As part of a proof of the Chinese-remainder equivalent of groups, I want to show that:
$$ \phi(a \bmod mn)= (a \bmod m, a \bmod n) $$ is a well-defined map
Note that $\gcd(m,n)=1$ so they are coprime.
We usually need to show that given: $a \equiv b \mod {mn}$, we are able to demonstrate that: $$ \phi(a)=\phi(b)$$
So let's start out writing what both this means. $$ (a \bmod m, a \bmod n)=(b \bmod m, b \bmod n)$$ We get the systems of equations: $$ a \bmod m = b \bmod m$$ $$ a \bmod n = b\bmod n$$
Why is this result true, if it is true I can work my way from bottom to top and arrive at the desired result.