Integral hull of a polyhedron Q is polyhedron

Question

Let $Q \subseteq R^n$ be a rational polyhedron and let $Q_I=Convexhull(Q \cap Z^n)$. By finite basis theorem, we have $Q=P+C$ for some rational polytope $P$ and finitely generated cone $C$ where $C=R_+ a_1+ \cdots+ R_+a_s$ and $a_i \in Z^n$ for $1 \leq i \leq s$. Now consider a linear map $T:R^s \rightarrow R^n $ where $T(a_i)=e_i$ and let $B=T([0,1]^s)$ be a polytope in $R^n$ whose elements have the form $\lambda_1 a_1+\cdots+\lambda_s a_s$ with $0 \leq \lambda_i \leq 1$.
I was reading the proof to show $Q_I$ is a polyhedron from Monomial Algebras by R.H.Villareal. I could not able to prove $$Q_I= Convexhull((P+B)\cap Z^n)+C.$$ Because if the above equality is proved then by finite basis theorem, $Q_I$ is a polyhedron.