Theorems · Theorem · group theory
AddSubgroup.add_mem
∀ {G : Type u_1} [inst : AddGroup G] (H : AddSubgroup G) {x y : G}, x ∈ H → y ∈ H → x + y ∈ HAn AddSubgroup is closed under addition.
- Defined in
- Mathlib.Algebra.Group.Subgroup.Defs
- Cited by
- 17 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
- AddMemClass.add_memproof · cited by 229
Cited by17
Results whose statement or proof uses this declaration.
- Submodule.span_int_eq_addSubgroupClosureproof · cited by 5
- AddSubgroup.isOpen_of_mem_nhdsproof · cited by 5
- AddSubgroup.addCommutator_le_rightproof · cited by 4
- AddSubgroup.add_mem_supproof · cited by 3
- controlled_sum_of_mem_closureproof · cited by 2
- TwoSidedIdeal.mem_span_iff_mem_addSubgroup_closure_absorbingproof · cited by 2
- AddSubgroup.exists_leftTransversal_of_FiniteIndexproof · cited by 1
- AddSubgroup.closure_add_image_add_eq_topproof · cited by 1
- AddSubgroup.closure_add_leproof · cited by 1
- AddGroup.fg_of_descentproof · cited by 1
- AddSubgroup.mem_sup_of_normal_rightproof · cited by 1
- FreeAddGroup.range_lift_leproof · cited by 1