no goals
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a : α
ha : Finite a a
⊢ a ^ 1 ∣ a
simp
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a : α
ha : Finite a a
x✝ : a ^ (1 + 1) ∣ a
b : α
hb : 1 = 1 * (a * 1 * b)
⊢ False
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a : α
ha : Finite a a
x✝ : a ^ (1 + 1) ∣ a
b : α
hb : a = a ^ (1 + 1) * b
⊢ False
rw [← mul_one a, pow_add, pow_one, mul_assoc, mul_assoc,
    mul_right_inj' (ne_zero_of_finite ha)] at hb
no goals
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a : α
ha : Finite a a
x✝ : a ^ (1 + 1) ∣ a
b : α
hb : 1 = 1 * (a * 1 * b)
⊢ False
exact
  mt isUnit_iff_dvd_one.2 (not_unit_of_finite ha) ⟨b, by simp_all⟩
no goals
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a : α
ha : Finite a a
x✝ : a ^ (1 + 1) ∣ a
b : α
hb : 1 = 1 * (a * 1 * b)
⊢ 1 = a * b
simp_all