Summation of Zero/Set

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Theorem

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

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:

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


Proof

By Support of Zero Mapping, the support of $0$ is empty.

By Empty Set is Finite, the support of $0$ is indeed finite.

By Summation over Empty Set:

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

$\blacksquare$