Theorems · Theorem · order theory

le_ciInf_iff

∀ {α : Type u_1} {ι : Sort u_4} [inst : ConditionallyCompleteLattice α] [Nonempty ι] {f : ι → α} {a : α},
  BddBelow (Set.range f) → (a ≤ iInf f ↔ ∀ (i : ι), a ≤ f i)

Defined in: Mathlib.Order.ConditionallyCompleteLattice.Indexed
Cited by: 5 results in Mathlib
Foundations: Depth 11 from the axioms · uses propext, Quot.sound
Assumes: ConditionallyCompleteLattice Nonempty

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Around this declaration

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

Cites7

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

Set.rangestatement and proof · cited by 4,705
iInfstatement · cited by 1,690
BddBelowstatement and proof · cited by 401
ConditionallyCompleteLatticestatement and proof · cited by 364
Set.forall_mem_rangeproof · cited by 135
le_isGLB_iffproof · cited by 17
isGLB_ciInfproof · cited by 3

Cited by5

Results whose statement or proof uses this declaration.

le_schnirelmannDensity_iffproof · cited by 3
Set.Iic_ciInfproof · cited by 2
le_ciInf_iff'proof · cited by 1
ENNReal.ofReal_iInfproof · cited by 0
ContinuousLinearMap.forall_norm_sub_apply_le_iff_adjoint_apply_sub_eq_zeroproof · cited by 0