mem_span_range_iff_exists_fun — Mathlib

English

Let α be finite. Then x ∈ span(Set.range v) iff there exists c : α → R with x = ∑ i c(i) v(i).

Русский

Пусть α конечен. Тогда x ∈ span(Set.range v), если существует c : α → R such that x = ∑ i c(i) v(i).

LaTeX

$$$x \\in \\operatorname{span}(\\operatorname{Set.range} v) \\iff \\exists c : \\alpha \\to R, \\sum i, c(i) \\cdot v(i) = x$$$

Lean4

/-- An element `x` lies in the span of `v` iff it can be written as sum `∑ cᵢ • vᵢ = x`.
-/
theorem mem_span_range_iff_exists_fun [Fintype α] {x : R} {v : α → R} :
    x ∈ Ideal.span (Set.range v) ↔ ∃ c : α → R, ∑ i, c i * v i = x :=
  Submodule.mem_span_range_iff_exists_fun _

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