closure_biInter_eq_biInter_closure_of_biUnion_indep

English

If A is nonempty and I_i are independent, then closure of the biInter equals the biInter of closures.

Русский

Если A непусто и I_i независимы, то cl(⋂ i∈A I_i) = ⋂ i∈A cl(I_i).

LaTeX

$$$A \neq \varnothing \Rightarrow \operatorname{cl}\left(\bigcap_{i\in A} I_i\right) = \bigcap_{i\in A} \operatorname{cl}(I_i)$$$

Lean4

theorem closure_biInter_eq_biInter_closure_of_biUnion_indep {ι : Type*} {A : Set ι} (hA : A.Nonempty) {I : ι → Set α}
    (h : M.Indep (⋃ i ∈ A, I i)) : M.closure (⋂ i ∈ A, I i) = ⋂ i ∈ A, M.closure (I i) :=
  by
  have := hA.coe_sort
  convert closure_iInter_eq_iInter_closure_of_iUnion_indep (Is := fun i : A ↦ I i) (by simpa) <;> simp

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