not_associated — Mathlib

English

Let p be a prime in a CancelCommMonoidWithZero M. Then q divides p^n iff there exists i ≤ n with q associated to p^i.

Русский

Пусть p — простое в CancelCommMonoidWithZero M. Тогда q делит p^n, если и только если существует i ≤ n such that q ассоциировано с p^i.

LaTeX

$$$ q \mid p^n \iff \exists i \le n,\ Associated(q, p^i) $$$

Lean4

theorem not_associated [CancelCommMonoidWithZero M] {p q : M} (h : DvdNotUnit p q) : ¬Associated p q :=
  by
  rintro ⟨a, rfl⟩
  obtain ⟨hp, x, hx, hx'⟩ := h
  rcases (mul_right_inj' hp).mp hx' with rfl
  exact hx a.isUnit

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