case pos
G₀ : Type u_1
inst✝ : GroupWithZero G₀
a : G₀
m n : ℤ
h : a ≠ 0 ∨ m + n ≠ 0 ∨ m = 0 ∧ n = 0
hm : m = 0
⊢ a ^ (m + n) = a ^ m * a ^ n

case neg
G₀ : Type u_1
inst✝ : GroupWithZero G₀
a : G₀
m n : ℤ
h : a ≠ 0 ∨ m + n ≠ 0 ∨ m = 0 ∧ n = 0
hm : ¬m = 0
⊢ a ^ (m + n) = a ^ m * a ^ n
G₀ : Type u_1
inst✝ : GroupWithZero G₀
a : G₀
m n : ℤ
h : a ≠ 0 ∨ m + n ≠ 0 ∨ m = 0 ∧ n = 0
⊢ a ^ (m + n) = a ^ m * a ^ n
by_cases hm : m = 0
case pos
G₀ : Type u_1
inst✝ : GroupWithZero G₀
a : G₀
m n : ℤ
h : a ≠ 0 ∨ m + n ≠ 0 ∨ m = 0 ∧ n = 0
hm : ¬m = 0
hn : n = 0
⊢ a ^ (m + n) = a ^ m * a ^ n

case neg
G₀ : Type u_1
inst✝ : GroupWithZero G₀
a : G₀
m n : ℤ
h : a ≠ 0 ∨ m + n ≠ 0 ∨ m = 0 ∧ n = 0
hm : ¬m = 0
hn : ¬n = 0
⊢ a ^ (m + n) = a ^ m * a ^ n
case neg
G₀ : Type u_1
inst✝ : GroupWithZero G₀
a : G₀
m n : ℤ
h : a ≠ 0 ∨ m + n ≠ 0 ∨ m = 0 ∧ n = 0
hm : ¬m = 0
⊢ a ^ (m + n) = a ^ m * a ^ n
by_cases hn : n = 0
case pos
G₀ : Type u_1
inst✝ : GroupWithZero G₀
a : G₀
m n : ℤ
h : a ≠ 0 ∨ m + n ≠ 0 ∨ m = 0 ∧ n = 0
hm : ¬m = 0
hn : ¬n = 0
ha : a = 0
⊢ a ^ (m + n) = a ^ m * a ^ n

case neg
G₀ : Type u_1
inst✝ : GroupWithZero G₀
a : G₀
m n : ℤ
h : a ≠ 0 ∨ m + n ≠ 0 ∨ m = 0 ∧ n = 0
hm : ¬m = 0
hn : ¬n = 0
ha : ¬a = 0
⊢ a ^ (m + n) = a ^ m * a ^ n
case neg
G₀ : Type u_1
inst✝ : GroupWithZero G₀
a : G₀
m n : ℤ
h : a ≠ 0 ∨ m + n ≠ 0 ∨ m = 0 ∧ n = 0
hm : ¬m = 0
hn : ¬n = 0
⊢ a ^ (m + n) = a ^ m * a ^ n
by_cases ha : a = 0
no goals
case pos
G₀ : Type u_1
inst✝ : GroupWithZero G₀
a : G₀
m n : ℤ
h : a ≠ 0 ∨ m + n ≠ 0 ∨ m = 0 ∧ n = 0
hm : m = 0
⊢ a ^ (m + n) = a ^ m * a ^ n
simp [hm]
no goals
case pos
G₀ : Type u_1
inst✝ : GroupWithZero G₀
a : G₀
m n : ℤ
h : a ≠ 0 ∨ m + n ≠ 0 ∨ m = 0 ∧ n = 0
hm : ¬m = 0
hn : n = 0
⊢ a ^ (m + n) = a ^ m * a ^ n
simp [hn]
case pos
G₀ : Type u_1
inst✝ : GroupWithZero G₀
m n : ℤ
hm : ¬m = 0
hn : ¬n = 0
h : 0 ≠ 0 ∨ m + n ≠ 0 ∨ m = 0 ∧ n = 0
⊢ 0 ^ (m + n) = 0 ^ m * 0 ^ n
case pos
G₀ : Type u_1
inst✝ : GroupWithZero G₀
a : G₀
m n : ℤ
h : a ≠ 0 ∨ m + n ≠ 0 ∨ m = 0 ∧ n = 0
hm : ¬m = 0
hn : ¬n = 0
ha : a = 0
⊢ a ^ (m + n) = a ^ m * a ^ n
subst a
case pos
G₀ : Type u_1
inst✝ : GroupWithZero G₀
m n : ℤ
hm : ¬m = 0
hn : ¬n = 0
h : ¬m + n = 0
⊢ 0 ^ (m + n) = 0 ^ m * 0 ^ n
case pos
G₀ : Type u_1
inst✝ : GroupWithZero G₀
m n : ℤ
hm : ¬m = 0
hn : ¬n = 0
h : 0 ≠ 0 ∨ m + n ≠ 0 ∨ m = 0 ∧ n = 0
⊢ 0 ^ (m + n) = 0 ^ m * 0 ^ n
simp only [false_or_iff, eq_self_iff_true, not_true, Ne.def, hm, hn, false_and_iff,
  or_false_iff] at h
no goals
case pos
G₀ : Type u_1
inst✝ : GroupWithZero G₀
m n : ℤ
hm : ¬m = 0
hn : ¬n = 0
h : ¬m + n = 0
⊢ 0 ^ (m + n) = 0 ^ m * 0 ^ n
rw [zero_zpow _ h, zero_zpow _ hm, zero_mul]
no goals
case neg
G₀ : Type u_1
inst✝ : GroupWithZero G₀
a : G₀
m n : ℤ
h : a ≠ 0 ∨ m + n ≠ 0 ∨ m = 0 ∧ n = 0
hm : ¬m = 0
hn : ¬n = 0
ha : ¬a = 0
⊢ a ^ (m + n) = a ^ m * a ^ n
exact zpow_add₀ ha m n