I know that the intersection of a regular language and a context free language is a context free language. I've seen this fact proven on here and other websites. However, after spending hours reading through proofs, I don't really understand the general idea of the proof itself. I'm not looking for a formal proof, but I am more concerned with understanding the high level idea. This is my basic idea for a proof, but there are some issues:
Given a regular language $R$ and a context free language $L$, the intersection of $R$ and $L$ is a context free language $L_1$. We know that there exists a DFA $D$ that recognizes $R$, as $R$ is a regular language. We also know that there exists a PDA $P$ that recognizes $L$, as $L$ is a context free language. Since it is a fact that $D$ can be converted into a PDA $P_1$, we know that $R$ is also a context free language. Next, we create a PDA with states as pairs of states from $P$ and $P_1$.
This is where I am getting stuck. How are the transitions modified to simulate both PDAs? And what happens to the stack?