Permutation of Indices of Summation/Proof

Theorem

$\ds \sum_{\map R j} a_j = \sum_{\map R {\map \pi j} } a_{\map \pi j}$

$\ds \sum_{\map R {\map \pi j} } a_{\map \pi j}$

$=$

$\ds \sum_{j \mathop \in \Z} a_{\map \pi j} \sqbrk {\map R {\map \pi j} }$

$\ds $

$=$

$\ds \sum_{j \mathop \in \Z} \sum_{i \mathop \in \Z} a_i \sqbrk {\map R i} \sqbrk {i = \map \pi j}$

$\ds $

$=$

$\ds \sum_{i \mathop \in \Z} a_i \sqbrk {\map R i} \sum_{j \mathop \in \Z} \sqbrk {i = \map \pi j}$

$\ds $

$=$

$\ds \sum_{i \mathop \in \Z} a_i \sqbrk {\map R i}$

$\ds $

$=$

$\ds \sum_{\map R i} a_i$

$\ds $

$=$

$\ds \sum_{\map R j} a_j$

$\blacksquare$