English
Suppose a,b ∈ ℤ satisfy a = 0 ↔ b = 0. Then a·b ∈ {0,1,2,3,4} if and only if the pair (a,b) lies in the explicit finite set of 17 pairs listed (including (0,0), (1,1), (2,2), (1,3), (3,1), etc.).
Русский
Пусть a,b ∈ ℤ удовлетворяют a = 0 ⇔ b = 0. Тогда ab ∈ {0,1,2,3,4} тогда и только тогда, когда пара (a,b) принадлежит перечисленному конечному множеству из 17 пар (включая (0,0), (1,1), (2,2), (1,3), (3,1) и т.д.).
LaTeX
$$$ a,b \in \mathbb Z,\ (a=0 \iff b=0) \implies a b \in \{0,1,2,3,4\} \iff (a,b) \in \{(0,0),(1,1),(-1,-1),(1,2),(2,1),(-1,-2),(-2,-1),(1,3),(3,1),(-1,-3),(-3,-1),(4,1),(1,4),(-4,-1),(-1,-4),(2,2),(-2,-2)\}.$$$
Lean4
/-- This lemma justifies its existence from its utility in crystallographic root system theory. -/
theorem mul_mem_one_two_three_iff {a b : ℤ} :
a * b ∈ ({ 1, 2, 3 } : Set ℤ) ↔
(a, b) ∈
({(1, 1), (-1, -1), (1, 2), (2, 1), (-1, -2), (-2, -1), (1, 3), (3, 1), (-1, -3), (-3, -1)} : Set (ℤ × ℤ)) :=
by
simp only [← Int.prodMk_mem_divisorsAntidiag, Set.mem_insert_iff, Set.mem_singleton_iff, ne_eq, one_ne_zero,
not_false_eq_true, OfNat.ofNat_ne_zero]
aesop
