toGCDDomain — Mathlib

English

In a principal ideal domain that is a domain, if a ≠ 0 and b is a factor of a, then b ≠ 0.

Русский

В областьном кольце с первым принципом, если a ≠ 0 и b делит a, то b ≠ 0.

LaTeX

$$$a \neq 0 \\\\Rightarrow b \\\\in \\\\text{factors}(a) \\\\Rightarrow b \\\\neq 0$$$

Lean4

/-- Any Bézout domain is a GCD domain. This is not an instance since `GCDMonoid` contains data,
and this might not be how we would like to construct it. -/
noncomputable def toGCDDomain [IsBezout R] [IsDomain R] [DecidableEq R] : GCDMonoid R :=
  gcdMonoidOfGCD (gcd · ·) (gcd_dvd_left · ·) (gcd_dvd_right · ·) dvd_gcd

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