Is the Kleene star of an intersection contained in the intersection of Kleene stars? I need to find if given two formal languages $L_1$ and $L_2$ $$(L_1 \cap L_2)^*\subseteq (L_1^* \cap L_2^*) $$
I think that it's true since this can be rewritten as $$ \bigcup^\infty_{i=0}(L_1 \cap L_2)^i \subseteq (\bigcup^\infty_{i=0}L_1^i \cap \bigcup^\infty_{i=0}L_2^i)$$ and in general, $$(A \cap B) \cup (C \cap D) \subseteq (A \cup C) \cap (B \cup D)$$
I may be wrong since it has been a long time since i have done anything with set theory, and this is taken as given.
Thanks
Yes, it is true. But  the argument to support the conjecture is incorrect.

The correct proof is as follows: $L_1 \cap L_2 \subseteq L_1 $ and therefore $(L_1 \cap L_2)^* \subseteq L_1^*$. (argument: if $A \subseteq B$ then $A^* \subseteq B^*$).
Similarly $(L_1 \cap L_2)^* \subseteq L_2^*$.

Therefore $(L_1 \cap L_2)^* \subseteq (L_1^* \cap L_2^*)$ (argument: if $A \subseteq B$ and $A \subseteq C$ then $A \subseteq B \cap C$).

Anyway you can not write $(A \cap B)^i \cup (C \cap D)^j = (A^i \cup C^j) \cap (B^i \cup D^j)$.

Correct relation is: $(A \cap B) \cup (C \cap D) = (A \cup C) \cap (B \cup D) \cap (A \cup D) \cap (B \cup C) $

However $\bigcup^\infty_{i=0}(L_1 \cap L_2)^i \subseteq (\bigcup^\infty_{i=0}L_1^i \cap \bigcup^\infty_{i=0}L_2^i)$ is correct because
$\bigcup^\infty_{i=0}(L_1 \cap L_2)^i \subseteq \bigcup^\infty_{i=0}L_1^i $  and $\bigcup^\infty_{i=0}(L_1 \cap L_2)^i \subseteq \bigcup^\infty_{i=0}L_2^i $.