Theorems · Theorem · order theory
le_sInf_iff
∀ {α : Type u_1} [inst : CompleteSemilatticeInf α] {s : Set α} {a : α}, a ≤ sInf s ↔ ∀ b ∈ s, a ≤ b- Defined in
- Mathlib.Order.CompleteLattice.Defs
- Cited by
- 13 results in Mathlib
- Foundations
- Depth 9 from the axioms · uses propext, Quot.sound
- Assumes
- CompleteSemilatticeInf
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites5
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
- InfSet.sInfstatement · cited by 935
- isGLB_sInfproof · cited by 23
- CompleteSemilatticeInfstatement and proof · cited by 19
- le_isGLB_iffproof · cited by 17
Cited by13
Results whose statement or proof uses this declaration.
- Ideal.exists_minimalPrimes_leproof · cited by 14
- Ideal.height_le_spanRank_toENat_of_mem_minimalPrimesproof · cited by 5
- Set.subset_sInter_iffproof · cited by 5
- OrderHom.map_le_lfpproof · cited by 3
- Ideal.eq_jacobson_iff_sInf_maximalproof · cited by 2
- Polynomial.mul_contentIdeal_le_radical_contentIdeal_mulproof · cited by 1
- lt_sInf_iffproof · cited by 1
- Ideal.iUnion_minimalPrimesproof · cited by 1
- DedekindCut.principal_sInf_rightproof · cited by 0
- IsArtinianRing.finite_of_compactSpace_of_t2Spaceproof · cited by 0
- CategoryTheory.Precoverage.toGrothendieck_eq_sInfproof · cited by 0
- LowerSet.Iic_sInfproof · cited by 0