Exchange of Order of Indexed Summations

Theorem

Let $\mathbb A$ be one of the standard number systems $\N, \Z, \Q, \R, \C$.

Let $a, b, c, d \in \Z$ be integers.

Let $\closedint a b$ denote the integer interval between $a$ and $b$.

Rectangular Domain

Let $D = \closedint a b \times \closedint c d$ be the cartesian product.

Let $f: D \to \mathbb A$ be a mapping

Then we have an equality of indexed summations:

$\displaystyle \sum_{i \mathop = a}^b \sum_{j \mathop = c}^d \map f {i, j} = \sum_{j \mathop = c}^d \sum_{i \mathop = a}^b \map f {i, j}$