of_orderEmbedding — Mathlib

English

If f is an order embedding and a1,a2 have codisjoint images in β with top elements, then a1,a2 are codisjoint in α.

Русский

Если имеется вложение по порядку и образы a1,a2 в β кодиодисjoint, то сами элементы в α тоже кодисjoint.

LaTeX

$$$\forall a_1,a_2:\alpha:\ Codisjoint( f(a_1), f(a_2)) \Rightarrow Codisjoint(a_1,a_2)$$$

Lean4

/-- If the images by an order embedding of two elements are codisjoint,
then they are themselves codisjoint. -/
theorem of_orderEmbedding [OrderTop α] [OrderTop β] {a₁ a₂ : α} : Codisjoint (f a₁) (f a₂) → Codisjoint a₁ a₂ :=
  Disjoint.of_orderEmbedding (α := αᵒᵈ) (β := βᵒᵈ) f.dual

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