contract_isBasis_of_disjoint_indep

English

If h is a basis I X and J is disjoint from X and I ∪ J is independent, then contracting by J preserves independence of I within X.

Русский

Если I X образуют базис и J дизjoint с X и I ∪ J независим, то сокращение по J сохраняет независимость I внутри X.

LaTeX

$$$(M IsBasis I X) \\rightarrow (Disjoint J X) \\rightarrow M.Indep(I \\cup J) \\Rightarrow (M ／ J).IsBasis' (I \\ J) (X \\ J)$$

Lean4

theorem contract_isBasis_of_disjoint_indep (h : M.IsBasis I X) (hdj : Disjoint J X) (h_ind : M.Indep (I ∪ J)) :
    (M ／ J).IsBasis I X :=
  by
  rw [← hdj.sdiff_eq_right, ← (hdj.mono_right h.subset).sdiff_eq_right]
  exact h.contract_isBasis_of_indep h_ind

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