Change of Index Variable of Product

Theorem

 * $\ds \prod_{\map R i} a_i = \prod_{\map R j} a_j$

where $\ds \prod_{\map R i} a_i$ denotes the product over $a_i$ for all $i$ that satisfy the propositional function $\map R i$.

Proof
Let $S = \set {i \in \Z: \map R i}$.

Let $T = \set {j \in \Z: \map R j}$.

Let $i \in S$.

Then $\map R i$.

Let $j = i$.

By Leibniz's Law, $\map R j$.

Thus $i \in T$.

By definition of subset, $S \subseteq T$.

Similarly, let $j \in T$.

Then $\map R j$.

Let $i = j$.

By Leibniz's Law, $\map R i$.

Thus $j \in S$.

By definition of subset, $T \subseteq S$.

Thus by definition of set equality: $S = T$

Thus: