Product Distributes over Modulo Operation

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

Let $x \,\bmod\, y$ denote the modulo operation.

Then:
 * $z \left({x \,\bmod\, y}\right) = \left({z x}\right) \bmod\, \left({z y}\right)$

Proof
Let $x \,\bmod\, y$.

From the definition of the modulo operation, we have:


 * $x \, \bmod \, y := \begin{cases}

x - y \left \lfloor {\dfrac x y}\right \rfloor & : y \ne 0 \\ x & : y = 0 \end{cases}$

If $y = 0$ we have immediately that:
 * $z \left({x \,\bmod\, 0}\right) = z x = \left({z x}\right) \,\bmod\, \left({z 0}\right)$

If $y \ne 0$ we have that: