Theorems · Theorem · group theory
Subsemigroup.mem_closure
∀ {M : Type u_1} [inst : Mul M] {s : Set M} {x : M}, x ∈ Subsemigroup.closure s ↔ ∀ (S : Subsemigroup M), s ⊆ ↑S → x ∈ S- Defined in
- Mathlib.Algebra.Group.Subsemigroup.Basic
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 14 from the axioms · uses propext, Quot.sound
- Assumes
- Mul
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites5
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 · cited by 8,199
- Subsemigroupstatement · cited by 323
- Subsemigroup.closurestatement · cited by 43
- Subsemigroup.mem_sInfproof · cited by 1
Cited by3
Results whose statement or proof uses this declaration.
- Subsemigroup.subset_closureproof · cited by 16
- NonUnitalSubsemiring.closure_subsemigroup_closureproof · cited by 1
- NonUnitalSubsemiring.closure_addSubmonoid_closureproof · cited by 0