Permutation of Indices of Summation/Proof

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


\(\ds \sum_{\map R {\map \pi j} } a_{\map \pi j}\) \(=\) \(\ds \sum_{j \mathop \in \Z} a_{\map \pi j} \sqbrk {\map R {\map \pi j} }\) 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