Theorems · Theorem · commutative algebra
Subsemiring.mem_sSup_of_directedOn
∀ {R : Type u} [inst : NonAssocSemiring R] {S : Set (Subsemiring R)},
S.Nonempty → DirectedOn (fun x1 x2 => x1 ≤ x2) S → ∀ {x : R}, x ∈ sSup S ↔ ∃ s ∈ S, x ∈ s- Defined in
- Mathlib.Algebra.Ring.Subsemiring.Basic
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 69 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.
Cites11
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
- Set.Elemproof · cited by 7,166
- Set.Nonemptystatement and proof · cited by 2,627
- SupSet.sSupstatement · cited by 954
- NonAssocSemiringstatement and proof · cited by 805
- Subsemiringstatement and proof · cited by 456
- DirectedOnstatement and proof · cited by 271
- Set.Nonempty.to_subtypeproof · cited by 55
- sSup_eq_iSup'proof · cited by 38
- DirectedOn.directed_valproof · cited by 36
- Subsemiring.mem_iSup_of_directedproof · cited by 2
Cited by1
Results whose statement or proof uses this declaration.
- Subsemiring.coe_sSup_of_directedOnproof · cited by 0