Theorems · Theorem · group theory
Submonoid.closure_mono
∀ {M : Type u_1} [inst : MulOneClass M] ⦃s t : Set M⦄, s ⊆ t → Submonoid.closure s ≤ Submonoid.closure tSubmonoid 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
- 6 results in Mathlib
- Foundations
- Depth 64 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.
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
- Submonoidstatement · cited by 3,086
- MulOneClassstatement and proof · cited by 1,018
- Set.Subset.transproof · cited by 218
- Submonoid.closurestatement · cited by 167
- Submonoid.subset_closureproof · cited by 46
- Submonoid.closure_leproof · cited by 27
Cited by6
Results whose statement or proof uses this declaration.
- Subgroup.closure_toSubmonoidproof · cited by 9
- Submonoid.closure_sdiff_eq_closureproof · cited by 2
- Submonoid.closure_image_isMulIndecomposable_baseOfproof · cited by 2
- Algebra.pow_smul_mem_of_smul_subset_of_mem_adjoinproof · cited by 2
- Submonoid.fg_of_divisiveproof · cited by 2
- Submonoid.closure_powproof · cited by 0