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Theorems · Theorem · order theory

isLUB_prod

∀ {α : Type u_1} {β : Type u_2} [inst : Preorder α] [inst_1 : Preorder β] {s : Set (α × β)} {p : α × β},
  IsLUB s p ↔ IsLUB (Prod.fst '' s) p.1 ∧ IsLUB (Prod.snd '' s) p.2
Defined in
Mathlib.Order.Bounds.Image
Cited by
2 results in Mathlib
Foundations
Depth 12 from the axioms · uses propext, Quot.sound
Assumes
PreorderPreorder
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Around this declaration

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

ScottContinuousOn.fromProd · cited by 1ScottContinuousOn.fromProdCountableSupClosed.prod · cited by 1CountableSupClosed.prodSet · cited by 53352SetPreorder · cited by 7952PreorderSet.image · cited by 5609Set.imageSet.mem_image_of_mem · cited by 371Set.mem_image_of_memIsLUB · cited by 280IsLUBupperBounds · cited by 263upperBoundsMonotone.mem_upperBounds_image · cited by 12Monotone.mem_upperBounds_…monotone_snd · cited by 9monotone_sndmonotone_fst · cited by 9monotone_fstisLUB_prodCITED BYCITES

Cites9

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Cited by2

Results whose statement or proof uses this declaration.