In showing that if $V=X \oplus Y$, then $X \cap Y = \{0\}$, is writing $v=0+v$ and $v=v+0$ for $v \in X \cap Y$ sufficient?

Question

$V=X \oplus Y$ if every vector in $V$ can be written uniquely as $v=x+y$ for $x \in X$ and $y \in Y$.

Suppose $V=X \oplus Y$. We want to show $X \cap Y =\{0\}$. Suppose not. Let $v \in X \cap Y$. Then $v=v+0$ where $v \in X$ and $0 \in Y$. Also, $v=0+v$ where $0 \in X$ and $v \in Y$. Thus, $v$ cannot be written uniquely.

Is this proof sufficient?

Answer 1

The only thing missing is specifying that $v \ne 0$, so instead of saying "Let $v \in X \cap Y$" I'd say "Pick a non-zero $v$ in $X \cap Y$". Other than that it's fine.