disjoint_of_disjoint_support — Mathlib

English

If every element of H has disjoint support from every element of K, then H and K are disjoint subgroups of Perm α (their intersection is trivial).

Русский

Если каждый элемент H имеет дисjointную поддержку с каждым элементом K, то H и K — дисjoint подгруппы группы перестановок.

LaTeX

$$$\\forall a \\in H,\\forall b \\in K,\\ \\operatorname{Disjoint}(a, b) \\Rightarrow \\operatorname{Disjoint}(H, K).$$$

Lean4

theorem disjoint_of_disjoint_support {H K : Subgroup (Perm α)}
    (h : ∀ a ∈ H, ∀ b ∈ K, _root_.Disjoint a.support b.support) : _root_.Disjoint H K :=
  by
  rw [disjoint_iff_inf_le]
  intro x ⟨hx1, hx2⟩
  specialize h x hx1 x hx2
  rwa [disjoint_self, Finset.bot_eq_empty, support_eq_empty_iff] at h

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