gal(K/k))
⊢ Membership.mem (α := Gal(↥L/k)) (↑L'.fixingSubgroup : Set Gal(↥L/k))
    ((AlgEquiv.restrictNormalHom ↥L : (a : Gal(K/k)) → Gal(↥L/k)) x : Gal(↥L/k))
have h : ∀ (hx : ∀ x_1 ∈ IntermediateField.lift L', (x : (a : K) → K) x_1 = x_1),
  ∀ x_1 ∈ L', (((AlgEquiv.restrictNormalHom ↥L : (a : Gal(K/k)) → Gal(↥L/k)) x : Gal(↥L/k)) : (a : ↥L) → ↥L) x_1 = x_1
Tactic: simp only [SetLike.mem_coe, IntermediateField.mem_fixingSubgroup_iff] at hx ⊢
Cleared hypotheses: hx
-- Lean 4 proof step from Mathlib.FieldTheory.Galois.Profinite (hypothesis)
Goal:
k : Type u_3
K : Type u_4
inst : Field k
inst_1 : Field K
inst_2 : Algebra k K
L : IntermediateField k K
inst_3 : Normal.{u_3, u_4} k ↥L
N : Set Gal(↥L/k)
L' : IntermediateField k ↥L
left : FiniteDimensional.{u_3, u_4} k ↥L'
hO : (↑L'.fixingSubgroup : Set Gal(↥L/k)) ⊆ N
this : Module.Finite.{u_3, u_4} k ↥(IntermediateField.lift L')
x : Gal(K/k)
hx : ∀ x_1 ∈ IntermediateField.lift L', (x : (a : K) → K) x_1 = x_1
y : ↥L
hy : y ∈ L'
⊢ (((AlgEquiv.restrictNormalHom ↥L : (a : Gal(K/k)) → Gal(↥L/k)) x : Gal(↥L/k)) : (a : ↥L) → ↥L) y = y
have this₁ : Eq (α := K) ((algebraMap (↥L) K : (a : ↥L) → K) ((x.restrictNormal ↥L : (a : ↥L) → ↥L) y : ↥L) : K)
  ((x : (a : K) → K) ((algebraMap (↥L) K : (a : ↥L) → K) y : K) : K)
Tactic: have := AlgEquiv.restrictNormal_commutes x L y
-- Lean 4 proof step from Mathlib.FieldTheory.Galois.Profinite (hypothesis)
Goal:
k : Type u_3
K : Type u_4
inst : Field k
inst_1 : Field K
inst_2 : Algebra k K
L : IntermediateField k K
inst_3 : Normal.{u_3, u_4} k ↥L
N : Set Gal(↥L/k)
L' : IntermediateField k ↥L
left : FiniteDimensional.{u_3, u_4} k ↥L'
hO : (↑L'.fixingSubgroup : Set Gal(↥L/k)) ⊆ N
this_1 : Module.Finite.{u_3, u_4} k ↥(IntermediateField.lift L')
x : Gal(K/k)
hx : ∀ x_1 ∈ IntermediateField.lift L', (x : (a : K) → K) x_1 = x_1
y : ↥L
hy : y ∈ L'
this : Eq (α := K) ((↑((x.restrictNormal ↥L : (a : ↥L) → ↥L) y) : K) : K) ((x : (a : K) → K) (↑y : K) : K)
⊢ (((AlgEquiv.restrictNormalHom ↥L : (a : Gal(K/k)) → Gal(↥L/k)) x : Gal(↥L/k)) : (a : ↥L) → ↥L) y = y
have this₁ : (↑((x.restrictNormal ↥L : (a : ↥L) → ↥L) y) : K) = (↑y : K)
Tactic: rw [hx y.1 ((IntermediateField.mem_lift y).mpr hy)] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.FieldTheory.Galois.Profinite (goal)
Goal:
k : Type u_3
K : Type u_4
inst : Field k
inst_1 : Field K
inst_2 : Algebra k K
⊢ Continuous (X := Gal(K/k)) (Y :=
    (↑((limit (J := (FiniteGaloisIntermediateField k K)ᵒᵖ) (asProfiniteGaloisGroupFunctor k K)).toProfinite.toTop (P :=
          fun (X : TopCat.{u_4}) ↦ TotallyDisconnectedSpace.{u_4} (↑X : Type u_4))) :
      Type u_4))
    (⇑(algEquivToLimit k K) :
      (a : Gal(K/k)) → (↑(limit (asProfiniteGaloisGroupFunctor k K)).toProfinite.toTop : Type u_4))
have rw : Continuous (X := Gal(K/k)) (Y :=
  (j : (FiniteGaloisIntermediateField k K)ᵒᵖ) →
    (↑((Functor.obj (asProfiniteGaloisGroupFunctor k K) j).toProfinite.toTop (P := fun (X : TopCat.{u_4}) ↦
          TotallyDisconnectedSpace.{u_4} (↑X : Type u_4))) :
      Type u_4))
  (Function.comp (β :=
    (↑((limit (J := (FiniteGaloisIntermediateField k K)ᵒᵖ) (asProfiniteGaloisGroupFunctor k K)).toProfinite.toTop (P :=
          fun (X : TopCat.{u_4}) ↦ TotallyDisconnectedSpace.{u_4} (↑X : Type u_4))) :
      Type u_4))
    (Subtype.val (p :=
      fun
        (x :
          (j : (FiniteGaloisIntermediateField k K)ᵒᵖ) →
            (↑((Functor.obj (asProfiniteGaloisGroupFunctor k K) j).toProfinite.toTop (P := fun (X : TopCat.{u_4}) ↦
                  TotallyDisconnectedSpace.{u_4} (↑X : Type u_4))) :
              Type u_4)) ↦
      Membership.mem (γ :=
        Subgroup
          ((j : (FiniteGaloisIntermediateField k K)ᵒᵖ) →
            (↑((Functor.obj (asProfiniteGaloisGroupFunctor k K) j).toProfinite.toTop (P := fun (X : TopCat.{u_4}) ↦
                  TotallyDisconnectedSpace.{u_4} (↑X : Type u_4))) :
              Type u_4)))
        (limitConePtAux (asProfiniteGaloisGroupFunctor k K)) x))
    (⇑(algEquivToLimit k K) :
      (a : Gal(K/k)) → (↑(limit (asProfiniteGaloisGroupFunctor k K)).toProfinite.toTop : Type u_4)))
Tactic: rw [continuous_induced_rng]
-- Lean 4 proof step from Mathlib.FieldTheory.Galois.Profinite (goal)
Goal:
k : Type u_3
K : Type u_4
inst : Field k
inst_1 : Field K
inst_2 : Algebra k K
⊢ Continuous (X := Gal(K/k)) (Y :=
    (j : (FiniteGaloisIntermediateField k K)ᵒᵖ) →
      (↑((Functor.obj (asProfiniteGaloisGroupFunctor k K) j).toProfinite.toTop (P := fun (X : TopCat.{u_4}) ↦
            TotallyDisconnectedSpace.{u_4} (↑X : Type u_4))) :
        Type u_4))
    (Function.comp (β :=
      (↑((limit (J := (FiniteGaloisIntermediateField k K)ᵒᵖ) (asProfiniteGaloisGroupFunctor k K)).toProfinite.toTop
            (P := fun (X : TopCat.{u_4}) ↦ TotallyDisconnectedSpace.{u_4} (↑X : Type u_4))) :
        Type u_4))
      (Subtype.val (p :=
        fun
          (x :
            (j : (FiniteGaloisIntermediateField k K)ᵒᵖ) →
              (↑((Functor.obj (asProfiniteGaloisGroupFunctor k K) j).toProfinite.toTop (P := fun (X : TopCat.{u_4}) ↦
                    TotallyDisconnectedSpace.{u_4} (↑X : Type u_4))) :
                Type u_4)) ↦
        Membership.mem (γ :=
          Subgroup
            ((j : (FiniteGaloisIntermediateField k K)ᵒᵖ) →
              (↑((Functor.obj (asProfiniteGaloisGroupFunctor k K) j).toProfinite.toTop (P := fun (X : TopCat.{u_4}) ↦
                    TotallyDisconnectedSpace.{u_4} (↑X : Type u_4))) :
                Type u_4)))
          (limitConePtAux (asProfiniteGaloisGroupFunctor k K)) x))
      (⇑(algEquivToLimit k K) :
        (a : Gal(K/k)) → (↑(limit (asProfiniteGaloisGroupFunctor k K)).toProfinite.toTop : Type u_4)))
have refine : ∀ (L : (FiniteGaloisIntermediateField k K)ᵒᵖ),
  Continuous (Y :=
    (↑((Functor.obj (asProfiniteGaloisGroupFunctor k K) L).toProfinite.toTop (P := fun (X : TopCat.{u_4}) ↦
          TotallyDisconnectedSpace.{u_4} (↑X : Type u_4))) :
      Type u_4))
    fun (a : Gal(K/k)) ↦
    Function.comp (β :=
      (↑((limit (J := (FiniteGaloisIntermediateField k K)ᵒᵖ) (asProfiniteGaloisGroupFunctor k K)).toProfinite.toTop
            (P := fun (X : TopCat.{u_4}) ↦ TotallyDisconnectedSpace.{u_4} (↑X : Type u_4))) :
        Type u_4))
      (Subtype.val (p :=
        fun
          (x :
            (j : (FiniteGaloisIntermediateField k K)ᵒᵖ) →
              (↑((Functor.obj (asProfiniteGaloisGroupFunctor k K) j).toProfinite.toTop (P := fun (X : TopCat.{u_4}) ↦
                    TotallyDisconnectedSpace.{u_4} (↑X : Type u_4))) :
                Type u_4)) ↦
        Membership.mem (γ :=
          Subgroup
            ((j : (FiniteGaloisIntermediateField k K)ᵒᵖ) →
              (↑((Functor.obj (asProfiniteGaloisGroupFunctor k K) j).toProfinite.toTop (P := fun (X : TopCat.{u_4}) ↦
                    TotallyDisconnectedSpace.{u_4} (↑X : Type u_4))) :
                Type u_4)))
          (limitConePtAux (asProfiniteGaloisGroupFunctor k K)) x))
      (⇑(algEquivToLimit k K) :
        (a : Gal(K/k)) → (↑(limit (asProfiniteGaloisGroupFunctor k K)).toProfinite.toTop : Type u_4))
      a L
Tactic: refine continuous_pi (fun L ↦ ?_)