Absolute Value Function is Completely Multiplicative/Proof 4

Theorem

The absolute value function on the real numbers $\R$ is completely multiplicative:

$\forall x, y \in \R: \left\vert{x y}\right\vert = \left\vert{x}\right\vert \, \left\vert{y}\right\vert$

where $\left \vert{a}\right \vert$ denotes the absolute value of $a$.

Proof

We have the result Multiplicative Group of Reals is Subgroup of Complex.

Therefore, any result applying to all complex numbers will also hold for all real numbers.

The result follows from Complex Modulus of Product of Complex Numbers.

$\blacksquare$