Theorems · Theorem · group theory
Submonoid.subset_closure
∀ {M : Type u_1} [inst : MulOneClass M] {s : Set M}, s ⊆ ↑(Submonoid.closure s)The submonoid generated by a set includes the set.
- Defined in
- Mathlib.Algebra.Group.Submonoid.Basic
- Cited by
- 46 results in Mathlib
- Foundations
- Depth 17 from the axioms · uses propext, Quot.sound
- Assumes
- MulOneClass
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
- Submonoidstatement and proof · cited by 3,086
- MulOneClassstatement and proof · cited by 1,018
- Submonoid.closurestatement · cited by 167
- Submonoid.mem_closureproof · cited by 4
Cited by46
Results whose statement or proof uses this declaration.
- Submonoid.closure_inductionstatement and proof · cited by 27
- Submonoid.closure_leproof · cited by 27
- Algebra.adjoin_eq_spanproof · cited by 13
- Subgroup.closure_toSubmonoidproof · cited by 9
- Submonoid.closure_monoproof · cited by 6
- Subgroup.fg_iff_submonoid_fgproof · cited by 5
- Submonoid.closure_induction_leftstatement and proof · cited by 4
- Submonoid.mem_closure_of_memproof · cited by 3
- FreeMonoid.closure_range_ofproof · cited by 3
- Submonoid.closure_eq_one_unionproof · cited by 2
- Submonoid.closure_image_isMulIndecomposable_baseOfproof · cited by 2
- Submonoid.closure_induction_rightstatement and proof · cited by 2