Summation of Zero

Theorem

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

Indexed Summation of Zero

Let $a, b$ be integers.

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

Let $f_0 : \closedint a b \to \mathbb A$ be the zero mapping.

Then the indexed summation of $0$ from $a$ to $b$ equals zero:

$\ds \sum_{i \mathop = a}^b \map {f_0} i = 0$

Finite Set

Let $S$ be a finite set.

Let $0 : S \to \mathbb A$ be the zero mapping.

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Then the summation of $0$ over $S$ equals zero:

$\ds \sum_{s \mathop \in S} 0 \left({s}\right) = 0$

Arbitrary Set

Let $S$ be a set.

Let $0: S \to \mathbb A$ be the zero mapping.

Then the summation with finite support of $0$ over $S$ equals zero:

$\ds \sum_{s \mathop \in S} \map 0 s = 0$