Theorems · Theorem · group theory
NegMemClass.neg_mem
∀ {S : Type u_3} {G : outParam (Type u_4)} {inst : Neg G} {inst_1 : SetLike S G} [self : NegMemClass S G] {s : S}
{x : G}, x ∈ s → -x ∈ ss is closed under negation
- Defined in
- Mathlib.Algebra.Group.Subgroup.Defs
- Cited by
- 63 results in Mathlib
- Foundations
- Depth 5 from the axioms · uses no axioms
- Assumes
- NegMemClass
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites2
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- SetLikestatement and proof · cited by 1,084
- NegMemClassstatement and proof · cited by 11
Cited by63
Results whose statement or proof uses this declaration.
- neg_mem_iffproof · cited by 42
- sub_memproof · cited by 40
- Submodule.neg_memproof · cited by 31
- AddSubgroup.neg_memproof · cited by 23
- zsmul_memproof · cited by 14
- AddSubgroup.closure_inductionstatement and proof · cited by 14
- add_mem_cancel_rightproof · cited by 10
- AddSubgroup.normalClosure_le_normalproof · cited by 9
- add_mem_cancel_leftproof · cited by 8
- Subalgebra.neg_memproof · cited by 7
- Subring.closure_inductionstatement and proof · cited by 4
- AddSubgroup.mem_supproof · cited by 4