= Congruence reflection for Frobenius substitution
{title2=$\phi^{-1}(p^kR)=p^kR$}
For $R=\mathbb Z_p[[T]]$ and $\phi(f)=f((1+T)^p-1)$, reduction modulo $p$ is the injective substitution $\overline f(T)\mapsto\overline f(T^p)$. Thus $\phi(f)\in pR$ implies $f\in pR$. Dividing by $p$ and repeating proves the displayed equality for every $k\geq1$. In particular, a <ring congruence> of <Frobenius substitutions on cyclotomic power series> reflects back to the original <formal power series>.
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