I often see the statement- "gibbs free energy of a reaction is minimized at equilibrium"
The reasoning given is that $ \Delta_rG<0$ before equilibrium and $ \Delta_r G=0$ at equilibrium. Thus, it is monotonically decreasing until equilibrium where it stabilizes and therefore, the value of $ \Delta_rG$ at equilibrium is the minimum value.
This makes sense mathematically, but considering the definition of $ \Delta_r G$ i.e. $ \Delta_r G$=$ \Sigma G_{products}-\Sigma G_{reactants}$=o, I don't understand what quantity is being "minimized". $ \Delta_r G$ doesn't seem to denote the change of the same function but rather the difference between gibbs free energies of products and reactants. In other words, the minimum value of a quantity $x$ can occur when function $ \Delta x$=0, but the quantity $x$ needs to have meaning in isolation. The quantity "$ _r G$" is meaningless so what exactly is being minimized here?