l3lab/ntp-mathlib huggingface dataset:
attribute [local simp] EndMonoidal in
instance (X : C) : MonoidalCategory (EndMonoidal X) where
  tensorObj f g := f ≫ g
  whiskerLeft {f g h} η := f ◁ η
  whiskerRight {f g} η h := η ▷ h
  tensorUnit := 𝟙 _
  associator f g h := α_ f g h
  leftUnitor f := λ_ f
  rightUnitor f := ρ_ f
  tensor_comp := by
    intros
    dsimp
    

Proof state:
C : Type u_1
inst✝ : Bicategory C
X : C
X₁✝ Y₁✝ Z₁✝ X₂✝ Y₂✝ Z₂✝ : EndMonoidal X
f₁✝ : X₁✝ ⟶ Y₁✝
f₂✝ : X₂✝ ⟶ Y₂✝
g₁✝ : Y₁✝ ⟶ Z₁✝
g₂✝ : Y₂✝ ⟶ Z₂✝
⊢ (f₁✝ ≫ g₁✝) ▷ X₂✝ ≫ Z₁✝ ◁ (f₂✝ ≫ g₂✝) = (f₁✝ ▷ X₂✝ ≫ Y₁✝ ◁ f₂✝) ≫ g₁✝ ▷ Y₂✝ ≫ Z₁✝ ◁ g₂✝

Next tactic:
rw [Bicategory.whiskerLeft_comp, Bicategory.comp_whiskerRight, Category.assoc, Category.assoc,
      Bicategory.whisker_exchange_assoc]
theorem commutatorElement_eq_one_iff_mul_comm : ⁅g₁, g₂⁆ = 1 ↔ g₁ * g₂ = g₂ * g₁ := by
  

Proof state:
G : Type u_1
G' : Type u_2
F : Type u_3
inst✝² : Group G
inst✝¹ : Group G'
inst✝ : MonoidHomClass F G G'
f : F
g₁ g₂ g₃ g : G
⊢ ⁅g₁, g₂⁆ = 1 ↔ g₁ * g₂ = g₂ * g₁

Next tactic:
rw [commutatorElement_def, mul_inv_eq_one, mul_inv_eq_iff_eq_mul]
@[simp]
theorem commutatorElement_inv : ⁅g₁, g₂⁆⁻¹ = ⁅g₂, g₁⁆ := by
  

Proof state:
G : Type u_1
G' : Type u_2
F : Type u_3
inst✝² : Group G
inst✝¹ : Group G'
inst✝ : MonoidHomClass F G G'
f : F
g₁ g₂ g₃ g : G
⊢ ⁅g₁, g₂⁆⁻¹ = ⁅g₂, g₁⁆

Next tactic:
simp_rw [commutatorElement_def, mul_inv_rev, inv_inv, mul_assoc]
theorem map_commutatorElement : (f ⁅g₁, g₂⁆ : G') = ⁅f g₁, f g₂⁆ := by
  

Proof state:
G : Type u_1
G' : Type u_2
F : Type u_3
inst✝² : Group G
inst✝¹ : Group G'
inst✝ : MonoidHomClass F G G'
f : F
g₁ g₂ g₃ g : G
⊢ f ⁅g₁, g₂⁆ = ⁅f g₁, f g₂⁆

Next tactic:
simp_rw [commutatorElement_def, map_mul f, map_inv f]
theorem commutator_eq_bot_iff_le_centralizer : ⁅H₁, H₂⁆ = ⊥ ↔ H₁ ≤ centralizer H₂ := by
  

Proof state:
G : Type u_1
G' : Type u_2
F : Type u_3
inst✝² : Group G
inst✝¹ : Group G'
inst✝ : MonoidHomClass F G G'
f : F
g₁ g₂ g₃ g : G
H₁ H₂ H₃ K₁ K₂ : Subgroup G
⊢ ⁅H₁, H₂⁆ = ⊥ ↔ H₁ ≤ centralizer ↑H₂

Next tactic:
rw [eq_bot_iff, commutator_le]
theorem commutator_eq_bot_iff_le_centralizer : ⁅H₁, H₂⁆ = ⊥ ↔ H₁ ≤ centralizer H₂ := by
  rw [eq_bot_iff, commutator_le]
  

Proof state:
G : Type u_1
G' : Type u_2
F : Type u_3
inst✝² : Group G
inst✝¹ : Group G'
inst✝ : MonoidHomClass F G G'
f : F
g₁ g₂ g₃ g : G
H₁ H₂ H₃ K₁ K₂ : Subgroup G
⊢ (∀ g₁ ∈ H₁, ∀ g₂ ∈ H₂, ⁅g₁, g₂⁆ ∈ ⊥) ↔ H₁ ≤ centralizer ↑H₂

Next tactic:
refine'
    forall_congr' fun p => forall_congr' fun _hp => forall_congr' fun q => forall_congr' fun hq => _
theorem commutator_eq_bot_iff_le_centralizer : ⁅H₁, H₂⁆ = ⊥ ↔ H₁ ≤ centralizer H₂ := by
  rw [eq_bot_iff, commutator_le]
  refine'
    forall_congr' fun p => forall_congr' fun _hp => forall_congr' fun q => forall_congr' fun hq => _
  

Proof state:
G : Type u_1
G' : Type u_2
F : Type u_3
inst✝² : Group G
inst✝¹ : Group G'
inst✝ : MonoidHomClass F G G'
f : F
g₁ g₂ g₃ g : G
H₁ H₂ H₃ K₁ K₂ : Subgroup G
p : G
_hp : p ∈ H₁
q : G
hq : q ∈ H₂
⊢ ⁅p, q⁆ ∈ ⊥ ↔ q * p = p * q

Next tactic:
rw [mem_bot, commutatorElement_eq_one_iff_mul_comm, eq_comm]
/-- **The Three Subgroups Lemma** (via the Hall-Witt identity) -/
theorem commutator_commutator_eq_bot_of_rotate (h1 : ⁅⁅H₂, H₃⁆, H₁⁆ = ⊥) (h2 : ⁅⁅H₃, H₁⁆, H₂⁆ = ⊥) :
    ⁅⁅H₁, H₂⁆, H₃⁆ = ⊥ := by
  

Proof state:
G : Type u_1
G' : Type u_2
F : Type u_3
inst✝² : Group G
inst✝¹ : Group G'
inst✝ : MonoidHomClass F G G'
f : F
g₁ g₂ g₃ g : G
H₁ H₂ H₃ K₁ K₂ : Subgroup G
h1 : ⁅⁅H₂, H₃⁆, H₁⁆ = ⊥
h2 : ⁅⁅H₃, H₁⁆, H₂⁆ = ⊥
⊢ ⁅⁅H₁, H₂⁆, H₃⁆ = ⊥

Next tactic:
simp_rw [commutator_eq_bot_iff_le_centralizer, commutator_le,
    mem_centralizer_iff_commutator_eq_one, ← commutatorElement_def] at h1 h2 ⊢
/-- **The Three Subgroups Lemma** (via the Hall-Witt identity) -/
theorem commutator_commutator_eq_bot_of_rotate (h1 : ⁅⁅H₂, H₃⁆, H₁⁆ = ⊥) (h2 : ⁅⁅H₃, H₁⁆, H₂⁆ = ⊥) :
    ⁅⁅H₁, H₂⁆, H₃⁆ = ⊥ := by
  simp_rw [commutator_eq_bot_iff_le_centralizer, commutator_le,
    mem_centralizer_iff_commutator_eq_one, ← commutatorElement_def] at h1 h2 ⊢
  

Proof state:
G : Type u_1
G' : Type u_2
F : Type u_3
inst✝² : Group G
inst✝¹ : Group G'
inst✝ : MonoidHomClass F G G'
f : F
g₁ g₂ g₃ g : G
H₁ H₂ H₃ K₁ K₂ : Subgroup G
h1 : ∀ g₁ ∈ H₂, ∀ g₂ ∈ H₃, ∀ h ∈ ↑H₁, ⁅h, ⁅g₁, g₂⁆⁆ = 1
h2 : ∀ g₁ ∈ H₃, ∀ g₂ ∈ H₁, ∀ h ∈ ↑H₂, ⁅h, ⁅g₁, g₂⁆⁆ = 1
⊢ ∀ g₁ ∈ H₁, ∀ g₂ ∈ H₂, ∀ h ∈ ↑H₃, ⁅h, ⁅g₁, g₂⁆⁆ = 1

Next tactic:
intro x hx y hy z hz
/-- **The Three Subgroups Lemma** (via the Hall-Witt identity) -/
theorem commutator_commutator_eq_bot_of_rotate (h1 : ⁅⁅H₂, H₃⁆, H₁⁆ = ⊥) (h2 : ⁅⁅H₃, H₁⁆, H₂⁆ = ⊥) :
    ⁅⁅H₁, H₂⁆, H₃⁆ = ⊥ := by
  simp_rw [commutator_eq_bot_iff_le_centralizer, commutator_le,
    mem_centralizer_iff_commutator_eq_one, ← commutatorElement_def] at h1 h2 ⊢
  intro x hx y hy z hz
  

Proof state:
G : Type u_1
G' : Type u_2
F : Type u_3
inst✝² : Group G
inst✝¹ : Group G'
inst✝ : MonoidHomClass F G G'
f : F
g₁ g₂ g₃ g : G
H₁ H₂ H₃ K₁ K₂ : Subgroup G
h1 : ∀ g₁ ∈ H₂, ∀ g₂ ∈ H₃, ∀ h ∈ ↑H₁, ⁅h, ⁅g₁, g₂⁆⁆ = 1
h2 : ∀ g₁ ∈ H₃, ∀ g₂ ∈ H₁, ∀ h ∈ ↑H₂, ⁅h, ⁅g₁, g₂⁆⁆ = 1
x : G
hx : x ∈ H₁
y : G
hy : y ∈ H₂
z : G
hz : z ∈ ↑H₃
⊢ ⁅z, ⁅x, y⁆⁆ = 1

Next tactic:
trans x * z * ⁅y, ⁅z⁻¹, x⁻¹⁆⁆⁻¹ * z⁻¹ * y * ⁅x⁻¹, ⁅y⁻¹, z⁆⁆⁻¹ * y⁻¹ * x⁻¹
/-- **The Three Subgroups Lemma** (via the Hall-Witt identity) -/
theorem commutator_commutator_eq_bot_of_rotate (h1 : ⁅⁅H₂, H₃⁆, H₁⁆ = ⊥) (h2 : ⁅⁅H₃, H₁⁆, H₂⁆ = ⊥) :
    ⁅⁅H₁, H₂⁆, H₃⁆ = ⊥ := by
  simp_rw [commutator_eq_bot_iff_le_centralizer, commutator_le,
    mem_centralizer_iff_commutator_eq_one, ← commutatorElement_def] at h1 h2 ⊢
  intro x hx y hy z hz
  trans x * z * ⁅y, ⁅z⁻¹, x⁻¹⁆⁆⁻¹ * z⁻¹ * y * ⁅x⁻¹, ⁅y⁻¹, z⁆⁆⁻¹ * y⁻¹ * x⁻¹
  · 

Proof state:
G : Type u_1
G' : Type u_2
F : Type u_3
inst✝² : Group G
inst✝¹ : Group G'
inst✝ : MonoidHomClass F G G'
f : F
g₁ g₂ g₃ g : G
H₁ H₂ H₃ K₁ K₂ : Subgroup G
h1 : ∀ g₁ ∈ H₂, ∀ g₂ ∈ H₃, ∀ h ∈ ↑H₁, ⁅h, ⁅g₁, g₂⁆⁆ = 1
h2 : ∀ g₁ ∈ H₃, ∀ g₂ ∈ H₁, ∀ h ∈ ↑H₂, ⁅h, ⁅g₁, g₂⁆⁆ = 1
x : G
hx : x ∈ H₁
y : G
hy : y ∈ H₂
z : G
hz : z ∈ ↑H₃
⊢ ⁅z, ⁅x, y⁆⁆ = x * z * ⁅y, ⁅z⁻¹, x⁻¹⁆⁆⁻¹ * z⁻¹ * y * ⁅x⁻¹, ⁅y⁻¹, z⁆⁆⁻¹ * y⁻¹ * x⁻¹

Next tactic:
group
/-- **The Three Subgroups Lemma** (via the Hall-Witt identity) -/
theorem commutator_commutator_eq_bot_of_rotate (h1 : ⁅⁅H₂, H₃⁆, H₁⁆ = ⊥) (h2 : ⁅⁅H₃, H₁⁆, H₂⁆ = ⊥) :
    ⁅⁅H₁, H₂⁆, H₃⁆ = ⊥ := by
  simp_rw [commutator_eq_bot_iff_le_centralizer, commutator_le,
    mem_centralizer_iff_commutator_eq_one, ← commutatorElement_def] at h1 h2 ⊢
  intro x hx y hy z hz
  trans x * z * ⁅y, ⁅z⁻¹, x⁻¹⁆⁆⁻¹ * z⁻¹ * y * ⁅x⁻¹, ⁅y⁻¹, z⁆⁆⁻¹ * y⁻¹ * x⁻¹
  · group
  · 

Proof state:
G : Type u_1
G' : Type u_2
F : Type u_3
inst✝² : Group G
inst✝¹ : Group G'
inst✝ : MonoidHomClass F G G'
f : F
g₁ g₂ g₃ g : G
H₁ H₂ H₃ K₁ K₂ : Subgroup G
h1 : ∀ g₁ ∈ H₂, ∀ g₂ ∈ H₃, ∀ h ∈ ↑H₁, ⁅h, ⁅g₁, g₂⁆⁆ = 1
h2 : ∀ g₁ ∈ H₃, ∀ g₂ ∈ H₁, ∀ h ∈ ↑H₂, ⁅h, ⁅g₁, g₂⁆⁆ = 1
x : G
hx : x ∈ H₁
y : G
hy : y ∈ H₂
z : G
hz : z ∈ ↑H₃
⊢ x * z * ⁅y, ⁅z⁻¹, x⁻¹⁆⁆⁻¹ * z⁻¹ * y * ⁅x⁻¹, ⁅y⁻¹, z⁆⁆⁻¹ * y⁻¹ * x⁻¹ = 1

Next tactic:
rw [h1 _ (H₂.inv_mem hy) _ hz _ (H₁.inv_mem hx), h2 _ (H₃.inv_mem hz) _ (H₁.inv_mem hx) _ hy]
/-- **The Three Subgroups Lemma** (via the Hall-Witt identity) -/
theorem commutator_commutator_eq_bot_of_rotate (h1 : ⁅⁅H₂, H₃⁆, H₁⁆ = ⊥) (h2 : ⁅⁅H₃, H₁⁆, H₂⁆ = ⊥) :
    ⁅⁅H₁, H₂⁆, H₃⁆ = ⊥ := by
  simp_rw [commutator_eq_bot_iff_le_centralizer, commutator_le,
    mem_centralizer_iff_commutator_eq_one, ← commutatorElement_def] at h1 h2 ⊢
  intro x hx y hy z hz
  trans x * z * ⁅y, ⁅z⁻¹, x⁻¹⁆⁆⁻¹ * z⁻¹ * y * ⁅x⁻¹, ⁅y⁻¹, z⁆⁆⁻¹ * y⁻¹ * x⁻¹
  · group
  · rw [h1 _ (H₂.inv_mem hy) _ hz _ (H₁.inv_mem hx), h2 _ (H₃.inv_mem hz) _ (H₁.inv_mem hx) _ hy]
    

Proof state:
G : Type u_1
G' : Type u_2
F : Type u_3
inst✝² : Group G
inst✝¹ : Group G'
inst✝ : MonoidHomClass F G G'
f : F
g₁ g₂ g₃ g : G
H₁ H₂ H₃ K₁ K₂ : Subgroup G
h1 : ∀ g₁ ∈ H₂, ∀ g₂ ∈ H₃, ∀ h ∈ ↑H₁, ⁅h, ⁅g₁, g₂⁆⁆ = 1
h2 : ∀ g₁ ∈ H₃, ∀ g₂ ∈ H₁, ∀ h ∈ ↑H₂, ⁅h, ⁅g₁, g₂⁆⁆ = 1
x : G
hx : x ∈ H₁
y : G
hy : y ∈ H₂
z : G
hz : z ∈ ↑H₃
⊢ x * z * 1⁻¹ * z⁻¹ * y * 1⁻¹ * y⁻¹ * x⁻¹ = 1

Next tactic:
group
instance commutator_normal [h₁ : H₁.Normal] [h₂ : H₂.Normal] : Normal ⁅H₁, H₂⁆ := by
  

Proof state:
G : Type u_1
G' : Type u_2
F : Type u_3
inst✝² : Group G
inst✝¹ : Group G'
inst✝ : MonoidHomClass F G G'
f : F
g₁ g₂ g₃ g : G
H₁ H₂ H₃ K₁ K₂ : Subgroup G
h₁ : Normal H₁
h₂ : Normal H₂
⊢ Normal ⁅H₁, H₂⁆

Next tactic:
let base : Set G := { x | ∃ g₁ ∈ H₁, ∃ g₂ ∈ H₂, ⁅g₁, g₂⁆ = x }
instance commutator_normal [h₁ : H₁.Normal] [h₂ : H₂.Normal] : Normal ⁅H₁, H₂⁆ := by
  let base : Set G := { x | ∃ g₁ ∈ H₁, ∃ g₂ ∈ H₂, ⁅g₁, g₂⁆ = x }
  

Proof state:
G : Type u_1
G' : Type u_2
F : Type u_3
inst✝² : Group G
inst✝¹ : Group G'
inst✝ : MonoidHomClass F G G'
f : F
g₁ g₂ g₃ g : G
H₁ H₂ H₃ K₁ K₂ : Subgroup G
h₁ : Normal H₁
h₂ : Normal H₂
base : Set G := {x | ∃ g₁ ∈ H₁, ∃ g₂ ∈ H₂, ⁅g₁, g₂⁆ = x}
⊢ Normal ⁅H₁, H₂⁆

Next tactic:
change (closure base).Normal
instance commutator_normal [h₁ : H₁.Normal] [h₂ : H₂.Normal] : Normal ⁅H₁, H₂⁆ := by
  let base : Set G := { x | ∃ g₁ ∈ H₁, ∃ g₂ ∈ H₂, ⁅g₁, g₂⁆ = x }
  change (closure base).Normal
  

Proof state:
G : Type u_1
G' : Type u_2
F : Type u_3
inst✝² : Group G
inst✝¹ : Group G'
inst✝ : MonoidHomClass F G G'
f : F
g₁ g₂ g₃ g : G
H₁ H₂ H₃ K₁ K₂ : Subgroup G
h₁ : Normal H₁
h₂ : Normal H₂
base : Set G := {x | ∃ g₁ ∈ H₁, ∃ g₂ ∈ H₂, ⁅g₁, g₂⁆ = x}
⊢ Normal (closure base)

Next tactic:
suffices h_base : base = Group.conjugatesOfSet base
instance commutator_normal [h₁ : H₁.Normal] [h₂ : H₂.Normal] : Normal ⁅H₁, H₂⁆ := by
  let base : Set G := { x | ∃ g₁ ∈ H₁, ∃ g₂ ∈ H₂, ⁅g₁, g₂⁆ = x }
  change (closure base).Normal
  suffices h_base : base = Group.conjugatesOfSet base
  · 

Proof state:
G : Type u_1
G' : Type u_2
F : Type u_3
inst✝² : Group G
inst✝¹ : Group G'
inst✝ : MonoidHomClass F G G'
f : F
g₁ g₂ g₃ g : G
H₁ H₂ H₃ K₁ K₂ : Subgroup G
h₁ : Normal H₁
h₂ : Normal H₂
base : Set G := {x | ∃ g₁ ∈ H₁, ∃ g₂ ∈ H₂, ⁅g₁, g₂⁆ = x}
h_base : base = Group.conjugatesOfSet base
⊢ Normal (closure base)

Next tactic:
rw [h_base]
instance commutator_normal [h₁ : H₁.Normal] [h₂ : H₂.Normal] : Normal ⁅H₁, H₂⁆ := by
  let base : Set G := { x | ∃ g₁ ∈ H₁, ∃ g₂ ∈ H₂, ⁅g₁, g₂⁆ = x }
  change (closure base).Normal
  suffices h_base : base = Group.conjugatesOfSet base
  · rw [h_base]
    

Proof state:
G : Type u_1
G' : Type u_2
F : Type u_3
inst✝² : Group G
inst✝¹ : Group G'
inst✝ : MonoidHomClass F G G'
f : F
g₁ g₂ g₃ g : G
H₁ H₂ H₃ K₁ K₂ : Subgroup G
h₁ : Normal H₁
h₂ : Normal H₂
base : Set G := {x | ∃ g₁ ∈ H₁, ∃ g₂ ∈ H₂, ⁅g₁, g₂⁆ = x}
h_base : base = Group.conjugatesOfSet base
⊢ Normal (closure (Group.conjugatesOfSet base))

Next tactic:
exact Subgroup.normalClosure_normal
instance commutator_normal [h₁ : H₁.Normal] [h₂ : H₂.Normal] : Normal ⁅H₁, H₂⁆ := by
  let base : Set G := { x | ∃ g₁ ∈ H₁, ∃ g₂ ∈ H₂, ⁅g₁, g₂⁆ = x }
  change (closure base).Normal
  suffices h_base : base = Group.conjugatesOfSet base
  · rw [h_base]
    exact Subgroup.normalClosure_normal
  

Proof state:
case h_base
G : Type u_1
G' : Type u_2
F : Type u_3
inst✝² : Group G
inst✝¹ : Group G'
inst✝ : MonoidHomClass F G G'
f : F
g₁ g₂ g₃ g : G
H₁ H₂ H₃ K₁ K₂ : Subgroup G
h₁ : Normal H₁
h₂ : Normal H₂
base : Set G := {x | ∃ g₁ ∈ H₁, ∃ g₂ ∈ H₂, ⁅g₁, g₂⁆ = x}
⊢ base = Group.conjugatesOfSet base

Next tactic:
refine' Set.Subset.antisymm Group.subset_conjugatesOfSet fun a h => _
instance commutator_normal [h₁ : H₁.Normal] [h₂ : H₂.Normal] : Normal ⁅H₁, H₂⁆ := by
  let base : Set G := { x | ∃ g₁ ∈ H₁, ∃ g₂ ∈ H₂, ⁅g₁, g₂⁆ = x }
  change (closure base).Normal
  suffices h_base : base = Group.conjugatesOfSet base
  · rw [h_base]
    exact Subgroup.normalClosure_normal
  refine' Set.Subset.antisymm Group.subset_conjugatesOfSet fun a h => _
  

Proof state:
case h_base
G : Type u_1
G' : Type u_2
F : Type u_3
inst✝² : Group G
inst✝¹ : Group G'
inst✝ : MonoidHomClass F G G'
f : F
g₁ g₂ g₃ g : G
H₁ H₂ H₃ K₁ K₂ : Subgroup G
h₁ : Normal H₁
h₂ : Normal H₂
base : Set G := {x | ∃ g₁ ∈ H₁, ∃ g₂ ∈ H₂, ⁅g₁, g₂⁆ = x}
a : G
h : a ∈ Group.conjugatesOfSet base
⊢ a ∈ base

Next tactic:
simp_rw [Group.mem_conjugatesOfSet_iff, isConj_iff] at h
instance commutator_normal [h₁ : H₁.Normal] [h₂ : H₂.Normal] : Normal ⁅H₁, H₂⁆ := by
  let base : Set G := { x | ∃ g₁ ∈ H₁, ∃ g₂ ∈ H₂, ⁅g₁, g₂⁆ = x }
  change (closure base).Normal
  suffices h_base : base = Group.conjugatesOfSet base
  · rw [h_base]
    exact Subgroup.normalClosure_normal
  refine' Set.Subset.antisymm Group.subset_conjugatesOfSet fun a h => _
  simp_rw [Group.mem_conjugatesOfSet_iff, isConj_iff] at h
  

Proof state:
case h_base
G : Type u_1
G' : Type u_2
F : Type u_3
inst✝² : Group G
inst✝¹ : Group G'
inst✝ : MonoidHomClass F G G'
f : F
g₁ g₂ g₃ g : G
H₁ H₂ H₃ K₁ K₂ : Subgroup G
h₁ : Normal H₁
h₂ : Normal H₂
base : Set G := {x | ∃ g₁ ∈ H₁, ∃ g₂ ∈ H₂, ⁅g₁, g₂⁆ = x}
a : G
h : ∃ a_1 ∈ {x | ∃ g₁ ∈ H₁, ∃ g₂ ∈ H₂, ⁅g₁, g₂⁆ = x}, ∃ c, c * a_1 * c⁻¹ = a
⊢ a ∈ base

Next tactic:
rcases h with ⟨b, ⟨c, hc, e, he, rfl⟩, d, rfl⟩
instance commutator_normal [h₁ : H₁.Normal] [h₂ : H₂.Normal] : Normal ⁅H₁, H₂⁆ := by
  let base : Set G := { x | ∃ g₁ ∈ H₁, ∃ g₂ ∈ H₂, ⁅g₁, g₂⁆ = x }
  change (closure base).Normal
  suffices h_base : base = Group.conjugatesOfSet base
  · rw [h_base]
    exact Subgroup.normalClosure_normal
  refine' Set.Subset.antisymm Group.subset_conjugatesOfSet fun a h => _
  simp_rw [Group.mem_conjugatesOfSet_iff, isConj_iff] at h
  rcases h with ⟨b, ⟨c, hc, e, he, rfl⟩, d, rfl⟩
  

Proof state:
case h_base.intro.intro.intro.intro.intro.intro.intro
G : Type u_1
G' : Type u_2
F : Type u_3
inst✝² : Group G
inst✝¹ : Group G'
inst✝ : MonoidHomClass F G G'
f : F
g₁ g₂ g₃ g : G
H₁ H₂ H₃ K₁ K₂ : Subgroup G
h₁ : Normal H₁
h₂ : Normal H₂
base : Set G := {x | ∃ g₁ ∈ H₁, ∃ g₂ ∈ H₂, ⁅g₁, g₂⁆ = x}
c : G
hc : c ∈ H₁
e : G
he : e ∈ H₂
d : G
⊢ d * ⁅c, e⁆ * d⁻¹ ∈ base

Next tactic:
exact ⟨_, h₁.conj_mem c hc d, _, h₂.conj_mem e he d, (conjugate_commutatorElement c e d).symm⟩
theorem map_commutator (f : G →* G') : map f ⁅H₁, H₂⁆ = ⁅map f H₁, map f H₂⁆ := by
  

Proof state:
G : Type u_1
G' : Type u_2
F : Type u_3
inst✝² : Group G
inst✝¹ : Group G'
inst✝ : MonoidHomClass F G G'
f✝ : F
g₁ g₂ g₃ g : G
H₁ H₂ H₃ K₁ K₂ : Subgroup G
f : G →* G'
⊢ map f ⁅H₁, H₂⁆ = ⁅map f H₁, map f H₂⁆

Next tactic:
simp_rw [le_antisymm_iff, map_le_iff_le_comap, commutator_le, mem_comap, map_commutatorElement]
theorem map_commutator (f : G →* G') : map f ⁅H₁, H₂⁆ = ⁅map f H₁, map f H₂⁆ := by
  simp_rw [le_antisymm_iff, map_le_iff_le_comap, commutator_le, mem_comap, map_commutatorElement]
  

Proof state:
G : Type u_1
G' : Type u_2
F : Type u_3
inst✝² : Group G
inst✝¹ : Group G'
inst✝ : MonoidHomClass F G G'
f✝ : F
g₁ g₂ g₃ g : G
H₁ H₂ H₃ K₁ K₂ : Subgroup G
f : G →* G'
⊢ (∀ g₁ ∈ H₁, ∀ g₂ ∈ H₂, ⁅f g₁, f g₂⁆ ∈ ⁅map f H₁, map f H₂⁆) ∧
    ∀ g₁ ∈ map f H₁, ∀ g₂ ∈ map f H₂, ⁅g₁, g₂⁆ ∈ map f ⁅H₁, H₂⁆

Next tactic:
constructor
theorem map_commutator (f : G →* G') : map f ⁅H₁, H₂⁆ = ⁅map f H₁, map f H₂⁆ := by
  simp_rw [le_antisymm_iff, map_le_iff_le_comap, commutator_le, mem_comap, map_commutatorElement]
  constructor
  · 

Proof state:
case left
G : Type u_1
G' : Type u_2
F : Type u_3
inst✝² : Group G
inst✝¹ : Group G'
inst✝ : MonoidHomClass F G G'
f✝ : F
g₁ g₂ g₃ g : G
H₁ H₂ H₃ K₁ K₂ : Subgroup G
f : G →* G'
⊢ ∀ g₁ ∈ H₁, ∀ g₂ ∈ H₂, ⁅f g₁, f g₂⁆ ∈ ⁅map f H₁, map f H₂⁆

Next tactic:
intro p hp q hq
theorem map_commutator (f : G →* G') : map f ⁅H₁, H₂⁆ = ⁅map f H₁, map f H₂⁆ := by
  simp_rw [le_antisymm_iff, map_le_iff_le_comap, commutator_le, mem_comap, map_commutatorElement]
  constructor
  · intro p hp q hq
    

Proof state:
case left
G : Type u_1
G' : Type u_2
F : Type u_3
inst✝² : Group G
inst✝¹ : Group G'
inst✝ : MonoidHomClass F G G'
f✝ : F
g₁ g₂ g₃ g : G
H₁ H₂ H₃ K₁ K₂ : Subgroup G
f : G →* G'
p : G
hp : p ∈ H₁
q : G
hq : q ∈ H₂
⊢ ⁅f p, f q⁆ ∈ ⁅map f H₁, map f H₂⁆

Next tactic:
exact commutator_mem_commutator (mem_map_of_mem _ hp) (mem_map_of_mem _ hq)
theorem map_commutator (f : G →* G') : map f ⁅H₁, H₂⁆ = ⁅map f H₁, map f H₂⁆ := by
  simp_rw [le_antisymm_iff, map_le_iff_le_comap, commutator_le, mem_comap, map_commutatorElement]
  constructor
  · intro p hp q hq
    exact commutator_mem_commutator (mem_map_of_mem _ hp) (mem_map_of_mem _ hq)
  · 

Proof state:
case right
G : Type u_1
G' : Type u_2
F : Type u_3
inst✝² : Group G
inst✝¹ : Group G'
inst✝ : MonoidHomClass F G G'
f✝ : F
g₁ g₂ g₃ g : G
H₁ H₂ H₃ K₁ K₂ : Subgroup G
f : G →* G'
⊢ ∀ g₁ ∈ map f H₁, ∀ g₂ ∈ map f H₂, ⁅g₁, g₂⁆ ∈ map f ⁅H₁, H₂⁆

Next tactic:
rintro _ ⟨p, hp, rfl⟩ _ ⟨q, hq, rfl⟩
theorem map_commutator (f : G →* G') : map f ⁅H₁, H₂⁆ = ⁅map f H₁, map f H₂⁆ := by
  simp_rw [le_antisymm_iff, map_le_iff_le_comap, commutator_le, mem_comap, map_commutatorElement]
  constructor
  · intro p hp q hq
    exact commutator_mem_commutator (mem_map_of_mem _ hp) (mem_map_of_mem _ hq)
  · rintro _ ⟨p, hp, rfl⟩ _ ⟨q, hq, rfl⟩
    

Proof state:
case right.intro.intro.intro.intro
G : Type u_1
G' : Type u_2
F : Type u_3
inst✝² : Group G
inst✝¹ : Group G'
inst✝ : MonoidHomClass F G G'
f✝ : F
g₁ g₂ g₃ g : G
H₁ H₂ H₃ K₁ K₂ : Subgroup G
f : G →* G'
p : G
hp : p ∈ ↑H₁
q : G
hq : q ∈ ↑H₂
⊢ ⁅f p, f q⁆ ∈ map f ⁅H₁, H₂⁆

Next tactic:
rw [← map_commutatorElement]
theorem map_commutator (f : G →* G') : map f ⁅H₁, H₂⁆ = ⁅map f H₁, map f H₂⁆ := by
  simp_rw [le_antisymm_iff, map_le_iff_le_comap, commutator_le, mem_comap, map_commutatorElement]
  constructor
  · intro p hp q hq
    exact commutator_mem_commutator (mem_map_of_mem _ hp) (mem_map_of_mem _ hq)
  · rintro _ ⟨p, hp, rfl⟩ _ ⟨q, hq, rfl⟩
    rw [← map_commutatorElement]
    

Proof state:
case right.intro.intro.intro.intro
G : Type u_1
G' : Type u_2
F : Type u_3
inst✝² : Group G
inst✝¹ : Group G'
inst✝ : MonoidHomClass F G G'
f✝ : F
g₁ g₂ g₃ g : G
H₁ H₂ H₃ K₁ K₂ : Subgroup G
f : G →* G'
p : G
hp : p ∈ ↑H₁
q : G
hq : q ∈ ↑H₂
⊢ f ⁅p, q⁆ ∈ map f ⁅H₁, H₂⁆

Next tactic:
exact mem_map_of_mem _ (commutator_mem_commutator hp hq)
theorem commutator_prod_prod (K₁ K₂ : Subgroup G') :
    ⁅H₁.prod K₁, H₂.prod K₂⁆ = ⁅H₁, H₂⁆.prod ⁅K₁, K₂⁆ := by
  

Proof state:
G : Type u_1
G' : Type u_2
F : Type u_3
inst✝² : Group G
inst✝¹ : Group G'
inst✝ : MonoidHomClass F G G'
f : F
g₁ g₂ g₃ g : G
H₁ H₂ H₃ K₁✝ K₂✝ : Subgroup G
K₁ K₂ : Subgroup G'
⊢ ⁅prod H₁ K₁, prod H₂ K₂⁆ = prod ⁅H₁, H₂⁆ ⁅K₁, K₂⁆

Next tactic:
apply le_antisymm
theorem commutator_prod_prod (K₁ K₂ : Subgroup G') :
    ⁅H₁.prod K₁, H₂.prod K₂⁆ = ⁅H₁, H₂⁆.prod ⁅K₁, K₂⁆ := by
  apply le_antisymm
  · 

Proof state:
case a
G : Type u_1
G' : Type u_2
F : Type u_3
inst✝² : Group G
inst✝¹ : Group G'
inst✝ : MonoidHomClass F G G'
f : F
g₁ g₂ g₃ g : G
H₁ H₂ H₃ K₁✝ K₂✝ : Subgroup G
K₁ K₂ : Subgroup G'
⊢ ⁅prod H₁ K₁, prod H₂ K₂⁆ ≤ prod ⁅H₁, H₂⁆ ⁅K₁, K₂⁆

Next tactic:
rw [commutator_le]
theorem commutator_prod_prod (K₁ K₂ : Subgroup G') :
    ⁅H₁.prod K₁, H₂.prod K₂⁆ = ⁅H₁, H₂⁆.prod ⁅K₁, K₂⁆ := by
  apply le_antisymm
  · rw [commutator_le]
    

Proof state:
case a
G : Type u_1
G' : Type u_2
F : Type u_3
inst✝² : Group G
inst✝¹ : Group G'
inst✝ : MonoidHomClass F G G'
f : F
g₁ g₂ g₃ g : G
H₁ H₂ H₃ K₁✝ K₂✝ : Subgroup G
K₁ K₂ : Subgroup G'
⊢ ∀ g₁ ∈ prod H₁ K₁, ∀ g₂ ∈ prod H₂ K₂, ⁅g₁, g₂⁆ ∈ prod ⁅H₁, H₂⁆ ⁅K₁, K₂⁆

Next tactic:
rintro ⟨p₁, p₂⟩ ⟨hp₁, hp₂⟩ ⟨q₁, q₂⟩ ⟨hq₁, hq₂⟩
theorem commutator_prod_prod (K₁ K₂ : Subgroup G') :
    ⁅H₁.prod K₁, H₂.prod K₂⁆ = ⁅H₁, H₂⁆.prod ⁅K₁, K₂⁆ := by
  apply le_antisymm
  · rw [commutator_le]
    rintro ⟨p₁, p₂⟩ ⟨hp₁, hp₂⟩ ⟨q₁, q₂⟩ ⟨hq₁, hq₂⟩
    

Proof state:
case a.mk.intro.mk.intro
G : Type u_1
G' : Type u_2
F : Type u_3
inst✝² : Group G
inst✝¹ : Group G'
inst✝ : MonoidHomClass F G G'
f : F
g₁ g₂ g₃ g : G
H₁ H₂ H₃ K₁✝ K₂✝ : Subgroup G
K₁ K₂ : Subgroup G'
p₁ : G
p₂ : G'
hp₁ : (p₁, p₂).1 ∈ ↑H₁.toSubmonoid
hp₂ : (p₁, p₂).2 ∈ ↑K₁.toSubmonoid
q₁ : G
q₂ : G'
hq₁ : (q₁, q₂).1 ∈ ↑H₂.toSubmonoid
hq₂ : (q₁, q₂).2 ∈ ↑K₂.toSubmonoid
⊢ ⁅(p₁, p₂), (q₁, q₂)⁆ ∈ prod ⁅H₁, H₂⁆ ⁅K₁, K₂⁆

Next tactic:
exact ⟨commutator_mem_commutator hp₁ hq₁, commutator_mem_commutator hp₂ hq₂⟩
theorem commutator_prod_prod (K₁ K₂ : Subgroup G') :
    ⁅H₁.prod K₁, H₂.prod K₂⁆ = ⁅H₁, H₂⁆.prod ⁅K₁, K₂⁆ := by
  apply le_antisymm
  · rw [commutator_le]
    rintro ⟨p₁, p₂⟩ ⟨hp₁, hp₂⟩ ⟨q₁, q₂⟩ ⟨hq₁, hq₂⟩
    exact ⟨commutator_mem_commutator hp₁ hq₁, commutator_mem_commutator hp₂ hq₂⟩
  · 

Proof state:
case a
G : Type u_1
G' : Type u_2
F : Type u_3
inst✝² : Group G
inst✝¹ : Group G'
inst✝ : MonoidHomClass F G G'
f : F
g₁ g₂ g₃ g : G
H₁ H₂ H₃ K₁✝ K₂✝ : Subgroup G
K₁ K₂ : Subgroup G'
⊢ prod ⁅H₁, H₂⁆ ⁅K₁, K₂⁆ ≤ ⁅prod H₁ K₁, prod H₂ K₂⁆

Next tactic:
rw [prod_le_iff]
theorem commutator_prod_prod (K₁ K₂ : Subgroup G') :
    ⁅H₁.prod K₁, H₂.prod K₂⁆ = ⁅H₁, H₂⁆.prod ⁅K₁, K₂⁆ := by
  apply le_antisymm
  · rw [commutator_le]
    rintro ⟨p₁, p₂⟩ ⟨hp₁, hp₂⟩ ⟨q₁, q₂⟩ ⟨hq₁, hq₂⟩
    exact ⟨commutator_mem_commutator hp₁ hq₁, commutator_mem_commutator hp₂ hq₂⟩
  · rw [prod_le_iff]
    

Proof state:
case a
G : Type u_1
G' : Type u_2
F : Type u_3
inst✝² : Group G
inst✝¹ : Group G'
inst✝ : MonoidHomClass F G G'
f : F
g₁ g₂ g₃ g : G
H₁ H₂ H₃ K₁✝ K₂✝ : Subgroup G
K₁ K₂ : Subgroup G'
⊢ map (MonoidHom.inl G G') ⁅H₁, H₂⁆ ≤ ⁅prod H₁ K₁, prod H₂ K₂⁆ ∧
    map (MonoidHom.inr G G') ⁅K₁, K₂⁆ ≤ ⁅prod H₁ K₁, prod H₂ K₂⁆

Next tactic:
constructor
theorem commutator_prod_prod (K₁ K₂ : Subgroup G') :
    ⁅H₁.prod K₁, H₂.prod K₂⁆ = ⁅H₁, H₂⁆.prod ⁅K₁, K₂⁆ := by
  apply le_antisymm
  · rw [commutator_le]
    rintro ⟨p₁, p₂⟩ ⟨hp₁, hp₂⟩ ⟨q₁, q₂⟩ ⟨hq₁, hq₂⟩
    exact ⟨commutator_mem_commutator hp₁ hq₁, commutator_mem_commutator hp₂ hq₂⟩
  · rw [prod_le_iff]
    constructor <;>
      · 

Proof state:
case a.left
G : Type u_1
G' : Type u_2
F : Type u_3
inst✝² : Group G
inst✝¹ : Group G'
inst✝ : MonoidHomClass F G G'
f : F
g₁ g₂ g₃ g : G
H₁ H₂ H₃ K₁✝ K₂✝ : Subgroup G
K₁ K₂ : Subgroup G'
⊢ map (MonoidHom.inl G G') ⁅H₁, H₂⁆ ≤ ⁅prod H₁ K₁, prod H₂ K₂⁆

Next tactic:
rw [map_commutator]
theorem commutator_prod_prod (K₁ K₂ : Subgroup G') :
    ⁅H₁.prod K₁, H₂.prod K₂⁆ = ⁅H₁, H₂⁆.prod ⁅K₁, K₂⁆ := by
  apply le_antisymm
  · rw [commutator_le]
    rintro ⟨p₁, p₂⟩ ⟨hp₁, hp₂⟩ ⟨q₁, q₂⟩ ⟨hq₁, hq₂⟩
    exact ⟨commutator_mem_commutator hp₁ hq₁, commutator_mem_commutator hp₂ hq₂⟩
  · rw [prod_le_iff]
    constructor <;>
      · rw [map_commutator]
        

Proof state:
case a.left
G : Type u_1
G' : Type u_2
F : Type u_3
inst✝² : Group G
inst✝¹ : Group G'
inst✝ : MonoidHomClass F G G'
f : F
g₁ g₂ g₃ g : G
H₁ H₂ H₃ K₁✝ K₂✝ : Subgroup G
K₁ K₂ : Subgroup G'
⊢ ⁅map (MonoidHom.inl G G') H₁, map (MonoidHom.inl G G') H₂⁆ ≤ ⁅prod H₁ K₁, prod H₂ K₂⁆

Next tactic:
apply commutator_mono
theorem commutator_prod_prod (K₁ K₂ : Subgroup G') :
    ⁅H₁.prod K₁, H₂.prod K₂⁆ = ⁅H₁, H₂⁆.prod ⁅K₁, K₂⁆ := by
  apply le_antisymm
  · rw [commutator_le]
    rintro ⟨p₁, p₂⟩ ⟨hp₁, hp₂⟩ ⟨q₁, q₂⟩ ⟨hq₁, hq₂⟩
    exact ⟨commutator_mem_commutator hp₁ hq₁, commutator_mem_commutator hp₂ hq₂⟩
  · rw [prod_le_iff]
    constructor <;>
      · rw [map_commutator]
        apply commutator_mono <;>
          

Proof state:
case a.left.h₁
G : Type u_1
G' : Type u_2
F : Type u_3
inst✝² : Group G
inst✝¹ : Group G'
inst✝ : MonoidHomClass F G G'
f : F
g₁ g₂ g₃ g : G
H₁ H₂ H₃ K₁✝ K₂✝ : Subgroup G
K₁ K₂ : Subgroup G'
⊢ map (MonoidHom.inl G G') H₁ ≤ prod H₁ K₁

Next tactic:
simp [le_prod_iff, map_map, MonoidHom.fst_comp_inl, MonoidHom.snd_comp_inl,
            MonoidHom.fst_comp_inr, MonoidHom.snd_comp_inr]
theorem commutator_prod_prod (K₁ K₂ : Subgroup G') :
    ⁅H₁.prod K₁, H₂.prod K₂⁆ = ⁅H₁, H₂⁆.prod ⁅K₁, K₂⁆ := by
  apply le_antisymm
  · rw [commutator_le]
    rintro ⟨p₁, p₂⟩ ⟨hp₁, hp₂⟩ ⟨q₁, q₂⟩ ⟨hq₁, hq₂⟩
    exact ⟨commutator_mem_commutator hp₁ hq₁, commutator_mem_commutator hp₂ hq₂⟩
  · rw [prod_le_iff]
    constructor <;>
      · rw [map_commutator]
        apply commutator_mono <;>
          

Proof state:
case a.left.h₂
G : Type u_1
G' : Type u_2
F : Type u_3
inst✝² : Group G
inst✝¹ : Group G'
inst✝ : MonoidHomClass F G G'
f : F
g₁ g₂ g₃ g : G
H₁ H₂ H₃ K₁✝ K₂✝ : Subgroup G
K₁ K₂ : Subgroup G'
⊢ map (MonoidHom.inl G G') H₂ ≤ prod H₂ K₂

Next tactic:
simp [le_prod_iff, map_map, MonoidHom.fst_comp_inl, MonoidHom.snd_comp_inl,
            MonoidHom.fst_comp_inr, MonoidHom.snd_comp_inr]