Theorems · Theorem · group theory
AddSubgroup.neg_mem
∀ {G : Type u_1} [inst : AddGroup G] (H : AddSubgroup G) {x : G}, x ∈ H → -x ∈ HAn AddSubgroup is closed under inverse.
- Defined in
- Mathlib.Algebra.Group.Subgroup.Defs
- Cited by
- 23 results in Mathlib
- Foundations
- Depth 18 from the axioms · uses propext
- Assumes
- AddGroup
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.
- AddGroupstatement and proof · cited by 4,410
- AddSubgroupstatement and proof · cited by 3,232
- NegMemClass.neg_memproof · cited by 63
Cited by23
Results whose statement or proof uses this declaration.
- AddSubgroup.le_normalizerproof · cited by 6
- AddSubgroup.isOpen_of_mem_nhdsproof · cited by 5
- AddSubgroup.addCommutator_le_rightproof · cited by 4
- QuotientAddGroup.preimage_image_mkproof · cited by 2
- AddSubgroup.addCommutator_top_left_le_iffproof · cited by 2
- AddSubgroup.disjoint_iff_add_eq_zeroproof · cited by 2
- controlled_sum_of_mem_closureproof · cited by 2
- TwoSidedIdeal.mem_span_iff_mem_addSubgroup_closure_absorbingproof · cited by 2
- compl_add_closure_zero_eqproof · cited by 1
- rightAddCoset_mem_rightAddCosetproof · cited by 1
- AddSubgroup.closure_add_image_add_eq_topproof · cited by 1
- AddSubgroup.mem_sup_of_normal_rightproof · cited by 1