Theorems · Theorem · group theory
Subgroup.closure_toSubmonoid
∀ {G : Type u_2} [inst : Group G] (S : Set G), (Subgroup.closure S).toSubmonoid = Submonoid.closure (S ∪ S⁻¹)- Defined in
- Mathlib.Algebra.Group.Subgroup.Pointwise
- Cited by
- 9 results in Mathlib
- Foundations
- Depth 71 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.
Cites19
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
- Groupstatement and proof · cited by 6,238
- Submonoidstatement · cited by 3,086
- le_antisymmproof · cited by 2,068
- inv_invproof · cited by 494
- Subgroup.closurestatement and proof · cited by 196
- Submonoid.closurestatement and proof · cited by 167
- Set.subset_union_leftproof · cited by 142
- Set.invstatement · cited by 132
- Subgroup.toSubmonoidstatement and proof · cited by 114
- Set.union_commproof · cited by 99
- Submonoid.subset_closureproof · cited by 46
Cited by9
Results whose statement or proof uses this declaration.
- Subgroup.fg_iff_submonoid_fgproof · cited by 5
- CoxeterSystem.submonoid_closure_range_simpleproof · cited by 4
- Subgroup.closure_induction_leftproof · cited by 2
- Subgroup.closure_invproof · cited by 2
- topologicalClosure_subgroupClosure_toSubmonoidproof · cited by 1
- RootPairing.weylGroup_toSubmonoidproof · cited by 1
- Subgroup.toSubmonoid_zpowersproof · cited by 1
- MonoidWithZeroHom.valueMonoid_eq_valueGroupproof · cited by 1
- Subgroup.closure_toSubmonoid_of_isOfFinOrderproof · cited by 1