Theorems · Theorem · group theory
AddSubmonoid.closure_le
∀ {M : Type u_1} [inst : AddZeroClass M] {s : Set M} {S : AddSubmonoid M}, AddSubmonoid.closure s ≤ S ↔ s ⊆ ↑SAn additive submonoid S includes closure s if and only if it includes s.
- Defined in
- Mathlib.Algebra.Group.Submonoid.Basic
- Cited by
- 35 results in Mathlib
- Foundations
- Depth 63 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- AddZeroClass
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
- AddZeroClassstatement and proof · cited by 1,237
- AddSubmonoidstatement and proof · cited by 1,178
- AddSubmonoid.closurestatement · cited by 224
- Set.Subset.transproof · cited by 218
- sInf_leproof · cited by 110
- AddSubmonoid.subset_closureproof · cited by 63
Cited by36
Results whose statement or proof uses this declaration.
- AddSubmonoid.closure_inductionproof · cited by 30
- AddSubmonoid.closure_monoproof · cited by 10
- AddSubgroup.closure_toAddSubmonoidproof · cited by 6
- AddSubgroup.fg_iff_addSubmonoid_fgproof · cited by 6
- AddSubgroup.le_closure_toAddSubmonoidproof · cited by 5
- AddSubmonoid.giproof · cited by 5
- AddSubmonoid.multiples_leproof · cited by 4
- AddSubmonoid.closure_eq_of_leproof · cited by 3
- AddSubmonoid.closure_singleton_le_iff_memproof · cited by 2
- Submodule.closure_subset_spanproof · cited by 2
- AddMonoidHom.eqOn_closureMproof · cited by 2
- AddSubmonoid.closure_image_isAddIndecomposable_baseOfproof · cited by 2