Theorems · Theorem · group theory
AddSubmonoid.closure_mono
∀ {M : Type u_1} [inst : AddZeroClass M] ⦃s t : Set M⦄, s ⊆ t → AddSubmonoid.closure s ≤ AddSubmonoid.closure tAdditive submonoid closure of a set is monotone in its argument: if s ⊆ t,
then closure s ≤ closure t.
- Defined in
- Mathlib.Algebra.Group.Submonoid.Basic
- Cited by
- 10 results in Mathlib
- Foundations
- Depth 64 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.
Cites7
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
- AddZeroClassstatement and proof · cited by 1,237
- AddSubmonoidstatement · cited by 1,178
- AddSubmonoid.closurestatement · cited by 224
- Set.Subset.transproof · cited by 218
- AddSubmonoid.subset_closureproof · cited by 63
- AddSubmonoid.closure_leproof · cited by 35
Cited by10
Results whose statement or proof uses this declaration.
- star_le_star_iffproof · cited by 6
- AddSubgroup.closure_toAddSubmonoidproof · cited by 6
- Polynomial.addSubmonoid_closure_setOfPred_eq_monomialproof · cited by 2
- AddSubmonoid.closure_image_isAddIndecomposable_baseOfproof · cited by 2
- AddSubmonoid.fg_of_subtractiveproof · cited by 2
- AddSubmonoid.closure_sdiff_eq_closureproof · cited by 2
- IsLinearSet.closureproof · cited by 1
- IsLinearSet.isProperSemilinearSetproof · cited by 1
- exists_frobeniusNumber_iffproof · cited by 0
- AddSubmonoid.closure_nsmulproof · cited by 0