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
From Absolute Value Function on Integers induces Equivalence Relation, $\RR$ is an equivalence relation.

Let:
 * $\size {x_1} = \size {x_2}$
 * $\size {y_1} = \size {y_2}$

Then by Absolute Value Function is Completely Multiplicative:

That is:
 * $\paren {x_1 y_1, x_2 y_2} \in \RR$

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

Also see

 * Absolute Value induces Equivalence not Compatible with Integer Addition