Let $\alpha=(\alpha_1,\alpha_2,\ldots,\alpha_n)$ is an n-tuple of nonnegative integers, $x\in\mathbb{R}^n$ and $|\alpha|=\sum_{i=1}^n\alpha_i.$ Let $\beta=(\beta_1,\beta_2,\ldots,\beta_n)$ is an n-tuple of nonnegative integers, $x\in\mathbb{R}^n$ and $|\beta|=\sum_{i=1}^n\beta_i.$ Let $u\in W^{m,p}(\Omega)$ where $\Omega\subset\mathbb{R}^n$ bounded domain, $m$ is a nonnegative integer, $k\in(0,m)$ and $1\leq p<+\infty.$ My question is, the following statement is true or false, and why?
$$\sum_{|\alpha|=k}\sum_{|\beta|=m-k}\|D^{\alpha+\beta}u\|^p_{L_{p}(\Omega)}=|u|_{W^{m,p}(\Omega)}^p.$$ Where $|\cdot|_{W^{m,p}(\Omega)}$ is the seminorm on Sobolev spaces and $$D^av=\frac{\partial^{|\alpha|}v}{\partial x_1^{\alpha_1}\cdots\partial x_n^{\alpha_n}}.$$