Theorems · Theorem · group theory
Submonoid.closure_le
∀ {M : Type u_1} [inst : MulOneClass M] {s : Set M} {S : Submonoid M}, Submonoid.closure s ≤ S ↔ s ⊆ ↑SA submonoid S includes closure s if and only if it includes s.
- Defined in
- Mathlib.Algebra.Group.Submonoid.Basic
- Cited by
- 27 results in Mathlib
- Foundations
- Depth 63 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- MulOneClass
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
- Submonoidstatement and proof · cited by 3,086
- MulOneClassstatement and proof · cited by 1,018
- Set.Subset.transproof · cited by 218
- Submonoid.closurestatement · cited by 167
- sInf_leproof · cited by 110
- Submonoid.subset_closureproof · cited by 46
Cited by28
Results whose statement or proof uses this declaration.
- Submonoid.closure_inductionproof · 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.giproof · cited by 5
- Subgroup.le_closure_toSubmonoidproof · cited by 4
- Submonoid.closure_eq_of_leproof · cited by 2
- Submonoid.closure_image_isMulIndecomposable_baseOfproof · cited by 2
- Submonoid.closure_sdiff_eq_closureproof · cited by 2
- Submonoid.closure_singleton_le_iff_memproof · cited by 2
- Algebra.FiniteType.of_span_eq_top_targetproof · cited by 2