Permutation of Indices of Supremum

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Let $\family {a_i}_{i \mathop \in I}$ be a family of elements of the non-negative real numbers $\R_{\ge 0}$ indexed by $I$.

Let $\map R i$ be a propositional functions of $i \in I$.

Let $\ds \sup_{\map R i} a_i$ be the indexed supremum on $\family {a_i}$.


$\ds \sum_{\map R i} a_i = \sum_{\map R {\map \pi i} } a_{\map \pi i}$

where $\pi$ is a permutation on the fiber of truth of $R$.


\(\ds \sup_{\map R {\map \pi j} } a_{\map \pi j}\) \(=\) \(\ds \sup_{j \mathop \in I} a_{\map \pi j} \sqbrk {\map R {\map \pi j} }\)
\(\ds \) \(=\) \(\ds \sup_{j \mathop \in I} \paren {\sup_{i \mathop \in I} a_i \sqbrk {\map R i} \sqbrk {i = \map \pi j} }\)
\(\ds \) \(=\) \(\ds \sup_{i \mathop \in I} a_i \sqbrk {\map R i} \sup_{j \mathop \in I} \sqbrk {i = \map \pi j}\)
\(\ds \) \(=\) \(\ds \sup_{i \mathop \in I} a_i \sqbrk {\map R i}\)
\(\ds \) \(=\) \(\ds \sup_{\map R i} a_i\)
\(\ds \) \(=\) \(\ds \sup_{\map R j} a_j\) Change of Index Variable of Supremum