coe_eq_zero_iff — Mathlib

English

The canonical map 𝓞K → K is injective in the sense that algebraMap x = 0 if and only if x = 0 in 𝓞K.

Русский

Каноническое отображение 𝓞K → K инъективно в том смысле, что algebraMap x = 0, если и только если x = 0 в 𝓞K.

LaTeX

$$$\\big( \\mathrm{algebraMap}_{\\mathcal{O}_K,K} (x) = 0 \\big) \\iff x = 0$$$

Lean4

/-- The canonical map from `𝓞 K` to `K` is injective.

This is a convenient abbreviation for `map_eq_zero_iff` applied to
`FaithfulSMul.algebraMap_injective`.
-/
theorem coe_eq_zero_iff {x : 𝓞 K} : algebraMap _ K x = 0 ↔ x = 0 :=
  map_eq_zero_iff _ coe_injective

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