Linear programming formulation for the single-source shortest path problem

Question

In this course lecture; section 5.1, single-source shortest path (SSSP) is formulated as the following linear program (LP):

\begin{align} \max &\sum d_u \\ \text{subject to} & \\ d_v &\le d_u + l_{uv} \quad \forall (u,v) \in E \\ d_s &= 0 \end{align}

The comment on the objective function is as follows (emphasis added):

The variables $d_u$ represent the distances from $s$ to each vertex $u$. Maximizing the sum of the $d_u$ is done by maximizing each one individually, since increasing any single $d_u$ never forces us to decrease some other $d_v$.

I can get its basic idea. However, how to argue that $(\max d_u \;\forall u \in V)$ is equivalent to $(\max \sum d_u)$ more rigorously? Specifically, why is that "increasing any single $d_u$ never forces us to decrease some other $d_v$"?