Theorems · Theorem · order theory

Set.Finite.iSup_biInf_of_monotone

∀ {ι : Type u_1} {ι' : Type u_2} {α : Type u_3} [inst : Preorder ι'] [Nonempty ι'] [IsDirectedOrder ι']
  [inst_3 : Order.Frame α] {s : Set ι},
  s.Finite → ∀ {f : ι → ι' → α}, (∀ i ∈ s, Monotone (f i)) → ⨆ j, ⨅ i ∈ s, f i j = ⨅ i ∈ s, ⨆ j, f i j

Defined in: Mathlib.Data.Set.Finite.Lattice
Cited by: 2 results in Mathlib
Foundations: Depth 68 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: Preorder Nonempty IsDirectedOrder Order.Frame

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Cites18

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
Top.topproof · cited by 9,680
Preorderstatement and proof · cited by 7,952
iSupstatement and proof · cited by 2,415
Set.Finitestatement and proof · cited by 1,814
iInfstatement and proof · cited by 1,690
Monotonestatement and proof · cited by 1,397
IsDirectedOrderstatement and proof · cited by 316
iInf_congr_Propproof · cited by 218
Order.Framestatement and proof · cited by 88
Set.Finite.induction_onproof · cited by 39
iInf_negproof · cited by 27

Cited by2

Results whose statement or proof uses this declaration.

iSup_iInf_of_monotoneproof · cited by 3
Set.Finite.iSup_biInf_of_antitoneproof · cited by 0