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Theorems · Theorem · order theory

le_csInf_iff

∀ {α : Type u_1} [inst : ConditionallyCompleteLattice α] {s : Set α} {a : α},
  BddBelow s → s.Nonempty → (a ≤ sInf s ↔ ∀ b ∈ s, a ≤ b)
Defined in
Mathlib.Order.ConditionallyCompleteLattice.Basic
Cited by
3 results in Mathlib
Foundations
Depth 10 from the axioms · uses propext, Quot.sound
Assumes
ConditionallyCompleteLattice
Mathlib docsSourceLoogleLeanSearch

Around this declaration

Dashed lines are statement dependencies; solid lines are citations in proofs.

ContinuousLinearMap.opNorm_eq_of_bounds · cited by 5ContinuousLinearMap.opNor…le_csInf_iff'' · cited by 1le_csInf_iff''NormedAddGroupHom.opNorm_eq_of_bounds · cited by 1NormedAddGroupHom.opNorm_…Set · cited by 53352SetSet.Nonempty · cited by 2627Set.NonemptyInfSet.sInf · cited by 935InfSet.sInfBddBelow · cited by 401BddBelowConditionallyCompleteLattice · cited by 364ConditionallyCompleteLatt…isGLB_csInf · cited by 24isGLB_csInfle_isGLB_iff · cited by 17le_isGLB_iffle_csInf_iffCITED BYCITES

Cites7

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Cited by3

Results whose statement or proof uses this declaration.