pair_add_smul_left_iff — Mathlib

English

If each f(i) is linearly independent with respect to id, and a certain finite-disjointness condition holds for all i and finite t, then the iUnion of f(i) is linearly independent.

Русский

Если каждый f(i) линейно независим и выполняется условие конечной раздельности, то iUnion f(i) линейно независим.

LaTeX

$$$\\text{linearIndepOn}_{i}\\; f(i) \\Rightarrow \\text{independence of union with finite disjointness}$$$

Lean4

@[simp]
theorem pair_add_smul_left_iff : LinearIndependent R ![x + b • y, y] ↔ LinearIndependent R ![x, y] :=
  by
  rcases subsingleton_or_nontrivial S with hS | hS; · simp [hS.elim b 0]
  nontriviality R
  simpa using pair_add_smul_add_smul_iff (x := x) (y := y) 1 b 0 1

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