irreducible — Mathlib

English

A prime element is irreducible in a CancelCommutativeMonoidWithZero.

Русский

Простая элемент является несводимым в CancelCommоdio With Zero.

LaTeX

$$Prime p → Irreducible p$$

Lean4

protected theorem irreducible (hp : Prime p) : Irreducible p :=
  ⟨hp.not_unit, fun a b ↦ by
    rintro rfl
    exact
      (hp.dvd_or_dvd dvd_rfl).symm.imp
        (isUnit_of_dvd_one <| (mul_dvd_mul_iff_right <| right_ne_zero_of_mul hp.ne_zero).mp <| dvd_mul_of_dvd_right · _)
        (isUnit_of_dvd_one <| (mul_dvd_mul_iff_left <| left_ne_zero_of_mul hp.ne_zero).mp <| dvd_mul_of_dvd_left · _)⟩

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