Properties of Natural Number Powers in a Field

Let $K$ be a field, with additive identity $0$ and multiplicative identity $1$, and let $c,d\in K$. Let $\mathbb{N}$ be the set of natural numbers with successor map $S$ as in that definition, and let $n\in\mathbb{N}$. Powers are those of Natural Number Power of an Element of a Field.

Then the following hold.

1. (Recursion) $c^{1}=c$, and $c^{S(n)}=c^{n}\,c$.

2. (Unit) $1^{n}=1$.

3. (Products) $(c\,d)^{n}=c^{n}\,d^{n}$.

4. (Vanishing) $c^{n}=0$ if and only if $c=0$.

5. (Nonnegative bases) Suppose $K$ is the field of real numbers, with the order $\le$ of its ordered field structure, and that $0\le c$. Then $0\le c^{n}$; and if in addition $c\le d$, then $c^{n}\le d^{n}$.