linearIndependent_iff_sum — Mathlib

English

The span of eval C equals the span of the sum of eval branches.

Русский

Замыкание образов eval C совпадает с замыканием суммы ветвей eval.

LaTeX

$$$\\mathrm{range}(\\mathrm{eval}\\ C) = \\mathrm{range}\\left(\\mathrm{Sum.elim}\\ (\\mathrm{lambda } l. \\mathrm{Products.eval}\\ C l.1) (\\mathrm{lambda } l.1. \\mathrm{Products.eval}\\ C l.1)\\right)$$$

Lean4

theorem linearIndependent_iff_sum : LinearIndependent ℤ (eval C) ↔ LinearIndependent ℤ (SumEval C ho) :=
  by
  rw [← linearIndependent_equiv (sum_equiv C hsC ho), SumEval, ← sum_equiv_comp_eval_eq_elim C hsC ho]
  exact Iff.rfl

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