Summation of Zero/Indexed Summation

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

Let $\left[{a \,.\,.\, b}\right]$ denote the integer interval between $a$ and $b$.

Let $0 : \left[{a \,.\,.\, b}\right] \to \mathbb A$ be the zero mapping.

Then the indexed summation of $0$ from $a$ to $b$ equals zero:
 * $\displaystyle \sum_{i \mathop = a}^b 0(i) = 0$

Proof
At least three proofs are possible:
 * by induction, using Identity Element of Addition on Numbers
 * using Indexed Summation of Multiple of Mapping
 * using Indexed Summation of Sum of Mappings