Theorems · Theorem · group theory
Submonoid.mem_closure
∀ {M : Type u_1} [inst : MulOneClass M] {s : Set M} {x : M},
x ∈ Submonoid.closure s ↔ ∀ (S : Submonoid M), s ⊆ ↑S → x ∈ S- Defined in
- Mathlib.Algebra.Group.Submonoid.Basic
- Cited by
- 4 results in Mathlib
- Foundations
- Depth 16 from the axioms · uses propext, Quot.sound
- Assumes
- MulOneClass
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites6
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
- Submonoidstatement · cited by 3,086
- MulOneClassstatement and proof · cited by 1,018
- Submonoid.closurestatement · cited by 167
- Submonoid.mem_sInfproof · cited by 1
Cited by4
Results whose statement or proof uses this declaration.
- Submonoid.subset_closureproof · cited by 46
- Submonoid.closure_sdiff_eq_closureproof · cited by 2
- Subsemiring.closure_submonoid_closureproof · cited by 1
- Subsemiring.closure_addSubmonoid_closureproof · cited by 0