Modulo Addition is Well-Defined

Theorem
The addition operation on $$\mathbb{Z}_m$$, the set of integers modulo $m$, defined by the rule:

$$\left[\left[{a}\right]\right]_m +_m \left[\left[{b}\right]\right]_m = \left[\left[{a + b}\right]\right]_m$$

is a well-defined operation.

This operation is called addition modulo $$m$$.

Proof
We need to show that if:


 * $$\left[\left[{x'}\right]\right]_m = \left[\left[{x}\right]\right]_m$$ and
 * $$\left[\left[{y'}\right]\right]_m = \left[\left[{y}\right]\right]_m$$

then $$\left[\left[{x' + y'}\right]\right]_m = \left[\left[{x + y}\right]\right]_m$$.

Since $$\left[\left[{x'}\right]\right]_m = \left[\left[{x}\right]\right]_m$$ and $$\left[\left[{y'}\right]\right]_m = \left[\left[{y}\right]\right]_m$$, it follows from the definition of congruence class modulo $m$ that $$x \equiv x' \left({\bmod\, m}\right)$$ and $$y \equiv y' \left({\bmod\, m}\right)$$.

By definition, we have:


 * $$x \equiv x' \left({\bmod\, m}\right) \Longrightarrow \exists k_1 \in \mathbb{Z}: x = x' + k_1 m$$
 * $$y \equiv y' \left({\bmod\, m}\right) \Longrightarrow \exists k_2 \in \mathbb{Z}: y = y' + k_2 m$$

which gives us $$x + y = x' + k_1 m + y' + k_2 m = x' + y' + \left({k_1 + k_2}\right) m$$.

Thus by definition $$x + y \equiv \left({x' + y'}\right) \left({\bmod\, m}\right)$$.

Therefore, by the definition of congruence class modulo $m$, $$\left[\left[{x' + y'}\right]\right]_m = \left[\left[{x + y}\right]\right]_m$$.

Comment
Although the operation of addition modulo $$m$$ is denoted by the symbol $$+_m$$, if there is no danger of confusion, the symbol $$+$$ is often used instead.

In fact, the notation for addition of two integers modulo $m$ is not usually $$\left[\left[{a}\right]\right]_m +_m \left[\left[{b}\right]\right]_m$$.

What is more normally seen is $$a + b \left({\bmod\, m}\right)$$.