G : Type u_1
inst✝¹ : Group G
H : Subgroup G
inst✝ : Normal H
hG : Group.IsNilpotent G
⊢ Nat.find (_ : ∃ n, lowerCentralSeries G n = ⊥) = Nat.find (_ : ∃ n H, IsDescendingCentralSeries H ∧ H n = ⊥)
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
inst✝ : Normal H
hG : Group.IsNilpotent G
⊢ Nat.find (_ : ∃ n, lowerCentralSeries G n = ⊥) = nilpotencyClass G
rw [← least_descending_central_series_length_eq_nilpotencyClass]
case refine'_1
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
inst✝ : Normal H
hG : Group.IsNilpotent G
⊢ ∀ (n : ℕ), (∃ H, IsDescendingCentralSeries H ∧ H n = ⊥) → lowerCentralSeries G n = ⊥

case refine'_2
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
inst✝ : Normal H
hG : Group.IsNilpotent G
⊢ ∀ (n : ℕ), lowerCentralSeries G n = ⊥ → ∃ H, IsDescendingCentralSeries H ∧ H n = ⊥
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
inst✝ : Normal H
hG : Group.IsNilpotent G
⊢ Nat.find (_ : ∃ n, lowerCentralSeries G n = ⊥) = Nat.find (_ : ∃ n H, IsDescendingCentralSeries H ∧ H n = ⊥)
refine' le_antisymm (Nat.find_mono _) (Nat.find_mono _)
case refine'_1.intro.intro
G : Type u_1
inst✝¹ : Group G
H✝ : Subgroup G
inst✝ : Normal H✝
hG : Group.IsNilpotent G
n : ℕ
H : ℕ → Subgroup G
hH : IsDescendingCentralSeries H
hn : H n = ⊥
⊢ lowerCentralSeries G n = ⊥
case refine'_1
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
inst✝ : Normal H
hG : Group.IsNilpotent G
⊢ ∀ (n : ℕ), (∃ H, IsDescendingCentralSeries H ∧ H n = ⊥) → lowerCentralSeries G n = ⊥
rintro n ⟨H, ⟨hH, hn⟩⟩
case refine'_1.intro.intro
G : Type u_1
inst✝¹ : Group G
H✝ : Subgroup G
inst✝ : Normal H✝
hG : Group.IsNilpotent G
n : ℕ
H : ℕ → Subgroup G
hH : IsDescendingCentralSeries H
hn : H n = ⊥
⊢ lowerCentralSeries G n ≤ H n
case refine'_1.intro.intro
G : Type u_1
inst✝¹ : Group G
H✝ : Subgroup G
inst✝ : Normal H✝
hG : Group.IsNilpotent G
n : ℕ
H : ℕ → Subgroup G
hH : IsDescendingCentralSeries H
hn : H n = ⊥
⊢ lowerCentralSeries G n = ⊥
rw [← le_bot_iff, ← hn]
no goals
case refine'_1.intro.intro
G : Type u_1
inst✝¹ : Group G
H✝ : Subgroup G
inst✝ : Normal H✝
hG : Group.IsNilpotent G
n : ℕ
H : ℕ → Subgroup G
hH : IsDescendingCentralSeries H
hn : H n = ⊥
⊢ lowerCentralSeries G n ≤ H n
exact descending_central_series_ge_lower H hH n
case refine'_2
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
inst✝ : Normal H
hG : Group.IsNilpotent G
n : ℕ
h : lowerCentralSeries G n = ⊥
⊢ ∃ H, IsDescendingCentralSeries H ∧ H n = ⊥
case refine'_2
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
inst✝ : Normal H
hG : Group.IsNilpotent G
⊢ ∀ (n : ℕ), lowerCentralSeries G n = ⊥ → ∃ H, IsDescendingCentralSeries H ∧ H n = ⊥
rintro n h
no goals
case refine'_2
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
inst✝ : Normal H
hG : Group.IsNilpotent G
n : ℕ
h : lowerCentralSeries G n = ⊥
⊢ ∃ H, IsDescendingCentralSeries H ∧ H n = ⊥
exact ⟨lowerCentralSeries G, ⟨lowerCentralSeries_isDescendingCentralSeries, h⟩⟩