Complex Modulus equals Complex Modulus of Conjugate

Theorem
Let $z \in \C$ be a complex number.

Let $\overline z$ denote the complex conjugate of $z$.

Let $\cmod z$ denote the modulus of $z$.

Then:
 * $\cmod z = \cmod {\overline z}$

Proof
Let $z = a + b i$.

Then:

and: