Let $$ E: y^2 = x^3+Ax+B $$ be an elliptic curve over a finite field $\mathbb{F}_q$ of characteristic $p\neq2,3$ and let $G$ be a finite subgroup of $E$ of order 2 or odd. Vélu's formulas for $$ \phi : E \rightarrow E' \\ (x,y) \mapsto (x',y')$$ can be written as \begin{align*} x'(P) &= x(P) + \sum_{Q\in G\setminus\{\infty\}} \big(x(P+Q)-x(Q)\big) \\ y'(P) &= y(P) + \sum_{Q\in G\setminus\{\infty\}} \big( y(P+Q) - y(Q)\big) \;, \end{align*} How do I see that the above formulas for $x',y'$ show that the points in the subgroup $G$ are exactly the points mapping to $\infty$? This is from "Elliptic Curves, Number Theory and Cryptography" by Washington (proof of lemma 12.17).
The curve $E'$ is given by $$ E' : y^2 = x^3 + (A-5v)x + B-7w \;,$$ where \begin{align*} g_Q^x &= 3x_Q^2+A \;, & g_Q^y &= -2y_Q \\ v_Q &= \begin{cases} g_Q^x & \text{if } |G|=2 \\ 2g_Q^x & \text{otherwise,} \end{cases} & u_Q &= (g_Q^y)^2 \\ v &= \sum_{Q\in R} v_Q \;, & w &= \sum_{Q\in R} u_Q+x_Qv_Q \; \end{align*} and I know that $x',y'$ are points on $E'$.
Suppose that $P\in G$. Then taking $Q=-P$ (negation in group of points on the elliptic curve) gives that $x(P+Q)$ and $y(P+Q)$ are infinite ($P+Q=O$ on $E$). All other terms in the formula are finite, so $x'(P)$ and $y'(P)$ are infinite, so $P$ maps to the basepoint of $E'$.