Theorems · Theorem · group theory
AddSubgroup.subset_closure
∀ {G : Type u_1} [inst : AddGroup G] {k : Set G}, k ⊆ ↑(AddSubgroup.closure k)The AddSubgroup generated by a set includes the set.
- Defined in
- Mathlib.Algebra.Group.Subgroup.Lattice
- Cited by
- 49 results in Mathlib
- Foundations
- Depth 68 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- AddGroup
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.
- Setstatement and proof · cited by 53,352
- SetLike.coestatement and proof · cited by 8,199
- AddGroupstatement and proof · cited by 4,410
- AddSubgroupstatement and proof · cited by 3,232
- AddSubgroup.closurestatement · cited by 156
- AddSubgroup.mem_closureproof · cited by 1
Cited by49
Results whose statement or proof uses this declaration.
- AddSubgroup.closure_leproof · cited by 30
- AddSubgroup.closure_inductionstatement and proof · cited by 14
- AddSubgroup.addCommutator_mem_addCommutatorproof · cited by 9
- AddSubgroup.mem_closure_singletonproof · cited by 6
- AddSubgroup.closure_toAddSubmonoidproof · cited by 6
- AddSubgroup.fg_iff_addSubmonoid_fgproof · cited by 6
- Submodule.span_int_eq_addSubgroupClosureproof · cited by 5
- AddSubgroup.le_closure_toAddSubmonoidproof · cited by 5
- FreeAddGroup.range_lift_eq_closureproof · cited by 5
- Subgroup.toAddSubgroup_closureproof · cited by 3
- AddSubgroup.closure_induction''statement and proof · cited by 3
- RootPairing.linearIndepOn_root_baseOfproof · cited by 3