exchange — Mathlib

English

If I and J are bases on X and e ∈ I \ J, then there exists f ∈ J \ I with a modified basis on X including f.

Русский

Если I и J — базы на X и e ∈ I \ J, то существует f ∈ J \ I such that новая база содержит f.

LaTeX

$$∃ f ∈ J \ I, M.IsBasis (insert f (I \ { e })) X$$

Lean4

theorem exchange (hIX : M.IsBasis I X) (hJX : M.IsBasis J X) (he : e ∈ I \ J) :
    ∃ f ∈ J \ I, M.IsBasis (insert f (I \ { e })) X :=
  by
  obtain ⟨y, hy, h⟩ := hIX.restrict_isBase.exchange hJX.restrict_isBase he
  exact ⟨y, hy, by rwa [isBase_restrict_iff] at h⟩

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