Absolute Value induces Equivalence Compatible with Integer Multiplication

Theorem

Let $\Z$ be the set of integers.

Let $\RR$ be the relation on $\Z$ defined as:

$\forall x, y \in \Z: \struct {x, y} \in \RR \iff \size x = \size y$

where $\size x$ denotes the absolute value of $x$.

Then $\RR$ is a congruence relation for integer multiplication.

Proof

Let:

$\size {x_1} = \size {x_2}$
$\size {y_1} = \size {y_2}$

Then by Absolute Value of Product:

 $\ds \size {x_1 y_1}$ $=$ $\ds \size {x_1} \size {y_1}$ $\ds$ $=$ $\ds \size {x_2} \size {y_2}$ $\ds$ $=$ $\ds \size {x_2 y_2}$

That is:

$\paren {x_1 y_1, x_2 y_2} \in \RR$

That is, $\RR$ is a congruence relation for integer multiplication.

$\blacksquare$