Mappings Between Residue Classes

Theorem
Let $$\left[\!\left[{a}\right]\!\right]_m$$ be the congruence class of $a$ (modulo $m$).

Let $$\phi: \Z_m \to \Z_n$$ be a mapping given by $$\phi \left({\left[\!\left[{x}\right]\!\right]_m}\right) = \left[\!\left[{x}\right]\!\right]_n$$.

Then $$\phi$$ is well defined when $$m \backslash n$$.

Proof
For $$\phi$$ to be well defined, we require that:

$$\forall x, y \in \Z_m: \left[\!\left[{x}\right]\!\right]_m = \left[\!\left[{y}\right]\!\right]_m \Longrightarrow \phi \left({\left[\!\left[{x}\right]\!\right]_m}\right) = \phi \left({\left[\!\left[{y}\right]\!\right]_m}\right)$$

Now $$\left[\!\left[{x}\right]\!\right]_m = \left[\!\left[{y}\right]\!\right]_m \Longrightarrow x - y \backslash m$$.

For $$\phi \left({\left[\!\left[{x}\right]\!\right]_m}\right) = \phi \left({\left[\!\left[{y}\right]\!\right]_m}\right)$$ we require that $$\left[\!\left[{x}\right]\!\right]_n = \left[\!\left[{y}\right]\!\right]_n \Longrightarrow x - y \backslash n$$.

Thus $$\phi$$ is well defined iff $$x - y \backslash m \Longrightarrow x - y \backslash n$$

That is, iff $$m \backslash n$$.