# Permutation of Indices of Summation/Proof

## Theorem

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

## Proof

 $\ds \sum_{\map R {\map \pi j} } a_{\map \pi j}$ $=$ $\ds \sum_{j \mathop \in \Z} a_{\map \pi j} \sqbrk {\map R {\map \pi j} }$ Definition of Summation by Iverson's Convention $\ds$ $=$ $\ds \sum_{j \mathop \in \Z} \sum_{i \mathop \in \Z} a_i \sqbrk {\map R i} \sqbrk {i = \map \pi j}$ $\ds$ $=$ $\ds \sum_{i \mathop \in \Z} a_i \sqbrk {\map R i} \sum_{j \mathop \in \Z} \sqbrk {i = \map \pi j}$ $\ds$ $=$ $\ds \sum_{i \mathop \in \Z} a_i \sqbrk {\map R i}$ $\ds$ $=$ $\ds \sum_{\map R i} a_i$ $\ds$ $=$ $\ds \sum_{\map R j} a_j$ Change of Index Variable of Summation

$\blacksquare$