awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.FieldTheory.IntermediateField.Adjoin.Basic (goal)
Goal:
K : Type u_3
L : Type u_4
inst : Field K
inst_1 : Field L
inst_2 : Algebra K L
ι : Type u_5
t : ι → IntermediateField K L
p : ι → K[X]
s : Finset ι
h0 : Ne.{u_3 + 1} (α := K[X]) (∏ i ∈ s, p i) 0
F : IntermediateField K L := ⨆ i ∈ s, t i
hF : ∀ i ∈ s, t i ≤ F
h : ∀ i ∈ s, Splits.{u_4} (R := ↥(t i)) (Polynomial.map (algebraMap K ↥(t i)) (p i)) ∧ t i = adjoin K ((p i).rootSet L)
⊢ Eq (α := IntermediateField K L) (⨆ i ∈ s, t i) (adjoin K (rootSet.{u_3, u_4} (T := K) (∏ i ∈ s, p i) L))
have simp : Eq.{u_4 + 1} (α := IntermediateField K L) (⨆ i ∈ s, t i) (⨆ i ∈ s, adjoin K ((p i).rootSet L))
Tactic: simp only [rootSet_prod p s h0, ← Set.iSup_eq_iUnion, (@gc K _ L _ _).l_iSup₂]
-- Lean 4 proof step from Mathlib.FieldTheory.IntermediateField.Adjoin.Basic (hypothesis)
Goal:
F : Type u_1
inst : Field F
E : Type u_2
inst_1 : Field E
inst_2 : Algebra F E
K : Type u_3
inst_3 : Field K
inst_4 : Algebra F K
inst_5 : Algebra E K
inst_6 : IsScalarTower F E K
L : IntermediateField F K
halg : Algebra.IsAlgebraic F E ∨ Algebra.IsAlgebraic.{u_1, u_3} F ↥L
⊢ Module.rank E ↥(adjoin E (↑L : Set K)) ≤ Module.rank F ↥L
have h : Eq (α := Subalgebra E K) (toSubalgebra (adjoin E (↑L.toSubalgebra : Set K)))
  (Algebra.adjoin E (↑L.toSubalgebra : Set K))
Tactic: have h : (adjoin E (L.toSubalgebra : Set K)).toSubalgebra =
      Algebra.adjoin E (L.toSubalgebra : Set K) :=
    L.adjoin_intermediateField_toSubalgebra_of_isAlgebraic E halg
-- Lean 4 proof step from Mathlib.FieldTheory.IntermediateField.Adjoin.Basic (hypothesis)
Goal:
F : Type u_1
inst : Field F
E : Type u_2
inst_1 : Field E
inst_2 : Algebra F E
K : Type u_3
inst_3 : Field K
inst_4 : Algebra F K
inst_5 : Algebra E K
inst_6 : IsScalarTower F E K
L : IntermediateField F K
halg : Algebra.IsAlgebraic F E ∨ Algebra.IsAlgebraic.{u_1, u_3} F ↥L
h :
  Eq (α := Subalgebra E K) (toSubalgebra (adjoin E (↑L.toSubalgebra : Set K)))
    (Algebra.adjoin E (↑L.toSubalgebra : Set K))
⊢ Module.rank E ↥(adjoin E (↑L : Set K)) ≤ Module.rank F ↥L
have this : Module.rank E ↥(Algebra.adjoin E (↑L.toSubalgebra : Set K)) ≤ Module.rank F ↥L.toSubalgebra
Tactic: have := L.toSubalgebra.adjoin_rank_le E
-- Lean 4 proof step from Mathlib.FieldTheory.IntermediateField.Adjoin.Basic (goal)
Goal:
F : Type u_1
inst : Field F
E : Type u_2
inst_1 : Field E
inst_2 : Algebra F E
K : Type u_3
inst_3 : Field K
inst_4 : Algebra F K
inst_5 : Algebra E K
inst_6 : IsScalarTower F E K
L : IntermediateField F K
halg : Algebra.IsAlgebraic F E ∨ Algebra.IsAlgebraic.{u_1, u_3} F ↥L
h :
  Eq (α := Subalgebra E K) (toSubalgebra (adjoin E (↑L.toSubalgebra : Set K)))
    (Algebra.adjoin E (↑L.toSubalgebra : Set K))
this : Module.rank E ↥(Algebra.adjoin E (↑L.toSubalgebra : Set K)) ≤ Module.rank F ↥L.toSubalgebra
⊢ Module.rank E ↥(adjoin E (↑L : Set K)) ≤ Module.rank F ↥L
have rwa : ∀ (this : Module.rank E ↥(toSubalgebra (adjoin E (↑L.toSubalgebra : Set K))) ≤ Module.rank F ↥L.toSubalgebra),
  Module.rank E ↥(adjoin E (↑L : Set K)) ≤ Module.rank F ↥L
Tactic: rwa [(Subalgebra.equivOfEq _ _ h).symm.toLinearEquiv.rank_eq] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.FieldTheory.IntermediateField.Adjoin.Basic (goal)
Goal:
F : Type u_1
inst : Field F
E : Type u_2
inst_1 : Field E
inst_2 : Algebra F E
x : E
⊢ IsCompactElement (α := IntermediateField F E) F⟮x⟯
have simp_rw : ∀ (s : Set (IntermediateField F E)),
  s.Nonempty →
    DirectedOn (fun (x1 x2 : IntermediateField F E) ↦ x1 ≤ x2) s →
      Membership.mem (γ := IntermediateField F E) (⨆ (a : (↑s : Type u_2)), (↑a : IntermediateField F E)) x →
        ∃ (x_1 : IntermediateField F E) (_ : x_1 ∈ s), x ∈ x_1
Tactic: simp_rw [isCompactElement_iff_le_of_directed_sSup_le,
    adjoin_simple_le_iff, sSup_eq_iSup', ← exists_prop]
-- Lean 4 proof step from Mathlib.FieldTheory.IntermediateField.Adjoin.Basic (hypothesis)
Goal:
F : Type u_1
inst : Field F
E : Type u_2
inst_1 : Field E
inst_2 : Algebra F E
x : E
s : Set (IntermediateField F E)
hne : s.Nonempty
hs : DirectedOn (fun (x1 x2 : IntermediateField F E) ↦ x1 ≤ x2) s
hx : Membership.mem (γ := IntermediateField F E) (⨆ (a : (↑s : Type u_2)), (↑a : IntermediateField F E)) x
⊢ ∃ (x_1 : IntermediateField F E) (_ : x_1 ∈ s), x ∈ x_1
have this : Nonempty.{u_2 + 1} (↑s : Type u_2)
Tactic: have := hne.to_subtype
-- Lean 4 proof step from Mathlib.FieldTheory.IntermediateField.Adjoin.Basic (goal)
Goal:
F : Type u_1
inst : Field F
E : Type u_2
inst_1 : Field E
inst_2 : Algebra F E
x : E
s : Set (IntermediateField F E)
hne : s.Nonempty
hs : DirectedOn (fun (x1 x2 : IntermediateField F E) ↦ x1 ≤ x2) s
hx : Membership.mem (γ := IntermediateField F E) (⨆ (a : (↑s : Type u_2)), (↑a : IntermediateField F E)) x
this : Nonempty.{u_2 + 1} (↑s : Type u_2)
⊢ ∃ (x_1 : IntermediateField F E) (_ : x_1 ∈ s), x ∈ x_1
have rwa : ∀ (hx : x ∈ (↑(⨆ (a : (↑s : Type u_2)), (↑a : IntermediateField F E)) : Set E))
  (this : Nonempty.{u_2 + 1} (↑s : Type u_2)), ∃ (x_1 : IntermediateField F E) (_ : x_1 ∈ s), x ∈ x_1
Tactic: rwa [← SetLike.mem_coe, coe_iSup_of_directed hs.directed_val, mem_iUnion, Subtype.exists] at hx
Cleared hypotheses: hx, this
-- Lean 4 proof step from Mathlib.FieldTheory.IntermediateField.Adjoin.Basic (goal)
Goal:
F : Type u_1
inst : Field F
E : Type u_2
inst_1 : Field E
inst_2 : Algebra F E
x : E
s : Set (IntermediateField F E)
hne : s.Nonempty
hs : DirectedOn (fun (x1 x2 : IntermediateField F E) ↦ x1 ≤ x2) s
hx : Membership.mem (γ := IntermediateField F E) (⨆ (a : (↑s : Type u_2)), (↑a : IntermediateField F E)) x
this : Nonempty.{u_2 + 1} (↑s : Type u_2)
rwa :
  ∀ (hx : x ∈ (↑(⨆ (a : (↑s : Type u_2)), (↑a : IntermediateField F E)) : Set E))
    (this : Nonempty.{u_2 + 1} (↑s : Type u_2)), ∃ (x_1 : IntermediateField F E) (_ : x_1 ∈ s), x ∈ x_1
⊢ ∃ (x_1 : IntermediateField F E) (_ : x_1 ∈ s), x ∈ x_1
have rwa₁ : ∀
  (hx :
    Membership.mem (γ := Set E)
      (⋃ (i : { x : IntermediateField F E // x ∈ s }), (↑(↑i : IntermediateField F E) : Set E)) x)
  (this : Nonempty.{u_2 + 1} (↑s : Type u_2)), ∃ (x_1 : IntermediateField F E) (_ : x_1 ∈ s), x ∈ x_1
Tactic: rwa [← SetLike.mem_coe, coe_iSup_of_directed hs.directed_val, mem_iUnion, Subtype.exists] at hx
Cleared hypotheses: hx, this
-- Lean 4 proof step from Mathlib.FieldTheory.IntermediateField.Adjoin.Basic (goal)
Goal:
F : Type u_1
inst : Field F
E : Type u_2
inst_1 : Field E
inst_2 : Algebra F E
x : E
s : Set (IntermediateField F E)
hne : s.Nonempty
hs : DirectedOn (fun (x1 x2 : IntermediateField F E) ↦ x1 ≤ x2) s
hx : Membership.mem (γ := IntermediateField F E) (⨆ (a : (↑s : Type u_2)), (↑a : IntermediateField F E)) x
this : Nonempty.{u_2 + 1} (↑s : Type u_2)
rwa :
  ∀ (hx : x ∈ (↑(⨆ (a : (↑s : Type u_2)), (↑a : IntermediateField F E)) : Set E))
    (this : Nonempty.{u_2 + 1} (↑s : Type u_2)), ∃ (x_1 : IntermediateField F E) (_ : x_1 ∈ s), x ∈ x_1
rwa₁ :
  ∀
    (hx :
      Membership.mem (γ := Set E)
        (⋃ (i : { x : IntermediateField F E // x ∈ s }), (↑(↑i : IntermediateField F E) : Set E)) x)
    (this : Nonempty.{u_2 + 1} (↑s : Type u_2)), ∃ (x_1 : IntermediateField F E) (_ : x_1 ∈ s), x ∈ x_1
⊢ ∃ (x_1 : IntermediateField F E) (_ : x_1 ∈ s), x ∈ x_1
have rwa₂ : ∀ (hx : ∃ (i : { x : IntermediateField F E // x ∈ s }), x ∈ (↑(↑i : IntermediateField F E) : Set E))
  (this : Nonempty.{u_2 + 1} (↑s : Type u_2)), ∃ (x_1 : IntermediateField F E) (_ : x_1 ∈ s), x ∈ x_1
Tactic: rwa [← SetLike.mem_coe, coe_iSup_of_directed hs.directed_val, mem_iUnion, Subtype.exists] at hx
Cleared hypotheses: hx, this
-- Lean 4 proof step from Mathlib.FieldTheory.IntermediateField.Adjoin.Basic (goal)
Goal:
F : Type u_1
inst : Field F
E : Type u_2
inst_1 : Field E
inst_2 : Algebra F E
x : E
s : Set (IntermediateField F E)
hne : s.Nonempty
hs : DirectedOn (fun (x1 x2 : IntermediateField F E) ↦ x1 ≤ x2) s
hx : Membership.mem (γ := IntermediateField F E) (⨆ (a : (↑s : Type u_2)), (↑a : IntermediateField F E)) x
this : Nonempty.{u_2 + 1} (↑s : Type u_2)
rwa :
  ∀ (hx : x ∈ (↑(⨆ (a : (↑s : Type u_2)), (↑a : IntermediateField F E)) : Set E))
    (this : Nonempty.{u_2 + 1} (↑s : Type u_2)), ∃ (x_1 : IntermediateField F E) (_ : x_1 ∈ s), x ∈ x_1
rwa₁ :
  ∀
    (hx :
      Membership.mem (γ := Set E)
        (⋃ (i : { x : IntermediateField F E // x ∈ s }), (↑(↑i : IntermediateField F E) : Set E)) x)
    (this : Nonempty.{u_2 + 1} (↑s : Type u_2)), ∃ (x_1 : IntermediateField F E) (_ : x_1 ∈ s), x ∈ x_1
rwa₂ :
  ∀ (hx : ∃ (i : { x : IntermediateField F E // x ∈ s }), x ∈ (↑(↑i : IntermediateField F E) : Set E))
    (this : Nonempty.{u_2 + 1} (↑s : Type u_2)), ∃ (x_1 : IntermediateField F E) (_ : x_1 ∈ s), x ∈ x_1
⊢ ∃ (x_1 : IntermediateField F E) (_ : x_1 ∈ s), x ∈ x_1
have rwa₃ : ∀
  (hx :
    ∃ (a : IntermediateField F E) (b : a ∈ s),
      x ∈ (↑(↑(Subtype.mk (p := fun (a : IntermediateField F E) ↦ a ∈ s) a b) : IntermediateField F E) : Set E))
  (this : Nonempty.{u_2 + 1} (↑s : Type u_2)), ∃ (x_1 : IntermediateField F E) (_ : x_1 ∈ s), x ∈ x_1
Tactic: rwa [← SetLike.mem_coe, coe_iSup_of_directed hs.directed_val, mem_iUnion, Subtype.exists] at hx
Cleared hypotheses: hx, this
-- Lean 4 proof step from Mathlib.FieldTheory.IntermediateField.Adjoin.Basic (goal)
Goal:
F : Type u_1
inst : Field F
E : Type u_2
inst_1 : Field E
inst_2 : Algebra F E
S : Finset E
⊢ IsCompactElement (α := IntermediateField F E) (adjoin F (↑S : Set E))
have rw : IsCompactElement.{u_2} (α := IntermediateField F E) (⨆ x ∈ (↑S : Set E), F⟮x⟯)
Tactic: rw [← biSup_adjoin_simple]
-- Lean 4 proof step from Mathlib.FieldTheory.IntermediateField.Adjoin.Basic (goal)
Goal:
F : Type u_1
inst : Field F
E : Type u_2
inst_1 : Field E
inst_2 : Algebra F E
S : Finset E
⊢ IsCompactElement.{u_2} (α := IntermediateField F E) (⨆ x ∈ (↑S : Set E), F⟮x⟯)
have simp_rw : IsCompactElement (α := IntermediateField F E) (S.sup fun (a : E) ↦ F⟮a⟯)
Tactic: simp_rw [Finset.mem_coe, ← Finset.sup_eq_iSup]
-- Lean 4 proof step from Mathlib.FieldTheory.IntermediateField.Adjoin.Basic (hypothesis)
Goal:
F : Type u_1
inst : Field F
E : Type u_2
inst_1 : Field E
inst_2 : Algebra F E
ι : Type u_3
f : ι → IntermediateField F E
x : E
hx : Membership.mem (γ := IntermediateField F E) (⨆ (i : ι), f i) x
⊢ ∃ (s : Finset ι), Membership.mem (γ := IntermediateField F E) (⨆ i ∈ s, f i) x
have this : ∀ (h : LE.le (α := IntermediateField F E) F⟮x⟯ (⨆ (i : ι), f i)),
  ∃ (s : Finset ι), LE.le (α := IntermediateField F E) F⟮x⟯ (⨆ i ∈ s, f i)
Tactic: have := (adjoin_simple_isCompactElement x).exists_finset_of_le_iSup (IntermediateField F E) f
-- Lean 4 proof step from Mathlib.FieldTheory.IntermediateField.Adjoin.Basic (hypothesis)
Goal:
F : Type u_1
inst : Field F
E : Type u_2
inst_1 : Field E
inst_2 : Algebra F E
ι : Type u_3
f : ι → IntermediateField F E
x : E
hx : Membership.mem (γ := IntermediateField F E) (⨆ (i : ι), f i) x
this :
  ∀ (h : LE.le (α := IntermediateField F E) F⟮x⟯ (⨆ (i : ι), f i)),
    ∃ (s : Finset ι), LE.le (α := IntermediateField F E) F⟮x⟯ (⨆ i ∈ s, f i)
⊢ ∃ (s : Finset ι), Membership.mem (γ := IntermediateField F E) (⨆ i ∈ s, f i) x
have this₁ : ∀ (h : Membership.mem (γ := IntermediateField F E) (⨆ (i : ι), f i) x),
  ∃ (s : Finset ι), Membership.mem (γ := IntermediateField F E) (⨆ i ∈ s, f i) x
Tactic: simp only [adjoin_simple_le_iff] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.FieldTheory.IntermediateField.Adjoin.Basic (goal)
Goal:
F : Type u_1
inst : Field F
E : Type u_2
inst_1 : Field E
inst_2 : Algebra F E
ι : Type u_3
f : ι → IntermediateField F E
x : E
hx : Membership.mem (γ := IntermediateField F E) (⨆ (i : ι), f i) x
⊢ ∃ (s : Finset ((i : ι) × ↥(f i))),
    Membership.mem (γ := IntermediateField F E) (⨆ i ∈ s, F⟮(↑(i.snd (β := fun (i : ι) ↦ ↥(f i))) : E)⟯) x
have refine : ∀ (i : ι) (x : E) (h : x ∈ f i),
  Membership.mem (γ := IntermediateField F E) (⨆ (i : (i : ι) × ↥(f i)), F⟮(↑(i.snd (β := fun (i : ι) ↦ ↥(f i))) : E)⟯)
    x
Tactic: refine exists_finset_of_mem_iSup (SetLike.le_def.mp (iSup_le fun i x h ↦ ?_) hx)
-- Lean 4 proof step from Mathlib.FieldTheory.IntermediateField.Adjoin.Basic (goal)
Goal:
F : Type u_1
inst : Field F
E : Type u_2
inst_1 : Field E
inst_2 : Algebra F E
ι : Type u_3
f : ι → IntermediateField F E
h : ∀ (i : ι), Algebra.IsAlgebraic.{u_1, u_2} F ↥(f i)
x : E
hx : Membership.mem (γ := IntermediateField F E) (⨆ (i : ι), f i) x
⊢ ∃ (s : Finset ((i : ι) × ↥(f i))),
    Membership.mem (γ := IntermediateField F E)
      (⨆ i ∈ s, adjoin F (rootSet (minpoly F (i.snd (β := fun (i : ι) ↦ ↥(f i)))) E)) x
have refine : ∀ (i : ι) (x1 : E) (hx1 : x1 ∈ f i),
  Membership.mem (γ := IntermediateField F E)
    (⨆ (i : (i : ι) × ↥(f i)), adjoin F (rootSet (minpoly F (i.snd (β := fun (i : ι) ↦ ↥(f i)))) E)) x1
Tactic: refine exists_finset_of_mem_iSup (SetLike.le_def.mp (iSup_le (fun i x1 hx1 => ?_)) hx)
-- Lean 4 proof step from Mathlib.FieldTheory.IntermediateField.Adjoin.Basic (goal)
Goal:
F : Type u_1
inst : Field F
E : Type u_2
inst_1 : Field E
inst_2 : Algebra F E
ι : Type u_3
f : ι → IntermediateField F E
h : ∀ (i : ι), Algebra.IsAlgebraic.{u_1, u_2} F ↥(f i)
x : E
hx : Membership.mem (γ := IntermediateField F E) (⨆ (i : ι), f i) x
i : ι
x1 : E
hx1 : x1 ∈ f i
⊢ Membership.mem (γ := IntermediateField F E)
    (⨆ (i : (i : ι) × ↥(f i)), adjoin F (rootSet (minpoly F (i.snd (β := fun (i : ι) ↦ ↥(f i)))) E)) x1
have refine_1 : LE.le (α := IntermediateField F E) (adjoin F (rootSet (minpoly F x1) E))
  (adjoin F
    (rootSet
      (minpoly.{u_1, u_2} F (B :=
        ↥(f
            (Sigma.fst (β := fun (i : ι) ↦ ↥(f i))
              (Sigma.mk (β := fun (i : ι) ↦ ↥(f i)) i (Subtype.mk (p := fun (x : E) ↦ x ∈ f i) x1 hx1)))))
        (Sigma.snd (β := fun (i : ι) ↦ ↥(f i))
            (Sigma.mk (β := fun (i : ι) ↦ ↥(f i)) i (Subtype.mk (p := fun (x : E) ↦ x ∈ f i) x1 hx1)) :
          ↥(f
              (Sigma.fst (β := fun (i : ι) ↦ ↥(f i))
                (Sigma.mk (β := fun (i : ι) ↦ ↥(f i)) i (Subtype.mk (p := fun (x : E) ↦ x ∈ f i) x1 hx1))))))
      E))
Tactic: refine SetLike.le_def.mp (le_iSup_of_le ⟨i, x1, hx1⟩ ?_)
    (subset_adjoin F (rootSet (minpoly F x1) E) ?_)
-- Lean 4 proof step from Mathlib.FieldTheory.IntermediateField.Adjoin.Basic (goal)
Goal:
F : Type u_1
inst : Field F
E : Type u_2
inst_1 : Field E
inst_2 : Algebra F E
ι : Type u_3
f : ι → IntermediateField F E
h : ∀ (i : ι), Algebra.IsAlgebraic.{u_1, u_2} F ↥(f i)
x : E
hx : Membership.mem (γ := IntermediateField F E) (⨆ (i : ι), f i) x
i : ι
x1 : E
hx1 : x1 ∈ f i
refine_1 :
  LE.le (α := IntermediateField F E) (adjoin F (rootSet (minpoly F x1) E))
    (adjoin F
      (rootSet
        (minpoly.{u_1, u_2} F (B :=
          ↥(f
              (Sigma.fst (β := fun (i : ι) ↦ ↥(f i))
                (Sigma.mk (β := fun (i : ι) ↦ ↥(f i)) i (Subtype.mk (p := fun (x : E) ↦ x ∈ f i) x1 hx1)))))
          (Sigma.snd (β := fun (i : ι) ↦ ↥(f i))
              (Sigma.mk (β := fun (i : ι) ↦ ↥(f i)) i (Subtype.mk (p := fun (x : E) ↦ x ∈ f i) x1 hx1)) :
            ↥(f
                (Sigma.fst (β := fun (i : ι) ↦ ↥(f i))
                  (Sigma.mk (β := fun (i : ι) ↦ ↥(f i)) i (Subtype.mk (p := fun (x : E) ↦ x ∈ f i) x1 hx1))))))
        E))
⊢ Membership.mem (γ := IntermediateField F E)
    (⨆ (i : (i : ι) × ↥(f i)), adjoin F (rootSet (minpoly F (i.snd (β := fun (i : ι) ↦ ↥(f i)))) E)) x1
have refine_2 : x1 ∈ rootSet (minpoly F x1) E
Tactic: refine SetLike.le_def.mp (le_iSup_of_le ⟨i, x1, hx1⟩ ?_)
    (subset_adjoin F (rootSet (minpoly F x1) E) ?_)
-- Lean 4 proof step from Mathlib.FieldTheory.IntermediateField.Adjoin.Basic (goal)
Goal:
F : Type u_1
inst : Field F
E : Type u_2
inst_1 : Field E
inst_2 : Algebra F E
ι : Type u_3
f : ι → IntermediateField F E
h : ∀ (i : ι), Algebra.IsAlgebraic.{u_1, u_2} F ↥(f i)
x : E
hx : Membership.mem (γ := IntermediateField F E) (⨆ (i : ι), f i) x
i : ι
x1 : E
hx1 : x1 ∈ f i
⊢ LE.le (α := IntermediateField F E) (adjoin F (rootSet (minpoly F x1) E))
    (adjoin F
      (rootSet
        (minpoly.{u_1, u_2} F (B :=
          ↥(f
              (Sigma.fst (β := fun (i : ι) ↦ ↥(f i))
                (Sigma.mk (β := fun (i : ι) ↦ ↥(f i)) i (Subtype.mk (p := fun (x : E) ↦ x ∈ f i) x1 hx1)))))
          (Sigma.snd (β := fun (i : ι) ↦ ↥(f i))
              (Sigma.mk (β := fun (i : ι) ↦ ↥(f i)) i (Subtype.mk (p := fun (x : E) ↦ x ∈ f i) x1 hx1)) :
            ↥(f
                (Sigma.fst (β := fun (i : ι) ↦ ↥(f i))
                  (Sigma.mk (β := fun (i : ι) ↦ ↥(f i)) i (Subtype.mk (p := fun (x : E) ↦ x ∈ f i) x1 hx1))))))
        E))
have refine_1 : LE.le (α := IntermediateField F E) (adjoin F (rootSet (minpoly F x1) E))
  (adjoin F
    (rootSet
      (minpoly.{u_1, u_2} F (B := E)
        (↑(Sigma.snd (β := fun (i : ι) ↦ ↥(f i))
              (Sigma.mk (β := fun (i : ι) ↦ ↥(f i)) i (Subtype.mk (p := fun (x : E) ↦ x ∈ f i) x1 hx1))) :
          E))
      E))
Tactic: rw [IntermediateField.minpoly_eq, Subtype.coe_mk]
-- Lean 4 proof step from Mathlib.FieldTheory.IntermediateField.Adjoin.Basic (goal)
Goal:
F : Type u_1
inst : Field F
E : Type u_2
inst_1 : Field E
inst_2 : Algebra F E
ι : Type u_3
f : ι → IntermediateField F E
h : ∀ (i : ι), Algebra.IsAlgebraic.{u_1, u_2} F ↥(f i)
x : E
hx : Membership.mem (γ := IntermediateField F E) (⨆ (i : ι), f i) x
i : ι
x1 : E
hx1 : x1 ∈ f i
refine_1 :
  LE.le (α := IntermediateField F E) (adjoin F (rootSet (minpoly F x1) E))
    (adjoin F
      (rootSet
        (minpoly.{u_1, u_2} F (B := E)
          (↑(Sigma.snd (β := fun (i : ι) ↦ ↥(f i))
                (Sigma.mk (β := fun (i : ι) ↦ ↥(f i)) i (Subtype.mk (p := fun (x : E) ↦ x ∈ f i) x1 hx1))) :
            E))
        E))
⊢ LE.le (α := IntermediateField F E) (adjoin F (rootSet (minpoly F x1) E))
    (adjoin F
      (rootSet
        (minpoly.{u_1, u_2} F (B :=
          ↥(f
              (Sigma.fst (β := fun (i : ι) ↦ ↥(f i))
                (Sigma.mk (β := fun (i : ι) ↦ ↥(f i)) i (Subtype.mk (p := fun (x : E) ↦ x ∈ f i) x1 hx1)))))
          (Sigma.snd (β := fun (i : ι) ↦ ↥(f i))
              (Sigma.mk (β := fun (i : ι) ↦ ↥(f i)) i (Subtype.mk (p := fun (x : E) ↦ x ∈ f i) x1 hx1)) :
            ↥(f
                (Sigma.fst (β := fun (i : ι) ↦ ↥(f i))
                  (Sigma.mk (β := fun (i : ι) ↦ ↥(f i)) i (Subtype.mk (p := fun (x : E) ↦ x ∈ f i) x1 hx1))))))
        E))
have refine_1₁ : LE.le (α := IntermediateField F E) (adjoin F (rootSet (minpoly F x1) E)) (adjoin F (rootSet (minpoly F x1) E))
Tactic: rw [IntermediateField.minpoly_eq, Subtype.coe_mk]
-- Lean 4 proof step from Mathlib.FieldTheory.IntermediateField.Adjoin.Basic (goal)
Goal:
F : Type u_1
inst : Field F
E : Type u_2
inst_1 : Field E
inst_2 : Algebra F E
ι : Type u_3
f : ι → IntermediateField F E
h : ∀ (i : ι), Algebra.IsAlgebraic.{u_1, u_2} F ↥(f i)
x : E
hx : Membership.mem (γ := IntermediateField F E) (⨆ (i : ι), f i) x
i : ι
x1 : E
hx1 : x1 ∈ f i
⊢ x1 ∈ rootSet (minpoly F x1) E
have refine_2 : Ne (α := F[X]) (minpoly F x1) 0
Tactic: rw [mem_rootSet_of_ne, minpoly.aeval]
-- Lean 4 proof step from Mathlib.FieldTheory.IntermediateField.Adjoin.Basic (hypothesis)
Goal:
F : Type u_1
inst : Field F
E : Type u_2
inst_1 : Field E
inst_2 : Algebra F E
S : Set E
x : E
hx : Membership.mem (γ := IntermediateField F E) (adjoin F S) x
⊢ ∃ (T : Finset E), (↑T : Set E) ⊆ S ∧ Membership.mem (γ := IntermediateField F E) (adjoin F (↑T : Set E)) x
have hx₁ : Membership.mem (γ := IntermediateField F E) (⨆ (i : (↑S : Type u_2)), F⟮(↑i : E)⟯) x
Tactic: simp_rw [← biSup_adjoin_simple S, ← iSup_subtype''] at hx
Cleared hypotheses: hx
-- Lean 4 proof step from Mathlib.FieldTheory.IntermediateField.Adjoin.Basic (goal)
Goal:
F : Type u_1
inst : Field F
E : Type u_2
inst_1 : Field E
inst_2 : Algebra F E
S : Set E
x : E
hx : Membership.mem (γ := IntermediateField F E) (⨆ (i : (↑S : Type u_2)), F⟮(↑i : E)⟯) x
⊢ ∃ (T : Finset E), (↑T : Set E) ⊆ S ∧ Membership.mem (γ := IntermediateField F E) (adjoin F (↑T : Set E)) x
have obtain : ∀ (s : Finset (↑S : Type u_2)) (hx' : Membership.mem (γ := IntermediateField F E) (⨆ i ∈ s, F⟮(↑i : E)⟯) x),
  ∃ (T : Finset E), (↑T : Set E) ⊆ S ∧ Membership.mem (γ := IntermediateField F E) (adjoin F (↑T : Set E)) x
Tactic: obtain ⟨s, hx'⟩ := exists_finset_of_mem_iSup hx
-- Lean 4 proof step from Mathlib.FieldTheory.IntermediateField.Adjoin.Basic (goal)
Goal:
F : Type u_1
inst : Field F
E : Type u_2
inst_1 : Field E
inst_2 : Algebra F E
S : Set E
x : E
hx : Membership.mem (γ := IntermediateField F E) (⨆ (i : (↑S : Type u_2)), F⟮(↑i : E)⟯) x
s : Finset (↑S : Type u_2)
hx' : Membership.mem (γ := IntermediateField F E) (⨆ i ∈ s, F⟮(↑i : E)⟯) x
⊢ ∃ (T : Finset E), (↑T : Set E) ⊆ S ∧ Membership.mem (γ := IntermediateField F E) (adjoin F (↑T : Set E)) x
have refine : LE.le (α := IntermediateField F E) (⨆ i ∈ s, F⟮(↑i : E)⟯)
  (adjoin F (↑(Finset.image (Subtype.val (p := fun (x : E) ↦ x ∈ S)) s) : Set E))
Tactic: refine ⟨s.image Subtype.val, by simp, SetLike.le_def.mp ?_ hx'⟩
-- Lean 4 proof step from Mathlib.FieldTheory.IntermediateField.Adjoin.Basic (goal)
Goal:
F : Type u_1
inst : Field F
E : Type u_2
inst_1 : Field E
inst_2 : Algebra F E
S : Set E
x : E
hx : Membership.mem (γ := IntermediateField F E) (⨆ (i : (↑S : Type u_2)), F⟮(↑i : E)⟯) x
s : Finset (↑S : Type u_2)
hx' : Membership.mem (γ := IntermediateField F E) (⨆ i ∈ s, F⟮(↑i : E)⟯) x
⊢ LE.le (α := IntermediateField F E) (⨆ i ∈ s, F⟮(↑i : E)⟯)
    (adjoin F (↑(Finset.image (Subtype.val (p := fun (x : E) ↦ x ∈ S)) s) : Set E))
have simp_rw : ∀ (i : (↑S : Type u_2)) (i_1 : i ∈ s),
  {(↑i : E)} ⊆ (↑(adjoin F (Subtype.val (p := fun (x : E) ↦ x ∈ S) '' (↑s : Set { x : E // x ∈ S }))) : Set E)
Tactic: simp_rw [Finset.coe_image, iSup_le_iff, adjoin_le_iff]
-- Lean 4 proof step from Mathlib.FieldTheory.IntermediateField.Adjoin.Basic (goal)
Goal:
F : Type u_1
inst : Field F
E : Type u_2
inst_1 : Field E
inst_2 : Algebra F E
S : Set E
x : E
hx : Membership.mem (γ := IntermediateField F E) (⨆ (i : (↑S : Type u_2)), F⟮(↑i : E)⟯) x
s : Finset (↑S : Type u_2)
hx' : Membership.mem (γ := IntermediateField F E) (⨆ i ∈ s, F⟮(↑i : E)⟯) x
⊢ ∃ (T : Finset E), (↑T : Set E) ⊆ S ∧ Membership.mem (γ := IntermediateField F E) (adjoin F (↑T : Set E)) x
have refine : LE.le (α := IntermediateField F E) (⨆ i ∈ s, F⟮(↑i : E)⟯)
  (adjoin F (↑(Finset.image (Subtype.val (p := fun (x : E) ↦ x ∈ S)) s) : Set E))
Tactic: classical
  refine ⟨s.image Subtype.val, by simp, SetLike.le_def.mp ?_ hx'⟩
  simp_rw [Finset.coe_image, iSup_le_iff, adjoin_le_iff]
  rintro _ h _ rfl
  exact subset_adjoin F _ ⟨_, h, rfl⟩
-- Lean 4 proof step from Mathlib.FieldTheory.IntermediateField.Adjoin.Basic (goal)
Goal:
F : Type u_1
inst : Field F
E : Type u_2
inst_1 : Field E
inst_2 : Algebra F E
S : Set E
x : E
hx : Membership.mem (γ := IntermediateField F E) (⨆ (i : (↑S : Type u_2)), F⟮(↑i : E)⟯) x
s : Finset (↑S : Type u_2)
hx' : Membership.mem (γ := IntermediateField F E) (⨆ i ∈ s, F⟮(↑i : E)⟯) x
refine :
  LE.le (α := IntermediateField F E) (⨆ i ∈ s, F⟮(↑i : E)⟯)
    (adjoin F (↑(Finset.image (Subtype.val (p := fun (x : E) ↦ x ∈ S)) s) : Set E))
⊢ ∃ (T : Finset E), (↑T : Set E) ⊆ S ∧ Membership.mem (γ := IntermediateField F E) (adjoin F (↑T : Set E)) x
have refine₁ : ∀ (i : (↑S : Type u_2)) (i_1 : i ∈ s),
  {(↑i : E)} ⊆ (↑(adjoin F (Subtype.val (p := fun (x : E) ↦ x ∈ S) '' (↑s : Set { x : E // x ∈ S }))) : Set E)
Tactic: classical
  refine ⟨s.image Subtype.val, by simp, SetLike.le_def.mp ?_ hx'⟩
  simp_rw [Finset.coe_image, iSup_le_iff, adjoin_le_iff]
  rintro _ h _ rfl
  exact subset_adjoin F _ ⟨_, h, rfl⟩