For context, A and B is a regular language, and related to Theory of Computation.
As the title says, it is so intuitive that $$ \text{if } {B \subseteq A} \implies A^*B^* = A^* $$
I don't know how to prove this formally. Is this an axiom or I'm missing some sort of key observation that can formally prove the statement?
The best I can do is:
We can see that $\{\varepsilon\} \in B^* \implies A^* \subseteq A^*B^*$.
We also know that $B \subseteq A \implies B^* \subseteq A^* \implies A^*B^* \subseteq A^*A^* = A^* $
This proofs $A^* \subseteq A^*B^* \wedge A^*B^* \subseteq A^* \implies A^*B^* = A^*$
But, I don't think that's good enough. Because I feel like it's missing something or it has something wrong. Is there any better way to prove this instead of the way I'm doing it right now?
The essence of the proof is that $$A^*=A^*\{\epsilon \}\subseteq A^*B^* \subseteq A^*A^*=A^*.$$ Is this formal enough or do you want to break it down further into definitions?