# Permutation of Indices of Product

## Theorem

Let $R: \Z \to \set {\mathrm T, \mathrm F}$ be a propositional function on the set of integers.

Let the fiber of truth of $R$ be finite.

Then:

$\displaystyle \prod_{\map R j} a_j = \prod_{\map R {\map \pi j} } a_{\map \pi j}$

where:

$\displaystyle \prod_{\map R j} a_j$ denotes the product over $a_j$ for all $j$ that satisfy the propositional function $\map R j$
$\pi$ is a permutation on the fiber of truth of $R$.

## Proof

 $\displaystyle \prod_{\map R {\map \pi j} } a_{\map \pi j}$ $=$ $\displaystyle \prod_{j \mathop \in \Z} {a_{\map \pi j} }^{\sqbrk {\map R {\map \pi j} } }$ Definition of Product by Iverson's Convention $\displaystyle$ $=$ $\displaystyle \prod_{j \mathop \in \Z} \prod_{i \mathop \in \Z} {a_i}^{\sqbrk {\map R i} \sqbrk {i = \map \pi j} }$ $\displaystyle$ $=$ $\displaystyle \paren {\prod_{i \mathop \in \Z} {a_i}^{\sqbrk {\map R i} } } \uparrow \paren {\prod_{j \mathop \in \Z} \sqbrk {i = \map \pi j} }$ using Knuth uparrow notation $\displaystyle$ $=$ $\displaystyle \prod_{i \mathop \in \Z} {a_i}^{\sqbrk {\map R i} }$ $\displaystyle$ $=$ $\displaystyle \prod_{\map R i} a_i$ $\displaystyle$ $=$ $\displaystyle \prod_{\map R j} a_j$ Change of Index Variable of Product

$\blacksquare$