Up-Complete Product/Lemma 2

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Theorem

Let $\struct {S, \preceq_1}$, $\struct {T, \preceq_2}$ be non-empty ordered sets.

Let $\struct {S \times T, \preceq}$ be Cartesian product of $\struct {S, \preceq_1}$ and $\struct {T, \preceq_2}$.


Let $X$ be a directed subset of $S \times T$.

Then

$\map {\pr_1^\to} X$ and $\map {\pr_2^\to} X$ are directed

where

$\pr_1$ denotes the first projection on $S \times T$
$\pr_2$ denotes the second projection on $S \times T$
$\map {\pr_1^\to} X$ denotes the image of $X$ under $\pr_1$


Proof

Let $x, y \in \map {\pr_1^\to} X$.

By definitions of image of set and projections:

$\exists x' \in T: \tuple {x, x'} \in X$

and

$\exists y' \in T: \tuple {y, y'} \in X$

By definition of directed:

$\exists \tuple {a, b} \in X: \tuple {x, x'} \preceq \tuple {a, b} \land \tuple {y, y'} \preceq \tuple {a, b}$

By definition of Cartesian product of ordered sets:

$\exists a \in \map {\pr_1^\to} X: x \preceq_1 a \land y \preceq_1 a$

Thus by definition

$\map {\pr_1^\to} X$ is directed.

By mutatis mutandis:

$\map {\pr_2^\to} X$ is directed.

$\blacksquare$


Sources