Translation of Index Variable of Summation

Theorem

 * $\displaystyle \sum_{\Phi \left({i}\right)} a_i = \sum_{\Phi \left({c \mathop + j}\right)} a_{c \mathop + j} = \sum_{\Phi \left({c \mathop - j}\right)} a_{c \mathop - j}$

where:
 * $\displaystyle \sum_{\Phi \left({i}\right)} a_i$ denotes the summation over $a_i$ for all $i$ that satisfy the propositional function $\Phi \left({i}\right)$
 * $c$ is an integer constant which is not dependent upon $j$.

Also known as
This operation is often referred to as Adjustment of Indices.

Some expositions, when invoking this result, refer to it as an instance of Permutation of Indices, otherwise known as permutation of the range.