What is the proof of the law of inertia?

This is actually not a very good example for Newton’s First Law (a.k.a the Law of Inertia). Newton’s First Law may be stated as: “An object at rest tends to stay at rest and an object in motion tends to stay in motion with constant velocity unless acted upon by a net external force.”

An orbiting satellite (in a circular orbit) has a constant net external force acting on it (the gravitational force of attraction on the satellite by the Earth), and the satellite is under a constant acceleration toward the center of the Earth. While the satellite’s speed may be constant, the velocity is not. (The direction of travel is constantly changing).

It is unfortunate in the extreme, but nonetheless I feel I must say this; universally, mass is a conserved quantity.

True, but velocity is a vector. Acceleration is a derivative of velocity. You can change a vector by changing its direction and leaving its speed intact.

Just pointing out that no one has brought up angular acceleration and velocity yet.