Number minus Modulo is Integer Multiple

Theorem
Let $x, y \in \R$ be real numbers.

Let $x \bmod y$ denote the modulo operation:
 * $x \bmod y := \begin{cases}

x - y \left \lfloor {\dfrac x y}\right \rfloor & : y \ne 0 \\ x & : y = 0 \end{cases}$ where $\left \lfloor {\dfrac x y}\right \rfloor$ denotes the floor of $\dfrac x y$.

Let $y < 0$.

Then:


 * $x - \left({x \bmod y}\right)$

is an integer multiple of $y$.

Proof
When $y = 0$ we have:
 * $x \bmod y := x$

Thus:
 * $x - \left({x \bmod y}\right) = 0$

From Zero is Integer Multiple of Zero it follows that:
 * $x - \left({x \bmod y}\right)$

is an integer multiple of $y$.

Let $y \ne 0$.

Then:

From Floor Function is Integer, $\left \lfloor {\dfrac x y}\right \rfloor$ is an integer.

Thus:
 * $\exists n \in \Z: x - \left({x \bmod y}\right) = n y$

where in the case $n = \left \lfloor {\dfrac x y}\right \rfloor$

Hence the result by definition of integer multiple.