Theorems · Theorem · group theory
div_mem
∀ {M : Type u_3} {S : Type u_4} [inst : DivInvMonoid M] [inst_1 : SetLike S M] [hSM : SubgroupClass S M] {H : S}
{x y : M}, x ∈ H → y ∈ H → x / y ∈ HA subgroup is closed under division.
- Defined in
- Mathlib.Algebra.Group.Subgroup.Defs
- Cited by
- 9 results in Mathlib
- Foundations
- Depth 7 from the axioms · uses no axioms
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites6
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
- div_eq_mul_invproof · cited by 715
- MulMemClass.mul_memproof · cited by 173
- DivInvMonoidstatement and proof · cited by 103
- InvMemClass.inv_memproof · cited by 52
- SubgroupClassstatement and proof · cited by 34
Cited by9
Results whose statement or proof uses this declaration.
- Subgroup.exists_isLeast_one_ltproof · cited by 2
- SubfieldClass.nnratCast_memproof · cited by 2
- SubfieldClass.ratCast_memproof · cited by 2
- IntermediateField.fg_top_iffproof · cited by 2
- Subgroup.mk_goursatFst_eq_iff_mk_goursatSnd_eqproof · cited by 1
- IsFractionRing.closure_range_algebraMapproof · cited by 1
- Subgroup.div_memproof · cited by 1
- Subfield.div_memproof · cited by 0
- IntermediateField.div_memproof · cited by 0