How to prove if $x \in\mathbb N$ and $y \in\mathbb N$, then $(x\mod y) \in\mathbb N$?

Question

The question says it all. It is obviously true, but how do you prove it with actual mathematical symbols?

Answer 1

I suspect this is your definition of operation 'mod' $$ x\text{ mod }y = \min\{ x-k\cdot y : k\in \mathbb{Z}\,\land\, x-k\cdot y\geq0\} $$ Directly from definition, $x\text{ mod }y\in\mathbb{Z}_+ = \mathbb{N}$.

Note that the minimum of that set exists, because for example $$ x\in \{ x-k\cdot y : k\in \mathbb{Z}\,\land\, x-k\cdot y\geq0\}$$ and natural numbers are a well-ordered set (every non-empty subset of them has a minimum).