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 \ \stackrel {\mathbf {def}} {=\!=} \ \begin{cases}

x - y \left \lfloor {\frac 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:

$$ $$ $$