Theorems · Theorem · group theory
Equiv.Perm.isCycle_iff_exists_isCycleOn
∀ {α : Type u_2} {f : Equiv.Perm α},
f.IsCycle ↔ ∃ s, s.Nontrivial ∧ f.IsCycleOn s ∧ ∀ ⦃x : α⦄, ¬Function.IsFixedPt (⇑f) x → x ∈ sThis lemma demonstrates the relation between Equiv.Perm.IsCycle and Equiv.Perm.IsCycleOn
in non-degenerate cases.
- Defined in
- Mathlib.GroupTheory.Perm.Cycle.Basic
- Cited by
- 0 results in Mathlib
- Foundations
- Depth 33 from the axioms · uses propext, Classical.choice, Quot.sound
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Cites13
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement and proof · cited by 62,936
- Setstatement and proof · cited by 53,352
- Set.ofPredproof · cited by 6,101
- Equiv.Permstatement and proof · cited by 1,375
- Equiv.injectiveproof · cited by 464
- Set.Nontrivialstatement and proof · cited by 145
- Equiv.Perm.SameCycleproof · cited by 116
- Equiv.Perm.IsCyclestatement and proof · cited by 108
- Function.IsFixedPtstatement and proof · cited by 84
- Equiv.Perm.IsCycleOnstatement and proof · cited by 41
- Set.Nontrivial.nonemptyproof · cited by 10
- Equiv.Perm.IsCycleOn.apply_neproof · cited by 4
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