l3lab/ntp-mathlib huggingface dataset:
/-- A generalization of **Sylow's third theorem**.
  If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
    card (Sylow p G) ≡ 1 [MOD p] := by
  refine' Sylow.nonempty.elim fun P : Sylow p G => _
  

Proof state:
p : ℕ
G : Type u_1
inst✝² : Group G
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Fintype (Sylow p G)
P : Sylow p G
⊢ card (Sylow p G) ≡ 1 [MOD p]

Next tactic:
have : fixedPoints P.1 (Sylow p G) = {P} :=
    Set.ext fun Q : Sylow p G =>
      calc
        Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
        _ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
        _ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
/-- A generalization of **Sylow's third theorem**.
  If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
    card (Sylow p G) ≡ 1 [MOD p] := by
  refine' Sylow.nonempty.elim fun P : Sylow p G => _
  have : fixedPoints P.1 (Sylow p G) = {P} :=
    Set.ext fun Q : Sylow p G =>
      calc
        Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
        _ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
        _ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm

  

Proof state:
p : ℕ
G : Type u_1
inst✝² : Group G
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Fintype (Sylow p G)
P : Sylow p G
this : fixedPoints (↥↑P) (Sylow p G) = {P}
⊢ card (Sylow p G) ≡ 1 [MOD p]

Next tactic:
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
    rw [this]
    infer_instance
/-- A generalization of **Sylow's third theorem**.
  If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
    card (Sylow p G) ≡ 1 [MOD p] := by
  refine' Sylow.nonempty.elim fun P : Sylow p G => _
  have : fixedPoints P.1 (Sylow p G) = {P} :=
    Set.ext fun Q : Sylow p G =>
      calc
        Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
        _ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
        _ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm

  have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
    

Proof state:
p : ℕ
G : Type u_1
inst✝² : Group G
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Fintype (Sylow p G)
P : Sylow p G
this : fixedPoints (↥↑P) (Sylow p G) = {P}
⊢ Fintype ↑(fixedPoints (↥↑P) (Sylow p G))

Next tactic:
rw [this]
/-- A generalization of **Sylow's third theorem**.
  If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
    card (Sylow p G) ≡ 1 [MOD p] := by
  refine' Sylow.nonempty.elim fun P : Sylow p G => _
  have : fixedPoints P.1 (Sylow p G) = {P} :=
    Set.ext fun Q : Sylow p G =>
      calc
        Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
        _ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
        _ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm

  have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
    rw [this]
    

Proof state:
p : ℕ
G : Type u_1
inst✝² : Group G
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Fintype (Sylow p G)
P : Sylow p G
this : fixedPoints (↥↑P) (Sylow p G) = {P}
⊢ Fintype ↑{P}

Next tactic:
infer_instance
/-- A generalization of **Sylow's third theorem**.
  If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
    card (Sylow p G) ≡ 1 [MOD p] := by
  refine' Sylow.nonempty.elim fun P : Sylow p G => _
  have : fixedPoints P.1 (Sylow p G) = {P} :=
    Set.ext fun Q : Sylow p G =>
      calc
        Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
        _ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
        _ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm

  have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
    rw [this]
    infer_instance
  

Proof state:
p : ℕ
G : Type u_1
inst✝² : Group G
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Fintype (Sylow p G)
P : Sylow p G
this : fixedPoints (↥↑P) (Sylow p G) = {P}
fin : Fintype ↑(fixedPoints (↥↑P) (Sylow p G))
⊢ card (Sylow p G) ≡ 1 [MOD p]

Next tactic:
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
/-- A generalization of **Sylow's third theorem**.
  If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
    card (Sylow p G) ≡ 1 [MOD p] := by
  refine' Sylow.nonempty.elim fun P : Sylow p G => _
  have : fixedPoints P.1 (Sylow p G) = {P} :=
    Set.ext fun Q : Sylow p G =>
      calc
        Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
        _ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
        _ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm

  have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
    rw [this]
    infer_instance
  have : card (fixedPoints P.1 (Sylow p G)) = 1 := by 

Proof state:
p : ℕ
G : Type u_1
inst✝² : Group G
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Fintype (Sylow p G)
P : Sylow p G
this : fixedPoints (↥↑P) (Sylow p G) = {P}
fin : Fintype ↑(fixedPoints (↥↑P) (Sylow p G))
⊢ card ↑(fixedPoints (↥↑P) (Sylow p G)) = 1

Next tactic:
simp [this]
/-- A generalization of **Sylow's third theorem**.
  If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
    card (Sylow p G) ≡ 1 [MOD p] := by
  refine' Sylow.nonempty.elim fun P : Sylow p G => _
  have : fixedPoints P.1 (Sylow p G) = {P} :=
    Set.ext fun Q : Sylow p G =>
      calc
        Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
        _ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
        _ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm

  have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
    rw [this]
    infer_instance
  have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
  

Proof state:
p : ℕ
G : Type u_1
inst✝² : Group G
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Fintype (Sylow p G)
P : Sylow p G
this✝ : fixedPoints (↥↑P) (Sylow p G) = {P}
fin : Fintype ↑(fixedPoints (↥↑P) (Sylow p G))
this : card ↑(fixedPoints (↥↑P) (Sylow p G)) = 1
⊢ card (Sylow p G) ≡ 1 [MOD p]

Next tactic:
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
/-- A generalization of **Sylow's third theorem**.
  If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
    card (Sylow p G) ≡ 1 [MOD p] := by
  refine' Sylow.nonempty.elim fun P : Sylow p G => _
  have : fixedPoints P.1 (Sylow p G) = {P} :=
    Set.ext fun Q : Sylow p G =>
      calc
        Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
        _ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
        _ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm

  have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
    rw [this]
    infer_instance
  have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
  exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by 

Proof state:
p : ℕ
G : Type u_1
inst✝² : Group G
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Fintype (Sylow p G)
P : Sylow p G
this✝ : fixedPoints (↥↑P) (Sylow p G) = {P}
fin : Fintype ↑(fixedPoints (↥↑P) (Sylow p G))
this : card ↑(fixedPoints (↥↑P) (Sylow p G)) = 1
⊢ card ↑(fixedPoints (↥↑P) (Sylow p G)) ≡ 1 [MOD p]

Next tactic:
rw [this]
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
  

Proof state:
p : ℕ
G : Type u_1
inst✝² : Group G
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P Q : Sylow p G
⊢ ↥↑P ≃* ↥↑Q

Next tactic:
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
  rw [← Classical.choose_spec (exists_smul_eq G P Q)]
  

Proof state:
p : ℕ
G : Type u_1
inst✝² : Group G
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P Q : Sylow p G
⊢ ↥↑P ≃* ↥↑(Classical.choose (_ : ∃ m, m • P = Q) • P)

Next tactic:
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
    stabilizer G P = (P : Subgroup G).normalizer := by
  

Proof state:
p : ℕ
G : Type u_1
inst✝ : Group G
P : Sylow p G
⊢ stabilizer G P = normalizer ↑P

Next tactic:
ext
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
    stabilizer G P = (P : Subgroup G).normalizer := by
  ext; 

Proof state:
case h
p : ℕ
G : Type u_1
inst✝ : Group G
P : Sylow p G
x✝ : G
⊢ x✝ ∈ stabilizer G P ↔ x✝ ∈ normalizer ↑P

Next tactic:
simp [Sylow.smul_eq_iff_mem_normalizer]
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
    (P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
    (hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
    ∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
  

Proof state:
p : ℕ
G : Type u_1
inst✝² : Group G
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
x g : G
hx : x ∈ centralizer ↑P
hy : g⁻¹ * x * g ∈ centralizer ↑P
⊢ ∃ n ∈ normalizer ↑P, g⁻¹ * x * g = n⁻¹ * x * n

Next tactic:
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
    (P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
    (hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
    ∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
  have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by 

Proof state:
p : ℕ
G : Type u_1
inst✝² : Group G
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
x g : G
hx : x ∈ centralizer ↑P
hy : g⁻¹ * x * g ∈ centralizer ↑P
⊢ ↑P ≤ centralizer ↑(zpowers x)

Next tactic:
rwa [le_centralizer_iff, zpowers_le]
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
    (P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
    (hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
    ∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
  have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
  

Proof state:
p : ℕ
G : Type u_1
inst✝² : Group G
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
x g : G
hx : x ∈ centralizer ↑P
hy : g⁻¹ * x * g ∈ centralizer ↑P
h1 : ↑P ≤ centralizer ↑(zpowers x)
⊢ ∃ n ∈ normalizer ↑P, g⁻¹ * x * g = n⁻¹ * x * n

Next tactic:
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
    rw [le_centralizer_iff, zpowers_le]
    rintro - ⟨z, hz, rfl⟩
    specialize hy z hz
    rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
      eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
    (P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
    (hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
    ∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
  have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
  have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
    

Proof state:
p : ℕ
G : Type u_1
inst✝² : Group G
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
x g : G
hx : x ∈ centralizer ↑P
hy : g⁻¹ * x * g ∈ centralizer ↑P
h1 : ↑P ≤ centralizer ↑(zpowers x)
⊢ ↑(g • P) ≤ centralizer ↑(zpowers x)

Next tactic:
rw [le_centralizer_iff, zpowers_le]
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
    (P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
    (hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
    ∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
  have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
  have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
    rw [le_centralizer_iff, zpowers_le]
    

Proof state:
p : ℕ
G : Type u_1
inst✝² : Group G
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
x g : G
hx : x ∈ centralizer ↑P
hy : g⁻¹ * x * g ∈ centralizer ↑P
h1 : ↑P ≤ centralizer ↑(zpowers x)
⊢ x ∈ centralizer ↑↑(g • P)

Next tactic:
rintro - ⟨z, hz, rfl⟩
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
    (P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
    (hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
    ∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
  have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
  have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
    rw [le_centralizer_iff, zpowers_le]
    rintro - ⟨z, hz, rfl⟩
    

Proof state:
case intro.intro
p : ℕ
G : Type u_1
inst✝² : Group G
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
x g : G
hx : x ∈ centralizer ↑P
hy : g⁻¹ * x * g ∈ centralizer ↑P
h1 : ↑P ≤ centralizer ↑(zpowers x)
z : G
hz : z ∈ ↑↑P
⊢ ((MulDistribMulAction.toMonoidEnd (MulAut G) G) (MulAut.conj g)) z * x =
    x * ((MulDistribMulAction.toMonoidEnd (MulAut G) G) (MulAut.conj g)) z

Next tactic:
specialize hy z hz
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
    (P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
    (hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
    ∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
  have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
  have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
    rw [le_centralizer_iff, zpowers_le]
    rintro - ⟨z, hz, rfl⟩
    specialize hy z hz
    

Proof state:
case intro.intro
p : ℕ
G : Type u_1
inst✝² : Group G
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
x g : G
hx : x ∈ centralizer ↑P
h1 : ↑P ≤ centralizer ↑(zpowers x)
z : G
hz : z ∈ ↑↑P
hy : z * (g⁻¹ * x * g) = g⁻¹ * x * g * z
⊢ ((MulDistribMulAction.toMonoidEnd (MulAut G) G) (MulAut.conj g)) z * x =
    x * ((MulDistribMulAction.toMonoidEnd (MulAut G) G) (MulAut.conj g)) z

Next tactic:
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
      eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
    (P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
    (hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
    ∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
  have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
  have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
    rw [le_centralizer_iff, zpowers_le]
    rintro - ⟨z, hz, rfl⟩
    specialize hy z hz
    rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
      eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
  

Proof state:
p : ℕ
G : Type u_1
inst✝² : Group G
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
x g : G
hx : x ∈ centralizer ↑P
hy : g⁻¹ * x * g ∈ centralizer ↑P
h1 : ↑P ≤ centralizer ↑(zpowers x)
h2 : ↑(g • P) ≤ centralizer ↑(zpowers x)
⊢ ∃ n ∈ normalizer ↑P, g⁻¹ * x * g = n⁻¹ * x * n

Next tactic:
obtain ⟨h, hh⟩ :=
    exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
    (P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
    (hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
    ∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
  have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
  have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
    rw [le_centralizer_iff, zpowers_le]
    rintro - ⟨z, hz, rfl⟩
    specialize hy z hz
    rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
      eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
  obtain ⟨h, hh⟩ :=
    exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
  

Proof state:
case intro
p : ℕ
G : Type u_1
inst✝² : Group G
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
x g : G
hx : x ∈ centralizer ↑P
hy : g⁻¹ * x * g ∈ centralizer ↑P
h1 : ↑P ≤ centralizer ↑(zpowers x)
h2 : ↑(g • P) ≤ centralizer ↑(zpowers x)
h : ↥(centralizer ↑(zpowers x))
hh : h • Sylow.subtype (g • P) h2 = Sylow.subtype P h1
⊢ ∃ n ∈ normalizer ↑P, g⁻¹ * x * g = n⁻¹ * x * n

Next tactic:
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
    (P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
    (hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
    ∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
  have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
  have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
    rw [le_centralizer_iff, zpowers_le]
    rintro - ⟨z, hz, rfl⟩
    specialize hy z hz
    rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
      eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
  obtain ⟨h, hh⟩ :=
    exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
  simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
  

Proof state:
case intro
p : ℕ
G : Type u_1
inst✝² : Group G
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
x g : G
hx : x ∈ centralizer ↑P
hy : g⁻¹ * x * g ∈ centralizer ↑P
h1 : ↑P ≤ centralizer ↑(zpowers x)
h2 : ↑(g • P) ≤ centralizer ↑(zpowers x)
h : ↥(centralizer ↑(zpowers x))
hh : Sylow.subtype ((↑h * g) • P) (_ : ↑((↑h * g) • P) ≤ centralizer ↑(zpowers x)) = Sylow.subtype P h1
⊢ ∃ n ∈ normalizer ↑P, g⁻¹ * x * g = n⁻¹ * x * n

Next tactic:
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
    (P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
    (hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
    ∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
  have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
  have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
    rw [le_centralizer_iff, zpowers_le]
    rintro - ⟨z, hz, rfl⟩
    specialize hy z hz
    rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
      eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
  obtain ⟨h, hh⟩ :=
    exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
  simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
  refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
  

Proof state:
case intro
p : ℕ
G : Type u_1
inst✝² : Group G
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
x g : G
hx : x ∈ centralizer ↑P
hy : g⁻¹ * x * g ∈ centralizer ↑P
h1 : ↑P ≤ centralizer ↑(zpowers x)
h2 : ↑(g • P) ≤ centralizer ↑(zpowers x)
h : ↥(centralizer ↑(zpowers x))
hh : Sylow.subtype ((↑h * g) • P) (_ : ↑((↑h * g) • P) ≤ centralizer ↑(zpowers x)) = Sylow.subtype P h1
⊢ g⁻¹ * x * g = (↑h * g)⁻¹ * x * (↑h * g)

Next tactic:
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
    (P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
  calc
    Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
    _ ≃ orbit G P := by 

Proof state:
p : ℕ
G : Type u_1
inst✝² : Group G
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Fintype (Sylow p G)
P : Sylow p G
⊢ ↑⊤ ≃ ↑(orbit G P)

Next tactic:
rw [P.orbit_eq_top]
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
    (P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
  calc
    Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
    _ ≃ orbit G P := by rw [P.orbit_eq_top]
    _ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
    _ ≃ G ⧸ (P : Subgroup G).normalizer := by 

Proof state:
p : ℕ
G : Type u_1
inst✝² : Group G
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Fintype (Sylow p G)
P : Sylow p G
⊢ G ⧸ stabilizer G P ≃ G ⧸ normalizer ↑P

Next tactic:
rw [P.stabilizer_eq_normalizer]
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
    [fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
  

Proof state:
p : ℕ
G : Type u_1
inst✝¹ : Group G
hp : Fact (Nat.Prime p)
P : Sylow p G
inst✝ : Normal ↑P
fP : FiniteIndex ↑P
⊢ ¬p ∣ index ↑P

Next tactic:
intro h
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
    [fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
  intro h
  

Proof state:
p : ℕ
G : Type u_1
inst✝¹ : Group G
hp : Fact (Nat.Prime p)
P : Sylow p G
inst✝ : Normal ↑P
fP : FiniteIndex ↑P
h : p ∣ index ↑P
⊢ False

Next tactic:
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
    [fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
  intro h
  letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
  

Proof state:
p : ℕ
G : Type u_1
inst✝¹ : Group G
hp : Fact (Nat.Prime p)
P : Sylow p G
inst✝ : Normal ↑P
fP : FiniteIndex ↑P
h : p ∣ index ↑P
this : Fintype (G ⧸ ↑P) := fintypeQuotientOfFiniteIndex ↑P
⊢ False

Next tactic:
rw [index_eq_card (P : Subgroup G)] at h
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
    [fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
  intro h
  letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
  rw [index_eq_card (P : Subgroup G)] at h
  

Proof state:
p : ℕ
G : Type u_1
inst✝¹ : Group G
hp : Fact (Nat.Prime p)
P : Sylow p G
inst✝ : Normal ↑P
fP : FiniteIndex ↑P
this : Fintype (G ⧸ ↑P) := fintypeQuotientOfFiniteIndex ↑P
h : p ∣ card (G ⧸ ↑P)
⊢ False

Next tactic:
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
    [fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
  intro h
  letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
  rw [index_eq_card (P : Subgroup G)] at h
  obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
  

Proof state:
case intro
p : ℕ
G : Type u_1
inst✝¹ : Group G
hp : Fact (Nat.Prime p)
P : Sylow p G
inst✝ : Normal ↑P
fP : FiniteIndex ↑P
this : Fintype (G ⧸ ↑P) := fintypeQuotientOfFiniteIndex ↑P
h : p ∣ card (G ⧸ ↑P)
x : G ⧸ ↑P
hx : orderOf x = p
⊢ False

Next tactic:
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)