Prime-inverted spectra are prime-divisible

Proof. We need to show that for every prime \(p\in P\), the map \(\cdot p\colon X[P^{-1}] \to X[P^{-1}]\) is an isomorphism of spectra. We may check this on homotopy groups, where it becomes the statement that the map \[ p\colon \pi _*(X)[P^{-1}] \to \pi _*(X)[P^{-1}] \] is an isomorphism. This holds by elementary properties of localization of abelian groups. □