isPrincipalIdealRing_pi_iff — Mathlib

English

For nonzero f, we have f = divXPowOrder f if and only if f is a unit.

Русский

Для непустого f верно: f = divXPowOrder f ⇔ f является единицей.

LaTeX

$$$ f \neq 0 \Rightarrow (f = divXPowOrder f \iff IsUnit f). $$$

Lean4

theorem isPrincipalIdealRing_pi_iff {ι} [Finite ι] {R : ι → Type*} [∀ i, Semiring (R i)] :
    IsPrincipalIdealRing (Π i, R i) ↔ ∀ i, IsPrincipalIdealRing (R i)
    where
  mp h i := h.of_surjective (Pi.evalRingHom R i) (Function.surjective_eval _)
  mpr _ := inferInstance

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