Theorems · Theorem · group theory
AddAction.pretransitive_iff_subsingleton_quotient
∀ (G : Type u_1) (α : Type u_2) [inst : AddGroup G] [inst_1 : AddAction G α], AddAction.IsPretransitive G α ↔ Subsingleton (AddAction.orbitRel.Quotient G α)
An additive action is pretransitive if and only if the quotient by
AddAction.orbitRel is a subsingleton.
- Defined in
- Mathlib.GroupTheory.GroupAction.Basic
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 67 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites7
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
- AddAction.IsPretransitivestatement and proof · cited by 56
- AddAction.orbitRelproof · cited by 47
- Quotient.eq''proof · cited by 44
- AddAction.exists_vadd_eqproof · cited by 20
- AddAction.orbitRel.Quotientstatement and proof · cited by 17
Cited by2
Results whose statement or proof uses this declaration.
- AddAction.pretransitive_iff_unique_quotient_of_nonemptyproof · cited by 0
- AddSubgroup.finite_quotient_of_pretransitive_of_index_ne_zeroproof · cited by 0