l3lab/ntp-mathlib huggingface dataset:
/-- If a finite group is the direct product of its Sylow groups, it is nilpotent -/
theorem isNilpotent_of_product_of_sylow_group
    (e : (∀ p : (Fintype.card G).primeFactors, ∀ P : Sylow p G, (↑P : Subgroup G)) ≃* G) :
    IsNilpotent G := by
  classical
    let ps := (Fintype.card G).primeFactors
    have : ∀ (p : ps) (P : Sylow p G), IsNilpotent (↑P : Subgroup G) := by
      intro p P
      

Proof state:
G : Type u_1
hG : Group G
inst✝ : Fintype G
e : ((p : { x // x ∈ (card G).primeFactors }) → (P : Sylow (↑p) G) → ↥↑P) ≃* G
ps : Finset ℕ := (card G).primeFactors
p : { x // x ∈ ps }
P : Sylow (↑p) G
⊢ Group.IsNilpotent ↥↑P

Next tactic:
haveI : Fact (Nat.Prime ↑p) := Fact.mk $ Nat.prime_of_mem_primeFactors p.2
/-- If a finite group is the direct product of its Sylow groups, it is nilpotent -/
theorem isNilpotent_of_product_of_sylow_group
    (e : (∀ p : (Fintype.card G).primeFactors, ∀ P : Sylow p G, (↑P : Subgroup G)) ≃* G) :
    IsNilpotent G := by
  classical
    let ps := (Fintype.card G).primeFactors
    have : ∀ (p : ps) (P : Sylow p G), IsNilpotent (↑P : Subgroup G) := by
      intro p P
      haveI : Fact (Nat.Prime ↑p) := Fact.mk $ Nat.prime_of_mem_primeFactors p.2
      

Proof state:
G : Type u_1
hG : Group G
inst✝ : Fintype G
e : ((p : { x // x ∈ (card G).primeFactors }) → (P : Sylow (↑p) G) → ↥↑P) ≃* G
ps : Finset ℕ := (card G).primeFactors
p : { x // x ∈ ps }
P : Sylow (↑p) G
this : Fact (Nat.Prime ↑p)
⊢ Group.IsNilpotent ↥↑P

Next tactic:
exact P.isPGroup'.isNilpotent
/-- If a finite group is the direct product of its Sylow groups, it is nilpotent -/
theorem isNilpotent_of_product_of_sylow_group
    (e : (∀ p : (Fintype.card G).primeFactors, ∀ P : Sylow p G, (↑P : Subgroup G)) ≃* G) :
    IsNilpotent G := by
  classical
    let ps := (Fintype.card G).primeFactors
    have : ∀ (p : ps) (P : Sylow p G), IsNilpotent (↑P : Subgroup G) := by
      intro p P
      haveI : Fact (Nat.Prime ↑p) := Fact.mk $ Nat.prime_of_mem_primeFactors p.2
      exact P.isPGroup'.isNilpotent
    

Proof state:
G : Type u_1
hG : Group G
inst✝ : Fintype G
e : ((p : { x // x ∈ (card G).primeFactors }) → (P : Sylow (↑p) G) → ↥↑P) ≃* G
ps : Finset ℕ := (card G).primeFactors
this : ∀ (p : { x // x ∈ ps }) (P : Sylow (↑p) G), Group.IsNilpotent ↥↑P
⊢ Group.IsNilpotent G

Next tactic:
exact nilpotent_of_mulEquiv e
/-- A finite group is nilpotent iff the normalizer condition holds, and iff all maximal groups are
normal and iff all sylow groups are normal and iff the group is the direct product of its sylow
groups. -/
theorem isNilpotent_of_finite_tFAE :
    List.TFAE
      [IsNilpotent G, NormalizerCondition G, ∀ H : Subgroup G, IsCoatom H → H.Normal,
        ∀ (p : ℕ) (_hp : Fact p.Prime) (P : Sylow p G), (↑P : Subgroup G).Normal,
        Nonempty
          ((∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Subgroup G)) ≃* G)] := by
  

Proof state:
G : Type u_1
hG : Group G
inst✝ : Fintype G
⊢ List.TFAE
    [Group.IsNilpotent G, NormalizerCondition G, ∀ (H : Subgroup G), IsCoatom H → Normal H,
      ∀ (p : ℕ), Fact (Nat.Prime p) → ∀ (P : Sylow p G), Normal ↑P,
      Nonempty (((p : { x // x ∈ (card G).primeFactors }) → (P : Sylow (↑p) G) → ↥↑P) ≃* G)]

Next tactic:
tfae_have 1 → 2
/-- A finite group is nilpotent iff the normalizer condition holds, and iff all maximal groups are
normal and iff all sylow groups are normal and iff the group is the direct product of its sylow
groups. -/
theorem isNilpotent_of_finite_tFAE :
    List.TFAE
      [IsNilpotent G, NormalizerCondition G, ∀ H : Subgroup G, IsCoatom H → H.Normal,
        ∀ (p : ℕ) (_hp : Fact p.Prime) (P : Sylow p G), (↑P : Subgroup G).Normal,
        Nonempty
          ((∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Subgroup G)) ≃* G)] := by
  tfae_have 1 → 2
  · 

Proof state:
case tfae_1_to_2
G : Type u_1
hG : Group G
inst✝ : Fintype G
⊢ Group.IsNilpotent G → NormalizerCondition G

Next tactic:
exact @normalizerCondition_of_isNilpotent _ _
/-- A finite group is nilpotent iff the normalizer condition holds, and iff all maximal groups are
normal and iff all sylow groups are normal and iff the group is the direct product of its sylow
groups. -/
theorem isNilpotent_of_finite_tFAE :
    List.TFAE
      [IsNilpotent G, NormalizerCondition G, ∀ H : Subgroup G, IsCoatom H → H.Normal,
        ∀ (p : ℕ) (_hp : Fact p.Prime) (P : Sylow p G), (↑P : Subgroup G).Normal,
        Nonempty
          ((∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Subgroup G)) ≃* G)] := by
  tfae_have 1 → 2
  · exact @normalizerCondition_of_isNilpotent _ _
  

Proof state:
G : Type u_1
hG : Group G
inst✝ : Fintype G
tfae_1_to_2 : Group.IsNilpotent G → NormalizerCondition G
⊢ List.TFAE
    [Group.IsNilpotent G, NormalizerCondition G, ∀ (H : Subgroup G), IsCoatom H → Normal H,
      ∀ (p : ℕ), Fact (Nat.Prime p) → ∀ (P : Sylow p G), Normal ↑P,
      Nonempty (((p : { x // x ∈ (card G).primeFactors }) → (P : Sylow (↑p) G) → ↥↑P) ≃* G)]

Next tactic:
tfae_have 2 → 3
/-- A finite group is nilpotent iff the normalizer condition holds, and iff all maximal groups are
normal and iff all sylow groups are normal and iff the group is the direct product of its sylow
groups. -/
theorem isNilpotent_of_finite_tFAE :
    List.TFAE
      [IsNilpotent G, NormalizerCondition G, ∀ H : Subgroup G, IsCoatom H → H.Normal,
        ∀ (p : ℕ) (_hp : Fact p.Prime) (P : Sylow p G), (↑P : Subgroup G).Normal,
        Nonempty
          ((∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Subgroup G)) ≃* G)] := by
  tfae_have 1 → 2
  · exact @normalizerCondition_of_isNilpotent _ _
  tfae_have 2 → 3
  · 

Proof state:
case tfae_2_to_3
G : Type u_1
hG : Group G
inst✝ : Fintype G
tfae_1_to_2 : Group.IsNilpotent G → NormalizerCondition G
⊢ NormalizerCondition G → ∀ (H : Subgroup G), IsCoatom H → Normal H

Next tactic:
exact fun h H => NormalizerCondition.normal_of_coatom H h
/-- A finite group is nilpotent iff the normalizer condition holds, and iff all maximal groups are
normal and iff all sylow groups are normal and iff the group is the direct product of its sylow
groups. -/
theorem isNilpotent_of_finite_tFAE :
    List.TFAE
      [IsNilpotent G, NormalizerCondition G, ∀ H : Subgroup G, IsCoatom H → H.Normal,
        ∀ (p : ℕ) (_hp : Fact p.Prime) (P : Sylow p G), (↑P : Subgroup G).Normal,
        Nonempty
          ((∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Subgroup G)) ≃* G)] := by
  tfae_have 1 → 2
  · exact @normalizerCondition_of_isNilpotent _ _
  tfae_have 2 → 3
  · exact fun h H => NormalizerCondition.normal_of_coatom H h
  

Proof state:
G : Type u_1
hG : Group G
inst✝ : Fintype G
tfae_1_to_2 : Group.IsNilpotent G → NormalizerCondition G
tfae_2_to_3 : NormalizerCondition G → ∀ (H : Subgroup G), IsCoatom H → Normal H
⊢ List.TFAE
    [Group.IsNilpotent G, NormalizerCondition G, ∀ (H : Subgroup G), IsCoatom H → Normal H,
      ∀ (p : ℕ), Fact (Nat.Prime p) → ∀ (P : Sylow p G), Normal ↑P,
      Nonempty (((p : { x // x ∈ (card G).primeFactors }) → (P : Sylow (↑p) G) → ↥↑P) ≃* G)]

Next tactic:
tfae_have 3 → 4
/-- A finite group is nilpotent iff the normalizer condition holds, and iff all maximal groups are
normal and iff all sylow groups are normal and iff the group is the direct product of its sylow
groups. -/
theorem isNilpotent_of_finite_tFAE :
    List.TFAE
      [IsNilpotent G, NormalizerCondition G, ∀ H : Subgroup G, IsCoatom H → H.Normal,
        ∀ (p : ℕ) (_hp : Fact p.Prime) (P : Sylow p G), (↑P : Subgroup G).Normal,
        Nonempty
          ((∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Subgroup G)) ≃* G)] := by
  tfae_have 1 → 2
  · exact @normalizerCondition_of_isNilpotent _ _
  tfae_have 2 → 3
  · exact fun h H => NormalizerCondition.normal_of_coatom H h
  tfae_have 3 → 4
  · 

Proof state:
case tfae_3_to_4
G : Type u_1
hG : Group G
inst✝ : Fintype G
tfae_1_to_2 : Group.IsNilpotent G → NormalizerCondition G
tfae_2_to_3 : NormalizerCondition G → ∀ (H : Subgroup G), IsCoatom H → Normal H
⊢ (∀ (H : Subgroup G), IsCoatom H → Normal H) → ∀ (p : ℕ), Fact (Nat.Prime p) → ∀ (P : Sylow p G), Normal ↑P

Next tactic:
intro h p _ P
/-- A finite group is nilpotent iff the normalizer condition holds, and iff all maximal groups are
normal and iff all sylow groups are normal and iff the group is the direct product of its sylow
groups. -/
theorem isNilpotent_of_finite_tFAE :
    List.TFAE
      [IsNilpotent G, NormalizerCondition G, ∀ H : Subgroup G, IsCoatom H → H.Normal,
        ∀ (p : ℕ) (_hp : Fact p.Prime) (P : Sylow p G), (↑P : Subgroup G).Normal,
        Nonempty
          ((∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Subgroup G)) ≃* G)] := by
  tfae_have 1 → 2
  · exact @normalizerCondition_of_isNilpotent _ _
  tfae_have 2 → 3
  · exact fun h H => NormalizerCondition.normal_of_coatom H h
  tfae_have 3 → 4
  · intro h p _ P; 

Proof state:
case tfae_3_to_4
G : Type u_1
hG : Group G
inst✝ : Fintype G
tfae_1_to_2 : Group.IsNilpotent G → NormalizerCondition G
tfae_2_to_3 : NormalizerCondition G → ∀ (H : Subgroup G), IsCoatom H → Normal H
h : ∀ (H : Subgroup G), IsCoatom H → Normal H
p : ℕ
_hp✝ : Fact (Nat.Prime p)
P : Sylow p G
⊢ Normal ↑P

Next tactic:
exact Sylow.normal_of_all_max_subgroups_normal h _
/-- A finite group is nilpotent iff the normalizer condition holds, and iff all maximal groups are
normal and iff all sylow groups are normal and iff the group is the direct product of its sylow
groups. -/
theorem isNilpotent_of_finite_tFAE :
    List.TFAE
      [IsNilpotent G, NormalizerCondition G, ∀ H : Subgroup G, IsCoatom H → H.Normal,
        ∀ (p : ℕ) (_hp : Fact p.Prime) (P : Sylow p G), (↑P : Subgroup G).Normal,
        Nonempty
          ((∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Subgroup G)) ≃* G)] := by
  tfae_have 1 → 2
  · exact @normalizerCondition_of_isNilpotent _ _
  tfae_have 2 → 3
  · exact fun h H => NormalizerCondition.normal_of_coatom H h
  tfae_have 3 → 4
  · intro h p _ P; exact Sylow.normal_of_all_max_subgroups_normal h _
  

Proof state:
G : Type u_1
hG : Group G
inst✝ : Fintype G
tfae_1_to_2 : Group.IsNilpotent G → NormalizerCondition G
tfae_2_to_3 : NormalizerCondition G → ∀ (H : Subgroup G), IsCoatom H → Normal H
tfae_3_to_4 : (∀ (H : Subgroup G), IsCoatom H → Normal H) → ∀ (p : ℕ), Fact (Nat.Prime p) → ∀ (P : Sylow p G), Normal ↑P
⊢ List.TFAE
    [Group.IsNilpotent G, NormalizerCondition G, ∀ (H : Subgroup G), IsCoatom H → Normal H,
      ∀ (p : ℕ), Fact (Nat.Prime p) → ∀ (P : Sylow p G), Normal ↑P,
      Nonempty (((p : { x // x ∈ (card G).primeFactors }) → (P : Sylow (↑p) G) → ↥↑P) ≃* G)]

Next tactic:
tfae_have 4 → 5
/-- A finite group is nilpotent iff the normalizer condition holds, and iff all maximal groups are
normal and iff all sylow groups are normal and iff the group is the direct product of its sylow
groups. -/
theorem isNilpotent_of_finite_tFAE :
    List.TFAE
      [IsNilpotent G, NormalizerCondition G, ∀ H : Subgroup G, IsCoatom H → H.Normal,
        ∀ (p : ℕ) (_hp : Fact p.Prime) (P : Sylow p G), (↑P : Subgroup G).Normal,
        Nonempty
          ((∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Subgroup G)) ≃* G)] := by
  tfae_have 1 → 2
  · exact @normalizerCondition_of_isNilpotent _ _
  tfae_have 2 → 3
  · exact fun h H => NormalizerCondition.normal_of_coatom H h
  tfae_have 3 → 4
  · intro h p _ P; exact Sylow.normal_of_all_max_subgroups_normal h _
  tfae_have 4 → 5
  · 

Proof state:
case tfae_4_to_5
G : Type u_1
hG : Group G
inst✝ : Fintype G
tfae_1_to_2 : Group.IsNilpotent G → NormalizerCondition G
tfae_2_to_3 : NormalizerCondition G → ∀ (H : Subgroup G), IsCoatom H → Normal H
tfae_3_to_4 : (∀ (H : Subgroup G), IsCoatom H → Normal H) → ∀ (p : ℕ), Fact (Nat.Prime p) → ∀ (P : Sylow p G), Normal ↑P
⊢ (∀ (p : ℕ), Fact (Nat.Prime p) → ∀ (P : Sylow p G), Normal ↑P) →
    Nonempty (((p : { x // x ∈ (card G).primeFactors }) → (P : Sylow (↑p) G) → ↥↑P) ≃* G)

Next tactic:
exact fun h => Nonempty.intro (Sylow.directProductOfNormal fun {p hp hP} => h p hp hP)
/-- A finite group is nilpotent iff the normalizer condition holds, and iff all maximal groups are
normal and iff all sylow groups are normal and iff the group is the direct product of its sylow
groups. -/
theorem isNilpotent_of_finite_tFAE :
    List.TFAE
      [IsNilpotent G, NormalizerCondition G, ∀ H : Subgroup G, IsCoatom H → H.Normal,
        ∀ (p : ℕ) (_hp : Fact p.Prime) (P : Sylow p G), (↑P : Subgroup G).Normal,
        Nonempty
          ((∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Subgroup G)) ≃* G)] := by
  tfae_have 1 → 2
  · exact @normalizerCondition_of_isNilpotent _ _
  tfae_have 2 → 3
  · exact fun h H => NormalizerCondition.normal_of_coatom H h
  tfae_have 3 → 4
  · intro h p _ P; exact Sylow.normal_of_all_max_subgroups_normal h _
  tfae_have 4 → 5
  · exact fun h => Nonempty.intro (Sylow.directProductOfNormal fun {p hp hP} => h p hp hP)
  

Proof state:
G : Type u_1
hG : Group G
inst✝ : Fintype G
tfae_1_to_2 : Group.IsNilpotent G → NormalizerCondition G
tfae_2_to_3 : NormalizerCondition G → ∀ (H : Subgroup G), IsCoatom H → Normal H
tfae_3_to_4 : (∀ (H : Subgroup G), IsCoatom H → Normal H) → ∀ (p : ℕ), Fact (Nat.Prime p) → ∀ (P : Sylow p G), Normal ↑P
tfae_4_to_5 :
  (∀ (p : ℕ), Fact (Nat.Prime p) → ∀ (P : Sylow p G), Normal ↑P) →
    Nonempty (((p : { x // x ∈ (card G).primeFactors }) → (P : Sylow (↑p) G) → ↥↑P) ≃* G)
⊢ List.TFAE
    [Group.IsNilpotent G, NormalizerCondition G, ∀ (H : Subgroup G), IsCoatom H → Normal H,
      ∀ (p : ℕ), Fact (Nat.Prime p) → ∀ (P : Sylow p G), Normal ↑P,
      Nonempty (((p : { x // x ∈ (card G).primeFactors }) → (P : Sylow (↑p) G) → ↥↑P) ≃* G)]

Next tactic:
tfae_have 5 → 1
/-- A finite group is nilpotent iff the normalizer condition holds, and iff all maximal groups are
normal and iff all sylow groups are normal and iff the group is the direct product of its sylow
groups. -/
theorem isNilpotent_of_finite_tFAE :
    List.TFAE
      [IsNilpotent G, NormalizerCondition G, ∀ H : Subgroup G, IsCoatom H → H.Normal,
        ∀ (p : ℕ) (_hp : Fact p.Prime) (P : Sylow p G), (↑P : Subgroup G).Normal,
        Nonempty
          ((∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Subgroup G)) ≃* G)] := by
  tfae_have 1 → 2
  · exact @normalizerCondition_of_isNilpotent _ _
  tfae_have 2 → 3
  · exact fun h H => NormalizerCondition.normal_of_coatom H h
  tfae_have 3 → 4
  · intro h p _ P; exact Sylow.normal_of_all_max_subgroups_normal h _
  tfae_have 4 → 5
  · exact fun h => Nonempty.intro (Sylow.directProductOfNormal fun {p hp hP} => h p hp hP)
  tfae_have 5 → 1
  · 

Proof state:
case tfae_5_to_1
G : Type u_1
hG : Group G
inst✝ : Fintype G
tfae_1_to_2 : Group.IsNilpotent G → NormalizerCondition G
tfae_2_to_3 : NormalizerCondition G → ∀ (H : Subgroup G), IsCoatom H → Normal H
tfae_3_to_4 : (∀ (H : Subgroup G), IsCoatom H → Normal H) → ∀ (p : ℕ), Fact (Nat.Prime p) → ∀ (P : Sylow p G), Normal ↑P
tfae_4_to_5 :
  (∀ (p : ℕ), Fact (Nat.Prime p) → ∀ (P : Sylow p G), Normal ↑P) →
    Nonempty (((p : { x // x ∈ (card G).primeFactors }) → (P : Sylow (↑p) G) → ↥↑P) ≃* G)
⊢ Nonempty (((p : { x // x ∈ (card G).primeFactors }) → (P : Sylow (↑p) G) → ↥↑P) ≃* G) → Group.IsNilpotent G

Next tactic:
rintro ⟨e⟩
/-- A finite group is nilpotent iff the normalizer condition holds, and iff all maximal groups are
normal and iff all sylow groups are normal and iff the group is the direct product of its sylow
groups. -/
theorem isNilpotent_of_finite_tFAE :
    List.TFAE
      [IsNilpotent G, NormalizerCondition G, ∀ H : Subgroup G, IsCoatom H → H.Normal,
        ∀ (p : ℕ) (_hp : Fact p.Prime) (P : Sylow p G), (↑P : Subgroup G).Normal,
        Nonempty
          ((∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Subgroup G)) ≃* G)] := by
  tfae_have 1 → 2
  · exact @normalizerCondition_of_isNilpotent _ _
  tfae_have 2 → 3
  · exact fun h H => NormalizerCondition.normal_of_coatom H h
  tfae_have 3 → 4
  · intro h p _ P; exact Sylow.normal_of_all_max_subgroups_normal h _
  tfae_have 4 → 5
  · exact fun h => Nonempty.intro (Sylow.directProductOfNormal fun {p hp hP} => h p hp hP)
  tfae_have 5 → 1
  · rintro ⟨e⟩; 

Proof state:
case tfae_5_to_1.intro
G : Type u_1
hG : Group G
inst✝ : Fintype G
tfae_1_to_2 : Group.IsNilpotent G → NormalizerCondition G
tfae_2_to_3 : NormalizerCondition G → ∀ (H : Subgroup G), IsCoatom H → Normal H
tfae_3_to_4 : (∀ (H : Subgroup G), IsCoatom H → Normal H) → ∀ (p : ℕ), Fact (Nat.Prime p) → ∀ (P : Sylow p G), Normal ↑P
tfae_4_to_5 :
  (∀ (p : ℕ), Fact (Nat.Prime p) → ∀ (P : Sylow p G), Normal ↑P) →
    Nonempty (((p : { x // x ∈ (card G).primeFactors }) → (P : Sylow (↑p) G) → ↥↑P) ≃* G)
e : ((p : { x // x ∈ (card G).primeFactors }) → (P : Sylow (↑p) G) → ↥↑P) ≃* G
⊢ Group.IsNilpotent G

Next tactic:
exact isNilpotent_of_product_of_sylow_group e
/-- A finite group is nilpotent iff the normalizer condition holds, and iff all maximal groups are
normal and iff all sylow groups are normal and iff the group is the direct product of its sylow
groups. -/
theorem isNilpotent_of_finite_tFAE :
    List.TFAE
      [IsNilpotent G, NormalizerCondition G, ∀ H : Subgroup G, IsCoatom H → H.Normal,
        ∀ (p : ℕ) (_hp : Fact p.Prime) (P : Sylow p G), (↑P : Subgroup G).Normal,
        Nonempty
          ((∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Subgroup G)) ≃* G)] := by
  tfae_have 1 → 2
  · exact @normalizerCondition_of_isNilpotent _ _
  tfae_have 2 → 3
  · exact fun h H => NormalizerCondition.normal_of_coatom H h
  tfae_have 3 → 4
  · intro h p _ P; exact Sylow.normal_of_all_max_subgroups_normal h _
  tfae_have 4 → 5
  · exact fun h => Nonempty.intro (Sylow.directProductOfNormal fun {p hp hP} => h p hp hP)
  tfae_have 5 → 1
  · rintro ⟨e⟩; exact isNilpotent_of_product_of_sylow_group e
  

Proof state:
G : Type u_1
hG : Group G
inst✝ : Fintype G
tfae_1_to_2 : Group.IsNilpotent G → NormalizerCondition G
tfae_2_to_3 : NormalizerCondition G → ∀ (H : Subgroup G), IsCoatom H → Normal H
tfae_3_to_4 : (∀ (H : Subgroup G), IsCoatom H → Normal H) → ∀ (p : ℕ), Fact (Nat.Prime p) → ∀ (P : Sylow p G), Normal ↑P
tfae_4_to_5 :
  (∀ (p : ℕ), Fact (Nat.Prime p) → ∀ (P : Sylow p G), Normal ↑P) →
    Nonempty (((p : { x // x ∈ (card G).primeFactors }) → (P : Sylow (↑p) G) → ↥↑P) ≃* G)
tfae_5_to_1 :
  Nonempty (((p : { x // x ∈ (card G).primeFactors }) → (P : Sylow (↑p) G) → ↥↑P) ≃* G) → Group.IsNilpotent G
⊢ List.TFAE
    [Group.IsNilpotent G, NormalizerCondition G, ∀ (H : Subgroup G), IsCoatom H → Normal H,
      ∀ (p : ℕ), Fact (Nat.Prime p) → ∀ (P : Sylow p G), Normal ↑P,
      Nonempty (((p : { x // x ∈ (card G).primeFactors }) → (P : Sylow (↑p) G) → ↥↑P) ≃* G)]

Next tactic:
tfae_finish
example : ¬ UnivLE.{u+1, u} := by
  

Proof state:
⊢ ¬UnivLE.{u + 1, u}

Next tactic:
simp only [Small_iff, not_forall, not_exists, not_nonempty_iff]
example : ¬ UnivLE.{u+1, u} := by
  simp only [Small_iff, not_forall, not_exists, not_nonempty_iff]
  

Proof state:
⊢ ∃ x, ∀ (x_1 : Type u), IsEmpty (x ≃ x_1)

Next tactic:
exact ⟨Type u, fun α => ⟨fun f => Function.not_surjective_Type.{u, u} f.symm f.symm.surjective⟩⟩
/-- The trace form of a Lie module is compatible with the action of the Lie algebra.

See also `LieModule.traceForm_apply_lie_apply'`. -/
lemma traceForm_apply_lie_apply (x y z : L) :
    traceForm R L M ⁅x, y⁆ z = traceForm R L M x ⁅y, z⁆ := by
  

Proof state:
R : Type u_1
K : Type u_2
L : Type u_3
M : Type u_4
inst✝¹³ : CommRing R
inst✝¹² : LieRing L
inst✝¹¹ : LieAlgebra R L
inst✝¹⁰ : AddCommGroup M
inst✝⁹ : Module R M
inst✝⁸ : LieRingModule L M
inst✝⁷ : LieModule R L M
inst✝⁶ : Module.Free R M
inst✝⁵ : Module.Finite R M
inst✝⁴ : Field K
inst✝³ : LieAlgebra K L
inst✝² : Module K M
inst✝¹ : LieModule K L M
inst✝ : FiniteDimensional K M
x y z : L
⊢ ((traceForm R L M) ⁅x, y⁆) z = ((traceForm R L M) x) ⁅y, z⁆

Next tactic:
calc traceForm R L M ⁅x, y⁆ z
      = trace R _ (φ ⁅x, y⁆ ∘ₗ φ z) := by simp only [traceForm_apply_apply]
    _ = trace R _ ((φ x * φ y - φ y * φ x) * φ z) := ?_
    _ = trace R _ (φ x * (φ y * φ z)) - trace R _ (φ y * (φ x * φ z)) := ?_
    _ = trace R _ (φ x * (φ y * φ z)) - trace R _ (φ x * (φ z * φ y)) := ?_
    _ = traceForm R L M x ⁅y, z⁆ := ?_
/-- The trace form of a Lie module is compatible with the action of the Lie algebra.

See also `LieModule.traceForm_apply_lie_apply'`. -/
lemma traceForm_apply_lie_apply (x y z : L) :
    traceForm R L M ⁅x, y⁆ z = traceForm R L M x ⁅y, z⁆ := by
  calc traceForm R L M ⁅x, y⁆ z
      = trace R _ (φ ⁅x, y⁆ ∘ₗ φ z) := by 

Proof state:
R : Type u_1
K : Type u_2
L : Type u_3
M : Type u_4
inst✝¹³ : CommRing R
inst✝¹² : LieRing L
inst✝¹¹ : LieAlgebra R L
inst✝¹⁰ : AddCommGroup M
inst✝⁹ : Module R M
inst✝⁸ : LieRingModule L M
inst✝⁷ : LieModule R L M
inst✝⁶ : Module.Free R M
inst✝⁵ : Module.Finite R M
inst✝⁴ : Field K
inst✝³ : LieAlgebra K L
inst✝² : Module K M
inst✝¹ : LieModule K L M
inst✝ : FiniteDimensional K M
x y z : L
⊢ ((traceForm R L M) ⁅x, y⁆) z = (trace R M) (φ ⁅x, y⁆ ∘ₗ φ z)

Next tactic:
simp only [traceForm_apply_apply]
/-- The trace form of a Lie module is compatible with the action of the Lie algebra.

See also `LieModule.traceForm_apply_lie_apply'`. -/
lemma traceForm_apply_lie_apply (x y z : L) :
    traceForm R L M ⁅x, y⁆ z = traceForm R L M x ⁅y, z⁆ := by
  calc traceForm R L M ⁅x, y⁆ z
      = trace R _ (φ ⁅x, y⁆ ∘ₗ φ z) := by simp only [traceForm_apply_apply]
    _ = trace R _ ((φ x * φ y - φ y * φ x) * φ z) := ?_
    _ = trace R _ (φ x * (φ y * φ z)) - trace R _ (φ y * (φ x * φ z)) := ?_
    _ = trace R _ (φ x * (φ y * φ z)) - trace R _ (φ x * (φ z * φ y)) := ?_
    _ = traceForm R L M x ⁅y, z⁆ := ?_
  · 

Proof state:
case calc_1
R : Type u_1
K : Type u_2
L : Type u_3
M : Type u_4
inst✝¹³ : CommRing R
inst✝¹² : LieRing L
inst✝¹¹ : LieAlgebra R L
inst✝¹⁰ : AddCommGroup M
inst✝⁹ : Module R M
inst✝⁸ : LieRingModule L M
inst✝⁷ : LieModule R L M
inst✝⁶ : Module.Free R M
inst✝⁵ : Module.Finite R M
inst✝⁴ : Field K
inst✝³ : LieAlgebra K L
inst✝² : Module K M
inst✝¹ : LieModule K L M
inst✝ : FiniteDimensional K M
x y z : L
⊢ (trace R M) (φ ⁅x, y⁆ ∘ₗ φ z) = (trace R M) ((φ x * φ y - φ y * φ x) * φ z)

Next tactic:
simp only [LieHom.map_lie, Ring.lie_def, ← LinearMap.mul_eq_comp]
/-- The trace form of a Lie module is compatible with the action of the Lie algebra.

See also `LieModule.traceForm_apply_lie_apply'`. -/
lemma traceForm_apply_lie_apply (x y z : L) :
    traceForm R L M ⁅x, y⁆ z = traceForm R L M x ⁅y, z⁆ := by
  calc traceForm R L M ⁅x, y⁆ z
      = trace R _ (φ ⁅x, y⁆ ∘ₗ φ z) := by simp only [traceForm_apply_apply]
    _ = trace R _ ((φ x * φ y - φ y * φ x) * φ z) := ?_
    _ = trace R _ (φ x * (φ y * φ z)) - trace R _ (φ y * (φ x * φ z)) := ?_
    _ = trace R _ (φ x * (φ y * φ z)) - trace R _ (φ x * (φ z * φ y)) := ?_
    _ = traceForm R L M x ⁅y, z⁆ := ?_
  · simp only [LieHom.map_lie, Ring.lie_def, ← LinearMap.mul_eq_comp]
  · 

Proof state:
case calc_2
R : Type u_1
K : Type u_2
L : Type u_3
M : Type u_4
inst✝¹³ : CommRing R
inst✝¹² : LieRing L
inst✝¹¹ : LieAlgebra R L
inst✝¹⁰ : AddCommGroup M
inst✝⁹ : Module R M
inst✝⁸ : LieRingModule L M
inst✝⁷ : LieModule R L M
inst✝⁶ : Module.Free R M
inst✝⁵ : Module.Finite R M
inst✝⁴ : Field K
inst✝³ : LieAlgebra K L
inst✝² : Module K M
inst✝¹ : LieModule K L M
inst✝ : FiniteDimensional K M
x y z : L
⊢ (trace R M) ((φ x * φ y - φ y * φ x) * φ z) = (trace R M) (φ x * (φ y * φ z)) - (trace R M) (φ y * (φ x * φ z))

Next tactic:
simp only [sub_mul, mul_sub, map_sub, mul_assoc]