awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.GroupTheory.Sylow (goal)
Goal:
p : ℕ
G : Type u_1
inst : Group G
P : Subgroup G
hP : IsPGroup.{u_1} p ↥P
c : Set (Subgroup G)
hc1 : c ⊆ {Q : Subgroup G | IsPGroup.{u_1} p ↥Q}
hc2 : IsChain (fun (x1 x2 : Subgroup G) ↦ x1 ≤ x2) c
Q : Subgroup G
hQ : Q ∈ c
x : ↥{ carrier := ⋃ (R : (↑c : Type u_1)), (↑(↑R : Subgroup G) : Set G), mul_mem' := ⋯, one_mem' := ⋯, inv_mem' := ⋯ }
g : G
S : (↑c : Type u_1)
hg : Membership.mem (γ := Set G) ((fun (R : (↑c : Type u_1)) ↦ (↑(↑R : Subgroup G) : Set G)) S) g
k : ℕ
hk : Subtype.mk (p := fun (x : G) ↦ x ∈ (↑S : Subgroup G)) g hg ^ p ^ k = 1
rwa :
  ∀ (hk : (↑(Subtype.mk (p := fun (x : G) ↦ x ∈ (↑S : Subgroup G)) g hg ^ p ^ k) : G) = (↑1 : G)),
    (↑(HPow.hPow (α :=
            ↥{ carrier := ⋃ (R : (↑c : Type u_1)), (↑(↑R : Subgroup G) : Set G), mul_mem' := ⋯, one_mem' := ⋯,
                inv_mem' := ⋯ : Subgroup G })
            (Subtype.mk (p := fun (x : G) ↦
              Membership.mem (γ := Subgroup G)
                { carrier := ⋃ (R : (↑c : Type u_1)), (↑(↑R : Subgroup G) : Set G), mul_mem' := ⋯, one_mem' := ⋯,
                  inv_mem' := ⋯ : Subgroup G }
                x)
              g ⋯)
            (p ^ k)) :
        G) =
      (↑1 : G)
⊢ HPow.hPow (α :=
      ↥{ carrier := ⋃ (R : (↑c : Type u_1)), (↑(↑R : Subgroup G) : Set G), mul_mem' := ⋯, one_mem' := ⋯, inv_mem' := ⋯ :
          Subgroup G })
      (Subtype.mk (p := fun (x : G) ↦
        Membership.mem (γ := Subgroup G)
          { carrier := ⋃ (R : (↑c : Type u_1)), (↑(↑R : Subgroup G) : Set G), mul_mem' := ⋯, one_mem' := ⋯,
            inv_mem' := ⋯ : Subgroup G }
          x)
        g ⋯)
      (p ^ k) =
    1
have rwa₁ : ∀ (hk : (↑(Subtype.mk (p := fun (x : G) ↦ x ∈ (↑S : Subgroup G)) g hg) : G) ^ p ^ k = (↑1 : G)),
  (↑(Subtype.mk (p := fun (x : G) ↦
            Membership.mem (γ := Subgroup G)
              { carrier := ⋃ (R : (↑c : Type u_1)), (↑(↑R : Subgroup G) : Set G), mul_mem' := sorry, one_mem' := sorry,
                inv_mem' := sorry : Subgroup G }
              x)
            g sorry) :
        G) ^
      p ^ k =
    (↑1 : G)
Tactic: rwa [Subtype.ext_iff, coe_pow] at hk ⊢
Cleared hypotheses: hk
-- Lean 4 proof step from Mathlib.GroupTheory.Sylow (goal)
Goal:
p : ℕ
G : Type u_1
inst : Group G
H : Type u_2
inst_1 : Group H
f : H →* G
hf : IsPGroup.{u_2} p ↥f.ker
inst_2 : Finite.{u_1 + 1} (Sylow p G)
h_exists : ∀ (P : Sylow p H), ∃ (Q : Sylow p G), comap f (↑Q : Subgroup G) = (↑P : Subgroup H) := ⋯
g : Sylow p H → Sylow p G :=
  fun (P : Sylow p H) ↦ Classical.choose (p := fun (x : Sylow p G) ↦ comap f (↑x : Subgroup G) = (↑P : Subgroup H)) ⋯
hg : ∀ (P : Sylow p H), comap f (↑(g P) : Subgroup G) = (↑P : Subgroup H)
P Q : Sylow p H
h : g P = g Q
⊢ (↑P : Subgroup H) = (↑Q : Subgroup H)
have rw : comap f (↑(g Q) : Subgroup G) = (↑Q : Subgroup H)
Tactic: rw [← hg, h]
-- Lean 4 proof step from Mathlib.GroupTheory.Sylow (goal)
Goal:
p : ℕ
G : Type u_1
inst : Group G
P : Sylow p G
inst_1 : (↑P : Subgroup G).FiniteIndex
⊢ Finite.{u_1 + 1} (Sylow p G)
have hf : IsPGroup.{u_1} p ↥(MonoidHom.ker (QuotientGroup.mk' (↑P : Subgroup G).normalCore))
Tactic: apply finite_of_ker_is_pGroup (f := QuotientGroup.mk' P.normalCore)
-- Lean 4 proof step from Mathlib.GroupTheory.Sylow (goal)
Goal:
p : ℕ
G : Type u_1
inst : Group G
P : Sylow p G
inst_1 : (↑P : Subgroup G).FiniteIndex
⊢ IsPGroup.{u_1} p ↥(MonoidHom.ker (QuotientGroup.mk' (↑P : Subgroup G).normalCore))
have hf : IsPGroup.{u_1} p ↥(↑P : Subgroup G).normalCore
Tactic: rw [QuotientGroup.ker_mk']
-- Lean 4 proof step from Mathlib.GroupTheory.Sylow (goal)
Goal:
p : ℕ
G : Type u_1
inst : Group G
g : G
P : Sylow p G
⊢ g • P = P ↔ Membership.mem (γ := Subgroup G) (normalizer (↑P : Set G)) g
have rw : (∀ (x : G), x ∈ P ↔ x ∈ g • P) ↔ ∀ (h : G), h ∈ (↑P : Set G) ↔ g⁻¹ * h * g ∈ (↑P : Set G)
Tactic: rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P),
      mem_set_normalizer_iff, inv_inv]
-- Lean 4 proof step from Mathlib.GroupTheory.Sylow (goal)
Goal:
p : ℕ
G : Type u_1
inst : Group G
H : Subgroup G
P : Sylow p G
⊢ Membership.mem (γ := Set (Sylow p G)) (fixedPoints (↥H) (Sylow p G)) P ↔ H ≤ normalizer (↑P : Set G)
have simp_rw : Membership.mem (γ := Set (Sylow p G)) (fixedPoints (↥H) (Sylow p G)) P ↔ ∀ ⦃x : G⦄, x ∈ H → x • P = P
Tactic: simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]
-- Lean 4 proof step from Mathlib.GroupTheory.Sylow (goal)
Goal:
p : ℕ
G : Type u_1
inst : Group G
P : Subgroup G
hP : IsPGroup.{u_1} p ↥P
Q : Sylow p G
⊢ Membership.mem (γ := Set (Sylow p G)) (fixedPoints (↥P) (Sylow p G)) Q ↔ P ≤ (↑Q : Subgroup G)
have rw : P ≤ normalizer (↑Q : Set G) ↔ P ≤ (↑Q : Subgroup G)
Tactic: rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
-- Lean 4 proof step from Mathlib.GroupTheory.Sylow (goal)
Goal:
p : ℕ
G : Type u_1
inst : Group G
P : Subgroup G
hP : IsPGroup.{u_1} p ↥P
Q : Sylow p G
rw : P ≤ normalizer (↑Q : Set G) ↔ P ≤ (↑Q : Subgroup G)
⊢ Membership.mem (γ := Set (Sylow p G)) (fixedPoints (↥P) (Sylow p G)) Q ↔ P ≤ (↑Q : Subgroup G)
have rw₁ : P ⊓ normalizer (↑Q : Set G) = P ↔ P ≤ (↑Q : Subgroup G)
Tactic: rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
-- Lean 4 proof step from Mathlib.GroupTheory.Sylow (goal)
Goal:
p : ℕ
G : Type u_1
inst : Group G
P : Subgroup G
hP : IsPGroup.{u_1} p ↥P
Q : Sylow p G
rw : P ≤ normalizer (↑Q : Set G) ↔ P ≤ (↑Q : Subgroup G)
rw₁ : P ⊓ normalizer (↑Q : Set G) = P ↔ P ≤ (↑Q : Subgroup G)
⊢ Membership.mem (γ := Set (Sylow p G)) (fixedPoints (↥P) (Sylow p G)) Q ↔ P ≤ (↑Q : Subgroup G)
have rw₂ : P ⊓ (↑Q : Subgroup G) = P ↔ P ≤ (↑Q : Subgroup G)
Tactic: rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
-- Lean 4 proof step from Mathlib.GroupTheory.Sylow (goal)
Goal:
p : ℕ
G : Type u_1
inst : Group G
P : Subgroup G
hP : IsPGroup.{u_1} p ↥P
Q : Sylow p G
rw : P ≤ normalizer (↑Q : Set G) ↔ P ≤ (↑Q : Subgroup G)
rw₁ : P ⊓ normalizer (↑Q : Set G) = P ↔ P ≤ (↑Q : Subgroup G)
rw₂ : P ⊓ (↑Q : Subgroup G) = P ↔ P ≤ (↑Q : Subgroup G)
⊢ Membership.mem (γ := Set (Sylow p G)) (fixedPoints (↥P) (Sylow p G)) Q ↔ P ≤ (↑Q : Subgroup G)
have rw₃ : P ≤ (↑Q : Subgroup G) ↔ P ≤ (↑Q : Subgroup G)
Tactic: rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
-- Lean 4 proof step from Mathlib.GroupTheory.Sylow (hypothesis)
Goal:
p : ℕ
G : Type u_1
inst : Group G
hp : Fact (Nat.Prime p)
inst_1 : Finite.{u_1 + 1} (Sylow p G)
P Q : Sylow p G
⊢ ∃ (g : G), g • P = Q
have H : ∀ {R : Sylow p G} {S : (↑(orbit G P) : Type u_1)},
  Membership.mem (γ := Set (↑(orbit G P) : Type u_1)) (fixedPoints ↥(↑R : Subgroup G) (↑(orbit G P) : Type u_1)) S ↔
    (↑(↑S : Sylow p G) : Subgroup G) = (↑R : Subgroup G)
Tactic: have H := fun {R : Sylow p G} {S : orbit G P} =>
        calc
          S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
            forall_congr' fun a => Subtype.ext_iff
          _ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
          _ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
-- Lean 4 proof step from Mathlib.GroupTheory.Sylow (goal)
Goal:
p : ℕ
G : Type u_1
inst : Group G
hp : Fact (Nat.Prime p)
inst_1 : Finite.{u_1 + 1} (Sylow p G)
P Q : Sylow p G
H :
  ∀ {R : Sylow p G} {S : (↑(orbit G P) : Type u_1)},
    Membership.mem (γ := Set (↑(orbit G P) : Type u_1)) (fixedPoints ↥(↑R : Subgroup G) (↑(orbit G P) : Type u_1)) S ↔
      (↑(↑S : Sylow p G) : Subgroup G) = (↑R : Subgroup G)
this : Set.Nonempty.{u_1} (α := (↑(orbit G P) : Type u_1)) (fixedPoints ↥(↑Q : Subgroup G) (↑(orbit G P) : Type u_1))
R : (↑(orbit G P) : Type u_1)
hR : Membership.mem (γ := Set (↑(orbit G P) : Type u_1)) (fixedPoints ↥(↑Q : Subgroup G) (↑(orbit G P) : Type u_1)) R
⊢ ∃ (g : G), g • P = Q
have rw : ∃ (g : G), g • P = (↑R : Sylow p G)
Tactic: rw [← Sylow.ext (H.mp hR)]
-- Lean 4 proof step from Mathlib.GroupTheory.Sylow (goal)
Goal:
p : ℕ
G : Type u_1
inst : Group G
hp : Fact (Nat.Prime p)
inst_1 : Finite.{u_1 + 1} (Sylow p G)
P Q : Sylow p G
H :
  ∀ {R : Sylow p G} {S : (↑(orbit G P) : Type u_1)},
    Membership.mem (γ := Set (↑(orbit G P) : Type u_1)) (fixedPoints ↥(↑R : Subgroup G) (↑(orbit G P) : Type u_1)) S ↔
      (↑(↑S : Sylow p G) : Subgroup G) = (↑R : Subgroup G)
⊢ ∃ (g : G), g • P = Q
have suffices : Set.Nonempty.{u_1} (α := (↑(orbit G P) : Type u_1)) (fixedPoints ↥(↑Q : Subgroup G) (↑(orbit G P) : Type u_1))
Tactic: suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
        exact Exists.elim this fun R hR => by
          rw [← Sylow.ext (H.mp hR)]
          exact R.2
-- Lean 4 proof step from Mathlib.GroupTheory.Sylow (goal)
Goal:
p : ℕ
G : Type u_1
inst : Group G
hp : Fact (Nat.Prime p)
inst_1 : Finite.{u_1 + 1} (Sylow p G)
P Q : Sylow p G
H :
  ∀ {R : Sylow p G} {S : (↑(orbit G P) : Type u_1)},
    Membership.mem (γ := Set (↑(orbit G P) : Type u_1)) (fixedPoints ↥(↑R : Subgroup G) (↑(orbit G P) : Type u_1)) S ↔
      (↑(↑S : Sylow p G) : Subgroup G) = (↑R : Subgroup G)
⊢ Set.Nonempty.{u_1} (α := (↑(orbit G P) : Type u_1)) (fixedPoints ↥(↑Q : Subgroup G) (↑(orbit G P) : Type u_1))
have hpα : ¬p ∣ Nat.card.{u_1} (↑(orbit G P) : Type u_1)
Tactic: apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
-- Lean 4 proof step from Mathlib.GroupTheory.Sylow (goal)
Goal:
p : ℕ
G : Type u_1
inst : Group G
hp : Fact (Nat.Prime p)
inst_1 : Finite.{u_1 + 1} (Sylow p G)
P Q : Sylow p G
H :
  ∀ {R : Sylow p G} {S : (↑(orbit G P) : Type u_1)},
    Membership.mem (γ := Set (↑(orbit G P) : Type u_1)) (fixedPoints ↥(↑R : Subgroup G) (↑(orbit G P) : Type u_1)) S ↔
      (↑(↑S : Sylow p G) : Subgroup G) = (↑R : Subgroup G)
⊢ ¬p ∣ Nat.card.{u_1} (↑(orbit G P) : Type u_1)
have hpα : ∀ (h : p ∣ Nat.card.{u_1} (↑(orbit G P) : Type u_1)), 1 ≡ 0 [MOD p]
Tactic: refine fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp ?_)
-- Lean 4 proof step from Mathlib.GroupTheory.Sylow (goal)
Goal:
p : ℕ
G : Type u_1
inst : Group G
hp : Fact (Nat.Prime p)
inst_1 : Finite.{u_1 + 1} (Sylow p G)
P Q : Sylow p G
H :
  ∀ {R : Sylow p G} {S : (↑(orbit G P) : Type u_1)},
    Membership.mem (γ := Set (↑(orbit G P) : Type u_1)) (fixedPoints ↥(↑R : Subgroup G) (↑(orbit G P) : Type u_1)) S ↔
      (↑(↑S : Sylow p G) : Subgroup G) = (↑R : Subgroup G)
h : p ∣ Nat.card.{u_1} (↑(orbit G P) : Type u_1)
⊢ 1 ≡ 0 [MOD p]
have hpα : 1 = Nat.card.{u_1} (↑(fixedPoints ↥(↑P : Subgroup G) (↑(orbit G P) : Type u_1)) : Type u_1)
Tactic: calc
        1 = Nat.card (fixedPoints P (orbit G P)) := ?_
        _ ≡ Nat.card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
        _ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
-- Lean 4 proof step from Mathlib.GroupTheory.Sylow (goal)
Goal:
p : ℕ
G : Type u_1
inst : Group G
hp : Fact (Nat.Prime p)
inst_1 : Finite.{u_1 + 1} (Sylow p G)
P Q : Sylow p G
H :
  ∀ {R : Sylow p G} {S : (↑(orbit G P) : Type u_1)},
    Membership.mem (γ := Set (↑(orbit G P) : Type u_1)) (fixedPoints ↥(↑R : Subgroup G) (↑(orbit G P) : Type u_1)) S ↔
      (↑(↑S : Sylow p G) : Subgroup G) = (↑R : Subgroup G)
h : p ∣ Nat.card.{u_1} (↑(orbit G P) : Type u_1)
⊢ 1 = Nat.card.{u_1} (↑(fixedPoints ↥(↑P : Subgroup G) (↑(orbit G P) : Type u_1)) : Type u_1)
have hpα : Nat.card.{u_1} (↑(fixedPoints ↥(↑P : Subgroup G) (↑(orbit G P) : Type u_1)) : Type u_1) =
  Nat.card.{u_1} (↑{Subtype.mk (p := fun (x : Sylow p G) ↦ x ∈ orbit G P) P sorry} : Type u_1)
Tactic: rw [← Nat.card_unique (α := ({⟨P, mem_orbit_self P⟩} : Set (orbit G P))), eq_comm]
-- Lean 4 proof step from Mathlib.GroupTheory.Sylow (goal)
Goal:
p : ℕ
G : Type u_1
inst : Group G
hp : Fact (Nat.Prime p)
inst_1 : Finite.{u_1 + 1} (Sylow p G)
P Q : Sylow p G
H :
  ∀ {R : Sylow p G} {S : (↑(orbit G P) : Type u_1)},
    Membership.mem (γ := Set (↑(orbit G P) : Type u_1)) (fixedPoints ↥(↑R : Subgroup G) (↑(orbit G P) : Type u_1)) S ↔
      (↑(↑S : Sylow p G) : Subgroup G) = (↑R : Subgroup G)
h : p ∣ Nat.card.{u_1} (↑(orbit G P) : Type u_1)
⊢ Nat.card.{u_1} (↑(fixedPoints ↥(↑P : Subgroup G) (↑(orbit G P) : Type u_1)) : Type u_1) =
    Nat.card.{u_1} (↑{Subtype.mk (p := fun (x : Sylow p G) ↦ x ∈ orbit G P) P ⋯} : Type u_1)
have hpα : Eq (α := Set (↑(orbit G P) : Type u_1)) (fixedPoints ↥(↑P : Subgroup G) (↑(orbit G P) : Type u_1))
  {Subtype.mk (p := fun (x : Sylow p G) ↦ x ∈ orbit G P) P sorry}
Tactic: congr
-- Lean 4 proof step from Mathlib.GroupTheory.Sylow (goal)
Goal:
p : ℕ
G : Type u_1
inst : Group G
hp : Fact (Nat.Prime p)
inst_1 : Finite.{u_1 + 1} (Sylow p G)
P Q : Sylow p G
H :
  ∀ {R : Sylow p G} {S : (↑(orbit G P) : Type u_1)},
    Membership.mem (γ := Set (↑(orbit G P) : Type u_1)) (fixedPoints ↥(↑R : Subgroup G) (↑(orbit G P) : Type u_1)) S ↔
      (↑(↑S : Sylow p G) : Subgroup G) = (↑R : Subgroup G)
h : p ∣ Nat.card.{u_1} (↑(orbit G P) : Type u_1)
⊢ Eq (α := Set (↑(orbit G P) : Type u_1)) (fixedPoints ↥(↑P : Subgroup G) (↑(orbit G P) : Type u_1))
    {Subtype.mk (p := fun (x : Sylow p G) ↦ x ∈ orbit G P) P ⋯}
have hpα : Membership.mem (γ := Set (↑(orbit G P) : Type u_1)) (fixedPoints ↥(↑P : Subgroup G) (↑(orbit G P) : Type u_1))
    (Subtype.mk (p := fun (x : Sylow p G) ↦ x ∈ orbit G P) P sorry) ∧
  ∀ (x : (↑(orbit G P) : Type u_1)),
    Membership.mem (γ := Set (↑(orbit G P) : Type u_1)) (fixedPoints ↥(↑P : Subgroup G) (↑(orbit G P) : Type u_1)) x →
      x = Subtype.mk (p := fun (x : Sylow p G) ↦ x ∈ orbit G P) P sorry
Tactic: rw [Set.eq_singleton_iff_unique_mem]
-- Lean 4 proof step from Mathlib.GroupTheory.Sylow (goal)
Goal:
p : ℕ
G : Type u_1
inst : Group G
hp : Fact (Nat.Prime p)
inst_1 : Finite.{u_1 + 1} (Sylow p G)
P Q : Sylow p G
⊢ ∃ (g : G), g • P = Q
have have : ∀
  (H :
    ∀ {R : Sylow p G} {S : (↑(orbit G P) : Type u_1)},
      Membership.mem (γ := Set (↑(orbit G P) : Type u_1)) (fixedPoints ↥(↑R : Subgroup G) (↑(orbit G P) : Type u_1)) S ↔
        (↑(↑S : Sylow p G) : Subgroup G) = (↑R : Subgroup G)),
  ∃ (g : G), g • P = Q
Tactic: classical
      have H := fun {R : Sylow p G} {S : orbit G P} =>
        calc
          S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
            forall_congr' fun a => Subtype.ext_iff
          _ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
          _ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
      suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
        exact Exists.elim this fun R hR => by
          rw [← Sylow.ext (H.mp hR)]
          exact R.2
      apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
      refine fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp ?_)
      calc
        1 = Nat.card (fixedPoints P (orbit G P)) := ?_
        _ ≡ Nat.card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
        _ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
      rw [← Nat.card_unique (α := ({⟨P, mem_orbit_self P⟩} : Set (orbit G P))), eq_comm]
      congr
      rw [Set.eq_singleton_iff_unique_mem]
      exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩
-- Lean 4 proof step from Mathlib.GroupTheory.Sylow (goal)
Goal:
p : ℕ
G : Type u_1
inst : Group G
hp : Fact (Nat.Prime p)
inst_1 : Finite.{u_1 + 1} (Sylow p G)
P Q : Sylow p G
have have₁ have₂ :
  ∀
    (H :
      ∀ {R : Sylow p G} {S : (↑(orbit G P) : Type u_1)},
        Membership.mem (γ := Set (↑(orbit G P) : Type u_1)) (fixedPoints ↥(↑R : Subgroup G) (↑(orbit G P) : Type u_1))
            S ↔
          (↑(↑S : Sylow p G) : Subgroup G) = (↑R : Subgroup G)),
    ∃ (g : G), g • P = Q
⊢ ∃ (g : G), g • P = Q
have have₃ : ∀
  (H :
    ∀ {R : Sylow p G} {S : (↑(orbit G P) : Type u_1)},
      Membership.mem (γ := Set (↑(orbit G P) : Type u_1)) (fixedPoints ↥(↑R : Subgroup G) (↑(orbit G P) : Type u_1)) S ↔
        (↑(↑S : Sylow p G) : Subgroup G) = (↑R : Subgroup G)),
  Set.Nonempty.{u_1} (α := (↑(orbit G P) : Type u_1)) (fixedPoints ↥(↑Q : Subgroup G) (↑(orbit G P) : Type u_1))
Tactic: classical
      have H := fun {R : Sylow p G} {S : orbit G P} =>
        calc
          S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
            forall_congr' fun a => Subtype.ext_iff
          _ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
          _ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
      suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
        exact Exists.elim this fun R hR => by
          rw [← Sylow.ext (H.mp hR)]
          exact R.2
      apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
      refine fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp ?_)
      calc
        1 = Nat.card (fixedPoints P (orbit G P)) := ?_
        _ ≡ Nat.card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
        _ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
      rw [← Nat.card_unique (α := ({⟨P, mem_orbit_self P⟩} : Set (orbit G P))), eq_comm]
      congr
      rw [Set.eq_singleton_iff_unique_mem]
      exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩
-- Lean 4 proof step from Mathlib.GroupTheory.Sylow (goal)
Goal:
p : ℕ
G : Type u_1
inst : Group G
hp : Fact (Nat.Prime p)
inst_1 : Finite.{u_1 + 1} (Sylow p G)
P Q : Sylow p G
have have₁ have₂ :
  ∀
    (H :
      ∀ {R : Sylow p G} {S : (↑(orbit G P) : Type u_1)},
        Membership.mem (γ := Set (↑(orbit G P) : Type u_1)) (fixedPoints ↥(↑R : Subgroup G) (↑(orbit G P) : Type u_1))
            S ↔
          (↑(↑S : Sylow p G) : Subgroup G) = (↑R : Subgroup G)),
    ∃ (g : G), g • P = Q
have₃ :
  ∀
    (H :
      ∀ {R : Sylow p G} {S : (↑(orbit G P) : Type u_1)},
        Membership.mem (γ := Set (↑(orbit G P) : Type u_1)) (fixedPoints ↥(↑R : Subgroup G) (↑(orbit G P) : Type u_1))
            S ↔
          (↑(↑S : Sylow p G) : Subgroup G) = (↑R : Subgroup G)),
    Set.Nonempty.{u_1} (α := (↑(orbit G P) : Type u_1)) (fixedPoints ↥(↑Q : Subgroup G) (↑(orbit G P) : Type u_1))
⊢ ∃ (g : G), g • P = Q
have hpα : ∀
  (H :
    ∀ {R : Sylow p G} {S : (↑(orbit G P) : Type u_1)},
      Membership.mem (γ := Set (↑(orbit G P) : Type u_1)) (fixedPoints ↥(↑R : Subgroup G) (↑(orbit G P) : Type u_1)) S ↔
        (↑(↑S : Sylow p G) : Subgroup G) = (↑R : Subgroup G)),
  ¬p ∣ Nat.card.{u_1} (↑(orbit G P) : Type u_1)
Tactic: classical
      have H := fun {R : Sylow p G} {S : orbit G P} =>
        calc
          S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
            forall_congr' fun a => Subtype.ext_iff
          _ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
          _ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
      suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
        exact Exists.elim this fun R hR => by
          rw [← Sylow.ext (H.mp hR)]
          exact R.2
      apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
      refine fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp ?_)
      calc
        1 = Nat.card (fixedPoints P (orbit G P)) := ?_
        _ ≡ Nat.card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
        _ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
      rw [← Nat.card_unique (α := ({⟨P, mem_orbit_self P⟩} : Set (orbit G P))), eq_comm]
      congr
      rw [Set.eq_singleton_iff_unique_mem]
      exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩
-- Lean 4 proof step from Mathlib.GroupTheory.Sylow (goal)
Goal:
p : ℕ
G : Type u_1
inst : Group G
hp : Fact (Nat.Prime p)
inst_1 : Finite.{u_1 + 1} (Sylow p G)
P Q : Sylow p G
have have₁ have₂ :
  ∀
    (H :
      ∀ {R : Sylow p G} {S : (↑(orbit G P) : Type u_1)},
        Membership.mem (γ := Set (↑(orbit G P) : Type u_1)) (fixedPoints ↥(↑R : Subgroup G) (↑(orbit G P) : Type u_1))
            S ↔
          (↑(↑S : Sylow p G) : Subgroup G) = (↑R : Subgroup G)),
    ∃ (g : G), g • P = Q
have₃ :
  ∀
    (H :
      ∀ {R : Sylow p G} {S : (↑(orbit G P) : Type u_1)},
        Membership.mem (γ := Set (↑(orbit G P) : Type u_1)) (fixedPoints ↥(↑R : Subgroup G) (↑(orbit G P) : Type u_1))
            S ↔
          (↑(↑S : Sylow p G) : Subgroup G) = (↑R : Subgroup G)),
    Set.Nonempty.{u_1} (α := (↑(orbit G P) : Type u_1)) (fixedPoints ↥(↑Q : Subgroup G) (↑(orbit G P) : Type u_1))
hpα :
  ∀
    (H :
      ∀ {R : Sylow p G} {S : (↑(orbit G P) : Type u_1)},
        Membership.mem (γ := Set (↑(orbit G P) : Type u_1)) (fixedPoints ↥(↑R : Subgroup G) (↑(orbit G P) : Type u_1))
            S ↔
          (↑(↑S : Sylow p G) : Subgroup G) = (↑R : Subgroup G)),
    ¬p ∣ Nat.card.{u_1} (↑(orbit G P) : Type u_1)
⊢ ∃ (g : G), g • P = Q
have hpα₁ : ∀
  (H :
    ∀ {R : Sylow p G} {S : (↑(orbit G P) : Type u_1)},
      Membership.mem (γ := Set (↑(orbit G P) : Type u_1)) (fixedPoints ↥(↑R : Subgroup G) (↑(orbit G P) : Type u_1)) S ↔
        (↑(↑S : Sylow p G) : Subgroup G) = (↑R : Subgroup G))
  (h : p ∣ Nat.card.{u_1} (↑(orbit G P) : Type u_1)), 1 ≡ 0 [MOD p]
Tactic: classical
      have H := fun {R : Sylow p G} {S : orbit G P} =>
        calc
          S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
            forall_congr' fun a => Subtype.ext_iff
          _ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
          _ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
      suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
        exact Exists.elim this fun R hR => by
          rw [← Sylow.ext (H.mp hR)]
          exact R.2
      apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
      refine fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp ?_)
      calc
        1 = Nat.card (fixedPoints P (orbit G P)) := ?_
        _ ≡ Nat.card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
        _ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
      rw [← Nat.card_unique (α := ({⟨P, mem_orbit_self P⟩} : Set (orbit G P))), eq_comm]
      congr
      rw [Set.eq_singleton_iff_unique_mem]
      exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩