Theorems · Theorem · group theory
Subgroup.closure_le
∀ {G : Type u_1} [inst : Group G] (K : Subgroup G) {k : Set G}, Subgroup.closure k ≤ K ↔ k ⊆ ↑KA subgroup K includes closure k if and only if it includes k.
- Defined in
- Mathlib.Algebra.Group.Subgroup.Lattice
- Cited by
- 31 results in Mathlib
- Foundations
- Depth 69 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- Group
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites8
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
- Groupstatement and proof · cited by 6,238
- Subgroupstatement and proof · cited by 3,593
- Set.Subset.transproof · cited by 218
- Subgroup.closurestatement · cited by 196
- sInf_leproof · cited by 110
- Subgroup.subset_closureproof · cited by 53
Cited by32
Results whose statement or proof uses this declaration.
- Subgroup.closure_inductionproof · cited by 14
- Subgroup.commutator_leproof · cited by 10
- Subgroup.zpowers_leproof · cited by 6
- Subgroup.giproof · cited by 6
- Subgroup.fg_iff_submonoid_fgproof · cited by 5
- FreeGroup.range_lift_eq_closureproof · cited by 5
- Abelianization.commutator_subset_kerproof · cited by 5
- Subgroup.closure_eq_of_leproof · cited by 4
- commutator_alternatingGroup_eq_topproof · cited by 3
- Subgroup.toAddSubgroup_closureproof · cited by 3
- Subgroup.cyclic_of_minproof · cited by 2
- Equiv.Perm.closure_cycleType_eq_two_two_eq_alternatingGroupproof · cited by 2