disjoint_cycleFactorsFinset — Mathlib

English

If f and g are disjoint, then their cycle factors are disjoint as well.

Русский

Если f и g дисjoint, то их цикловые факторы также дисjoint.

LaTeX

$$If f.Disjoint g then Disjoint (cycleFactorsFinset f) (cycleFactorsFinset g)$$

Lean4

theorem disjoint_cycleFactorsFinset {f g : Perm α} (h : Disjoint f g) :
    _root_.Disjoint (cycleFactorsFinset f) (cycleFactorsFinset g) :=
  by
  rw [disjoint_iff_disjoint_support] at h
  rw [Finset.disjoint_left]
  intro x hx hy
  simp only [mem_cycleFactorsFinset_iff, mem_support] at hx hy
  obtain ⟨⟨⟨a, ha, -⟩, hf⟩, -, hg⟩ := hx, hy
  have := h.le_bot (by simp [ha, ← hf a ha, ← hg a ha] : a ∈ f.support ∩ g.support)
  tauto

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