I am looking for a reference for a proof of the following:

Let $n$ and $a,b, \ldots ,z$ be non-negative integers with $a + b + \ldots + z = n$, and let $p$ be a prime. Write $n = n_0 + n_1 p + \ldots + n_m p^m$ in $p$-ary notation, similarly for $a, b, \ldots , z$. Then, modulo $p$, the multinomial coefficient ${n \choose a,b, \ldots, z}$ is zero if there is some "carrying" in computing the sum $a + b + \ldots + z$ (i.e., $a_i + b_i + \ldots + z_i \geq p$ for some $i$). Otherwise, it is the product of the multinomial coefficients of the individual $p$-digits, that is,

${ n \choose a, b, \ldots, z} = {n_0 \choose a_0, b_0, \ldots, z_0}{n_1 \choose a_1, b_1, \ldots, z_1} \ldots {n_m \choose a_m, b_m, \ldots , z_m}$

I've found plenty of references for this theorem restricted to *binomial* coefficients, otherwise known as Lucas' theorem, but can't seem to find one for arbitrary multinomials.

Thanks in advance for the help.