In linear measurement with noise, we denote the measurement as $$y = Ax+v$$ where $x$ is the measure we want to estimate, $y$ is the measurement, $A$ characterizes some sensor, and $v$ is sensor noise. We assume that $$x \sim \mathcal{N}(\bar{x}, \Sigma_x), v \sim \mathcal{N}(\bar{v}, \Sigma_v)$$ and that $x$ and $v$ are independent. Then in the proof, it takes $x$ and $v$ and does the following: $$\begin{bmatrix} x \\v \end{bmatrix} \sim \mathcal{N} \Big( \begin{bmatrix} \bar{x} \\ \bar{v} \end{bmatrix}, \begin{bmatrix} \Sigma_x, 0 \\ 0, \Sigma_v \end{bmatrix} \Big)$$
How do we know that combining the two in such a way results in a normal distribution? What exactly is this operation? Any link to a proof would be appreciated.