Theorems · Theorem · group theory
Subgroup.subset_closure
∀ {G : Type u_1} [inst : Group G] {k : Set G}, k ⊆ ↑(Subgroup.closure k)The subgroup generated by a set includes the set.
- Defined in
- Mathlib.Algebra.Group.Subgroup.Lattice
- Cited by
- 53 results in Mathlib
- Foundations
- Depth 68 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.
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
- Groupstatement and proof · cited by 6,238
- Subgroupstatement and proof · cited by 3,593
- Subgroup.closurestatement · cited by 196
- Subgroup.mem_closureproof · cited by 2
Cited by53
Results whose statement or proof uses this declaration.
- Subgroup.closure_leproof · cited by 31
- Subgroup.closure_inductionstatement and proof · cited by 14
- Subgroup.commutator_mem_commutatorproof · cited by 13
- Subgroup.closure_induction''statement and proof · cited by 8
- RootPairing.reflection_mem_weylGroupproof · cited by 7
- Subgroup.fg_iff_submonoid_fgproof · cited by 5
- FreeGroup.range_lift_eq_closureproof · cited by 5
- Subgroup.le_closure_toSubmonoidproof · cited by 4
- Subgroup.toAddSubgroup_closureproof · cited by 3
- Subgroup.mem_closure_of_memproof · cited by 3
- Subgroup.mem_closure_singletonproof · cited by 3
- Subgroup.Normal.quotient_commutative_iff_commutator_leproof · cited by 2