Theorems · Theorem · order theory
Order.Ideal.sup_mem
∀ {P : Type u_1} [inst : SemilatticeSup P] {x y : P} {s : Order.Ideal P}, x ∈ s → y ∈ s → x ⊔ y ∈ sA specific witness of I.directed when P has joins.
- Defined in
- Mathlib.Order.Ideal
- Cited by
- 5 results in Mathlib
- Foundations
- Depth 13 from the axioms · uses no axioms
- Assumes
- SemilatticeSup
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.
- SetLike.coeproof · cited by 8,199
- SemilatticeSupstatement and proof · cited by 785
- sup_leproof · cited by 159
- Order.Idealstatement and proof · cited by 102
- Order.Ideal.lowerproof · cited by 11
- Order.Ideal.directedproof · cited by 5
Cited by5
Results whose statement or proof uses this declaration.
- Order.PFilter.inf_memproof · cited by 4
- Order.Ideal.sup_mem_iffproof · cited by 1
- Order.Ideal.isPrime_of_mem_or_compl_memproof · cited by 1
- Order.Ideal.IsProper.notMem_of_compl_memproof · cited by 1
- DistribLattice.prime_ideal_of_disjoint_filter_idealproof · cited by 0