This proposition is from Silverman's Arithmetic of Elliptic Curves.The proof is a bit long to type so I've included the image.
Here $K$ is a local field with residue field $k$ of its ring of integers $R$.$\tilde{E}(k)$ is the reduction of the elliptic curve modulo the maximal ideal.And,$E_1(K):=\{P\in E(K) : \tilde{P}=\tilde{O}\}$ where $\tilde{P}$ is the reduction of $P$ modulo maximal ideal.
I do not understand why the highlighted text is true.I can see that $\nu(y)<0$ but why is it true that $\nu(x)<0$?Also,it is not obvious to me that the map defined(2nd highlighted text) is injective.Can someone help me?