Theorems · Theorem · group theory
AddAction.pretransitive_iff_unique_quotient_of_nonempty
∀ (G : Type u_1) (α : Type u_2) [inst : AddGroup G] [inst_1 : AddAction G α] [Nonempty α], AddAction.IsPretransitive G α ↔ Nonempty (Unique (AddAction.orbitRel.Quotient G α))
If α is non-empty, an additive action is pretransitive if and only if the
quotient has exactly one element.
- Defined in
- Mathlib.GroupTheory.GroupAction.Basic
- Cited by
- 0 results in Mathlib
- Foundations
- Depth 68 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites9
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- AddGroupstatement and proof · cited by 4,410
- AddActionstatement and proof · cited by 820
- Uniquestatement · cited by 400
- AddAction.IsPretransitivestatement and proof · cited by 56
- AddAction.orbitRelproof · cited by 47
- AddAction.orbitRel.Quotientstatement and proof · cited by 17
- unique_iff_subsingleton_and_nonemptyproof · cited by 6
- nonempty_quotient_iffproof · cited by 3
- AddAction.pretransitive_iff_subsingleton_quotientproof · cited by 2
Cited by0
Results whose statement or proof uses this declaration.
Nothing cites this yet.