Theorems · Theorem · group theory
AddSubsemigroup.closure_le
∀ {M : Type u_1} [inst : Add M] {s : Set M} {S : AddSubsemigroup M}, AddSubsemigroup.closure s ≤ S ↔ s ⊆ ↑SAn additive subsemigroup S includes closure s if and only if it includes s
- Defined in
- Mathlib.Algebra.Group.Subsemigroup.Basic
- Cited by
- 11 results in Mathlib
- Foundations
- Depth 63 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- Add
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
- SetLike.coestatement and proof · cited by 8,199
- AddSubsemigroupstatement and proof · cited by 262
- Set.Subset.transproof · cited by 218
- sInf_leproof · cited by 110
- AddSubsemigroup.closurestatement · cited by 34
- AddSubsemigroup.subset_closureproof · cited by 12
Cited by12
Results whose statement or proof uses this declaration.
- AddSubsemigroup.closure_inductionproof · cited by 5
- AddSubsemigroup.giproof · cited by 5
- AddHom.eqOn_closureproof · cited by 1
- AddSubsemigroup.closure_eq_of_leproof · cited by 1
- AddSubsemigroup.closure_le_centralizer_centralizerproof · cited by 1
- AddSubsemigroup.closure_singleton_le_iff_memproof · cited by 1
- Subsemigroup.toAddSubsemigroup'_closureproof · cited by 0
- Subsemigroup.toAddSubsemigroup_closureproof · cited by 0
- AddSubsemigroup.toSubsemigroup'_closureproof · cited by 0
- AddSubsemigroup.closure_monoproof · cited by 0
- AddSubsemigroup.toSubsemigroup_closureproof · cited by 0
- AddHom.mclosure_preimage_leproof · cited by 0