Let $(X_1,d_1,m_1)$ and $(X_2,d_2,m_2)$ be complete separable geodesic metric measure spaces. Assume that $m_i$ is a locally finite Borel measure on $X_i$ with full support for each $i\in\{1,2\}$. Let $K\in\mathbb R$, and let $N_1,N_2\in(1,\infty]$. Suppose that, for each $i\in\{1,2\}$, the metric measure space $(X_i,d_i,m_i)$ satisfies the weak Lott-Sturm-Villani curvature-dimension condition $CD(K,N_i)$.

Let $X:=X_1\times X_2$, let $m:=m_1\otimes m_2$, and define $d:X\times X\to[0,\infty)$ by

\begin{align*} d((x_1,x_2),(y_1,y_2))=\left(d_1(x_1,y_1)^2+d_2(x_2,y_2)^2\right)^{1/2}. \end{align*}

Then $(X,d,m)$ satisfies the weak Lott-Sturm-Villani curvature-dimension condition $CD(K,N_1+N_2)$, where $N+\infty=\infty$ for every $N\in(1,\infty]$.