Theorems · Theorem · ring theory
NonUnitalSubsemiring.closure_le
∀ {R : Type u} [inst : NonUnitalNonAssocSemiring R] {s : Set R} {t : NonUnitalSubsemiring R},
NonUnitalSubsemiring.closure s ≤ t ↔ s ⊆ ↑tA non-unital subsemiring S includes closure s if and only if it includes s.
- Cited by
- 8 results in Mathlib
- Foundations
- Depth 67 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- NonUnitalNonAssocSemiring
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
- NonUnitalNonAssocSemiringstatement and proof · cited by 1,081
- Set.Subset.transproof · cited by 218
- NonUnitalSubsemiringstatement and proof · cited by 201
- sInf_leproof · cited by 110
- NonUnitalSubsemiring.closurestatement · cited by 31
- NonUnitalSubsemiring.subset_closureproof · cited by 10
Cited by9
Results whose statement or proof uses this declaration.
- NonUnitalSubsemiring.giproof · cited by 6
- NonUnitalAlgebra.gcproof · cited by 5
- NonUnitalSubsemiring.closure_monoproof · cited by 2
- NonUnitalSubsemiring.closure_eq_of_leproof · cited by 1
- NonUnitalSubsemiring.closure_inductionproof · cited by 1
- NonUnitalRingHom.eqOn_sclosureproof · cited by 1
- NonUnitalSubsemiring.closure_le_centralizer_centralizerproof · cited by 1
- NonUnitalSubsemiring.closure_subsemigroup_closureproof · cited by 1
- NonUnitalRingHom.sclosure_preimage_leproof · cited by 0