English
Let η be a index set, with ιs(j) a family of index types and f(j) vectors; if each f(j) is linearly independent and a finite-disjointness condition holds, then the iUnion over j of f(j) is linearly independent.
Русский
Пусть η индексирует семейство векторов f(j); если каждый f(j) линейно независим и выполняется конечная диск joint-условие, то объединение по iJa независим.
LaTeX
$$$\\forall j,\\ \\mathrm{LinearIndependent}_R (f(j)) \\land \\text{finite-hard-disjointness} \\Rightarrow \\mathrm{LinearIndependent}_R (\\bigcup_j f(j))$$$
Lean4
@[simp]
theorem pair_add_right_iff : LinearIndependent R ![x, x + y] ↔ LinearIndependent R ![x, y] :=
by
suffices ∀ x y : M, LinearIndependent R ![x, x + y] → LinearIndependent R ![x, y] from
⟨this x y, fun h ↦ by simpa using this (-x) (x + y) (by simpa)⟩
simp only [LinearIndependent.pair_iff]
intro x y h s t h'
obtain ⟨h₁, h₂⟩ := h (s - t) t (by rw [sub_smul, smul_add, ← h']; abel)
rw [h₂, sub_zero] at h₁
tauto
