disjoint_span_singleton — Mathlib

English

Disjointness between s and the span of a singleton is characterized by whether the vector lies outside s.

Русский

Дисjoint-множество между s и span{x} определяется тем, лежит ли x вне s.

LaTeX

$$$Disjoint\; s\; (\operatorname{span}\;\{x\}) \iff x \in s \rightarrow x = 0$$$

Lean4

theorem disjoint_span_singleton : Disjoint s (K ∙ x) ↔ x ∈ s → x = 0 :=
  by
  refine disjoint_def.trans ⟨fun H hx => H x hx <| subset_span <| mem_singleton x, ?_⟩
  intro H y hy hyx
  obtain ⟨c, rfl⟩ := mem_span_singleton.1 hyx
  by_cases hc : c = 0
  · rw [hc, zero_smul]
  · rw [s.smul_mem_iff hc] at hy
    rw [H hy, smul_zero]

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