 Frobenius substitution on cyclotomic power series

On $R={\mathbb{Z}}_p[[T]]$, this injective substitution reduces modulo $p$ to $f(T)\mapsto f(T^p)$. The ring is a free module of rank $p$ over $\phi (R)$, with basis $1,(1+T),\dots ,(1+T{)}^{p-1}$. Prove this first modulo $p$, where the change of basis from $1,T,\dots ,T^{p-1}$ is triangular and invertible, and then lift coefficients successively modulo powers of $p$.