Theorem: IsSimpleGroup.comm_iff_isSolvable
File: Mathlib/GroupTheory/Solvable.lean

tactic: cases n
state_before: G : Type u_1
G' : Type u_2
inst✝² : Group G
inst✝¹ : Group G'
f : G →* G'
inst✝ : IsSimpleGroup G
x✝ : IsSolvable G
n : ℕ
hn : derivedSeries G n = ⊥
⊢ ∀ (a b : G), a * b = b * a
state_after: case zero
G : Type u_1
G' : Type u_2
inst✝² : Group G
inst✝¹ : Group G'
f : G →* G'
inst✝ : IsSimpleGroup G
x✝ : IsSolvable G
hn : derivedSeries G 0 = ⊥
⊢ ∀ (a b : G), a * b = b * a

case succ
G : Type u_1
G' : Type u_2
inst✝² : Group G
inst✝¹ : Group G'
f : G →* G'
inst✝ : IsSimpleGroup G
x✝ : IsSolvable G
n✝ : ℕ
hn : derivedSeries G (n✝ + 1) = ⊥
⊢ ∀ (a b : G), a * b = b * a

tactic: intro a b
state_before: case zero
G : Type u_1
G' : Type u_2
inst✝² : Group G
inst✝¹ : Group G'
f : G →* G'
inst✝ : IsSimpleGroup G
x✝ : IsSolvable G
hn : derivedSeries G 0 = ⊥
⊢ ∀ (a b : G), a * b = b * a
state_after: case zero
G : Type u_1
G' : Type u_2
inst✝² : Group G
inst✝¹ : Group G'
f : G →* G'
inst✝ : IsSimpleGroup G
x✝ : IsSolvable G
hn : derivedSeries G 0 = ⊥
a b : G
⊢ a * b = b * a

tactic: rw [← hn]
state_before: case zero.refine_2
G : Type u_1
G' : Type u_2
inst✝² : Group G
inst✝¹ : Group G'
f : G →* G'
inst✝ : IsSimpleGroup G
x✝ : IsSolvable G
hn : derivedSeries G 0 = ⊥
a b : G
⊢ b * a ∈ ⊥
state_after: case zero.refine_2
G : Type u_1
G' : Type u_2
inst✝² : Group G
inst✝¹ : Group G'
f : G →* G'
inst✝ : IsSimpleGroup G
x✝ : IsSolvable G
hn : derivedSeries G 0 = ⊥
a b : G
⊢ b * a ∈ derivedSeries G 0

tactic: exact mem_top _
state_before: case zero.refine_2
G : Type u_1
G' : Type u_2
inst✝² : Group G
inst✝¹ : Group G'
f : G →* G'
inst✝ : IsSimpleGroup G
x✝ : IsSolvable G
hn : derivedSeries G 0 = ⊥
a b : G
⊢ b * a ∈ derivedSeries G 0
state_after: no goals

tactic: rw [IsSimpleGroup.derivedSeries_succ] at hn
state_before: case succ
G : Type u_1
G' : Type u_2
inst✝² : Group G
inst✝¹ : Group G'
f : G →* G'
inst✝ : IsSimpleGroup G
x✝ : IsSolvable G
n✝ : ℕ
hn : derivedSeries G (n✝ + 1) = ⊥
⊢ ∀ (a b : G), a * b = b * a
state_after: case succ
G : Type u_1
G' : Type u_2
inst✝² : Group G
inst✝¹ : Group G'
f : G →* G'
inst✝ : IsSimpleGroup G
x✝ : IsSolvable G
n✝ : ℕ
hn : _root_.commutator G = ⊥
⊢ ∀ (a b : G), a * b = b * a

tactic: intro a b
state_before: case succ
G : Type u_1
G' : Type u_2
inst✝² : Group G
inst✝¹ : Group G'
f : G →* G'
inst✝ : IsSimpleGroup G
x✝ : IsSolvable G
n✝ : ℕ
hn : _root_.commutator G = ⊥
⊢ ∀ (a b : G), a * b = b * a
state_after: case succ
G : Type u_1
G' : Type u_2
inst✝² : Group G
inst✝¹ : Group G'
f : G →* G'
inst✝ : IsSimpleGroup G
x✝ : IsSolvable G
n✝ : ℕ
hn : _root_.commutator G = ⊥
a b : G
⊢ a * b = b * a

tactic: rw [← mul_inv_eq_one, mul_inv_rev, ← mul_assoc, ← mem_bot, ← hn, commutator_eq_closure]
state_before: case succ
G : Type u_1
G' : Type u_2
inst✝² : Group G
inst✝¹ : Group G'
f : G →* G'
inst✝ : IsSimpleGroup G
x✝ : IsSolvable G
n✝ : ℕ
hn : _root_.commutator G = ⊥
a b : G
⊢ a * b = b * a
state_after: case succ
G : Type u_1
G' : Type u_2
inst✝² : Group G
inst✝¹ : Group G'
f : G →* G'
inst✝ : IsSimpleGroup G
x✝ : IsSolvable G
n✝ : ℕ
hn : _root_.commutator G = ⊥
a b : G
⊢ a * b * a⁻¹ * b⁻¹ ∈ closure (commutatorSet G)

tactic: exact subset_closure ⟨a, b, rfl⟩
state_before: case succ
G : Type u_1
G' : Type u_2
inst✝² : Group G
inst✝¹ : Group G'
f : G →* G'
inst✝ : IsSimpleGroup G
x✝ : IsSolvable G
n✝ : ℕ
hn : _root_.commutator G = ⊥
a b : G
⊢ a * b * a⁻¹ * b⁻¹ ∈ closure (commutatorSet G)
state_after: no goals