Theorems · Theorem · group theory
AddSubgroup.mem_closure_singleton
∀ {G : Type u_1} [inst : AddGroup G] {x y : G}, y ∈ AddSubgroup.closure {x} ↔ ∃ n, n • x = yThe AddSubgroup generated by an element of an AddGroup equals the set of
natural number multiples of the element.
- Defined in
- Mathlib.Algebra.Group.Subgroup.Lattice
- Cited by
- 6 results in Mathlib
- Foundations
- Depth 71 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.
- Setstatement · cited by 53,352
- AddGroupstatement and proof · cited by 4,410
- AddSubgroupstatement · cited by 3,232
- Set.mem_singletonproof · cited by 183
- AddSubgroup.closurestatement and proof · cited by 156
- one_zsmulproof · cited by 59
- AddSubgroup.subset_closureproof · cited by 49
- neg_zsmulproof · cited by 41
- Set.eq_of_mem_singletonproof · cited by 22
- add_zsmulproof · cited by 21
- zero_zsmulproof · cited by 19
- zsmul_memproof · cited by 14
Cited by6
Results whose statement or proof uses this declaration.
- AddSubgroup.zmultiples_eq_closureproof · cited by 9
- AddSubgroup.cyclic_of_minproof · cited by 2
- AddSubgroup.closure_singleton_zeroproof · cited by 2
- AddSubgroup.isLeast_of_closure_iff_eq_absproof · cited by 0
- AddSubgroup.mem_closure_pairproof · cited by 0
- AddSubgroup.mem_closure_singleton_iff_existsUnique_zsmulproof · cited by 0