Mathlib Map

Theorems · Theorem · commutative algebra

Subsemiring.mem_closure

∀ {R : Type u} [inst : NonAssocSemiring R] {x : R} {s : Set R},
  x ∈ Subsemiring.closure s ↔ ∀ (S : Subsemiring R), s ⊆ ↑S → x ∈ S
Defined in
Mathlib.Algebra.Ring.Subsemiring.Basic
Cited by
2 results in Mathlib
Foundations
Depth 65 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes
NonAssocSemiring
Mathlib docsSourceLoogleLeanSearch

Around this declaration

Dashed lines are statement dependencies; solid lines are citations in proofs.

Subsemiring.subset_closure · cited by 13Subsemiring.subset_closureSubmonoid.subsemiringClosure_eq_closure · cited by 1Submonoid.subsemiringClos…Set · cited by 53352SetSetLike.coe · cited by 8199SetLike.coeNonAssocSemiring · cited by 805NonAssocSemiringSubsemiring · cited by 456SubsemiringSubsemiring.closure · cited by 53Subsemiring.closureSubsemiring.mem_sInf · cited by 1Subsemiring.mem_sInfSubsemiring.mem_closureCITED BYCITES

Cites6

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

Cited by2

Results whose statement or proof uses this declaration.