I don't understand a detail in the proof of theorem X.4.2.b) in Silverman's book on elliptic curves (2nd edition), see below.
Here is the context. We have some point $P \in E(\overline{K_v})$, where $E$ is an elliptic curve over a number field $K$ and $v$ is a place of $K$ such that $E$ has good reduction at $v$. We have an element $\sigma$ of the inertia group $I_v$. We have an isogeny $\phi : E \to E'$ defined over $K$ of degree $m$, and it stands that $P^{\sigma} - P \in E[\phi] = \ker(\phi) \subset E[m] \subset E(\overline K)$ and $P^{\sigma} - P$ lies in the kernel of the reduction map $E \to \widetilde E_v$. I've understood the proof until there.
Moreover, the reduction map $$E(K)[m] \to \widetilde{E}(k_v)$$ is injective (this is VIII.1.4). Then, Silverman states that this implies that $\{ P^{\sigma} - P \} = 0$, but I don't see why. I would agree if $$ P^{\sigma} - P \in E(K)[m]$$ would hold, but we only know $P^{\sigma} - P \in E[m]$. How do I solve this?
Here is the caption:
