l3lab/ntp-mathlib huggingface dataset:
/-- 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'`
  apply card_dvd_exponent_pow_rank'
  

Proof state:
case neg.hG
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)
⊢ ∀ (g : ↥(subgroupOf (center G) (_root_.commutator G))), g ^ index (center G) = 1

Next tactic:
intro 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')`
  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'`
  apply card_dvd_exponent_pow_rank'
  intro g
  -- `Z(G)` is abelian, so `g ∈ Z(G) ∩ G' ≤ G' ≤ ker (transfer : G → Z(G))`
  

Proof state:
case neg.hG
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)
g : ↥(subgroupOf (center G) (_root_.commutator G))
⊢ g ^ index (center G) = 1

Next tactic:
have := Abelianization.commutator_subset_ker (MonoidHom.transferCenterPow G) g.1.2
/-- 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'`
  apply card_dvd_exponent_pow_rank'
  intro g
  -- `Z(G)` is abelian, so `g ∈ Z(G) ∩ G' ≤ G' ≤ ker (transfer : G → Z(G))`
  have := Abelianization.commutator_subset_ker (MonoidHom.transferCenterPow G) g.1.2
  -- `transfer g` is defeq to `g ^ [G : Z(G)]`, so we are done
  

Proof state:
case neg.hG
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✝¹ : 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)
g : ↥(subgroupOf (center G) (_root_.commutator G))
this : ↑↑g ∈ MonoidHom.ker (MonoidHom.transferCenterPow G)
⊢ g ^ index (center G) = 1

Next tactic:
simpa only [MonoidHom.mem_ker, Subtype.ext_iff] using this
/-- A theorem of Schur: The size of the commutator subgroup is bounded in terms of the number of
  commutators. -/
theorem card_commutator_le_of_finite_commutatorSet [Finite (commutatorSet G)] :
    Nat.card (_root_.commutator G) ≤ cardCommutatorBound (Nat.card (commutatorSet G)) := by
  

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) ≤ cardCommutatorBound (Nat.card ↑(commutatorSet G))

Next tactic:
have h1 := index_center_le_pow (closureCommutatorRepresentatives G)
/-- A theorem of Schur: The size of the commutator subgroup is bounded in terms of the number of
  commutators. -/
theorem card_commutator_le_of_finite_commutatorSet [Finite (commutatorSet G)] :
    Nat.card (_root_.commutator G) ≤ cardCommutatorBound (Nat.card (commutatorSet G)) := by
  have h1 := index_center_le_pow (closureCommutatorRepresentatives G)
  

Proof state:
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
R S : Set G
inst✝ : Finite ↑(commutatorSet G)
h1 :
  index (center ↥(closureCommutatorRepresentatives G)) ≤
    Nat.card ↑(commutatorSet ↥(closureCommutatorRepresentatives G)) ^ Group.rank ↥(closureCommutatorRepresentatives G)
⊢ Nat.card ↥(_root_.commutator G) ≤ cardCommutatorBound (Nat.card ↑(commutatorSet G))

Next tactic:
have h2 := card_commutator_dvd_index_center_pow (closureCommutatorRepresentatives G)
/-- A theorem of Schur: The size of the commutator subgroup is bounded in terms of the number of
  commutators. -/
theorem card_commutator_le_of_finite_commutatorSet [Finite (commutatorSet G)] :
    Nat.card (_root_.commutator G) ≤ cardCommutatorBound (Nat.card (commutatorSet G)) := by
  have h1 := index_center_le_pow (closureCommutatorRepresentatives G)
  have h2 := card_commutator_dvd_index_center_pow (closureCommutatorRepresentatives G)
  

Proof state:
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
R S : Set G
inst✝ : Finite ↑(commutatorSet G)
h1 :
  index (center ↥(closureCommutatorRepresentatives G)) ≤
    Nat.card ↑(commutatorSet ↥(closureCommutatorRepresentatives G)) ^ Group.rank ↥(closureCommutatorRepresentatives G)
h2 :
  Nat.card ↥(_root_.commutator ↥(closureCommutatorRepresentatives G)) ∣
    index (center ↥(closureCommutatorRepresentatives G)) ^
      (index (center ↥(closureCommutatorRepresentatives G)) *
          Nat.card ↑(commutatorSet ↥(closureCommutatorRepresentatives G)) +
        1)
⊢ Nat.card ↥(_root_.commutator G) ≤ cardCommutatorBound (Nat.card ↑(commutatorSet G))

Next tactic:
rw [card_commutatorSet_closureCommutatorRepresentatives] at h1 h2
/-- A theorem of Schur: The size of the commutator subgroup is bounded in terms of the number of
  commutators. -/
theorem card_commutator_le_of_finite_commutatorSet [Finite (commutatorSet G)] :
    Nat.card (_root_.commutator G) ≤ cardCommutatorBound (Nat.card (commutatorSet G)) := by
  have h1 := index_center_le_pow (closureCommutatorRepresentatives G)
  have h2 := card_commutator_dvd_index_center_pow (closureCommutatorRepresentatives G)
  rw [card_commutatorSet_closureCommutatorRepresentatives] at h1 h2
  

Proof state:
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
R S : Set G
inst✝ : Finite ↑(commutatorSet G)
h1 :
  index (center ↥(closureCommutatorRepresentatives G)) ≤
    Nat.card ↑(commutatorSet G) ^ Group.rank ↥(closureCommutatorRepresentatives G)
h2 :
  Nat.card ↥(_root_.commutator ↥(closureCommutatorRepresentatives G)) ∣
    index (center ↥(closureCommutatorRepresentatives G)) ^
      (index (center ↥(closureCommutatorRepresentatives G)) * Nat.card ↑(commutatorSet G) + 1)
⊢ Nat.card ↥(_root_.commutator G) ≤ cardCommutatorBound (Nat.card ↑(commutatorSet G))

Next tactic:
rw [card_commutator_closureCommutatorRepresentatives] at h2
/-- A theorem of Schur: The size of the commutator subgroup is bounded in terms of the number of
  commutators. -/
theorem card_commutator_le_of_finite_commutatorSet [Finite (commutatorSet G)] :
    Nat.card (_root_.commutator G) ≤ cardCommutatorBound (Nat.card (commutatorSet G)) := by
  have h1 := index_center_le_pow (closureCommutatorRepresentatives G)
  have h2 := card_commutator_dvd_index_center_pow (closureCommutatorRepresentatives G)
  rw [card_commutatorSet_closureCommutatorRepresentatives] at h1 h2
  rw [card_commutator_closureCommutatorRepresentatives] at h2
  

Proof state:
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
R S : Set G
inst✝ : Finite ↑(commutatorSet G)
h1 :
  index (center ↥(closureCommutatorRepresentatives G)) ≤
    Nat.card ↑(commutatorSet G) ^ Group.rank ↥(closureCommutatorRepresentatives G)
h2 :
  Nat.card ↥(_root_.commutator G) ∣
    index (center ↥(closureCommutatorRepresentatives G)) ^
      (index (center ↥(closureCommutatorRepresentatives G)) * Nat.card ↑(commutatorSet G) + 1)
⊢ Nat.card ↥(_root_.commutator G) ≤ cardCommutatorBound (Nat.card ↑(commutatorSet G))

Next tactic:
replace h1 :=
    h1.trans
      (Nat.pow_le_pow_of_le_right Finite.card_pos (rank_closureCommutatorRepresentatives_le G))
/-- A theorem of Schur: The size of the commutator subgroup is bounded in terms of the number of
  commutators. -/
theorem card_commutator_le_of_finite_commutatorSet [Finite (commutatorSet G)] :
    Nat.card (_root_.commutator G) ≤ cardCommutatorBound (Nat.card (commutatorSet G)) := by
  have h1 := index_center_le_pow (closureCommutatorRepresentatives G)
  have h2 := card_commutator_dvd_index_center_pow (closureCommutatorRepresentatives G)
  rw [card_commutatorSet_closureCommutatorRepresentatives] at h1 h2
  rw [card_commutator_closureCommutatorRepresentatives] at h2
  replace h1 :=
    h1.trans
      (Nat.pow_le_pow_of_le_right Finite.card_pos (rank_closureCommutatorRepresentatives_le G))
  

Proof state:
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
R S : Set G
inst✝ : Finite ↑(commutatorSet G)
h2 :
  Nat.card ↥(_root_.commutator G) ∣
    index (center ↥(closureCommutatorRepresentatives G)) ^
      (index (center ↥(closureCommutatorRepresentatives G)) * Nat.card ↑(commutatorSet G) + 1)
h1 :
  index (center ↥(closureCommutatorRepresentatives G)) ≤ Nat.card ↑(commutatorSet G) ^ (2 * Nat.card ↑(commutatorSet G))
⊢ Nat.card ↥(_root_.commutator G) ≤ cardCommutatorBound (Nat.card ↑(commutatorSet G))

Next tactic:
replace h2 := h2.trans (pow_dvd_pow _ (add_le_add_right (mul_le_mul_right' h1 _) 1))
/-- A theorem of Schur: The size of the commutator subgroup is bounded in terms of the number of
  commutators. -/
theorem card_commutator_le_of_finite_commutatorSet [Finite (commutatorSet G)] :
    Nat.card (_root_.commutator G) ≤ cardCommutatorBound (Nat.card (commutatorSet G)) := by
  have h1 := index_center_le_pow (closureCommutatorRepresentatives G)
  have h2 := card_commutator_dvd_index_center_pow (closureCommutatorRepresentatives G)
  rw [card_commutatorSet_closureCommutatorRepresentatives] at h1 h2
  rw [card_commutator_closureCommutatorRepresentatives] at h2
  replace h1 :=
    h1.trans
      (Nat.pow_le_pow_of_le_right Finite.card_pos (rank_closureCommutatorRepresentatives_le G))
  replace h2 := h2.trans (pow_dvd_pow _ (add_le_add_right (mul_le_mul_right' h1 _) 1))
  

Proof state:
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
R S : Set G
inst✝ : Finite ↑(commutatorSet G)
h1 :
  index (center ↥(closureCommutatorRepresentatives G)) ≤ Nat.card ↑(commutatorSet G) ^ (2 * Nat.card ↑(commutatorSet G))
h2 :
  Nat.card ↥(_root_.commutator G) ∣
    index (center ↥(closureCommutatorRepresentatives G)) ^
      (Nat.card ↑(commutatorSet G) ^ (2 * Nat.card ↑(commutatorSet G)) * Nat.card ↑(commutatorSet G) + 1)
⊢ Nat.card ↥(_root_.commutator G) ≤ cardCommutatorBound (Nat.card ↑(commutatorSet G))

Next tactic:
rw [← pow_succ'] at h2
/-- A theorem of Schur: The size of the commutator subgroup is bounded in terms of the number of
  commutators. -/
theorem card_commutator_le_of_finite_commutatorSet [Finite (commutatorSet G)] :
    Nat.card (_root_.commutator G) ≤ cardCommutatorBound (Nat.card (commutatorSet G)) := by
  have h1 := index_center_le_pow (closureCommutatorRepresentatives G)
  have h2 := card_commutator_dvd_index_center_pow (closureCommutatorRepresentatives G)
  rw [card_commutatorSet_closureCommutatorRepresentatives] at h1 h2
  rw [card_commutator_closureCommutatorRepresentatives] at h2
  replace h1 :=
    h1.trans
      (Nat.pow_le_pow_of_le_right Finite.card_pos (rank_closureCommutatorRepresentatives_le G))
  replace h2 := h2.trans (pow_dvd_pow _ (add_le_add_right (mul_le_mul_right' h1 _) 1))
  rw [← pow_succ'] at h2
  

Proof state:
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
R S : Set G
inst✝ : Finite ↑(commutatorSet G)
h1 :
  index (center ↥(closureCommutatorRepresentatives G)) ≤ Nat.card ↑(commutatorSet G) ^ (2 * Nat.card ↑(commutatorSet G))
h2 :
  Nat.card ↥(_root_.commutator G) ∣
    index (center ↥(closureCommutatorRepresentatives G)) ^
      (Nat.card ↑(commutatorSet G) ^ (2 * Nat.card ↑(commutatorSet G) + 1) + 1)
⊢ Nat.card ↥(_root_.commutator G) ≤ cardCommutatorBound (Nat.card ↑(commutatorSet G))

Next tactic:
refine' (Nat.le_of_dvd _ h2).trans (Nat.pow_le_pow_left h1 _)
/-- A theorem of Schur: The size of the commutator subgroup is bounded in terms of the number of
  commutators. -/
theorem card_commutator_le_of_finite_commutatorSet [Finite (commutatorSet G)] :
    Nat.card (_root_.commutator G) ≤ cardCommutatorBound (Nat.card (commutatorSet G)) := by
  have h1 := index_center_le_pow (closureCommutatorRepresentatives G)
  have h2 := card_commutator_dvd_index_center_pow (closureCommutatorRepresentatives G)
  rw [card_commutatorSet_closureCommutatorRepresentatives] at h1 h2
  rw [card_commutator_closureCommutatorRepresentatives] at h2
  replace h1 :=
    h1.trans
      (Nat.pow_le_pow_of_le_right Finite.card_pos (rank_closureCommutatorRepresentatives_le G))
  replace h2 := h2.trans (pow_dvd_pow _ (add_le_add_right (mul_le_mul_right' h1 _) 1))
  rw [← pow_succ'] at h2
  refine' (Nat.le_of_dvd _ h2).trans (Nat.pow_le_pow_left h1 _)
  

Proof state:
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
R S : Set G
inst✝ : Finite ↑(commutatorSet G)
h1 :
  index (center ↥(closureCommutatorRepresentatives G)) ≤ Nat.card ↑(commutatorSet G) ^ (2 * Nat.card ↑(commutatorSet G))
h2 :
  Nat.card ↥(_root_.commutator G) ∣
    index (center ↥(closureCommutatorRepresentatives G)) ^
      (Nat.card ↑(commutatorSet G) ^ (2 * Nat.card ↑(commutatorSet G) + 1) + 1)
⊢ 0 <
    index (center ↥(closureCommutatorRepresentatives G)) ^
      (Nat.card ↑(commutatorSet G) ^ (2 * Nat.card ↑(commutatorSet G) + 1) + 1)

Next tactic:
exact pow_pos (Nat.pos_of_ne_zero FiniteIndex.finiteIndex) _
/-- A theorem of Schur: A group with finitely many commutators has finite commutator subgroup. -/
instance [Finite (commutatorSet G)] : Finite (_root_.commutator G) := by
  

Proof state:
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
R S : Set G
inst✝ : Finite ↑(commutatorSet G)
⊢ Finite ↥(_root_.commutator G)

Next tactic:
have h2 := card_commutator_dvd_index_center_pow (closureCommutatorRepresentatives G)
/-- A theorem of Schur: A group with finitely many commutators has finite commutator subgroup. -/
instance [Finite (commutatorSet G)] : Finite (_root_.commutator G) := by
  have h2 := card_commutator_dvd_index_center_pow (closureCommutatorRepresentatives G)
  

Proof state:
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
R S : Set G
inst✝ : Finite ↑(commutatorSet G)
h2 :
  Nat.card ↥(_root_.commutator ↥(closureCommutatorRepresentatives G)) ∣
    index (center ↥(closureCommutatorRepresentatives G)) ^
      (index (center ↥(closureCommutatorRepresentatives G)) *
          Nat.card ↑(commutatorSet ↥(closureCommutatorRepresentatives G)) +
        1)
⊢ Finite ↥(_root_.commutator G)

Next tactic:
refine' Nat.finite_of_card_ne_zero fun h => _
/-- A theorem of Schur: A group with finitely many commutators has finite commutator subgroup. -/
instance [Finite (commutatorSet G)] : Finite (_root_.commutator G) := by
  have h2 := card_commutator_dvd_index_center_pow (closureCommutatorRepresentatives G)
  refine' Nat.finite_of_card_ne_zero fun h => _
  

Proof state:
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
R S : Set G
inst✝ : Finite ↑(commutatorSet G)
h2 :
  Nat.card ↥(_root_.commutator ↥(closureCommutatorRepresentatives G)) ∣
    index (center ↥(closureCommutatorRepresentatives G)) ^
      (index (center ↥(closureCommutatorRepresentatives G)) *
          Nat.card ↑(commutatorSet ↥(closureCommutatorRepresentatives G)) +
        1)
h : Nat.card ↥(_root_.commutator G) = 0
⊢ False

Next tactic:
rw [card_commutator_closureCommutatorRepresentatives, h, zero_dvd_iff] at h2
/-- A theorem of Schur: A group with finitely many commutators has finite commutator subgroup. -/
instance [Finite (commutatorSet G)] : Finite (_root_.commutator G) := by
  have h2 := card_commutator_dvd_index_center_pow (closureCommutatorRepresentatives G)
  refine' Nat.finite_of_card_ne_zero fun h => _
  rw [card_commutator_closureCommutatorRepresentatives, h, zero_dvd_iff] at h2
  

Proof state:
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
R S : Set G
inst✝ : Finite ↑(commutatorSet G)
h2 :
  index (center ↥(closureCommutatorRepresentatives G)) ^
      (index (center ↥(closureCommutatorRepresentatives G)) *
          Nat.card ↑(commutatorSet ↥(closureCommutatorRepresentatives G)) +
        1) =
    0
h : Nat.card ↥(_root_.commutator G) = 0
⊢ False

Next tactic:
exact FiniteIndex.finiteIndex (pow_eq_zero h2)
@[simp, mfld_simps]
theorem target_eq : I.target = range (I : H → E) := by
  

Proof state:
𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
H : Type u_3
inst✝ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
⊢ I.target = range ↑I

Next tactic:
rw [← image_univ, ← I.source_eq]
@[simp, mfld_simps]
theorem target_eq : I.target = range (I : H → E) := by
  rw [← image_univ, ← I.source_eq]
  

Proof state:
𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
H : Type u_3
inst✝ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
⊢ I.target = ↑I '' I.source

Next tactic:
exact I.image_source_eq_target.symm
@[simp, mfld_simps]
protected theorem left_inv (x : H) : I.symm (I x) = x := by 

Proof state:
𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
H : Type u_3
inst✝ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
x : H
⊢ ↑(ModelWithCorners.symm I) (↑I x) = x

Next tactic:
refine' I.left_inv' _
@[simp, mfld_simps]
protected theorem left_inv (x : H) : I.symm (I x) = x := by refine' I.left_inv' _; 

Proof state:
𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
H : Type u_3
inst✝ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
x : H
⊢ x ∈ I.source

Next tactic:
simp
protected theorem image_eq (s : Set H) : I '' s = I.symm ⁻¹' s ∩ range I := by
  

Proof state:
𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
H : Type u_3
inst✝ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
s : Set H
⊢ ↑I '' s = ↑(ModelWithCorners.symm I) ⁻¹' s ∩ range ↑I

Next tactic:
refine' (I.toPartialEquiv.image_eq_target_inter_inv_preimage _).trans _
protected theorem image_eq (s : Set H) : I '' s = I.symm ⁻¹' s ∩ range I := by
  refine' (I.toPartialEquiv.image_eq_target_inter_inv_preimage _).trans _
  · 

Proof state:
case refine'_1
𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
H : Type u_3
inst✝ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
s : Set H
⊢ s ⊆ I.source

Next tactic:
rw [I.source_eq]
protected theorem image_eq (s : Set H) : I '' s = I.symm ⁻¹' s ∩ range I := by
  refine' (I.toPartialEquiv.image_eq_target_inter_inv_preimage _).trans _
  · rw [I.source_eq]; 

Proof state:
case refine'_1
𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
H : Type u_3
inst✝ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
s : Set H
⊢ s ⊆ univ

Next tactic:
exact subset_univ _