Integer Divided by Divisor is Integer

Theorem
Let $a, b \in \N$.

Then:
 * $b \mathrel \backslash a \implies \dfrac 1 b \times a \in \N$

where $\backslash$ denotes divisibilty.

Proof
Let $b \mathop \backslash a$.

By definition of divisibilty:
 * $\exists c \in \N: c \times b = a$

Then also:
 * $c \times 1 = c$

So by :
 * $1 : b = c : a$

Hence the result.