Basic computation in Steenrod algebra In Homology operations for $H_\infty$ and $H_n$ spectra (pdf), Steinberger makes the computation of the Dyer-Lashof operations in $H\mathbb F_p$, and at some point uses the following "basic fact"

Lemma 6.1. The following equalities hold in $\mathcal A_*$. For $p\geq2$ and $i\geq0$,
$$P^r_*\chi\xi_i=\left\{\begin{array}{ll}-\chi\xi_{i-k}^{p^k}&\text{if }r=\frac{p^k-1}{p-1}\\0&\text{otherwise}\end{array}\right.$$
[...]

I have tried my best to show it by hands but I couldn't manage to make the computation myself. Also I've been looking for a reference without success...
Would someone know a reference where this is made explicit ?
 A: Let me give this a shot.  I'm bound to mess up signs, so assume $p = 2$.  Write $\langle \cdot, \cdot \rangle$ for the pairing between $\mathcal{A}_*$ and $\mathcal{A}^*$.

Lemma. $\newcommand{\Sq}{\mathrm{Sq}}\langle \chi \xi_k, \Sq^r \rangle = \delta_{r = 2^k-1}$.

Proof. We have
\begin{align}
\langle \chi \xi_k, \Sq^r \rangle &= \langle \xi_1^{2^k-1} \bmod{(\xi_2, \xi_3, \ldots)}, \Sq^r \rangle \\
&= \langle \xi_1^{2^k-1}, \Sq^r \rangle \\
&= \delta_{r = 2^k-1}.
\end{align}

Proposition. $\Sq^r_* \chi \xi_i = \begin{cases} \chi \xi_{i-k}^{2^k}, & \text{if } r = 2^k - 1 \\ 0, & \text{otherwise}. \end{cases}$

Proof. For any $x \in \mathcal{A}^*$, we have
\begin{align}
\langle \Sq^r_* \chi \xi_i, x \rangle &= \langle \chi \xi_i, \Sq^r x \rangle \\
&= \langle \chi \xi_i, \mu(\Sq^r \otimes x) \rangle \\
&= \langle \xi_i, \chi \mu(\Sq^r \otimes x) \rangle \\
&= \langle \xi_i, \mu(\chi x \otimes \chi \Sq^r) \rangle \\
&= \langle \Delta \xi_i, \chi x \otimes \chi \Sq^r \rangle \\
&= \langle \sum_k \xi_{i-k}^{2^k} \otimes \xi_k, \chi x \otimes \chi \Sq^r \rangle \\
&= \sum_k \langle \xi_{i-k}^{2^k}, \chi x \rangle \cdot \langle \xi_k, \chi \Sq^r \rangle \\
&= \sum_k \langle \chi \xi_{i-k}^{2^k}, x \rangle \cdot \langle \chi \xi_k, \Sq^r \rangle \\
&= \sum_k \langle \chi \xi_{i-k}^{2^k}, x \rangle \cdot \delta_{r=2^k-1} \\
&= \begin{cases} \langle \chi \xi_{i-k}^{2^k}, x \rangle, & \text{if } r = 2^k - 1 \\ 0, & \text{otherwise}. \end{cases}
\end{align}
