linearIndepOn_id_union_iff — Mathlib

English

If s and t are disjoint subsets of M, then LinearIndepOn R id (s ∪ t) is equivalent to LI on s and t with disjoint spans.

Русский

Если s и t — дизjoint подмножества M, то LinearIndepOn R id (s ∪ t) эквивалентно независимости на s и t с дизjoint спанами.

LaTeX

$$$\\text{LinearIndepOn } R id (s\\cup t) \\iff \\text{LinearIndepOn } R id s \\land \\text{LinearIndepOn } R id t \\land \\text{Disjoint}(\\operatorname{span} s, \\operatorname{span} t)$$$

Lean4

theorem linearIndepOn_id_union_iff {s t : Set M} (hdj : Disjoint s t) :
    LinearIndepOn R id (s ∪ t) ↔ LinearIndepOn R id s ∧ LinearIndepOn R id t ∧ Disjoint (span R s) (span R t) := by
  rw [linearIndepOn_union_iff hdj, image_id, image_id]

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