Theorems · Theorem · group theory
Subgroup.inv_mem
∀ {G : Type u_1} [inst : Group G] (H : Subgroup G) {x : G}, x ∈ H → x⁻¹ ∈ HA subgroup is closed under inverse.
- Defined in
- Mathlib.Algebra.Group.Subgroup.Defs
- Cited by
- 40 results in Mathlib
- Foundations
- Depth 18 from the axioms · uses propext
- Assumes
- Group
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites3
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Groupstatement and proof · cited by 6,238
- Subgroupstatement and proof · cited by 3,593
- InvMemClass.inv_memproof · cited by 52
Cited by40
Results whose statement or proof uses this declaration.
- Subgroup.le_normalizerproof · cited by 19
- Matrix.GeneralLinearGroup.center_eq_range_scalarproof · cited by 6
- DoubleCoset.iUnion_quotToDoubleCosetproof · cited by 4
- MulAction.IwasawaStructure.commutator_leproof · cited by 4
- Subgroup.commutator_le_rightproof · cited by 4
- GrpCat.SurjectiveOfEpiAuxs.fromCoset_eq_of_mem_rangeproof · cited by 3
- GrpCat.SurjectiveOfEpiAuxs.fromCoset_ne_of_nin_rangeproof · cited by 3
- Subgroup.isOpen_of_mem_nhdsproof · cited by 3
- Subgroup.disjoint_iff_mul_eq_oneproof · cited by 2
- Sylow.normalizer_sup_eq_topproof · cited by 2
- IsPGroup.exists_le_sylowproof · cited by 2
- Subgroup.commutator_commutator_eq_bot_of_rotateproof · cited by 2