-- Module: Mathlib.GroupTheory.Sylow
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)
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]
hpα₂ :
  ∀
    (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)
hpα₃ :
  ∀
    (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)
hpα₄ :
  ∀
    (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 ⋯}
hpα₅ :
  ∀
    (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)),
    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 ⋯) ∧
      ∀ (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 ⋯
⊢ ∃ (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))⟩

Premises used: Sylow.isPGroup', Sylow.toSubgroup, Membership.mem, Set.Elem, Subtype, Nat.card, IsPGroup.card_modEq_card_fixedPoints, eq_comm, MulAction.orbit, Nat.ModEq.symm, Iff.mp, Nat.Prime.not_dvd_one, Nat.card_unique, Set.eq_singleton_iff_unique_mem, MulAction.fixedPoints, Nat.modEq_zero_iff_dvd, Fact.out, IsPGroup.nonempty_fixed_point_of_prime_not_dvd_card