Field Norm of Complex Number is Positive Definite

Theorem
Let $\C$ denote the set of complex numbers.

Let $N: \C \to \R_{\ge 0}$ denote the norm on complex numbers:
 * $\forall z \in \C: \map N z = \cmod z^2$

where $\cmod z$ denotes the complex modulus of $z$.

Then $N$ is a multiplicative norm on $\C$.

Proof
To show that $N$ is a multiplicative norm, the multiplicative norm axioms need to be shown to be fulfilled:

Proof of $N1$
Let $z = x + i y$.

So axiom $N1$ holds for $N$.

Proof of $N2$
So axiom $N2$ holds for $N$.

Proof of $N3$
So axiom $N3$ holds for $N$.

All multiplicative norm axioms are seen to be fulfilled.

Hence the result.