Theorems · Theorem · commutative algebra
Subsemiring.closure_le
∀ {R : Type u} [inst : NonAssocSemiring R] {s : Set R} {t : Subsemiring R}, Subsemiring.closure s ≤ t ↔ s ⊆ ↑tA subsemiring S includes closure s if and only if it includes s.
- Defined in
- Mathlib.Algebra.Ring.Subsemiring.Basic
- Cited by
- 12 results in Mathlib
- Foundations
- Depth 67 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- NonAssocSemiring
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites8
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
- NonAssocSemiringstatement and proof · cited by 805
- Subsemiringstatement and proof · cited by 456
- Set.Subset.transproof · cited by 218
- sInf_leproof · cited by 110
- Subsemiring.closurestatement · cited by 53
- Subsemiring.subset_closureproof · cited by 13
Cited by13
Results whose statement or proof uses this declaration.
- Algebra.gcproof · cited by 7
- Subsemiring.giproof · cited by 6
- Subsemiring.closure_monoproof · cited by 4
- Algebra.adjoin_algebraMap_image_union_eq_adjoin_adjoinproof · cited by 3
- Subsemiring.closure_inductionproof · cited by 3
- Subsemiring.closure_singleton_natCastproof · cited by 3
- Algebra.adjoin_natproof · cited by 1
- RingHom.eqOn_sclosureproof · cited by 1
- Subsemiring.closure_le_centralizer_centralizerproof · cited by 1
- Subsemiring.closure_submonoid_closureproof · cited by 1
- Subsemiring.closure_eq_of_leproof · cited by 0
- Subsemiring.toSubsemiring_homogeneousCore_leproof · cited by 0