  We prove two basic structural properties of the algebraic $K$-theory of rings
after $K(1)$-localization at an implicit prime $p$. Our first result (also
recently obtained by Land--Meier--Tamme by different methods) states that
$L_{K(1)} K(R)$ is insensitive to inverting $p$ on $R$; we deduce this from
recent advances in prismatic cohomology and $\mathrm{TC}$. Our second result
yields a K\"unneth formula in $K(1)$-local $K$-theory for adding $p$-power
roots of unity to $R$.
