Modulo Addition has Inverses

Theorem
Addition modulo $m$ has inverses:

For each element $$\left[\!\left[{x}\right]\!\right]_m \in \Z_m$$, there exists the element $$\left[\!\left[{-x}\right]\!\right]_m \in \Z_m$$ 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, $$\left[\!\left[{-x}\right]\!\right]_m \in \Z_m$$, whatever it may be.