Theorem: Subgroup.card_commutator_dvd_index_center_pow
File: Mathlib/GroupTheory/Schreier.lean

Proof steps:
state_before: G : Type u_1
inst✝¹ : Group G
H : Subgroup G
R S : Set G
inst✝ : Finite ↑(commutatorSet G)
⊢ Nat.card ↥(_root_.commutator G) ∣ (center G).index ^ ((center G).index * Nat.card ↑(commutatorSet G) + 1)
state_after: case pos
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
R S : Set G
inst✝ : Finite ↑(commutatorSet G)
hG : (center G).index = 0
⊢ Nat.card ↥(_root_.commutator G) ∣ (center G).index ^ ((center G).index * Nat.card ↑(commutatorSet G) + 1)

case neg
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
R S : Set G
inst✝ : Finite ↑(commutatorSet G)
hG : ¬(center G).index = 0
⊢ Nat.card ↥(_root_.commutator G) ∣ (center G).index ^ ((center G).index * Nat.card ↑(commutatorSet G) + 1)
tactic: by_cases hG : (center G).index = 0
state_before: case neg
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
R S : Set G
inst✝ : Finite ↑(commutatorSet G)
hG : ¬(center G).index = 0
⊢ Nat.card ↥(_root_.commutator G) ∣ (center G).index ^ ((center G).index * Nat.card ↑(commutatorSet G) + 1)
state_after: case neg
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
R S : Set G
inst✝ : Finite ↑(commutatorSet G)
hG : ¬(center G).index = 0
this : (center G).FiniteIndex
⊢ Nat.card ↥(_root_.commutator G) ∣ (center G).index ^ ((center G).index * Nat.card ↑(commutatorSet G) + 1)
tactic: haveI : FiniteIndex (center G) := ⟨hG⟩
state_before: case neg
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
R S : Set G
inst✝ : Finite ↑(commutatorSet G)
hG : ¬(center G).index = 0
this : (center G).FiniteIndex
⊢ Nat.card ↥(_root_.commutator G) ∣ (center G).index ^ ((center G).index * Nat.card ↑(commutatorSet G) + 1)
state_after: case neg
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
R S : Set G
inst✝ : Finite ↑(commutatorSet G)
hG : ¬(center G).index = 0
this : (center G).FiniteIndex
⊢ Nat.card ↥((center G).subgroupOf (_root_.commutator G)) * ((center G).subgroupOf (_root_.commutator G)).index ∣
    (center G).index ^ ((center G).index * Nat.card ↑(commutatorSet G)) * (center G).index
tactic: rw [← ((center G).subgroupOf (_root_.commutator G)).card_mul_index, pow_succ]
state_before: case neg
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
R S : Set G
inst✝ : Finite ↑(commutatorSet G)
hG : ¬(center G).index = 0
this : (center G).FiniteIndex
⊢ Nat.card ↥((center G).subgroupOf (_root_.commutator G)) * ((center G).subgroupOf (_root_.commutator G)).index ∣
    (center G).index ^ ((center G).index * Nat.card ↑(commutatorSet G)) * (center G).index
state_after: case neg
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
R S : Set G
inst✝ : Finite ↑(commutatorSet G)
hG : ¬(center G).index = 0
this : (center G).FiniteIndex
h1 : (center G).relindex (_root_.commutator G) ∣ (center G).index
⊢ Nat.card ↥((center G).subgroupOf (_root_.commutator G)) * ((center G).subgroupOf (_root_.commutator G)).index ∣
    (center G).index ^ ((center G).index * Nat.card ↑(commutatorSet G)) * (center G).index
tactic: have h1 := relindex_dvd_index_of_normal (center G) (_root_.commutator G)
state_before: case neg
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
R S : Set G
inst✝ : Finite ↑(commutatorSet G)
hG : ¬(center G).index = 0
this : (center G).FiniteIndex
h1 : (center G).relindex (_root_.commutator G) ∣ (center G).index
⊢ Nat.card ↥((center G).subgroupOf (_root_.commutator G)) * ((center G).subgroupOf (_root_.commutator G)).index ∣
    (center G).index ^ ((center G).index * Nat.card ↑(commutatorSet G)) * (center G).index
state_after: case neg
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
R S : Set G
inst✝ : Finite ↑(commutatorSet G)
hG : ¬(center G).index = 0
this : (center G).FiniteIndex
h1 : (center G).relindex (_root_.commutator G) ∣ (center G).index
⊢ Nat.card ↥((center G).subgroupOf (_root_.commutator G)) ∣
    (center G).index ^ ((center G).index * Nat.card ↑(commutatorSet G))
tactic: refine mul_dvd_mul ?_ h1
state_before: case neg
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
R S : Set G
inst✝ : Finite ↑(commutatorSet G)
hG : ¬(center G).index = 0
this : (center G).FiniteIndex
h1 : (center G).relindex (_root_.commutator G) ∣ (center G).index
⊢ Nat.card ↥((center G).subgroupOf (_root_.commutator G)) ∣
    (center G).index ^ ((center G).index * Nat.card ↑(commutatorSet G))
state_after: case neg
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
R S : Set G
inst✝ : Finite ↑(commutatorSet G)
hG : ¬(center G).index = 0
this✝ : (center G).FiniteIndex
h1 : (center G).relindex (_root_.commutator G) ∣ (center G).index
this : ((center G).subgroupOf (_root_.commutator G)).FiniteIndex
⊢ Nat.card ↥((center G).subgroupOf (_root_.commutator G)) ∣
    (center G).index ^ ((center G).index * Nat.card ↑(commutatorSet G))
tactic: haveI : FiniteIndex ((center G).subgroupOf (_root_.commutator G)) :=
    ⟨ne_zero_of_dvd_ne_zero hG h1⟩
state_before: case neg
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
R S : Set G
inst✝ : Finite ↑(commutatorSet G)
hG : ¬(center G).index = 0
this✝ : (center G).FiniteIndex
h1 : (center G).relindex (_root_.commutator G) ∣ (center G).index
this : ((center G).subgroupOf (_root_.commutator G)).FiniteIndex
⊢ Nat.card ↥((center G).subgroupOf (_root_.commutator G)) ∣
    (center G).index ^ ((center G).index * Nat.card ↑(commutatorSet G))
state_after: case neg
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
R S : Set G
inst✝ : Finite ↑(commutatorSet G)
hG : ¬(center G).index = 0
this✝ : (center G).FiniteIndex
h1 : (center G).relindex (_root_.commutator G) ∣ (center G).index
this : ((center G).subgroupOf (_root_.commutator G)).FiniteIndex
h2 :
  Group.rank ↥((center G).subgroupOf (_root_.commutator G)) ≤
    ((center G).subgroupOf (_root_.commutator G)).index * Group.rank ↥(_root_.commutator G)
⊢ Nat.card ↥((center G).subgroupOf (_root_.commutator G)) ∣
    (center G).index ^ ((center G).index * Nat.card ↑(commutatorSet G))
tactic: have h2 := rank_le_index_mul_rank ((center G).subgroupOf (_root_.commutator G))
state_before: case neg
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
R S : Set G
inst✝ : Finite ↑(commutatorSet G)
hG : ¬(center G).index = 0
this✝ : (center G).FiniteIndex
h1 : (center G).relindex (_root_.commutator G) ∣ (center G).index
this : ((center G).subgroupOf (_root_.commutator G)).FiniteIndex
h2 :
  Group.rank ↥((center G).subgroupOf (_root_.commutator G)) ≤
    ((center G).subgroupOf (_root_.commutator G)).index * Group.rank ↥(_root_.commutator G)
⊢ Nat.card ↥((center G).subgroupOf (_root_.commutator G)) ∣
    (center G).index ^ ((center G).index * Nat.card ↑(commutatorSet G))
state_after: case neg
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
R S : Set G
inst✝ : Finite ↑(commutatorSet G)
hG : ¬(center G).index = 0
this✝ : (center G).FiniteIndex
h1 : (center G).relindex (_root_.commutator G) ∣ (center G).index
this : ((center G).subgroupOf (_root_.commutator G)).FiniteIndex
h2 :
  Group.rank ↥((center G).subgroupOf (_root_.commutator G)) ≤
    ((center G).subgroupOf (_root_.commutator G)).index * Group.rank ↥(_root_.commutator G)
h3 :
  (center G).relindex (_root_.commutator G) * Group.rank ↥(_root_.commutator G) ≤
    (center G).index * Nat.card ↑(commutatorSet G)
⊢ Nat.card ↥((center G).subgroupOf (_root_.commutator G)) ∣
    (center G).index ^ ((center G).index * Nat.card ↑(commutatorSet G))
tactic: have h3 := Nat.mul_le_mul (Nat.le_of_dvd (Nat.pos_of_ne_zero hG) h1) (rank_commutator_le_card G)
state_before: case neg
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
R S : Set G
inst✝ : Finite ↑(commutatorSet G)
hG : ¬(center G).index = 0
this✝ : (center G).FiniteIndex
h1 : (center G).relindex (_root_.commutator G) ∣ (center G).index
this : ((center G).subgroupOf (_root_.commutator G)).FiniteIndex
h2 :
  Group.rank ↥((center G).subgroupOf (_root_.commutator G)) ≤
    ((center G).subgroupOf (_root_.commutator G)).index * Group.rank ↥(_root_.commutator G)
h3 :
  (center G).relindex (_root_.commutator G) * Group.rank ↥(_root_.commutator G) ≤
    (center G).index * Nat.card ↑(commutatorSet G)
⊢ Nat.card ↥((center G).subgroupOf (_root_.commutator G)) ∣
    (center G).index ^ ((center G).index * Nat.card ↑(commutatorSet G))
state_after: case neg
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
R S : Set G
inst✝ : Finite ↑(commutatorSet G)
hG : ¬(center G).index = 0
this✝ : (center G).FiniteIndex
h1 : (center G).relindex (_root_.commutator G) ∣ (center G).index
this : ((center G).subgroupOf (_root_.commutator G)).FiniteIndex
h2 :
  Group.rank ↥((center G).subgroupOf (_root_.commutator G)) ≤
    ((center G).subgroupOf (_root_.commutator G)).index * Group.rank ↥(_root_.commutator G)
h3 :
  (center G).relindex (_root_.commutator G) * Group.rank ↥(_root_.commutator G) ≤
    (center G).index * Nat.card ↑(commutatorSet G)
⊢ Nat.card ↥((center G).subgroupOf (_root_.commutator G)) ∣
    (center G).index ^ Group.rank ↥((center G).subgroupOf (_root_.commutator G))
tactic: refine dvd_trans ?_ (pow_dvd_pow (center G).index (h2.trans h3))
state_before: case neg
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
R S : Set G
inst✝ : Finite ↑(commutatorSet G)
hG : ¬(center G).index = 0
this✝ : (center G).FiniteIndex
h1 : (center G).relindex (_root_.commutator G) ∣ (center G).index
this : ((center G).subgroupOf (_root_.commutator G)).FiniteIndex
h2 :
  Group.rank ↥((center G).subgroupOf (_root_.commutator G)) ≤
    ((center G).subgroupOf (_root_.commutator G)).index * Group.rank ↥(_root_.commutator G)
h3 :
  (center G).relindex (_root_.commutator G) * Group.rank ↥(_root_.commutator G) ≤
    (center G).index * Nat.card ↑(commutatorSet G)
⊢ Nat.card ↥((center G).subgroupOf (_root_.commutator G)) ∣
    (center G).index ^ Group.rank ↥((center G).subgroupOf (_root_.commutator G))
state_after: case neg.hG
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
R S : Set G
inst✝ : Finite ↑(commutatorSet G)
hG : ¬(center G).index = 0
this✝ : (center G).FiniteIndex
h1 : (center G).relindex (_root_.commutator G) ∣ (center G).index
this : ((center G).subgroupOf (_root_.commutator G)).FiniteIndex
h2 :
  Group.rank ↥((center G).subgroupOf (_root_.commutator G)) ≤
    ((center G).subgroupOf (_root_.commutator G)).index * Group.rank ↥(_root_.commutator G)
h3 :
  (center G).relindex (_root_.commutator G) * Group.rank ↥(_root_.commutator G) ≤
    (center G).index * Nat.card ↑(commutatorSet G)
⊢ ∀ (g : ↥((center G).subgroupOf (_root_.commutator G))), g ^ (center G).index = 1
tactic: apply card_dvd_exponent_pow_rank'
state_before: case neg.hG
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
R S : Set G
inst✝ : Finite ↑(commutatorSet G)
hG : ¬(center G).index = 0
this✝ : (center G).FiniteIndex
h1 : (center G).relindex (_root_.commutator G) ∣ (center G).index
this : ((center G).subgroupOf (_root_.commutator G)).FiniteIndex
h2 :
  Group.rank ↥((center G).subgroupOf (_root_.commutator G)) ≤
    ((center G).subgroupOf (_root_.commutator G)).index * Group.rank ↥(_root_.commutator G)
h3 :
  (center G).relindex (_root_.commutator G) * Group.rank ↥(_root_.commutator G) ≤
    (center G).index * Nat.card ↑(commutatorSet G)
⊢ ∀ (g : ↥((center G).subgroupOf (_root_.commutator G))), g ^ (center G).index = 1
state_after: case neg.hG
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
R S : Set G
inst✝ : Finite ↑(commutatorSet G)
hG : ¬(center G).index = 0
this✝ : (center G).FiniteIndex
h1 : (center G).relindex (_root_.commutator G) ∣ (center G).index
this : ((center G).subgroupOf (_root_.commutator G)).FiniteIndex
h2 :
  Group.rank ↥((center G).subgroupOf (_root_.commutator G)) ≤
    ((center G).subgroupOf (_root_.commutator G)).index * Group.rank ↥(_root_.commutator G)
h3 :
  (center G).relindex (_root_.commutator G) * Group.rank ↥(_root_.commutator G) ≤
    (center G).index * Nat.card ↑(commutatorSet G)
g : ↥((center G).subgroupOf (_root_.commutator G))
⊢ g ^ (center G).index = 1
tactic: intro g
state_before: case neg.hG
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
R S : Set G
inst✝ : Finite ↑(commutatorSet G)
hG : ¬(center G).index = 0
this✝ : (center G).FiniteIndex
h1 : (center G).relindex (_root_.commutator G) ∣ (center G).index
this : ((center G).subgroupOf (_root_.commutator G)).FiniteIndex
h2 :
  Group.rank ↥((center G).subgroupOf (_root_.commutator G)) ≤
    ((center G).subgroupOf (_root_.commutator G)).index * Group.rank ↥(_root_.commutator G)
h3 :
  (center G).relindex (_root_.commutator G) * Group.rank ↥(_root_.commutator G) ≤
    (center G).index * Nat.card ↑(commutatorSet G)
g : ↥((center G).subgroupOf (_root_.commutator G))
⊢ g ^ (center G).index = 1
state_after: case neg.hG
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
R S : Set G
inst✝ : Finite ↑(commutatorSet G)
hG : ¬(center G).index = 0
this✝¹ : (center G).FiniteIndex
h1 : (center G).relindex (_root_.commutator G) ∣ (center G).index
this✝ : ((center G).subgroupOf (_root_.commutator G)).FiniteIndex
h2 :
  Group.rank ↥((center G).subgroupOf (_root_.commutator G)) ≤
    ((center G).subgroupOf (_root_.commutator G)).index * Group.rank ↥(_root_.commutator G)
h3 :
  (center G).relindex (_root_.commutator G) * Group.rank ↥(_root_.commutator G) ≤
    (center G).index * Nat.card ↑(commutatorSet G)
g : ↥((center G).subgroupOf (_root_.commutator G))
this : ↑↑g ∈ (MonoidHom.transferCenterPow G).ker
⊢ g ^ (center G).index = 1
tactic: have := Abelianization.commutator_subset_ker (MonoidHom.transferCenterPow G) g.1.2
state_before: case neg.hG
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
R S : Set G
inst✝ : Finite ↑(commutatorSet G)
hG : ¬(center G).index = 0
this✝¹ : (center G).FiniteIndex
h1 : (center G).relindex (_root_.commutator G) ∣ (center G).index
this✝ : ((center G).subgroupOf (_root_.commutator G)).FiniteIndex
h2 :
  Group.rank ↥((center G).subgroupOf (_root_.commutator G)) ≤
    ((center G).subgroupOf (_root_.commutator G)).index * Group.rank ↥(_root_.commutator G)
h3 :
  (center G).relindex (_root_.commutator G) * Group.rank ↥(_root_.commutator G) ≤
    (center G).index * Nat.card ↑(commutatorSet G)
g : ↥((center G).subgroupOf (_root_.commutator G))
this : ↑↑g ∈ (MonoidHom.transferCenterPow G).ker
⊢ g ^ (center G).index = 1
state_after: no goals
tactic: simpa only [MonoidHom.mem_ker, Subtype.ext_iff] using this