noZeroDivisors — Mathlib

English

If x ≠ 0 in the localization S, then the first component of its secant (sec M x) is nonzero.

Русский

Если x ≠ 0 в локализации S, то первая компонента секанта (sec M x) непустая (не равна нулю).

LaTeX

$$$\\text{sec }M\\; x = (\\text{fst}, \\text{snd})\\; \\Rightarrow\\; x \\neq 0 \\Rightarrow (\\text{fst }(\\text{sec }M\\; x)) \\neq 0.$$$

Lean4

/-- Any localization of a commutative semiring without zero-divisors also has no zero-divisors. -/
theorem noZeroDivisors [NoZeroDivisors R] : NoZeroDivisors S :=
  (toLocalizationMap M S).noZeroDivisors

Похожие утверждения