Suppose that $m$ and $n$ are positive prime number, and $r$ and $s$ are some integers.
we can define the following: $rm + sn = 1$.
The question is, how do we lead to the following?
$rm + sn = 1 \mod {mn}$
$sn = 1 \mod {m}$
Also,
the function f : $\mathbb{Z}/m$ × $\mathbb{Z}/n$ -> $\mathbb{Z}/mn$ defined by $f(x, y) = y · rm + x · sn$ is a bijection. The inverse map $f^{−1} : \mathbb{Z}/mn -> \mathbb{Z}/m × \mathbb{Z}/n$ is $f^{−1}(z) = $ ($x$-mod-m, $y$-mod-n).
What does this function imply in the usage of Chinese remainder theorem? Also, what does $x$-mod-m exactly mean?