disjoint_normalizedFactors — Mathlib

English

For nonzero x, the multiset of normalized factors is nonempty exactly when x is not a unit.

Русский

Для ненулевого x мультимножество нормализованных факторов не пусто тогда и только тогда, когда x не единица.

LaTeX

$$normalizedFactors_pos(x) \iff (x \text{ is not a unit})$$

Lean4

/-- Relatively prime elements have disjoint prime factors (as multisets). -/
theorem disjoint_normalizedFactors {a b : α} (hc : IsRelPrime a b) :
    Disjoint (normalizedFactors a) (normalizedFactors b) :=
  by
  rw [Multiset.disjoint_left]
  intro x hxa hxb
  have x_dvd_a := dvd_of_mem_normalizedFactors hxa
  have x_dvd_b := dvd_of_mem_normalizedFactors hxb
  exact (prime_of_normalized_factor x hxa).not_unit (hc x_dvd_a x_dvd_b)

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