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 $\left\vert{z}\right\vert$ denote the modulus of $z$.

Then:
 * $\left\vert{z}\right\vert = \left\vert{\overline z}\right\vert$

Proof
Let $z = a + b i$.

Then:

and: