Theorems · Theorem · group theory
Subgroup.mem_closure_singleton
∀ {G : Type u_1} [inst : Group G] {x y : G}, y ∈ Subgroup.closure {x} ↔ ∃ n, x ^ n = yThe subgroup generated by an element of a group equals the set of integer number powers of the element.
- Defined in
- Mathlib.Algebra.Group.Subgroup.Lattice
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 71 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- Group
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
- Groupstatement and proof · cited by 6,238
- Subgroupstatement · cited by 3,593
- zpow_negproof · cited by 198
- Subgroup.closurestatement and proof · cited by 196
- Set.mem_singletonproof · cited by 183
- Subgroup.subset_closureproof · cited by 53
- zpow_zeroproof · cited by 52
- zpow_oneproof · cited by 41
- zpow_addproof · cited by 40
- Set.eq_of_mem_singletonproof · cited by 22
- Subgroup.closure_inductionproof · cited by 14
Cited by3
Results whose statement or proof uses this declaration.
- Subgroup.zpowers_eq_closureproof · cited by 5
- Subgroup.mem_closure_singleton_iff_existsUnique_zpowproof · cited by 0
- Subgroup.isLeast_of_closure_iff_eq_mabsproof · cited by 0