Binomial expansion of (u+p^rv)^{p^n} modulo p^{n+r+1}

Let $A$ be a commutative ring, let $p$ be a prime, and let $n, r$ be natural numbers with $1 \le r$ and such that either $p \ne 2$ or $2 \le r$. Then for all $u, v \in A$ there exists $w \in A$ with

where $p$ denotes the image of the natural number $p$ under the canonical map $\mathbb{N} \to A$ and $p^n - 1$ is truncated subtraction in $\mathbb{N}$ (harmless, since $p^n \ge 1$). In other words, the binomial expansion of $(u+p^rv)^{p^n}$ agrees with its first two terms modulo the ideal generated by $p^{n+r+1}$. No hypothesis of torsion-freeness, flatness or characteristic is imposed on $A$; the assertion is the existence of a witness $w$, not a formula for it. The case $p = 2$, $r = 1$ is genuinely excluded: there the term of index $2$ contributes $2^{n+1}u^{2^n-2}v^2$, which need not lie in $2^{n+2}A$.

This is the elementary Kummer-type estimate $v_p\binom{p^n}{m} = n - v_p(m)$ packaged as a congruence: the $m$-th binomial term of $(u+p^rv)^{p^n}$ is divisible by $p^{rm+n-v_p(m)}$, and $rm - v_p(m) \ge r+1$ for every $m \ge 2$ precisely under the stated hypothesis on $p$ and $r$. It is used in the deformation-theoretic part of the development, in the analysis of the partial sums of the $w$-series attached to a $p$-adic evaluation (Deformation.PLoc.wPartialSum_adicEval_add_sub_sub_algebraMap_mul_sum_mem_powSub), where it supplies the approximate additivity of $x \mapsto \sum_n p^{-n} a_n(x)^{p^n}$ under $x \mapsto x + p^r y$.