For the Chinese Remainder Theorem for rings we have:
$$ A/(I\cap J) \cong A/I\times A/J $$
So far I have proven that there is a ring homomorphism from $\phi :A \rightarrow A/I\times A/J $ and the $ker(\phi)=I\cap J$. Additionally, since I and J are comaximal, $I + J = A$, implies $I\cap J=IJ $
My question is, how does this last fact prove isomorphism in the last step of the proof?