l3lab/ntp-mathlib huggingface dataset:
theorem exists_finset_card_le_mul [FiniteIndex H] {S : Finset G} (hS : closure (S : Set G) = ⊤) :
    ∃ T : Finset H, T.card ≤ H.index * S.card ∧ closure (T : Set H) = ⊤ := by
  letI := H.fintypeQuotientOfFiniteIndex
  haveI : DecidableEq G := Classical.decEq G
  obtain ⟨R₀, hR, hR1⟩ := H.exists_right_transversal 1
  haveI : Fintype R₀ := Fintype.ofEquiv _ (toEquiv hR)
  let R : Finset G := Set.toFinset R₀
  replace hR : (R : Set G) ∈ rightTransversals (H : Set G) := by rwa [Set.coe_toFinset]
  replace hR1 : (1 : G) ∈ R := by rwa [Set.mem_toFinset]
  refine' ⟨_, _, closure_mul_image_eq_top' hR hR1 hS⟩
  

Proof state:
case intro.intro
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
R✝ S✝ : Set G
inst✝ : FiniteIndex H
S : Finset G
hS : closure ↑S = ⊤
this✝¹ : Fintype (G ⧸ H) := fintypeQuotientOfFiniteIndex H
this✝ : DecidableEq G
R₀ : Set G
this : Fintype ↑R₀
R : Finset G := Set.toFinset R₀
hR : ↑R ∈ rightTransversals ↑H
hR1 : 1 ∈ R
⊢ Finset.card
      (Finset.image (fun g => { val := g * (↑(toFun hR g))⁻¹, property := (_ : g * (↑(toFun hR g))⁻¹ ∈ H) }) (R * S)) ≤
    index H * Finset.card S

Next tactic:
calc
    _ ≤ (R * S).card := Finset.card_image_le
    _ ≤ (R ×ˢ S).card := Finset.card_image_le
    _ = R.card * S.card := (R.card_product S)
    _ = H.index * S.card := congr_arg (· * S.card) ?_
theorem exists_finset_card_le_mul [FiniteIndex H] {S : Finset G} (hS : closure (S : Set G) = ⊤) :
    ∃ T : Finset H, T.card ≤ H.index * S.card ∧ closure (T : Set H) = ⊤ := by
  letI := H.fintypeQuotientOfFiniteIndex
  haveI : DecidableEq G := Classical.decEq G
  obtain ⟨R₀, hR, hR1⟩ := H.exists_right_transversal 1
  haveI : Fintype R₀ := Fintype.ofEquiv _ (toEquiv hR)
  let R : Finset G := Set.toFinset R₀
  replace hR : (R : Set G) ∈ rightTransversals (H : Set G) := by rwa [Set.coe_toFinset]
  replace hR1 : (1 : G) ∈ R := by rwa [Set.mem_toFinset]
  refine' ⟨_, _, closure_mul_image_eq_top' hR hR1 hS⟩
  calc
    _ ≤ (R * S).card := Finset.card_image_le
    _ ≤ (R ×ˢ S).card := Finset.card_image_le
    _ = R.card * S.card := (R.card_product S)
    _ = H.index * S.card := congr_arg (· * S.card) ?_
  

Proof state:
case intro.intro
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
R✝ S✝ : Set G
inst✝ : FiniteIndex H
S : Finset G
hS : closure ↑S = ⊤
this✝¹ : Fintype (G ⧸ H) := fintypeQuotientOfFiniteIndex H
this✝ : DecidableEq G
R₀ : Set G
this : Fintype ↑R₀
R : Finset G := Set.toFinset R₀
hR : ↑R ∈ rightTransversals ↑H
hR1 : 1 ∈ R
⊢ Finset.card R = index H

Next tactic:
calc
    R.card = Fintype.card R := (Fintype.card_coe R).symm
    _ = _ := (Fintype.card_congr (toEquiv hR)).symm
    _ = Fintype.card (G ⧸ H) := (QuotientGroup.card_quotient_rightRel H)
    _ = H.index := H.index_eq_card.symm
/-- **Schreier's Lemma**: A finite index subgroup of a finitely generated
  group is finitely generated. -/
instance fg_of_index_ne_zero [hG : Group.FG G] [FiniteIndex H] : Group.FG H := by
  

Proof state:
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
R S : Set G
hG : Group.FG G
inst✝ : FiniteIndex H
⊢ Group.FG ↥H

Next tactic:
obtain ⟨S, hS⟩ := hG.1
/-- **Schreier's Lemma**: A finite index subgroup of a finitely generated
  group is finitely generated. -/
instance fg_of_index_ne_zero [hG : Group.FG G] [FiniteIndex H] : Group.FG H := by
  obtain ⟨S, hS⟩ := hG.1
  

Proof state:
case intro
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
R S✝ : Set G
hG : Group.FG G
inst✝ : FiniteIndex H
S : Finset G
hS : closure ↑S = ⊤
⊢ Group.FG ↥H

Next tactic:
obtain ⟨T, -, hT⟩ := exists_finset_card_le_mul H hS
/-- **Schreier's Lemma**: A finite index subgroup of a finitely generated
  group is finitely generated. -/
instance fg_of_index_ne_zero [hG : Group.FG G] [FiniteIndex H] : Group.FG H := by
  obtain ⟨S, hS⟩ := hG.1
  obtain ⟨T, -, hT⟩ := exists_finset_card_le_mul H hS
  

Proof state:
case intro.intro.intro
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
R S✝ : Set G
hG : Group.FG G
inst✝ : FiniteIndex H
S : Finset G
hS : closure ↑S = ⊤
T : Finset ↥H
hT : closure ↑T = ⊤
⊢ Group.FG ↥H

Next tactic:
exact ⟨⟨T, hT⟩⟩
theorem rank_le_index_mul_rank [hG : Group.FG G] [FiniteIndex H] :
    Group.rank H ≤ H.index * Group.rank G := by
  

Proof state:
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
R S : Set G
hG : Group.FG G
inst✝ : FiniteIndex H
⊢ Group.rank ↥H ≤ index H * Group.rank G

Next tactic:
haveI := H.fg_of_index_ne_zero
theorem rank_le_index_mul_rank [hG : Group.FG G] [FiniteIndex H] :
    Group.rank H ≤ H.index * Group.rank G := by
  haveI := H.fg_of_index_ne_zero
  

Proof state:
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
R S : Set G
hG : Group.FG G
inst✝ : FiniteIndex H
this : Group.FG ↥H
⊢ Group.rank ↥H ≤ index H * Group.rank G

Next tactic:
obtain ⟨S, hS₀, hS⟩ := Group.rank_spec G
theorem rank_le_index_mul_rank [hG : Group.FG G] [FiniteIndex H] :
    Group.rank H ≤ H.index * Group.rank G := by
  haveI := H.fg_of_index_ne_zero
  obtain ⟨S, hS₀, hS⟩ := Group.rank_spec G
  

Proof state:
case intro.intro
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
R S✝ : Set G
hG : Group.FG G
inst✝ : FiniteIndex H
this : Group.FG ↥H
S : Finset G
hS₀ : Finset.card S = Group.rank G
hS : closure ↑S = ⊤
⊢ Group.rank ↥H ≤ index H * Group.rank G

Next tactic:
obtain ⟨T, hT₀, hT⟩ := exists_finset_card_le_mul H hS
theorem rank_le_index_mul_rank [hG : Group.FG G] [FiniteIndex H] :
    Group.rank H ≤ H.index * Group.rank G := by
  haveI := H.fg_of_index_ne_zero
  obtain ⟨S, hS₀, hS⟩ := Group.rank_spec G
  obtain ⟨T, hT₀, hT⟩ := exists_finset_card_le_mul H hS
  

Proof state:
case intro.intro.intro.intro
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
R S✝ : Set G
hG : Group.FG G
inst✝ : FiniteIndex H
this : Group.FG ↥H
S : Finset G
hS₀ : Finset.card S = Group.rank G
hS : closure ↑S = ⊤
T : Finset ↥H
hT₀ : Finset.card T ≤ index H * Finset.card S
hT : closure ↑T = ⊤
⊢ Group.rank ↥H ≤ index H * Group.rank G

Next tactic:
calc
    Group.rank H ≤ T.card := Group.rank_le H hT
    _ ≤ H.index * S.card := hT₀
    _ = H.index * Group.rank G := congr_arg (H.index * ·) hS₀
/-- If `G` has `n` commutators `[g₁, g₂]`, then `|G'| ∣ [G : Z(G)] ^ ([G : Z(G)] * n + 1)`,
where `G'` denotes the commutator of `G`. -/
theorem card_commutator_dvd_index_center_pow [Finite (commutatorSet G)] :
    Nat.card (_root_.commutator G) ∣
      (center G).index ^ ((center G).index * Nat.card (commutatorSet G) + 1) := by
  -- First handle the case when `Z(G)` has infinite index and `[G : Z(G)]` is defined to be `0`
  

Proof state:
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
R S : Set G
inst✝ : Finite ↑(commutatorSet G)
⊢ Nat.card ↥(_root_.commutator G) ∣ index (center G) ^ (index (center G) * Nat.card ↑(commutatorSet G) + 1)

Next tactic:
by_cases hG : (center G).index = 0
/-- If `G` has `n` commutators `[g₁, g₂]`, then `|G'| ∣ [G : Z(G)] ^ ([G : Z(G)] * n + 1)`,
where `G'` denotes the commutator of `G`. -/
theorem card_commutator_dvd_index_center_pow [Finite (commutatorSet G)] :
    Nat.card (_root_.commutator G) ∣
      (center G).index ^ ((center G).index * Nat.card (commutatorSet G) + 1) := by
  -- First handle the case when `Z(G)` has infinite index and `[G : Z(G)]` is defined to be `0`
  by_cases hG : (center G).index = 0
  · 

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

Next tactic:
simp_rw [hG, zero_mul, zero_add, pow_one, dvd_zero]
/-- If `G` has `n` commutators `[g₁, g₂]`, then `|G'| ∣ [G : Z(G)] ^ ([G : Z(G)] * n + 1)`,
where `G'` denotes the commutator of `G`. -/
theorem card_commutator_dvd_index_center_pow [Finite (commutatorSet G)] :
    Nat.card (_root_.commutator G) ∣
      (center G).index ^ ((center G).index * Nat.card (commutatorSet G) + 1) := by
  -- First handle the case when `Z(G)` has infinite index and `[G : Z(G)]` is defined to be `0`
  by_cases hG : (center G).index = 0
  · simp_rw [hG, zero_mul, zero_add, pow_one, dvd_zero]
  

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

Next tactic:
haveI : FiniteIndex (center G) := ⟨hG⟩
/-- If `G` has `n` commutators `[g₁, g₂]`, then `|G'| ∣ [G : Z(G)] ^ ([G : Z(G)] * n + 1)`,
where `G'` denotes the commutator of `G`. -/
theorem card_commutator_dvd_index_center_pow [Finite (commutatorSet G)] :
    Nat.card (_root_.commutator G) ∣
      (center G).index ^ ((center G).index * Nat.card (commutatorSet G) + 1) := by
  -- First handle the case when `Z(G)` has infinite index and `[G : Z(G)]` is defined to be `0`
  by_cases hG : (center G).index = 0
  · simp_rw [hG, zero_mul, zero_add, pow_one, dvd_zero]
  haveI : FiniteIndex (center G) := ⟨hG⟩
  -- Rewrite as `|Z(G) ∩ G'| * [G' : Z(G) ∩ G'] ∣ [G : Z(G)] ^ ([G : Z(G)] * n) * [G : Z(G)]`
  

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

Next tactic:
rw [← ((center G).subgroupOf (_root_.commutator G)).card_mul_index, pow_succ']
/-- If `G` has `n` commutators `[g₁, g₂]`, then `|G'| ∣ [G : Z(G)] ^ ([G : Z(G)] * n + 1)`,
where `G'` denotes the commutator of `G`. -/
theorem card_commutator_dvd_index_center_pow [Finite (commutatorSet G)] :
    Nat.card (_root_.commutator G) ∣
      (center G).index ^ ((center G).index * Nat.card (commutatorSet G) + 1) := by
  -- First handle the case when `Z(G)` has infinite index and `[G : Z(G)]` is defined to be `0`
  by_cases hG : (center G).index = 0
  · simp_rw [hG, zero_mul, zero_add, pow_one, dvd_zero]
  haveI : FiniteIndex (center G) := ⟨hG⟩
  -- Rewrite as `|Z(G) ∩ G'| * [G' : Z(G) ∩ G'] ∣ [G : Z(G)] ^ ([G : Z(G)] * n) * [G : Z(G)]`
  rw [← ((center G).subgroupOf (_root_.commutator G)).card_mul_index, pow_succ']
  -- We have `h1 : [G' : Z(G) ∩ G'] ∣ [G : Z(G)]`
  

Proof state:
case neg
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
R S : Set G
inst✝ : Finite ↑(commutatorSet G)
hG : ¬index (center G) = 0
this : FiniteIndex (center G)
⊢ Nat.card ↥(subgroupOf (center G) (_root_.commutator G)) * index (subgroupOf (center G) (_root_.commutator G)) ∣
    index (center G) ^ (index (center G) * Nat.card ↑(commutatorSet G)) * index (center G)

Next tactic:
have h1 := relindex_dvd_index_of_normal (center G) (_root_.commutator G)
/-- If `G` has `n` commutators `[g₁, g₂]`, then `|G'| ∣ [G : Z(G)] ^ ([G : Z(G)] * n + 1)`,
where `G'` denotes the commutator of `G`. -/
theorem card_commutator_dvd_index_center_pow [Finite (commutatorSet G)] :
    Nat.card (_root_.commutator G) ∣
      (center G).index ^ ((center G).index * Nat.card (commutatorSet G) + 1) := by
  -- First handle the case when `Z(G)` has infinite index and `[G : Z(G)]` is defined to be `0`
  by_cases hG : (center G).index = 0
  · simp_rw [hG, zero_mul, zero_add, pow_one, dvd_zero]
  haveI : FiniteIndex (center G) := ⟨hG⟩
  -- Rewrite as `|Z(G) ∩ G'| * [G' : Z(G) ∩ G'] ∣ [G : Z(G)] ^ ([G : Z(G)] * n) * [G : Z(G)]`
  rw [← ((center G).subgroupOf (_root_.commutator G)).card_mul_index, pow_succ']
  -- We have `h1 : [G' : Z(G) ∩ G'] ∣ [G : Z(G)]`
  have h1 := relindex_dvd_index_of_normal (center G) (_root_.commutator G)
  -- So we can reduce to proving `|Z(G) ∩ G'| ∣ [G : Z(G)] ^ ([G : Z(G)] * n)`
  

Proof state:
case neg
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
R S : Set G
inst✝ : Finite ↑(commutatorSet G)
hG : ¬index (center G) = 0
this : FiniteIndex (center G)
h1 : relindex (center G) (_root_.commutator G) ∣ index (center G)
⊢ Nat.card ↥(subgroupOf (center G) (_root_.commutator G)) * index (subgroupOf (center G) (_root_.commutator G)) ∣
    index (center G) ^ (index (center G) * Nat.card ↑(commutatorSet G)) * index (center G)

Next tactic:
refine' mul_dvd_mul _ h1
/-- If `G` has `n` commutators `[g₁, g₂]`, then `|G'| ∣ [G : Z(G)] ^ ([G : Z(G)] * n + 1)`,
where `G'` denotes the commutator of `G`. -/
theorem card_commutator_dvd_index_center_pow [Finite (commutatorSet G)] :
    Nat.card (_root_.commutator G) ∣
      (center G).index ^ ((center G).index * Nat.card (commutatorSet G) + 1) := by
  -- First handle the case when `Z(G)` has infinite index and `[G : Z(G)]` is defined to be `0`
  by_cases hG : (center G).index = 0
  · simp_rw [hG, zero_mul, zero_add, pow_one, dvd_zero]
  haveI : FiniteIndex (center G) := ⟨hG⟩
  -- Rewrite as `|Z(G) ∩ G'| * [G' : Z(G) ∩ G'] ∣ [G : Z(G)] ^ ([G : Z(G)] * n) * [G : Z(G)]`
  rw [← ((center G).subgroupOf (_root_.commutator G)).card_mul_index, pow_succ']
  -- We have `h1 : [G' : Z(G) ∩ G'] ∣ [G : Z(G)]`
  have h1 := relindex_dvd_index_of_normal (center G) (_root_.commutator G)
  -- So we can reduce to proving `|Z(G) ∩ G'| ∣ [G : Z(G)] ^ ([G : Z(G)] * n)`
  refine' mul_dvd_mul _ h1
  -- We know that `[G' : Z(G) ∩ G'] < ∞` by `h1` and `hG`
  

Proof state:
case neg
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
R S : Set G
inst✝ : Finite ↑(commutatorSet G)
hG : ¬index (center G) = 0
this : FiniteIndex (center G)
h1 : relindex (center G) (_root_.commutator G) ∣ index (center G)
⊢ Nat.card ↥(subgroupOf (center G) (_root_.commutator G)) ∣
    index (center G) ^ (index (center G) * Nat.card ↑(commutatorSet G))

Next tactic:
haveI : FiniteIndex ((center G).subgroupOf (_root_.commutator G)) :=
    ⟨ne_zero_of_dvd_ne_zero hG h1⟩
/-- If `G` has `n` commutators `[g₁, g₂]`, then `|G'| ∣ [G : Z(G)] ^ ([G : Z(G)] * n + 1)`,
where `G'` denotes the commutator of `G`. -/
theorem card_commutator_dvd_index_center_pow [Finite (commutatorSet G)] :
    Nat.card (_root_.commutator G) ∣
      (center G).index ^ ((center G).index * Nat.card (commutatorSet G) + 1) := by
  -- First handle the case when `Z(G)` has infinite index and `[G : Z(G)]` is defined to be `0`
  by_cases hG : (center G).index = 0
  · simp_rw [hG, zero_mul, zero_add, pow_one, dvd_zero]
  haveI : FiniteIndex (center G) := ⟨hG⟩
  -- Rewrite as `|Z(G) ∩ G'| * [G' : Z(G) ∩ G'] ∣ [G : Z(G)] ^ ([G : Z(G)] * n) * [G : Z(G)]`
  rw [← ((center G).subgroupOf (_root_.commutator G)).card_mul_index, pow_succ']
  -- We have `h1 : [G' : Z(G) ∩ G'] ∣ [G : Z(G)]`
  have h1 := relindex_dvd_index_of_normal (center G) (_root_.commutator G)
  -- So we can reduce to proving `|Z(G) ∩ G'| ∣ [G : Z(G)] ^ ([G : Z(G)] * n)`
  refine' mul_dvd_mul _ h1
  -- We know that `[G' : Z(G) ∩ G'] < ∞` by `h1` and `hG`
  haveI : FiniteIndex ((center G).subgroupOf (_root_.commutator G)) :=
    ⟨ne_zero_of_dvd_ne_zero hG h1⟩
  -- We have `h2 : rank (Z(G) ∩ G') ≤ [G' : Z(G) ∩ G'] * rank G'` by Schreier's lemma
  

Proof state:
case neg
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
R S : Set G
inst✝ : Finite ↑(commutatorSet G)
hG : ¬index (center G) = 0
this✝ : FiniteIndex (center G)
h1 : relindex (center G) (_root_.commutator G) ∣ index (center G)
this : FiniteIndex (subgroupOf (center G) (_root_.commutator G))
⊢ Nat.card ↥(subgroupOf (center G) (_root_.commutator G)) ∣
    index (center G) ^ (index (center G) * Nat.card ↑(commutatorSet G))

Next tactic:
have h2 := rank_le_index_mul_rank ((center G).subgroupOf (_root_.commutator G))
/-- If `G` has `n` commutators `[g₁, g₂]`, then `|G'| ∣ [G : Z(G)] ^ ([G : Z(G)] * n + 1)`,
where `G'` denotes the commutator of `G`. -/
theorem card_commutator_dvd_index_center_pow [Finite (commutatorSet G)] :
    Nat.card (_root_.commutator G) ∣
      (center G).index ^ ((center G).index * Nat.card (commutatorSet G) + 1) := by
  -- First handle the case when `Z(G)` has infinite index and `[G : Z(G)]` is defined to be `0`
  by_cases hG : (center G).index = 0
  · simp_rw [hG, zero_mul, zero_add, pow_one, dvd_zero]
  haveI : FiniteIndex (center G) := ⟨hG⟩
  -- Rewrite as `|Z(G) ∩ G'| * [G' : Z(G) ∩ G'] ∣ [G : Z(G)] ^ ([G : Z(G)] * n) * [G : Z(G)]`
  rw [← ((center G).subgroupOf (_root_.commutator G)).card_mul_index, pow_succ']
  -- We have `h1 : [G' : Z(G) ∩ G'] ∣ [G : Z(G)]`
  have h1 := relindex_dvd_index_of_normal (center G) (_root_.commutator G)
  -- So we can reduce to proving `|Z(G) ∩ G'| ∣ [G : Z(G)] ^ ([G : Z(G)] * n)`
  refine' mul_dvd_mul _ h1
  -- We know that `[G' : Z(G) ∩ G'] < ∞` by `h1` and `hG`
  haveI : FiniteIndex ((center G).subgroupOf (_root_.commutator G)) :=
    ⟨ne_zero_of_dvd_ne_zero hG h1⟩
  -- We have `h2 : rank (Z(G) ∩ G') ≤ [G' : Z(G) ∩ G'] * rank G'` by Schreier's lemma
  have h2 := rank_le_index_mul_rank ((center G).subgroupOf (_root_.commutator G))
  -- We have `h3 : [G' : Z(G) ∩ G'] * rank G' ≤ [G : Z(G)] * n` by `h1` and `rank G' ≤ n`
  

Proof state:
case neg
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
R S : Set G
inst✝ : Finite ↑(commutatorSet G)
hG : ¬index (center G) = 0
this✝ : FiniteIndex (center G)
h1 : relindex (center G) (_root_.commutator G) ∣ index (center G)
this : FiniteIndex (subgroupOf (center G) (_root_.commutator G))
h2 :
  Group.rank ↥(subgroupOf (center G) (_root_.commutator G)) ≤
    index (subgroupOf (center G) (_root_.commutator G)) * Group.rank ↥(_root_.commutator G)
⊢ Nat.card ↥(subgroupOf (center G) (_root_.commutator G)) ∣
    index (center G) ^ (index (center G) * Nat.card ↑(commutatorSet G))

Next tactic:
have h3 := Nat.mul_le_mul (Nat.le_of_dvd (Nat.pos_of_ne_zero hG) h1) (rank_commutator_le_card G)
/-- If `G` has `n` commutators `[g₁, g₂]`, then `|G'| ∣ [G : Z(G)] ^ ([G : Z(G)] * n + 1)`,
where `G'` denotes the commutator of `G`. -/
theorem card_commutator_dvd_index_center_pow [Finite (commutatorSet G)] :
    Nat.card (_root_.commutator G) ∣
      (center G).index ^ ((center G).index * Nat.card (commutatorSet G) + 1) := by
  -- First handle the case when `Z(G)` has infinite index and `[G : Z(G)]` is defined to be `0`
  by_cases hG : (center G).index = 0
  · simp_rw [hG, zero_mul, zero_add, pow_one, dvd_zero]
  haveI : FiniteIndex (center G) := ⟨hG⟩
  -- Rewrite as `|Z(G) ∩ G'| * [G' : Z(G) ∩ G'] ∣ [G : Z(G)] ^ ([G : Z(G)] * n) * [G : Z(G)]`
  rw [← ((center G).subgroupOf (_root_.commutator G)).card_mul_index, pow_succ']
  -- We have `h1 : [G' : Z(G) ∩ G'] ∣ [G : Z(G)]`
  have h1 := relindex_dvd_index_of_normal (center G) (_root_.commutator G)
  -- So we can reduce to proving `|Z(G) ∩ G'| ∣ [G : Z(G)] ^ ([G : Z(G)] * n)`
  refine' mul_dvd_mul _ h1
  -- We know that `[G' : Z(G) ∩ G'] < ∞` by `h1` and `hG`
  haveI : FiniteIndex ((center G).subgroupOf (_root_.commutator G)) :=
    ⟨ne_zero_of_dvd_ne_zero hG h1⟩
  -- We have `h2 : rank (Z(G) ∩ G') ≤ [G' : Z(G) ∩ G'] * rank G'` by Schreier's lemma
  have h2 := rank_le_index_mul_rank ((center G).subgroupOf (_root_.commutator G))
  -- We have `h3 : [G' : Z(G) ∩ G'] * rank G' ≤ [G : Z(G)] * n` by `h1` and `rank G' ≤ n`
  have h3 := Nat.mul_le_mul (Nat.le_of_dvd (Nat.pos_of_ne_zero hG) h1) (rank_commutator_le_card G)
  -- So we can reduce to proving `|Z(G) ∩ G'| ∣ [G : Z(G)] ^ rank (Z(G) ∩ G')`
  

Proof state:
case neg
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
R S : Set G
inst✝ : Finite ↑(commutatorSet G)
hG : ¬index (center G) = 0
this✝ : FiniteIndex (center G)
h1 : relindex (center G) (_root_.commutator G) ∣ index (center G)
this : FiniteIndex (subgroupOf (center G) (_root_.commutator G))
h2 :
  Group.rank ↥(subgroupOf (center G) (_root_.commutator G)) ≤
    index (subgroupOf (center G) (_root_.commutator G)) * Group.rank ↥(_root_.commutator G)
h3 :
  relindex (center G) (_root_.commutator G) * Group.rank ↥(_root_.commutator G) ≤
    index (center G) * Nat.card ↑(commutatorSet G)
⊢ Nat.card ↥(subgroupOf (center G) (_root_.commutator G)) ∣
    index (center G) ^ (index (center G) * Nat.card ↑(commutatorSet G))

Next tactic:
refine' dvd_trans _ (pow_dvd_pow (center G).index (h2.trans h3))
/-- If `G` has `n` commutators `[g₁, g₂]`, then `|G'| ∣ [G : Z(G)] ^ ([G : Z(G)] * n + 1)`,
where `G'` denotes the commutator of `G`. -/
theorem card_commutator_dvd_index_center_pow [Finite (commutatorSet G)] :
    Nat.card (_root_.commutator G) ∣
      (center G).index ^ ((center G).index * Nat.card (commutatorSet G) + 1) := by
  -- First handle the case when `Z(G)` has infinite index and `[G : Z(G)]` is defined to be `0`
  by_cases hG : (center G).index = 0
  · simp_rw [hG, zero_mul, zero_add, pow_one, dvd_zero]
  haveI : FiniteIndex (center G) := ⟨hG⟩
  -- Rewrite as `|Z(G) ∩ G'| * [G' : Z(G) ∩ G'] ∣ [G : Z(G)] ^ ([G : Z(G)] * n) * [G : Z(G)]`
  rw [← ((center G).subgroupOf (_root_.commutator G)).card_mul_index, pow_succ']
  -- We have `h1 : [G' : Z(G) ∩ G'] ∣ [G : Z(G)]`
  have h1 := relindex_dvd_index_of_normal (center G) (_root_.commutator G)
  -- So we can reduce to proving `|Z(G) ∩ G'| ∣ [G : Z(G)] ^ ([G : Z(G)] * n)`
  refine' mul_dvd_mul _ h1
  -- We know that `[G' : Z(G) ∩ G'] < ∞` by `h1` and `hG`
  haveI : FiniteIndex ((center G).subgroupOf (_root_.commutator G)) :=
    ⟨ne_zero_of_dvd_ne_zero hG h1⟩
  -- We have `h2 : rank (Z(G) ∩ G') ≤ [G' : Z(G) ∩ G'] * rank G'` by Schreier's lemma
  have h2 := rank_le_index_mul_rank ((center G).subgroupOf (_root_.commutator G))
  -- We have `h3 : [G' : Z(G) ∩ G'] * rank G' ≤ [G : Z(G)] * n` by `h1` and `rank G' ≤ n`
  have h3 := Nat.mul_le_mul (Nat.le_of_dvd (Nat.pos_of_ne_zero hG) h1) (rank_commutator_le_card G)
  -- So we can reduce to proving `|Z(G) ∩ G'| ∣ [G : Z(G)] ^ rank (Z(G) ∩ G')`
  refine' dvd_trans _ (pow_dvd_pow (center G).index (h2.trans h3))
  -- `Z(G) ∩ G'` is abelian, so it enough to prove that `g ^ [G : Z(G)] = 1` for `g ∈ Z(G) ∩ G'`
  

Proof state:
case neg
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
R S : Set G
inst✝ : Finite ↑(commutatorSet G)
hG : ¬index (center G) = 0
this✝ : FiniteIndex (center G)
h1 : relindex (center G) (_root_.commutator G) ∣ index (center G)
this : FiniteIndex (subgroupOf (center G) (_root_.commutator G))
h2 :
  Group.rank ↥(subgroupOf (center G) (_root_.commutator G)) ≤
    index (subgroupOf (center G) (_root_.commutator G)) * Group.rank ↥(_root_.commutator G)
h3 :
  relindex (center G) (_root_.commutator G) * Group.rank ↥(_root_.commutator G) ≤
    index (center G) * Nat.card ↑(commutatorSet G)
⊢ Nat.card ↥(subgroupOf (center G) (_root_.commutator G)) ∣
    index (center G) ^ Group.rank ↥(subgroupOf (center G) (_root_.commutator G))

Next tactic:
apply card_dvd_exponent_pow_rank'