Powers of Natural Numbers with Zero

In the setting of Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices, let powers in $\mathbb{N}_{0}$ be formed for the multiplication of $\mathbb{N}_{0}$ as in Powers with Exponents in the Natural Numbers with Zero (i) and Powers with Exponents in the Natural Numbers with Zero (ii), and let $m,n,j,k\in\mathbb{N}_{0}$.

(i) Closure. $m^{k}\in\mathbb{N}_{0}$.

(ii) Recursion. $m^{0}=1$ and $m^{k+1}=m^{k}\cdot m$.

(iii) Sum of exponents. $m^{j+k}=m^{j}\,m^{k}$.

(iv) Product of bases. $(mn)^{k}=m^{k}\,n^{k}$.

(v) Bases one and zero. $1^{k}=1$ and $0^{k+1}=0$.

(vi) Positivity. If $1\le m$, then $1\le m^{k}$.