disjoint_support_closure_of_disjoint_support

English

If every pair of elements a ∈ S and b ∈ T has disjoint supports, then the closures of S and T are disjoint.

Русский

Если каждый элемент a из S и b из T имеют взаимноdisjoint поддержки, то замыкания S и T дисjoint.

LaTeX

$$$\\forall a \\in \\overline{S}, \\forall b \\in \\overline{T}, \\operatorname{Disjoint}(a.\\operatorname{support}, b.\\operatorname{support}).$$$

Lean4

theorem disjoint_support_closure_of_disjoint_support {S T : Set (Perm α)}
    (h : ∀ a ∈ S, ∀ b ∈ T, _root_.Disjoint a.support b.support) :
    ∀ a ∈ closure S, ∀ b ∈ closure T, _root_.Disjoint a.support b.support :=
  by
  intro a ha b hb
  have key1 := support_closure_subset_union S a ha
  have key2 := support_closure_subset_union T b hb
  have key := Set.disjoint_of_subset key1 key2
  simp_rw [Set.disjoint_iUnion_left, Set.disjoint_iUnion_right, Finset.disjoint_coe] at key
  exact key h

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