I'm learning about the construction of the dual isogeny in Silverman's Arithmetic of Elliptic Curves. In particular, I'm reading the proof for Theorem 6.1 from Chapter III. There is a (probably easy) fact used in Line 4 of the proof for Theorem 6.1(b) that I'm not seeing immediately. I was wondering if someone can help me see why it's true.
Let $\phi:E_1\to E_2$ be a non-constant isogeny. Let $Q\in E_2$. Then (why is it true that) $$ \sum_{P\in\phi^{-1}(Q)}P = [\#\phi^{-1}(Q)] P \tag{for any $P\in \phi^{-1}(Q)$} $$ Here, addition is for points on an elliptic curve, and $[k]: E_1\to E_1$ is the multiplication by $k$ map for any $k\in\mathbb{Z}$.
The summation they are subtracting in the proof matters too; fix $P$ and prove then that $\phi^{-1}(Q)=\{P+T\mid T\in\phi^{-1}(O)\}$