span_generators — Mathlib

English

The submodule p is exactly the span of its generating set: span_R(generators(p)) = p.

Русский

Подмодуль p равен порождаемому множеству generators(p): span_R(generators(p)) = p.

LaTeX

$$$\operatorname{span}_R(\mathrm{generators}(p)) = p.$$$

Lean4

/-- The span of the generators equals the submodule. -/
theorem span_generators (p : Submodule R M) : span R (generators p) = p :=
  (Classical.choose_spec (exists_span_set_card_eq_spanRank p)).2

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