disjoint_span_image — Mathlib

English

If v and w agree on s, then the linear independence of v on s is equivalent to that of w on s.

Русский

Если v и w совпадают на s, независимость на s сохраняется: независимость v на s эквивалентна независимости w на s.

LaTeX

$$LinearIndepOn R v s ↔ LinearIndepOn R w s$$

Lean4

theorem disjoint_span_image (hv : LinearIndependent R v) {s t : Set ι} (hs : Disjoint s t) :
    Disjoint (Submodule.span R <| v '' s) (Submodule.span R <| v '' t) :=
  by
  simp only [disjoint_def, Finsupp.mem_span_image_iff_linearCombination]
  rintro _ ⟨l₁, hl₁, rfl⟩ ⟨l₂, hl₂, H⟩
  rw [hv.finsuppLinearCombination_injective.eq_iff] at H; subst l₂
  have : l₁ = 0 := Submodule.disjoint_def.mp (Finsupp.disjoint_supported_supported hs) _ hl₁ hl₂
  simp [this]

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