Modulo Addition has Inverses

Theorem
Addition modulo $m$ has inverses:

For each element $$\left[\left[{x}\right]\right]_m \in \mathbb{Z}_m$$, there exists the element $$\left[\left[{-x}\right]\right]_m \in \mathbb{Z}_m$$ with the property:

$$\left[\left[{x}\right]\right]_m +_m \left[\left[{-x}\right]\right]_m = \left[\left[{0}\right]\right]_m = \left[\left[{-x}\right]\right]_m +_m \left[\left[{x}\right]\right]_m$$

Proof
Follows directly from the definition of addition modulo $m$:

As $$-x$$ is a perfectly good integer, $$\left[\left[{-x}\right]\right]_m \in \mathbb{Z}_m$$, whatever it may be.