Theorems · Theorem · group theory
mem_zmultiples_zsmul_iff
∀ {G : Type u_1} [inst : AddGroup G] {g : G} {k : ℤ}, g ∈ AddSubgroup.zmultiples (k • g) ↔ k.gcd ↑(addOrderOf g) = 1- Defined in
- Mathlib.GroupTheory.OrderOfElement
- Cited by
- 1 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.
Cites12
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
- AddSubgroupstatement · cited by 3,232
- Nat.cast_oneproof · cited by 2,501
- AddSubgroup.zmultiplesstatement and proof · cited by 493
- addOrderOfstatement and proof · cited by 208
- one_zsmulproof · cited by 59
- Int.modEq_iff_dvdproof · cited by 16
- mul_zsmul'proof · cited by 16
- AddSubgroup.mem_zmultiples_iffproof · cited by 11
- zsmul_eq_zsmul_iff_modEqproof · cited by 4
- dvd_defproof · cited by 3
- Int.gcd_dvd_iffproof · cited by 3
Cited by1
Results whose statement or proof uses this declaration.
- mem_zmultiples_nsmul_iffproof · cited by 1