eq_cycleOf_of_mem_cycleFactorsFinset_iff

English

Repeat of the equivalence for cycle containment and support membership.

Русский

Повторение эквивалентности для принадлежности цикла окружности и опоры.

LaTeX

$$Same as 124016$$

Lean4

theorem eq_cycleOf_of_mem_cycleFactorsFinset_iff (g c : Perm α) (hc : c ∈ g.cycleFactorsFinset) (x : α) :
    c = g.cycleOf x ↔ x ∈ c.support := by
  refine ⟨?_, (cycle_is_cycleOf · hc)⟩
  rintro rfl
  rw [mem_support, cycleOf_apply_self, ne_eq, ← cycleOf_eq_one_iff]
  exact (mem_cycleFactorsFinset_iff.mp hc).left.ne_one

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