Vector space property proof verification My problem is from Linear Algebra by Friedberg, Insel, Spence:
In any vector space $V$, show that $(a+b)(x+y)=ax+ay+bx+by$ for any $x,y$ in $V$, and any $a,b$ in F.
My approach is:
$$\begin{eqnarray*}
(a+b)(x+y) &=& (a+b)x+(a+b)y \\ \\
&=& ax+bx+ay+by \\ \\
&=& ax+ay+bx+by
\end{eqnarray*}$$
However, this seems too obvious to me and I'm wondering if I'm missing anything?

After being pointed out what I need to do, here's my answer:
Let $(a+b)=c\in F$ (By closure of addition in fields)
$\Rightarrow c(x+y)=cx+cy$ (By scalar distribution)
i.e. $(a+b)\cdot(x+y)=(a+b)x+(a+b)y$
$\Rightarrow (a+b)\cdot(x+y)=ax+bx+ay+by$ (By vector distribution)
$\Rightarrow (a+b)\cdot(x+y)=ax+ay+bx+by$ (By commutativity of addition in vector space $V$)
