Theorems · Theorem · group theory
Subsemigroup.closure_le
∀ {M : Type u_1} [inst : Mul M] {s : Set M} {S : Subsemigroup M}, Subsemigroup.closure s ≤ S ↔ s ⊆ ↑SA subsemigroup S includes closure s if and only if it includes s.
- Defined in
- Mathlib.Algebra.Group.Subsemigroup.Basic
- Cited by
- 13 results in Mathlib
- Foundations
- Depth 63 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- Mul
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
- Subsemigroupstatement and proof · cited by 323
- Set.Subset.transproof · cited by 218
- sInf_leproof · cited by 110
- Subsemigroup.closurestatement · cited by 43
- Subsemigroup.subset_closureproof · cited by 16
Cited by14
Results whose statement or proof uses this declaration.
- Subsemigroup.closure_inductionproof · cited by 6
- Subsemigroup.giproof · cited by 5
- Submonoid.closure_eq_one_unionproof · cited by 2
- NonUnitalAlgebra.adjoin_eq_spanproof · cited by 2
- Subsemigroup.closure_singleton_le_iff_memproof · cited by 1
- MulHom.eqOn_closureproof · cited by 1
- Subsemigroup.closure_le_centralizer_centralizerproof · cited by 1
- Subsemigroup.closure_monoproof · cited by 0
- MulHom.mclosure_preimage_leproof · cited by 0
- Subsemigroup.toAddSubsemigroup'_closureproof · cited by 0
- Subsemigroup.toAddSubsemigroup_closureproof · cited by 0
- AddSubsemigroup.toSubsemigroup'_closureproof · cited by 0