Using closure properties to show that $L_1=\{a^lb^mc^m|l,m\ge 0\} \cup L(b^*c^*)$ is regular or not
You are not starting with the right proof idea. It's this:


I want to show $L$ is not regular.
Assume towards a contradiction that $L$ is regular.
Since REG is closed against $f$, I conclude that $L' = f(L)$ is also regular.
But I already know that $L'$ is not regular (from elsewhere). Contradiction; $L$ can't be regular.


See here for complete examples.