Theorems · Theorem · group theory
injective_zsmul_iff_not_isOfFinAddOrder
∀ {G : Type u_1} [inst : AddGroup G] {x : G}, (Function.Injective fun n => n • x) ↔ ¬IsOfFinAddOrder x- Defined in
- Mathlib.GroupTheory.OrderOfElement
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 53 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- AddGroup
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites13
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
- Nat.cast_zeroproof · cited by 1,870
- LT.lt.ne'proof · cited by 1,417
- Int.ModEqproof · cited by 147
- zero_nsmulproof · cited by 137
- natCast_zsmulproof · cited by 118
- Nat.cast_ne_zeroproof · cited by 113
- IsOfFinAddOrderstatement and proof · cited by 105
- isOfFinAddOrder_iff_nsmul_eq_zeroproof · cited by 27
- ofNat_zsmulproof · cited by 24
- addOrderOf_eq_zero_iffproof · cited by 9
- zsmul_eq_zsmul_iff_modEqproof · cited by 4
Cited by2
Results whose statement or proof uses this declaration.
- AddSubgroup.zmultiples_eq_zmultiples_iffproof · cited by 3
- LinearOrderedAddCommGroup.isAddCyclic_iff_nonempty_equiv_intproof · cited by 2