Proof of sum of powerset?
Hint: If I understand you correctly, you are after
$$ \sum_{B \subseteq A} \sum_{x \in B} x. $$
Try reversing the order of summation:
$$ \sum_{B \subseteq A} \sum_{x \in B} x = \sum_{x \in A} \sum_? x, $$
where $?$ is some condition that you will have to come up with. See if that helps you to find a closed formula for the original double sum.