exists_insert_of_not_isBasis — Mathlib

English

If I and J are bases for M on X (IsBasis'), then their encard are equal: I.encard = J.encard.

Русский

Если I и J являются базами M на X (IsBasis'), то их encard равны: I.encard = J.encard.

LaTeX

$$I.encard = J.encard$$

Lean4

theorem exists_insert_of_not_isBasis (hI : M.Indep I) (hIX : I ⊆ X) (hI' : ¬M.IsBasis I X) (hJ : M.IsBasis J X) :
    ∃ e ∈ J \ I, M.Indep (insert e I) := by
  rw [← isBase_restrict_iff] at hI'; rw [← isBase_restrict_iff] at hJ
  obtain ⟨e, he, hi⟩ := (hI.indep_restrict_of_subset hIX).exists_insert_of_not_isBase hI' hJ
  exact ⟨e, he, (restrict_indep_iff.mp hi).1⟩

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