mem_span_repr_support — Mathlib

English

For a basis b and a vector m, the support of b.repr m is contained in any set of indices s that spans m; equivalently m lies in the span of the corresponding basis elements.

Русский

Для базиса b и вектора m множество индексов, на которых_repr m, поддержка которого ⊆ s, если m принадлежит span R (b '' s).

LaTeX

$$$m\in \operatorname{span}\!\big(b''(\operatorname{repr}(m).\operatorname{support})\big).$$$

Lean4

theorem mem_span_repr_support (m : M) : m ∈ span R (b '' (b.repr m).support) :=
  (Finsupp.mem_span_image_iff_linearCombination _).2 ⟨b.repr m, by simp [Finsupp.mem_supported_support]⟩

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