For k independent variables, if each one is independent of $Y_1$,...,$Y_p$, how to formally prove their sum is also independent of each $Y_p$? SUppose I have $X_1,...,X_k$ independent of each other. I also have $Y_1,...,Y_p$ is independent of each other. If each one in $X_1$,...,$X_k$ is independent of each one in $Y_1$,...,$Y_p$, how to formally prove that $\sum_k X_k$ is independent of each $Y_p$?
Intuitively, this must be true, as what $Y_p$ turns out to be contains no information about the value of each $X_k$, thus it also contains no information about $\sum_k X_k$. How to formally prove this?

Counterexample: Suppose that $X_1$, $X_2$ and $Y$ take the values
$$
\begin{matrix}
X_1 & X_2 & Y \\
\hline
0 & 0 & 0 \\
1 & 1 & 0 \\
1 & 0 & 1 \\
0 & 1 & 1 \\
\end{matrix}
$$
with equal probability.  Then $X_1$ and $X_2$ are both independent of $Y$ and $X_1$ and $X_2$ are also independent of each other.  But if $Y=1$, $X_1+X_2=1$ with probability one whereas if $Y=0$, $X_1+X_2$ takes values of $0$ and $2$ with equal probabilty.  So $X_1+X_2$ is not independent of $Y$.
