pairwise_disjoint_of_mem_zpowers — Mathlib

English

Distinct cycle factors have pairwise disjoint z-powers.

Русский

Различные факторные циклы имеют попарно дисjoint z-пowers.

LaTeX

$$Let c,d ∈ f.cycleFactorsFinset. For any x ∈ Subgroup.zpowers c, y ∈ Subgroup.zpowers d, we have Disjoint x y.$$

Lean4

theorem pairwise_disjoint_of_mem_zpowers :
    Pairwise fun (i j : f.cycleFactorsFinset) ↦
      ∀ (x y : Perm α), x ∈ Subgroup.zpowers ↑i → y ∈ Subgroup.zpowers ↑j → Disjoint x y :=
  fun c d hcd ↦ fun x y hx hy ↦ by
  obtain ⟨m, hm⟩ := hx; obtain ⟨n, hn⟩ := hy
  simp only [← hm, ← hn]
  apply Disjoint.zpow_disjoint_zpow
  exact f.cycleFactorsFinset_pairwise_disjoint c.prop d.prop (Subtype.coe_ne_coe.mpr hcd)

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