We place ourselves in the ring of formal power series.
Why is it that :
$$\sum_{(k_{1},...,k_{d})\in \Bbb{N}^{d}}\prod_{i=1}^dT^{k_{i}m_{i}}=\prod_{i=1}^d\sum_{k\in \Bbb{N}}T^{km_{i}}$$
I guess that developing everything works but is there an easier method to see why?
A formal rule to go from LHS to RHS ?
Expand the RHS: $$ \begin{align} \prod_{i = 1}^d \sum_{k \in \Bbb{N}} T^{km_{i}} &= \prod_{i = 1}^d \left( 1 + T^{m_i} + T^{2m_i} + \cdots \right) \\ &= \left( 1 + T^{m_1} + T^{2m_1} + \cdots \right) \left( 1 + T^{m_2} + T^{2m_2} + \cdots \right) \\ &\quad \cdots \left( 1 + T^{m_d} + T^{2m_d} + \cdots \right) \end{align} $$
Let's chew on this final expression for a minute. When we open the parentheses, each term is going to be a $d$-fold product of powers of $T^{m_1}, \ldots, T^{m_d}$. Every possible power appears, or in other words, every term of the form $$ \prod_{i = 1}^d T^{k_i m_i} = T^{k_1 m_1} \cdots T^{k_d m_d} $$ appears exactly once. Therefore, we have the sum on the LHS.