Theorems · Theorem · order theory

Set.Finite.iInf_biSup_of_antitone

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

Defined in: Mathlib.Data.Set.Finite.Lattice
Cited by: 0 results in Mathlib
Foundations: Depth 69 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: Preorder Nonempty IsDirectedOrder Order.Coframe

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Setstatement and proof · cited by 53,352
Preorderstatement and proof · cited by 7,952
iSupstatement · cited by 2,415
Set.Finitestatement and proof · cited by 1,814
iInfstatement · cited by 1,690
Antitonestatement and proof · cited by 563
IsDirectedOrderstatement and proof · cited by 316
Order.Coframestatement and proof · cited by 38
Antitone.dual_leftproof · cited by 33
Set.Finite.iInf_biSup_of_monotoneproof · cited by 2

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