Change of Index Variable of Product

Theorem

 * $\displaystyle \prod_{R \left({i}\right)} a_i = \prod_{R \left({j}\right)} a_j$

where $\displaystyle \prod_{R \left({i}\right)} a_i$ denotes the product over $a_i$ for all $i$ that satisfy the propositional function $R \left({i}\right)$.

Proof
Let $S = \left\{ {i \in \Z: R \left({i}\right)}\right\}$.

Let $T = \left\{ {j \in \Z: R \left({j}\right)}\right\}$.

Let $i \in S$.

Then $R \left({i}\right)$.

Let $j = i$.

By Leibniz's Law, $R \left({j}\right)$.

Thus $i \in T$.

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

Similarly, let $j \in T$.

Then $R \left({j}\right)$.

Let $i = j$.

By Leibniz's Law, $R \left({i}\right)$.

Thus $j \in S$.

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

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

Thus: