Permutation of Indices of Product

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Theorem

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

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


Then:

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

where:

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

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

$\blacksquare$


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