I'm studying tensors right now and there are lots of sums involved and I've seen they do these sorts of things:
$(\sum_i x)(\sum_j y) (\sum_k z)=\sum_k \sum_j \sum_i x y z$
I've shown mechanically that this is true due to the distributive property when I consider two split sums. But I want a general proof that this way of joining the sums is valid and makes sense. I guess I'm not specifically looking for an equivalence to the distributive property, but just a statement that justifies such a step.
Thanks a whole lot.