Permutation of Indices of Supremum

Theorem

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 $\family {a_i}$.

Then:

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

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 $

$=$

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

$\blacksquare$