Theorems · Theorem · group theory
LinearOrderedAddCommGroup.isAddCyclic_iff_nonempty_equiv_int
∀ {A : Type u_4} [inst : AddCommGroup A] [inst_1 : LinearOrder A] [IsOrderedAddMonoid A] [Nontrivial A],
IsAddCyclic A ↔ Nonempty (A ≃+o ℤ)A linearly-ordered additive abelian group is cyclic iff it is isomorphic to ℤ as an ordered
additive monoid.
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 54 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites19
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- AddCommGroupstatement and proof · cited by 12,871
- LinearOrderstatement and proof · cited by 8,572
- Equivproof · cited by 8,337
- Nontrivialstatement and proof · cited by 2,416
- IsOrderedAddMonoidstatement and proof · cited by 1,659
- smul_negproof · cited by 181
- exists_neproof · cited by 101
- Equiv.ofBijectiveproof · cited by 70
- OrderAddMonoidIsostatement and proof · cited by 58
- IsAddCyclicstatement and proof · cited by 55
- OrderAddMonoidIso.symmproof · cited by 23
- add_zsmulproof · cited by 21
Cited by2
Results whose statement or proof uses this declaration.
- LinearOrderedAddCommGroup.isAddCyclic_iff_not_denselyOrderedproof · cited by 1
- LinearOrderedCommGroup.isCyclic_iff_nonempty_equiv_intproof · cited by 0