prove the following inequality for integral

Let $eta$ be the smooth function supported in $B_1(0)$.such that $inteta = 1$

Let $u$ be a smooth function defined on an open set $Vsubset Bbb{R}^n$ that is $uin C_c^infty(V)$.

Prove the following inequality holds：

$$int_{B_1(0)}eta(y)int_0^1int_V|Du(x-epsilon ty)|dxdtdyle int_V|Du(z)|dz$$

I do it as follows extend the integral domain $Vto Bbb{R^n}$ then the inequality $int_V|Du(x-epsilon ty)dxle int_mathbb{R^n}|Du(x-epsilon ty)|dx = int_V|Du(z)|dz$.Is my proof correct?If not how to do it?