not_dvd_lcm — Mathlib

English

If p is prime and p does not divide a and p does not divide b, then p does not divide lcm(a,b).

Русский

Если p простое и p не делит a и p не делит b, то p не делит НОК(a,b).

LaTeX

$$$\forall {p,a,b} \in \mathbb{N}, \text{Prime}(p) \land (p \nmid a) \land (p \nmid b) \Rightarrow (p \nmid \operatorname{lcm}(a,b))$$$

Lean4

theorem not_dvd_lcm (ha : ¬p ∣ a) (hb : ¬p ∣ b) : ¬p ∣ lcm a b :=
  hp.dvd_lcm.not.mpr <| not_or.mpr ⟨ha, hb⟩

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