awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- 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 : (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
have h : (↑((Functor.obj (asProfiniteGaloisGroupFunctor k K) L).toProfinite.toTop (P := fun (X : TopCat.{u_4}) ↦
          TotallyDisconnectedSpace.{u_4} (↑X : Type u_4))) :
      Type u_4) =
    Gal(↥(unop L).toIntermediateField/k) →
  Eq.{u_4 + 1} (α :=
    TopologicalSpace
      (↑((Functor.obj (asProfiniteGaloisGroupFunctor k K) L).toProfinite.toTop (P := fun (X : TopCat.{u_4}) ↦
            TotallyDisconnectedSpace.{u_4} (↑X : Type u_4))) :
        Type u_4))
    (Functor.obj (asProfiniteGaloisGroupFunctor k K) L).toProfinite.toTop (P := fun (X : TopCat.{u_4}) ↦
        TotallyDisconnectedSpace.{u_4} (↑X : Type u_4)).str
    (krullTopology k ↥(unop L).toIntermediateField)
Tactic: convert! restrictNormalHom_continuous L.unop.1
-- 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
inst_3 : IsGalois k K
g :
  (↑((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)
x : K
L : FiniteGaloisIntermediateField k K
hx' : toAlgEquivAux g x ∈ L.toIntermediateField
hx : x ∈ L.toIntermediateField
⊢ Subtype.mk (p := fun (x : K) ↦ x ∈ L.toIntermediateField) (toAlgEquivAux g x) hx' =
    (((proj L :
              (a : (↑(limit (asProfiniteGaloisGroupFunctor k K)).toProfinite.toTop : Type u_4)) →
                Gal(↥L.toIntermediateField/k))
            g :
          Gal(↥L.toIntermediateField/k)) :
        (a : ↥L.toIntermediateField) → ↥L.toIntermediateField)
      (Subtype.mk (p := fun (x : K) ↦ x ∈ L.toIntermediateField) x hx)
have rw : (↑(Subtype.mk (p := fun (x : K) ↦ x ∈ L.toIntermediateField) (toAlgEquivAux g x) hx') : K) =
  (↑((((proj L :
                (a : (↑(limit (asProfiniteGaloisGroupFunctor k K)).toProfinite.toTop : Type u_4)) →
                  Gal(↥L.toIntermediateField/k))
              g :
            Gal(↥L.toIntermediateField/k)) :
          (a : ↥L.toIntermediateField) → ↥L.toIntermediateField)
        (Subtype.mk (p := fun (x : K) ↦ x ∈ L.toIntermediateField) x hx)) :
    K)
Tactic: rw [Subtype.ext_iff, Subtype.coe_mk, toAlgEquivAux_eq_proj_of_mem]
-- 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
inst_3 : IsGalois k K
g :
  (↑((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)
x : K
L : FiniteGaloisIntermediateField k K
hx' : toAlgEquivAux g x ∈ L.toIntermediateField
hx : x ∈ L.toIntermediateField
rw :
  (↑(Subtype.mk (p := fun (x : K) ↦ x ∈ L.toIntermediateField) (toAlgEquivAux g x) hx') : K) =
    (↑((((proj L :
                  (a : (↑(limit (asProfiniteGaloisGroupFunctor k K)).toProfinite.toTop : Type u_4)) →
                    Gal(↥L.toIntermediateField/k))
                g :
              Gal(↥L.toIntermediateField/k)) :
            (a : ↥L.toIntermediateField) → ↥L.toIntermediateField)
          (Subtype.mk (p := fun (x : K) ↦ x ∈ L.toIntermediateField) x hx)) :
      K)
⊢ Subtype.mk (p := fun (x : K) ↦ x ∈ L.toIntermediateField) (toAlgEquivAux g x) hx' =
    (((proj L :
              (a : (↑(limit (asProfiniteGaloisGroupFunctor k K)).toProfinite.toTop : Type u_4)) →
                Gal(↥L.toIntermediateField/k))
            g :
          Gal(↥L.toIntermediateField/k)) :
        (a : ↥L.toIntermediateField) → ↥L.toIntermediateField)
      (Subtype.mk (p := fun (x : K) ↦ x ∈ L.toIntermediateField) x hx)
have rw₁ : toAlgEquivAux g x =
  (↑((((proj L :
                (a : (↑(limit (asProfiniteGaloisGroupFunctor k K)).toProfinite.toTop : Type u_4)) →
                  Gal(↥L.toIntermediateField/k))
              g :
            Gal(↥L.toIntermediateField/k)) :
          (a : ↥L.toIntermediateField) → ↥L.toIntermediateField)
        (Subtype.mk (p := fun (x : K) ↦ x ∈ L.toIntermediateField) x hx)) :
    K)
Tactic: rw [Subtype.ext_iff, Subtype.coe_mk, toAlgEquivAux_eq_proj_of_mem]
-- 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
inst_3 : IsGalois k K
g :
  (↑((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)
x : K
L : FiniteGaloisIntermediateField k K
hx' : toAlgEquivAux g x ∈ L.toIntermediateField
hx : x ∈ L.toIntermediateField
rw :
  (↑(Subtype.mk (p := fun (x : K) ↦ x ∈ L.toIntermediateField) (toAlgEquivAux g x) hx') : K) =
    (↑((((proj L :
                  (a : (↑(limit (asProfiniteGaloisGroupFunctor k K)).toProfinite.toTop : Type u_4)) →
                    Gal(↥L.toIntermediateField/k))
                g :
              Gal(↥L.toIntermediateField/k)) :
            (a : ↥L.toIntermediateField) → ↥L.toIntermediateField)
          (Subtype.mk (p := fun (x : K) ↦ x ∈ L.toIntermediateField) x hx)) :
      K)
rw₁ :
  toAlgEquivAux g x =
    (↑((((proj L :
                  (a : (↑(limit (asProfiniteGaloisGroupFunctor k K)).toProfinite.toTop : Type u_4)) →
                    Gal(↥L.toIntermediateField/k))
                g :
              Gal(↥L.toIntermediateField/k)) :
            (a : ↥L.toIntermediateField) → ↥L.toIntermediateField)
          (Subtype.mk (p := fun (x : K) ↦ x ∈ L.toIntermediateField) x hx)) :
      K)
⊢ Subtype.mk (p := fun (x : K) ↦ x ∈ L.toIntermediateField) (toAlgEquivAux g x) hx' =
    (((proj L :
              (a : (↑(limit (asProfiniteGaloisGroupFunctor k K)).toProfinite.toTop : Type u_4)) →
                Gal(↥L.toIntermediateField/k))
            g :
          Gal(↥L.toIntermediateField/k)) :
        (a : ↥L.toIntermediateField) → ↥L.toIntermediateField)
      (Subtype.mk (p := fun (x : K) ↦ x ∈ L.toIntermediateField) x hx)
have rw₂ : Eq.{u_4 + 1} (α := K)
  (↑((((proj L :
                (a : (↑(limit (asProfiniteGaloisGroupFunctor k K)).toProfinite.toTop : Type u_4)) →
                  Gal(↥L.toIntermediateField/k))
              g :
            Gal(↥L.toIntermediateField/k)) :
          (a : ↥L.toIntermediateField) → ↥L.toIntermediateField)
        (Subtype.mk (p := fun (x : K) ↦ x ∈ L.toIntermediateField) x hx)) :
    K)
  (↑((((proj L :
                (a : (↑(limit (asProfiniteGaloisGroupFunctor k K)).toProfinite.toTop : Type u_4)) →
                  Gal(↥L.toIntermediateField/k))
              g :
            Gal(↥L.toIntermediateField/k)) :
          (a : ↥L.toIntermediateField) → ↥L.toIntermediateField)
        (Subtype.mk (p := fun (x : K) ↦ x ∈ L.toIntermediateField) x hx)) :
    K)
Tactic: rw [Subtype.ext_iff, Subtype.coe_mk, toAlgEquivAux_eq_proj_of_mem]
-- 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
inst_3 : IsGalois k K
g :
  (↑((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)
x : K
L : FiniteGaloisIntermediateField k K
hx : x ∈ L.toIntermediateField
⊢ toAlgEquivAux g x =
    (AlgEquiv.liftNormal (K₁ := ↥L.toIntermediateField) (K₂ := ↥L.toIntermediateField)
          ((proj L :
              (a : (↑(limit (asProfiniteGaloisGroupFunctor k K)).toProfinite.toTop : Type u_4)) →
                Gal(↥L.toIntermediateField/k))
            g)
          K :
        (a : K) → K)
      x
have rw : Eq (α := K)
  ((↑((((proj L :
                  (a : (↑(limit (asProfiniteGaloisGroupFunctor k K)).toProfinite.toTop : Type u_4)) →
                    Gal(↥L.toIntermediateField/k))
                g :
              Gal(↥L.toIntermediateField/k)) :
            (a : ↥L.toIntermediateField) → ↥L.toIntermediateField)
          (Subtype.mk (p := fun (x : K) ↦ x ∈ L.toIntermediateField) x hx)) :
      K) :
    K)
  ((AlgEquiv.liftNormal (K₁ := ↥L.toIntermediateField) (K₂ := ↥L.toIntermediateField)
          ((proj L :
              (a : (↑(limit (asProfiniteGaloisGroupFunctor k K)).toProfinite.toTop : Type u_4)) →
                Gal(↥L.toIntermediateField/k))
            g)
          K :
        (a : K) → K)
      x :
    K)
Tactic: rw [toAlgEquivAux_eq_proj_of_mem g x L hx]
-- 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
inst_6 : IsGalois k K
g :
  (↑((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)
x : K
L : FiniteGaloisIntermediateField k K := adjoin k {x, toAlgEquivAux g x}
⊢ toAlgEquivAux g⁻¹ (toAlgEquivAux g x) = x
have hx : x ∈ L.toIntermediateField
Tactic: have hx : x ∈ L.1 := subset_adjoin _ _ (Set.mem_insert x {toAlgEquivAux g x})
-- 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
inst_6 : IsGalois k K
g :
  (↑((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)
x : K
L : FiniteGaloisIntermediateField k K := adjoin k {x, toAlgEquivAux g x}
hx : x ∈ L.toIntermediateField
⊢ toAlgEquivAux g⁻¹ (toAlgEquivAux g x) = x
have hx' : toAlgEquivAux g x ∈ L.toIntermediateField
Tactic: have hx' : toAlgEquivAux g x ∈ L.1 := subset_adjoin _ _ (Set.mem_insert_of_mem x 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
inst_6 : IsGalois k K
g :
  (↑((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)
x : K
L : FiniteGaloisIntermediateField k K := adjoin k {x, toAlgEquivAux g⁻¹ x}
⊢ toAlgEquivAux g (toAlgEquivAux g⁻¹ x) = x
have hx : x ∈ L.toIntermediateField
Tactic: have hx : x ∈ L.1 := subset_adjoin _ _ (Set.mem_insert x {toAlgEquivAux g⁻¹ x})
-- 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
inst_6 : IsGalois k K
g :
  (↑((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)
x : K
L : FiniteGaloisIntermediateField k K := adjoin k {x, toAlgEquivAux g⁻¹ x}
hx : x ∈ L.toIntermediateField
⊢ toAlgEquivAux g (toAlgEquivAux g⁻¹ x) = x
have hx' : toAlgEquivAux g⁻¹ x ∈ L.toIntermediateField
Tactic: have hx' : toAlgEquivAux g⁻¹ x ∈ L.1 := subset_adjoin _ _ (Set.mem_insert_of_mem x 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
inst_6 : IsGalois k K
g :
  (↑((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)
x y : K
⊢ toAlgEquivAux g (x * y) = toAlgEquivAux g x * toAlgEquivAux g y
have hx : Membership.mem (γ := IntermediateField k K) (toIntermediateField (adjoin k {x, y})) x
Tactic: have hx : x ∈ (adjoin k {x, y}).1 := subset_adjoin _ _ (Set.mem_insert x {y})
-- 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
inst_6 : IsGalois k K
g :
  (↑((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)
x y : K
hx : Membership.mem (γ := IntermediateField k K) (toIntermediateField (adjoin k {x, y})) x
⊢ toAlgEquivAux g (x * y) = toAlgEquivAux g x * toAlgEquivAux g y
have hy : Membership.mem (γ := IntermediateField k K) (toIntermediateField (adjoin k {x, y})) y
Tactic: have hy : y ∈ (adjoin k {x, y}).1 := subset_adjoin _ _ (Set.mem_insert_of_mem x 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
inst_6 : IsGalois k K
g :
  (↑((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)
x y : K
hx : Membership.mem (γ := IntermediateField k K) (toIntermediateField (adjoin k {x, y})) x
hy : Membership.mem (γ := IntermediateField k K) (toIntermediateField (adjoin k {x, y})) y
⊢ toAlgEquivAux g (x * y) = toAlgEquivAux g x * toAlgEquivAux g y
have rw : toAlgEquivAux g (x * y) =
  HMul.hMul (α := K)
    ((AlgEquiv.liftNormal (K₁ := ↥(toIntermediateField (adjoin k {x, y}))) (K₂ :=
            ↥(toIntermediateField (adjoin k {x, y})))
            ((proj (adjoin k {x, y}) :
                (a : (↑(limit (asProfiniteGaloisGroupFunctor k K)).toProfinite.toTop : Type u_4)) →
                  Gal(↥(adjoin k {x, y}).toIntermediateField/k))
              g)
            K :
          (a : K) → K)
        x :
      K)
    (toAlgEquivAux g y)
Tactic: rw [toAlgEquivAux_eq_liftNormal g x (adjoin k {x, y}) hx,
      toAlgEquivAux_eq_liftNormal g y (adjoin k {x, y}) hy,
      toAlgEquivAux_eq_liftNormal g (x * y) (adjoin k {x, y}) (mul_mem hx hy), map_mul]
-- 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
inst_6 : IsGalois k K
g :
  (↑((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)
x y : K
hx : Membership.mem (γ := IntermediateField k K) (toIntermediateField (adjoin k {x, y})) x
hy : Membership.mem (γ := IntermediateField k K) (toIntermediateField (adjoin k {x, y})) y
rw :
  toAlgEquivAux g (x * y) =
    HMul.hMul (α := K)
      ((AlgEquiv.liftNormal (K₁ := ↥(toIntermediateField (adjoin k {x, y}))) (K₂ :=
              ↥(toIntermediateField (adjoin k {x, y})))
              ((proj (adjoin k {x, y}) :
                  (a : (↑(limit (asProfiniteGaloisGroupFunctor k K)).toProfinite.toTop : Type u_4)) →
                    Gal(↥(adjoin k {x, y}).toIntermediateField/k))
                g)
              K :
            (a : K) → K)
          x :
        K)
      (toAlgEquivAux g y)
⊢ toAlgEquivAux g (x * y) = toAlgEquivAux g x * toAlgEquivAux g y
have rw₁ : toAlgEquivAux g (x * y) =
  HMul.hMul (α := K) (β := K)
    ((AlgEquiv.liftNormal (K₁ := ↥(toIntermediateField (adjoin k {x, y}))) (K₂ :=
            ↥(toIntermediateField (adjoin k {x, y})))
            ((proj (adjoin k {x, y}) :
                (a : (↑(limit (asProfiniteGaloisGroupFunctor k K)).toProfinite.toTop : Type u_4)) →
                  Gal(↥(adjoin k {x, y}).toIntermediateField/k))
              g)
            K :
          (a : K) → K)
        x :
      K)
    ((AlgEquiv.liftNormal (K₁ := ↥(toIntermediateField (adjoin k {x, y}))) (K₂ :=
            ↥(toIntermediateField (adjoin k {x, y})))
            ((proj (adjoin k {x, y}) :
                (a : (↑(limit (asProfiniteGaloisGroupFunctor k K)).toProfinite.toTop : Type u_4)) →
                  Gal(↥(adjoin k {x, y}).toIntermediateField/k))
              g)
            K :
          (a : K) → K)
        y :
      K)
Tactic: rw [toAlgEquivAux_eq_liftNormal g x (adjoin k {x, y}) hx,
      toAlgEquivAux_eq_liftNormal g y (adjoin k {x, y}) hy,
      toAlgEquivAux_eq_liftNormal g (x * y) (adjoin k {x, y}) (mul_mem hx hy), map_mul]
-- 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
inst_6 : IsGalois k K
g :
  (↑((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)
x y : K
⊢ toAlgEquivAux g (x + y) = toAlgEquivAux g x + toAlgEquivAux g y
have hx : Membership.mem (γ := IntermediateField k K) (toIntermediateField (adjoin k {x, y})) x
Tactic: have hx : x ∈ (adjoin k {x, y}).1 := subset_adjoin _ _ (Set.mem_insert x {y})
-- 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
inst_6 : IsGalois k K
g :
  (↑((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)
x y : K
hx : Membership.mem (γ := IntermediateField k K) (toIntermediateField (adjoin k {x, y})) x
⊢ toAlgEquivAux g (x + y) = toAlgEquivAux g x + toAlgEquivAux g y
have hy : Membership.mem (γ := IntermediateField k K) (toIntermediateField (adjoin k {x, y})) y
Tactic: have hy : y ∈ (adjoin k {x, y}).1 := subset_adjoin _ _ (Set.mem_insert_of_mem x 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
inst_6 : IsGalois k K
g :
  (↑((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))
      (limitToAlgEquiv g) =
    g
have a : (↑((algEquivToLimit k K : (a : Gal(K/k)) → (↑(limit (asProfiniteGaloisGroupFunctor k K)).toProfinite.toTop : Type u_4))
        (limitToAlgEquiv g)) :
    (j : (FiniteGaloisIntermediateField k K)ᵒᵖ) →
      (↑((asProfiniteGaloisGroupFunctor k K).obj j).toProfinite.toTop : Type u_4)) =
  (↑g :
    (j : (FiniteGaloisIntermediateField k K)ᵒᵖ) →
      (↑((asProfiniteGaloisGroupFunctor k K).obj j).toProfinite.toTop : Type u_4))
Tactic: apply Subtype.val_injective
-- 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
inst_6 : IsGalois k K
g :
  (↑((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))
          (limitToAlgEquiv g)) :
      (j : (FiniteGaloisIntermediateField k K)ᵒᵖ) →
        (↑((asProfiniteGaloisGroupFunctor k K).obj j).toProfinite.toTop : Type u_4)) =
    (↑g :
      (j : (FiniteGaloisIntermediateField k K)ᵒᵖ) →
        (↑((asProfiniteGaloisGroupFunctor k K).obj j).toProfinite.toTop : Type u_4))
have a : ∀ (L : (FiniteGaloisIntermediateField k K)ᵒᵖ),
  (↑((algEquivToLimit k K :
              (a : Gal(K/k)) → (↑(limit (asProfiniteGaloisGroupFunctor k K)).toProfinite.toTop : Type u_4))
            (limitToAlgEquiv g)) :
        (j : (FiniteGaloisIntermediateField k K)ᵒᵖ) →
          (↑((asProfiniteGaloisGroupFunctor k K).obj j).toProfinite.toTop : Type u_4))
      L =
    (↑g :
        (j : (FiniteGaloisIntermediateField k K)ᵒᵖ) →
          (↑((asProfiniteGaloisGroupFunctor k K).obj j).toProfinite.toTop : Type u_4))
      L
Tactic: ext L
-- 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
inst_6 : IsGalois k K
g :
  (↑((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)
L : (FiniteGaloisIntermediateField k K)ᵒᵖ
⊢ (limitToAlgEquiv g).restrictNormal ↥(unop L).toIntermediateField =
    (↑g :
        (j : (FiniteGaloisIntermediateField k K)ᵒᵖ) →
          (↑((asProfiniteGaloisGroupFunctor k K).obj j).toProfinite.toTop : Type u_4))
      L
have a : ∀ (x : ↥(unop L).toIntermediateField),
  Eq.{u_4 + 1} (α := K)
    (↑(((limitToAlgEquiv g).restrictNormal ↥(unop L).toIntermediateField :
            (a : ↥(unop L).toIntermediateField) → ↥(unop L).toIntermediateField)
          x) :
      K)
    (↑(((↑g :
                (j : (FiniteGaloisIntermediateField k K)ᵒᵖ) →
                  (↑((asProfiniteGaloisGroupFunctor k K).obj j).toProfinite.toTop : Type u_4))
              L :
            (a : ↥(unop L).toIntermediateField) → ↥(unop L).toIntermediateField)
          x) :
      K)
Tactic: ext x