If \((a ≡ b \bmod m)\) and \((a ≡ b \bmod n)\):
$$\begin{gather} m|(b-a) \\ n|(b-a) \end{gather}$$
Since \(\gcd(m, n) = 1\), we can use this lemma here:
$$ mn|(b-a) $$
This means \((a ≡ b \bmod mn)\).
If \((a ≡ b \bmod m)\) and \((a ≡ b \bmod n)\):
Since \(\gcd(m, n) = 1\), we can use this lemma here:
This means \((a ≡ b \bmod mn)\).