Proving non-regualarity from the trivial case

Question

The language $L = \{a^nb^n \mid n \geqslant 0\}$ is a well-known non-regular language. Suppose I have a similar language $L_2 = \{a^{2n}b^{2n} \mid n \geqslant 0\}$, is there any way to formally logically prove $L_2$ non-regularity from $L$?

I know:

1. A subset of a non-regular language is not necessarily non-regular
2. A homomorphism of a non-regular language is not necessarily non-regular

Answer 1

Hint. Let $f:\{a,b\}^* \to \{a,b\}^*$ be the homomorphism defined by $f(a) = a^2$ and $f(b) = b^2$. Show that $f^{-1}(L_2) = L$ and use the fact that regular languages are closed under inverses of homomorphisms.