awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.FieldTheory.JacobsonNoether (goal)
Goal:
L : Type u_2
D : Type u_3
inst : Field L
inst_1 : DivisionRing D
inst_2 : Algebra L D
inst_3 : Algebra.IsAlgebraic L D
inst_4 : IsCentral L D
hneq : Ne.{u_3 + 1} (α := Subalgebra L D) ⊥ ⊤
hcenter : Eq (α := Subalgebra L D) (Subalgebra.center L D) ⊥
ntrivial : Ne (α := Subring D) k ⊤
φ : ↥⊥ ≃ₐ[L] ↥(Subalgebra.center L D) := Subalgebra.equivOfEq ⊥ (Subalgebra.center L D) ⋯
equiv : RingEquiv.{u_2, u_3} L ↥k := (AlgEquiv.trans (AlgEquiv.symm (botEquiv L D)) φ).toRingEquiv
x_1 : Algebra.{u_2, u_3} L ↥k := equiv.toRingHom.toAlgebra
x_2 : Algebra.{u_3, u_2} (↥k) L := equiv.symm.toRingHom.toAlgebra
h : IsScalarTower L (↥k) D
h_1 : IsScalarTower (↥k) L D
h_2 : Algebra.IsAlgebraic (↥k) D
x : D
hxd : x ∉ k
hx : IsSeparable (↥k) x
⊢ x ∉ ⊥
have rwa : ∀ (hxd : x ∉ Subalgebra.toSubring (Subalgebra.center L D)) (hx : IsSeparable (↥k) x), x ∉ ⊥
Tactic: rwa [← Subalgebra.center_toSubring L, hcenter] at hxd
Cleared hypotheses: hxd, hx
-- Lean 4 proof step from Mathlib.FieldTheory.JacobsonNoether (goal)
Goal:
L : Type u_2
D : Type u_3
inst : Field L
inst_1 : DivisionRing D
inst_2 : Algebra L D
inst_3 : Algebra.IsAlgebraic L D
inst_4 : IsCentral L D
hneq : Ne.{u_3 + 1} (α := Subalgebra L D) ⊥ ⊤
hcenter : Eq (α := Subalgebra L D) (Subalgebra.center L D) ⊥
ntrivial : Ne (α := Subring D) k ⊤
φ : ↥⊥ ≃ₐ[L] ↥(Subalgebra.center L D) := Subalgebra.equivOfEq ⊥ (Subalgebra.center L D) ⋯
equiv : RingEquiv.{u_2, u_3} L ↥k := (AlgEquiv.trans (AlgEquiv.symm (botEquiv L D)) φ).toRingEquiv
x_1 : Algebra.{u_2, u_3} L ↥k := equiv.toRingHom.toAlgebra
x_2 : Algebra.{u_3, u_2} (↥k) L := equiv.symm.toRingHom.toAlgebra
h : IsScalarTower L (↥k) D
h_1 : IsScalarTower (↥k) L D
h_2 : Algebra.IsAlgebraic (↥k) D
x : D
hxd : x ∉ k
hx : IsSeparable (↥k) x
rwa : ∀ (hxd : x ∉ Subalgebra.toSubring (Subalgebra.center L D)) (hx : IsSeparable (↥k) x), x ∉ ⊥
⊢ x ∉ ⊥
have rwa₁ : ∀ (hxd : x ∉ Subalgebra.toSubring.{u_2, u_3} (R := L) ⊥) (hx : IsSeparable (↥k) x), x ∉ ⊥
Tactic: rwa [← Subalgebra.center_toSubring L, hcenter] at hxd
Cleared hypotheses: hxd, hx
-- Lean 4 proof step from Mathlib.FieldTheory.KrullTopology (hypothesis)
Goal:
K : Type u_1
L : Type u_2
inst : Field K
inst_1 : Field L
inst_2 : Algebra K L
E1 : IntermediateField K L
h_E1 : Membership.mem (γ := Set (IntermediateField K L)) (finiteExts K L) E1
E2 : IntermediateField K L
h_E2 : Membership.mem (γ := Set (IntermediateField K L)) (finiteExts K L) E2
⊢ ∃ (z : Set Gal(L/K)),
    Membership.mem (γ := Set (Set Gal(L/K))) ((fun (g : Subgroup Gal(L/K)) ↦ g.carrier) '' fixedByFinite K L) z ∧
      z ⊆
        (fun (g : Subgroup Gal(L/K)) ↦ g.carrier) E1.fixingSubgroup ∩
          (fun (g : Subgroup Gal(L/K)) ↦ g.carrier) E2.fixingSubgroup
have this : FiniteDimensional.{u_1, u_2} K ↥E1
Tactic: have : FiniteDimensional K E1 := h_E1
-- Lean 4 proof step from Mathlib.FieldTheory.KrullTopology (hypothesis)
Goal:
K : Type u_1
L : Type u_2
inst : Field K
inst_1 : Field L
inst_2 : Algebra K L
E1 : IntermediateField K L
h_E1 : Membership.mem (γ := Set (IntermediateField K L)) (finiteExts K L) E1
E2 : IntermediateField K L
h_E2 : Membership.mem (γ := Set (IntermediateField K L)) (finiteExts K L) E2
this : FiniteDimensional.{u_1, u_2} K ↥E1
⊢ ∃ (z : Set Gal(L/K)),
    Membership.mem (γ := Set (Set Gal(L/K))) ((fun (g : Subgroup Gal(L/K)) ↦ g.carrier) '' fixedByFinite K L) z ∧
      z ⊆
        (fun (g : Subgroup Gal(L/K)) ↦ g.carrier) E1.fixingSubgroup ∩
          (fun (g : Subgroup Gal(L/K)) ↦ g.carrier) E2.fixingSubgroup
have this₁ : FiniteDimensional.{u_1, u_2} K ↥E2
Tactic: have : FiniteDimensional K E2 := h_E2
-- Lean 4 proof step from Mathlib.FieldTheory.KrullTopology (goal)
Goal:
K : Type u_1
L : Type u_2
inst : Field K
inst_1 : Field L
inst_2 : Algebra K L
E1 : IntermediateField K L
h_E1 : Membership.mem (γ := Set (IntermediateField K L)) (finiteExts K L) E1
E2 : IntermediateField K L
h_E2 : Membership.mem (γ := Set (IntermediateField K L)) (finiteExts K L) E2
this_1 : FiniteDimensional.{u_1, u_2} K ↥E1
this : FiniteDimensional.{u_1, u_2} K ↥E2
⊢ ∃ (z : Set Gal(L/K)),
    Membership.mem (γ := Set (Set Gal(L/K))) ((fun (g : Subgroup Gal(L/K)) ↦ g.carrier) '' fixedByFinite K L) z ∧
      z ⊆
        (fun (g : Subgroup Gal(L/K)) ↦ g.carrier) E1.fixingSubgroup ∩
          (fun (g : Subgroup Gal(L/K)) ↦ g.carrier) E2.fixingSubgroup
have refine : (E1 ⊔ E2).fixingSubgroup.carrier ⊆
  (fun (g : Subgroup Gal(L/K)) ↦ g.carrier) E1.fixingSubgroup ∩
    (fun (g : Subgroup Gal(L/K)) ↦ g.carrier) E2.fixingSubgroup
Tactic: refine ⟨(E1 ⊔ E2).fixingSubgroup.carrier, ⟨_, ⟨_, E1.finiteDimensional_sup E2, rfl⟩, rfl⟩, ?_⟩
-- Lean 4 proof step from Mathlib.FieldTheory.KrullTopology (goal)
Goal:
K : Type u_1
L : Type u_2
inst : Field K
inst_1 : Field L
inst_2 : Algebra K L
U : Set Gal(L/K)
hU : Membership.mem (γ := Set (Set Gal(L/K))) (FilterBasis.sets (galBasis K L)) U
⊢ U * U ⊆ U
have rcases : ∀ (H : Subgroup Gal(L/K)),
  Membership.mem (γ := Set (Subgroup Gal(L/K))) (fixedByFinite K L) H →
    HMul.hMul (α := Set Gal(L/K)) ((fun (g : Subgroup Gal(L/K)) ↦ g.carrier) H)
        ((fun (g : Subgroup Gal(L/K)) ↦ g.carrier) H) ⊆
      (fun (g : Subgroup Gal(L/K)) ↦ g.carrier) H
Tactic: rcases hU with ⟨H, _, rfl⟩
Cleared hypotheses: U, hU
-- Lean 4 proof step from Mathlib.FieldTheory.KrullTopology (goal)
Goal:
K : Type u_1
L : Type u_2
inst : Field K
inst_1 : Field L
inst_2 : Algebra K L
U : Set Gal(L/K)
hU : Membership.mem (γ := Set (Set Gal(L/K))) (FilterBasis.sets (galBasis K L)) U
⊢ U ⊆ (fun (x : Gal(L/K)) ↦ x⁻¹) ⁻¹' U
have rcases : ∀ (H : Subgroup Gal(L/K)),
  Membership.mem (γ := Set (Subgroup Gal(L/K))) (fixedByFinite K L) H →
    Subset.{u_2} (α := Set Gal(L/K)) ((fun (g : Subgroup Gal(L/K)) ↦ g.carrier) H)
      (Set.preimage (β := Gal(L/K)) (fun (x : Gal(L/K)) ↦ x⁻¹) ((fun (g : Subgroup Gal(L/K)) ↦ g.carrier) H))
Tactic: rcases hU with ⟨H, _, rfl⟩
Cleared hypotheses: U, hU
-- Lean 4 proof step from Mathlib.FieldTheory.KrullTopology (goal)
Goal:
K : Type u_1
L : Type u_2
inst : Field K
inst_1 : Field L
inst_2 : Algebra K L
σ : Gal(L/K)
E : IntermediateField K L
hE : Membership.mem (γ := Set (IntermediateField K L)) (finiteExts K L) E
F : IntermediateField K L := IntermediateField.map (↑σ.symm : L →ₐ[K] L) E
⊢ ∃ (V : Set Gal(L/K)),
    Membership.mem (γ := Set (Set Gal(L/K))) (FilterBasis.sets (galBasis K L)) V ∧
      V ⊆
        Set.preimage (β := Gal(L/K)) (fun (x : Gal(L/K)) ↦ σ * x * σ⁻¹)
          ((fun (g : Subgroup Gal(L/K)) ↦ g.carrier) E.fixingSubgroup)
have refine_1 : Membership.mem (γ := Set (IntermediateField K L)) (finiteExts K L) F
Tactic: refine ⟨F.fixingSubgroup.carrier, ⟨⟨F.fixingSubgroup, ⟨F, ?_, rfl⟩, rfl⟩, fun g hg => ?_⟩⟩
-- Lean 4 proof step from Mathlib.FieldTheory.KrullTopology (goal)
Goal:
K : Type u_1
L : Type u_2
inst : Field K
inst_1 : Field L
inst_2 : Algebra K L
σ : Gal(L/K)
E : IntermediateField K L
hE : Membership.mem (γ := Set (IntermediateField K L)) (finiteExts K L) E
F : IntermediateField K L := IntermediateField.map (↑σ.symm : L →ₐ[K] L) E
refine_1 : Membership.mem (γ := Set (IntermediateField K L)) (finiteExts K L) F
⊢ ∃ (V : Set Gal(L/K)),
    Membership.mem (γ := Set (Set Gal(L/K))) (FilterBasis.sets (galBasis K L)) V ∧
      V ⊆
        Set.preimage (β := Gal(L/K)) (fun (x : Gal(L/K)) ↦ σ * x * σ⁻¹)
          ((fun (g : Subgroup Gal(L/K)) ↦ g.carrier) E.fixingSubgroup)
have refine_2 : ∀ (g : Gal(L/K)) (hg : g ∈ F.fixingSubgroup.carrier),
  Membership.mem (γ := Set Gal(L/K))
    (Set.preimage (β := Gal(L/K)) (fun (x : Gal(L/K)) ↦ σ * x * σ⁻¹)
      ((fun (g : Subgroup Gal(L/K)) ↦ g.carrier) E.fixingSubgroup))
    g
Tactic: refine ⟨F.fixingSubgroup.carrier, ⟨⟨F.fixingSubgroup, ⟨F, ?_, rfl⟩, rfl⟩, fun g hg => ?_⟩⟩
-- Lean 4 proof step from Mathlib.FieldTheory.KrullTopology (hypothesis)
Goal:
K : Type u_1
L : Type u_2
inst : Field K
inst_1 : Field L
inst_2 : Algebra K L
σ : Gal(L/K)
E : IntermediateField K L
hE : Membership.mem (γ := Set (IntermediateField K L)) (finiteExts K L) E
F : IntermediateField K L := IntermediateField.map (↑σ.symm : L →ₐ[K] L) E
⊢ Membership.mem (γ := Set (IntermediateField K L)) (finiteExts K L) F
have this : FiniteDimensional.{u_1, u_2} K ↥E
Tactic: have : FiniteDimensional K E := hE
-- Lean 4 proof step from Mathlib.FieldTheory.KrullTopology (goal)
Goal:
K : Type u_1
L : Type u_2
inst : Field K
inst_1 : Field L
inst_2 : Algebra K L
σ : Gal(L/K)
E : IntermediateField K L
hE : Membership.mem (γ := Set (IntermediateField K L)) (finiteExts K L) E
F : IntermediateField K L := IntermediateField.map (↑σ.symm : L →ₐ[K] L) E
g : Gal(L/K)
hg : g ∈ F.fixingSubgroup.carrier
⊢ Membership.mem (γ := Set Gal(L/K))
    (Set.preimage (β := Gal(L/K)) (fun (x : Gal(L/K)) ↦ σ * x * σ⁻¹)
      ((fun (g : Subgroup Gal(L/K)) ↦ g.carrier) E.fixingSubgroup))
    g
have refine_2 : σ * g * σ⁻¹ ∈ E.fixingSubgroup
Tactic: change σ * g * σ⁻¹ ∈ E.fixingSubgroup
-- Lean 4 proof step from Mathlib.FieldTheory.KrullTopology (goal)
Goal:
K : Type u_1
L : Type u_2
inst : Field K
inst_1 : Field L
inst_2 : Algebra K L
σ : Gal(L/K)
E : IntermediateField K L
hE : Membership.mem (γ := Set (IntermediateField K L)) (finiteExts K L) E
F : IntermediateField K L := IntermediateField.map (↑σ.symm : L →ₐ[K] L) E
g : Gal(L/K)
hg : g ∈ F.fixingSubgroup.carrier
⊢ σ * g * σ⁻¹ ∈ E.fixingSubgroup
have refine_2 : ∀ x ∈ E, (σ * g * σ⁻¹ : (a : L) → L) x = x
Tactic: rw [IntermediateField.mem_fixingSubgroup_iff]
-- Lean 4 proof step from Mathlib.FieldTheory.KrullTopology (goal)
Goal:
K : Type u_1
L : Type u_2
inst : Field K
inst_1 : Field L
inst_2 : Algebra K L
σ : Gal(L/K)
E : IntermediateField K L
hE : Membership.mem (γ := Set (IntermediateField K L)) (finiteExts K L) E
F : IntermediateField K L := IntermediateField.map (↑σ.symm : L →ₐ[K] L) E
g : Gal(L/K)
hg : g ∈ F.fixingSubgroup.carrier
x : L
hx : x ∈ E
⊢ (σ * g * σ⁻¹ : (a : L) → L) x = x
have refine_2 : (σ : (a : L) → L) ((g : (a : L) → L) ((σ⁻¹ : (a : L) → L) x : L) : L) = x
Tactic: change σ (g (σ⁻¹ x)) = x
-- Lean 4 proof step from Mathlib.FieldTheory.KrullTopology (hypothesis)
Goal:
K : Type u_1
L : Type u_2
inst : Field K
inst_1 : Field L
inst_2 : Algebra K L
σ : Gal(L/K)
E : IntermediateField K L
hE : Membership.mem (γ := Set (IntermediateField K L)) (finiteExts K L) E
F : IntermediateField K L := IntermediateField.map (↑σ.symm : L →ₐ[K] L) E
g : Gal(L/K)
hg : g ∈ F.fixingSubgroup.carrier
x : L
hx : x ∈ E
⊢ (σ : (a : L) → L) ((g : (a : L) → L) ((σ⁻¹ : (a : L) → L) x : L) : L) = x
have h_in_F : Membership.mem (α := L) F ((σ⁻¹ : (a : L) → L) x : L)
Tactic: have h_in_F : σ⁻¹ x ∈ F := ⟨x, hx, by dsimp⟩
-- Lean 4 proof step from Mathlib.FieldTheory.KrullTopology (goal)
Goal:
K : Type u_1
L : Type u_2
inst : Field K
inst_1 : Field L
inst_2 : Algebra K L
σ : Gal(L/K)
E : IntermediateField K L
hE : Membership.mem (γ := Set (IntermediateField K L)) (finiteExts K L) E
F : IntermediateField K L := IntermediateField.map (↑σ.symm : L →ₐ[K] L) E
g : Gal(L/K)
hg : g ∈ F.fixingSubgroup.carrier
x : L
hx : x ∈ E
h_in_F : Membership.mem (α := L) F ((σ⁻¹ : (a : L) → L) x : L)
⊢ Eq (α := L) ((g : (a : L) → L) ((σ⁻¹ : (a : L) → L) x : L) : L) ((σ⁻¹ : (a : L) → L) x : L)
have rw : ∀ (hg : ∀ x ∈ F, (g : (a : L) → L) x = x) (x : L) (hx : x ∈ E)
  (h_in_F : Membership.mem (α := L) F ((σ⁻¹ : (a : L) → L) x : L)),
  Eq (α := L) ((g : (a : L) → L) ((σ⁻¹ : (a : L) → L) x : L) : L) ((σ⁻¹ : (a : L) → L) x : L)
Tactic: rw [Subgroup.mem_carrier, IntermediateField.mem_fixingSubgroup_iff F g] at hg
Cleared hypotheses: hg, x, hx, h_in_F
-- Lean 4 proof step from Mathlib.FieldTheory.KrullTopology (hypothesis)
Goal:
K : Type u_1
L : Type u_2
inst : Field K
inst_1 : Field L
inst_2 : Algebra K L
σ : Gal(L/K)
E : IntermediateField K L
hE : Membership.mem (γ := Set (IntermediateField K L)) (finiteExts K L) E
F : IntermediateField K L := IntermediateField.map (↑σ.symm : L →ₐ[K] L) E
g : Gal(L/K)
hg : g ∈ F.fixingSubgroup.carrier
x : L
hx : x ∈ E
h_in_F : Membership.mem (α := L) F ((σ⁻¹ : (a : L) → L) x : L)
⊢ (σ : (a : L) → L) ((g : (a : L) → L) ((σ⁻¹ : (a : L) → L) x : L) : L) = x
have h_g_fix : Eq (α := L) ((g : (a : L) → L) ((σ⁻¹ : (a : L) → L) x : L) : L) ((σ⁻¹ : (a : L) → L) x : L)
Tactic: have h_g_fix : g (σ⁻¹ x) = σ⁻¹ x := by
      rw [Subgroup.mem_carrier, IntermediateField.mem_fixingSubgroup_iff F g] at hg
      exact hg (σ⁻¹ x) h_in_F
-- Lean 4 proof step from Mathlib.FieldTheory.KrullTopology (goal)
Goal:
K : Type u_1
L : Type u_2
inst : Field K
inst_1 : Field L
inst_2 : Algebra K L
σ : Gal(L/K)
E : IntermediateField K L
hE : Membership.mem (γ := Set (IntermediateField K L)) (finiteExts K L) E
F : IntermediateField K L := IntermediateField.map (↑σ.symm : L →ₐ[K] L) E
g : Gal(L/K)
hg : g ∈ F.fixingSubgroup.carrier
x : L
hx : x ∈ E
h_in_F : Membership.mem (α := L) F ((σ⁻¹ : (a : L) → L) x : L)
h_g_fix : Eq (α := L) ((g : (a : L) → L) ((σ⁻¹ : (a : L) → L) x : L) : L) ((σ⁻¹ : (a : L) → L) x : L)
⊢ (σ : (a : L) → L) ((g : (a : L) → L) ((σ⁻¹ : (a : L) → L) x : L) : L) = x
have refine_2 : (σ : (a : L) → L) ((σ⁻¹ : (a : L) → L) x : L) = x
Tactic: rw [h_g_fix]
-- Lean 4 proof step from Mathlib.FieldTheory.KrullTopology (goal)
Goal:
K : Type u_1
L : Type u_2
inst : Field K
inst_1 : Field L
inst_2 : Algebra K L
s : Set Gal(L/K)
⊢ Membership.mem (γ := Filter Gal(L/K)) (𝓝 1) s ↔
    ∃ (E : IntermediateField K L), FiniteDimensional.{u_1, u_2} K ↥E ∧ (↑E.fixingSubgroup : Set Gal(L/K)) ⊆ s
have rw : Membership.mem (γ := Filter Gal(L/K)) (FilterBasis.filter (@GroupFilterBasis.toFilterBasis _ AlgEquiv.aut _)) s ↔
  ∃ (E : IntermediateField K L), FiniteDimensional.{u_1, u_2} K ↥E ∧ (↑E.fixingSubgroup : Set Gal(L/K)) ⊆ s
Tactic: rw [GroupFilterBasis.nhds_one_eq]
-- Lean 4 proof step from Mathlib.FieldTheory.KrullTopology (goal)
Goal:
K : Type u_1
L : Type u_2
inst : Field K
inst_1 : Field L
inst_2 : Algebra K L
s : Set Gal(L/K)
⊢ Membership.mem (γ := Filter Gal(L/K)) (FilterBasis.filter (@GroupFilterBasis.toFilterBasis _ AlgEquiv.aut _)) s ↔
    ∃ (E : IntermediateField K L), FiniteDimensional.{u_1, u_2} K ↥E ∧ (↑E.fixingSubgroup : Set Gal(L/K)) ⊆ s
have mp : Membership.mem (γ := Filter Gal(L/K)) (FilterBasis.filter (@GroupFilterBasis.toFilterBasis _ AlgEquiv.aut _)) s →
  ∃ (E : IntermediateField K L), FiniteDimensional.{u_1, u_2} K ↥E ∧ (↑E.fixingSubgroup : Set Gal(L/K)) ⊆ s
Tactic: constructor
-- Lean 4 proof step from Mathlib.FieldTheory.KrullTopology (goal)
Goal:
K : Type u_1
L : Type u_2
inst : Field K
inst_1 : Field L
inst_2 : Algebra K L
s : Set Gal(L/K)
mp :
  Membership.mem (γ := Filter Gal(L/K)) (FilterBasis.filter (@GroupFilterBasis.toFilterBasis _ AlgEquiv.aut _)) s →
    ∃ (E : IntermediateField K L), FiniteDimensional.{u_1, u_2} K ↥E ∧ (↑E.fixingSubgroup : Set Gal(L/K)) ⊆ s
⊢ Membership.mem (γ := Filter Gal(L/K)) (FilterBasis.filter (@GroupFilterBasis.toFilterBasis _ AlgEquiv.aut _)) s ↔
    ∃ (E : IntermediateField K L), FiniteDimensional.{u_1, u_2} K ↥E ∧ (↑E.fixingSubgroup : Set Gal(L/K)) ⊆ s
have mpr : (∃ (E : IntermediateField K L), FiniteDimensional.{u_1, u_2} K ↥E ∧ (↑E.fixingSubgroup : Set Gal(L/K)) ⊆ s) →
  Membership.mem (γ := Filter Gal(L/K)) (FilterBasis.filter (@GroupFilterBasis.toFilterBasis _ AlgEquiv.aut _)) s
Tactic: constructor
-- Lean 4 proof step from Mathlib.FieldTheory.KrullTopology (goal)
Goal:
K : Type u_1
L : Type u_2
inst : Field K
inst_1 : Field L
inst_2 : Algebra K L
inst_3 : Normal K L
s : Set Gal(L/K)
⊢ Membership.mem (γ := Filter Gal(L/K)) (𝓝 1) s ↔
    ∃ (E : IntermediateField K L),
      FiniteDimensional.{u_1, u_2} K ↥E ∧ Normal.{u_1, u_2} K ↥E ∧ (↑E.fixingSubgroup : Set Gal(L/K)) ⊆ s
have rw : (∃ (E : IntermediateField K L), FiniteDimensional.{u_1, u_2} K ↥E ∧ (↑E.fixingSubgroup : Set Gal(L/K)) ⊆ s) ↔
  ∃ (E : IntermediateField K L),
    FiniteDimensional.{u_1, u_2} K ↥E ∧ Normal.{u_1, u_2} K ↥E ∧ (↑E.fixingSubgroup : Set Gal(L/K)) ⊆ s
Tactic: rw [krullTopology_mem_nhds_one_iff]
-- Lean 4 proof step from Mathlib.FieldTheory.KrullTopology (goal)
Goal:
K : Type u_1
L : Type u_2
inst : Field K
inst_1 : Field L
inst_2 : Algebra K L
inst_3 : Normal K L
s : Set Gal(L/K)
⊢ (∃ (E : IntermediateField K L), FiniteDimensional.{u_1, u_2} K ↥E ∧ (↑E.fixingSubgroup : Set Gal(L/K)) ⊆ s) ↔
    ∃ (E : IntermediateField K L),
      FiniteDimensional.{u_1, u_2} K ↥E ∧ Normal.{u_1, u_2} K ↥E ∧ (↑E.fixingSubgroup : Set Gal(L/K)) ⊆ s
have refine : (∃ (E : IntermediateField K L), FiniteDimensional.{u_1, u_2} K ↥E ∧ (↑E.fixingSubgroup : Set Gal(L/K)) ⊆ s) →
  ∀ (E : IntermediateField K L),
    FiniteDimensional.{u_1, u_2} K ↥E →
      ∀ (hE : (↑E.fixingSubgroup : Set Gal(L/K)) ⊆ s),
        ∃ (E : IntermediateField K L),
          FiniteDimensional.{u_1, u_2} K ↥E ∧ Normal.{u_1, u_2} K ↥E ∧ (↑E.fixingSubgroup : Set Gal(L/K)) ⊆ s
Tactic: refine ⟨fun ⟨E, _, hE⟩ ↦ ?_, fun ⟨E, hE⟩ ↦ ⟨E, hE.1, hE.2.2⟩⟩
-- Lean 4 proof step from Mathlib.FieldTheory.KrullTopology (goal)
Goal:
K : Type u_1
L : Type u_2
inst : Field K
inst_1 : Field L
inst_2 : Algebra K L
inst_3 : Normal K L
s : Set Gal(L/K)
h : ∃ (E : IntermediateField K L), FiniteDimensional.{u_1, u_2} K ↥E ∧ (↑E.fixingSubgroup : Set Gal(L/K)) ⊆ s
E : IntermediateField K L
left : FiniteDimensional.{u_1, u_2} K ↥E
hE : (↑E.fixingSubgroup : Set Gal(L/K)) ⊆ s
⊢ ∃ (E : IntermediateField K L),
    FiniteDimensional.{u_1, u_2} K ↥E ∧ Normal.{u_1, u_2} K ↥E ∧ (↑E.fixingSubgroup : Set Gal(L/K)) ⊆ s
have h : FiniteDimensional.{u_1, u_2} K ↥(IntermediateField.normalClosure K (↥E) L) ∧
  Normal.{u_1, u_2} K ↥(IntermediateField.normalClosure K (↥E) L) ∧
    (↑(IntermediateField.fixingSubgroup (IntermediateField.normalClosure K (↥E) L)) : Set Gal(L/K)) ⊆ s
Tactic: use (IntermediateField.normalClosure K E L)
-- Lean 4 proof step from Mathlib.FieldTheory.KrullTopology (goal)
Goal:
K : Type u_1
L : Type u_2
inst : Field K
inst_1 : Field L
inst_2 : Algebra K L
inst_3 : Normal K L
s : Set Gal(L/K)
h : ∃ (E : IntermediateField K L), FiniteDimensional.{u_1, u_2} K ↥E ∧ (↑E.fixingSubgroup : Set Gal(L/K)) ⊆ s
E : IntermediateField K L
left : FiniteDimensional.{u_1, u_2} K ↥E
hE : (↑E.fixingSubgroup : Set Gal(L/K)) ⊆ s
⊢ FiniteDimensional.{u_1, u_2} K ↥(IntermediateField.normalClosure K (↥E) L) ∧
    Normal.{u_1, u_2} K ↥(IntermediateField.normalClosure K (↥E) L) ∧
      (↑(IntermediateField.fixingSubgroup (IntermediateField.normalClosure K (↥E) L)) : Set Gal(L/K)) ⊆ s
have h : (↑(IntermediateField.fixingSubgroup (IntermediateField.normalClosure K (↥E) L)) : Set Gal(L/K)) ⊆ s
Tactic: simp only [normalClosure.is_finiteDimensional K E L, normalClosure.normal K E L, true_and]
-- Lean 4 proof step from Mathlib.FieldTheory.KrullTopology (hypothesis)
Goal:
K : Type u_1
L : Type u_2
inst : Field K
inst_1 : Field L
inst_2 : Algebra K L
E : IntermediateField K L
inst_3 : FiniteDimensional.{u_1, u_2} K ↥E
⊢ IsOpen (↑E.fixingSubgroup : Set Gal(L/K))
have h_basis : Membership.mem (γ := GroupFilterBasis Gal(L/K)) (galGroupBasis K L) E.fixingSubgroup.carrier
Tactic: have h_basis : E.fixingSubgroup.carrier ∈ galGroupBasis K L :=
    ⟨E.fixingSubgroup, ⟨E, ‹_›, rfl⟩, rfl⟩
-- Lean 4 proof step from Mathlib.FieldTheory.KrullTopology (hypothesis)
Goal:
K : Type u_1
L : Type u_2
inst : Field K
inst_1 : Field L
inst_2 : Algebra K L
E : IntermediateField K L
inst_3 : FiniteDimensional.{u_1, u_2} K ↥E
h_basis : Membership.mem (γ := GroupFilterBasis Gal(L/K)) (galGroupBasis K L) E.fixingSubgroup.carrier
⊢ IsOpen (↑E.fixingSubgroup : Set Gal(L/K))
have h_nhds : Membership.mem (γ := Filter Gal(L/K)) (𝓝 1) E.fixingSubgroup.carrier
Tactic: have h_nhds := GroupFilterBasis.mem_nhds_one (galGroupBasis K L) h_basis
-- Lean 4 proof step from Mathlib.FieldTheory.KrullTopology (goal)
Goal:
K : Type u_1
L : Type u_2
inst : Field K
inst_1 : Field L
inst_2 : Algebra K L
inst_3 : Algebra.IsIntegral K L
f g : Gal(L/K)
hfg : f ≠ g
φ : Gal(L/K) := f⁻¹ * g
⊢ (fun (x y : Gal(L/K)) ↦ ∃ (u : Set Gal(L/K)) (v : Set Gal(L/K)), IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ y ∈ v ∧ Disjoint u v) f
    g
have obtain : ∀ (x : L) (hx : Ne (α := L) ((f : (a : L) → L) x : L) ((g : (a : L) → L) x : L)),
  (fun (x y : Gal(L/K)) ↦ ∃ (u : Set Gal(L/K)) (v : Set Gal(L/K)), IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ y ∈ v ∧ Disjoint u v) f
    g
Tactic: obtain ⟨x, hx⟩ := DFunLike.exists_ne hfg
-- Lean 4 proof step from Mathlib.FieldTheory.KrullTopology (goal)
Goal:
K : Type u_1
L : Type u_2
inst : Field K
inst_1 : Field L
inst_2 : Algebra K L
inst_3 : Algebra.IsIntegral K L
f g : Gal(L/K)
hfg : f ≠ g
φ : Gal(L/K) := f⁻¹ * g
x : L
hx : Ne (α := L) ((f : (a : L) → L) x : L) ((g : (a : L) → L) x : L)
⊢ (φ : (a : L) → L) x ≠ x
have apply : Ne (α := L) ((f : (a : L) → L) ((φ : (a : L) → L) x : L) : L) ((f : (a : L) → L) x : L)
Tactic: apply ne_of_apply_ne f
-- Lean 4 proof step from Mathlib.FieldTheory.KrullTopology (goal)
Goal:
K : Type u_1
L : Type u_2
inst : Field K
inst_1 : Field L
inst_2 : Algebra K L
inst_3 : Algebra.IsIntegral K L
f g : Gal(L/K)
hfg : f ≠ g
φ : Gal(L/K) := f⁻¹ * g
x : L
hx : Ne (α := L) ((f : (a : L) → L) x : L) ((g : (a : L) → L) x : L)
⊢ Ne (α := L) ((f : (a : L) → L) ((φ : (a : L) → L) x : L) : L) ((f : (a : L) → L) x : L)
have change : Ne (α := L) ((f : (a : L) → L) ((f.symm : (a : L) → L) ((g : (a : L) → L) x : L) : L) : L) ((f : (a : L) → L) x : L)
Tactic: change f (f.symm (g x)) ≠ f x
-- Lean 4 proof step from Mathlib.FieldTheory.KrullTopology (hypothesis)
Goal:
K : Type u_1
L : Type u_2
inst : Field K
inst_1 : Field L
inst_2 : Algebra K L
inst_3 : Algebra.IsIntegral K L
f g : Gal(L/K)
hfg : f ≠ g
φ : Gal(L/K) := f⁻¹ * g
x : L
hx : Ne (α := L) ((f : (a : L) → L) x : L) ((g : (a : L) → L) x : L)
⊢ (fun (x y : Gal(L/K)) ↦ ∃ (u : Set Gal(L/K)) (v : Set Gal(L/K)), IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ y ∈ v ∧ Disjoint u v) f
    g
have hφx : (φ : (a : L) → L) x ≠ x
Tactic: have hφx : φ x ≠ x := by
        apply ne_of_apply_ne f
        change f (f.symm (g x)) ≠ f x
        rw [AlgEquiv.apply_symm_apply f (g x), ne_comm]
        exact hx
-- Lean 4 proof step from Mathlib.FieldTheory.KrullTopology (hypothesis)
Goal:
K : Type u_1
L : Type u_2
inst : Field K
inst_1 : Field L
inst_2 : Algebra K L
inst_3 : Algebra.IsIntegral K L
f g : Gal(L/K)
hfg : f ≠ g
φ : Gal(L/K) := f⁻¹ * g
x : L
hx : Ne (α := L) ((f : (a : L) → L) x : L) ((g : (a : L) → L) x : L)
hφx : (φ : (a : L) → L) x ≠ x
E : IntermediateField K L := K⟮x⟯
h_findim : FiniteDimensional.{u_1, u_2} K ↥E := ⋯
H : Subgroup Gal(L/K) := E.fixingSubgroup
⊢ (fun (x y : Gal(L/K)) ↦ ∃ (u : Set Gal(L/K)) (v : Set Gal(L/K)), IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ y ∈ v ∧ Disjoint u v) f
    g
have h_basis : Membership.mem (γ := GroupFilterBasis Gal(L/K)) (galGroupBasis K L) (↑H : Set Gal(L/K))
Tactic: have h_basis : (H : Set Gal(L/K)) ∈ galGroupBasis K L := ⟨H, ⟨E, ⟨h_findim, rfl⟩⟩, rfl⟩
-- Lean 4 proof step from Mathlib.FieldTheory.KrullTopology (hypothesis)
Goal:
K : Type u_1
L : Type u_2
inst : Field K
inst_1 : Field L
inst_2 : Algebra K L
inst_3 : Algebra.IsIntegral K L
f g : Gal(L/K)
hfg : f ≠ g
φ : Gal(L/K) := f⁻¹ * g
x : L
hx : Ne (α := L) ((f : (a : L) → L) x : L) ((g : (a : L) → L) x : L)
hφx : (φ : (a : L) → L) x ≠ x
E : IntermediateField K L := K⟮x⟯
h_findim : FiniteDimensional.{u_1, u_2} K ↥E := ⋯
H : Subgroup Gal(L/K) := E.fixingSubgroup
h_basis : Membership.mem (γ := GroupFilterBasis Gal(L/K)) (galGroupBasis K L) (↑H : Set Gal(L/K))
⊢ (fun (x y : Gal(L/K)) ↦ ∃ (u : Set Gal(L/K)) (v : Set Gal(L/K)), IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ y ∈ v ∧ Disjoint u v) f
    g
have h_nhds : Membership.mem (γ := Filter Gal(L/K)) (𝓝 1) (↑H : Set Gal(L/K))
Tactic: have h_nhds := GroupFilterBasis.mem_nhds_one (galGroupBasis K L) h_basis
-- Lean 4 proof step from Mathlib.FieldTheory.KrullTopology (hypothesis)
Goal:
K : Type u_1
L : Type u_2
inst : Field K
inst_1 : Field L
inst_2 : Algebra K L
inst_3 : Algebra.IsIntegral K L
f g : Gal(L/K)
hfg : f ≠ g
φ : Gal(L/K) := f⁻¹ * g
x : L
hx : Ne (α := L) ((f : (a : L) → L) x : L) ((g : (a : L) → L) x : L)
hφx : (φ : (a : L) → L) x ≠ x
E : IntermediateField K L := K⟮x⟯
h_findim : FiniteDimensional.{u_1, u_2} K ↥E := ⋯
H : Subgroup Gal(L/K) := E.fixingSubgroup
h_basis : Membership.mem (γ := GroupFilterBasis Gal(L/K)) (galGroupBasis K L) (↑H : Set Gal(L/K))
h_nhds : Membership.mem (γ := Filter Gal(L/K)) (𝓝 1) (↑H : Set Gal(L/K))
⊢ (fun (x y : Gal(L/K)) ↦ ∃ (u : Set Gal(L/K)) (v : Set Gal(L/K)), IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ y ∈ v ∧ Disjoint u v) f
    g
have h_nhds₁ : ∃ t ⊆ (↑H : Set Gal(L/K)), IsOpen t ∧ 1 ∈ t
Tactic: rw [mem_nhds_iff] at h_nhds
Cleared hypotheses: h_nhds