In aggregate a force is applied to all the air in a way that is equal and opposite to the force applied on the plane. However "Ergo, air is moving downwards so that the plane will move upwards." is not quite complete. One need only look at an airfoil in a windtunnel to see that the action of an airfoil is not the simple "air based rocket" you would have us envision.
If what you said were true, the most efficient wing design would just be a flat plank strapped in at a 45 degree angle of attack.