pair_add_smul_right_iff — Mathlib

English

There exists a maximal independent set for the family v, i.e., a subset s such that v|_s is independent and inclusion-maximal with this property.

Русский

Существует максимальное по включению независимое подмножество семейства v.

LaTeX

$$$\\exists s\\; (\\mathrm{LinearIndepOn}_R(v,s) \\land \\forall t\\supseteq s,\\ \\mathrm{LinearIndepOn}_R(v,t) \\Rightarrow t=s)$$$

Lean4

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

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