Is there a proof of the first law of thermodynamics?

Just like according to Is there any proof for the 2nd law of thermodynamics?, there might not be a proof of the second law of thermodynamics, I'm wondering if there isn't a proof of the first law either. I know how it can be proven according to Newtonian physics that energy is always conserved because of the electrostatic potential energy. Although I don't know of a proof, I trust that it has also been proven that energy is always conserved according to non quantum mechanical general relativity. Since atoms don't follow classical relativity, I don't see how to prove that there even exists a way of defining the amount of energy per mass of each substance at any temperature in such a way that the total amount of energy in an isolated system never changes.

Answer

I don't see how to prove that there even exists a way of defining the amount of energy per mass of each substance at any temperature in such a way that the total amount of energy in an isolated system never changes.

Physics theories are mathematical models for specific frameworks where measurements and observations are made.

These mathematical models are strict and axiomatic, and physics enters with "laws" or "postulates" in order to pick the subset of mathematical solutions that are relevant to data, i.e. fit existing data and are predictive of new data.

The underlying level of nature is quantum mechanical, and at this level conservation laws are important and axiomatic, because together with the mathematical model of quantum mechanics the theories developed are descriptive, predictive and have been continuously validated.

Since the quantum mechanical is the underlying level of the physics frameworks, there is a way to show how classical frameworks emerge from quantum frameworks , i.e. how in the limit of h_bar being zero Newtonian mechanics emerges , and classical electrodynamics emerges, see for example here. It needs quantum electrodynamics background to work things out.

The thermodynamic framework emerges from the statistical mechanics framework. Statistical mechanics solves the many body problem of Newtonian mechanics. Thus the laws of Newtonian mechanics are incorporated in the laws of thermodynamics.

So there is a mathematical way to go from the underlying "laws" in the inner framework of quantum mechanics to the thermodynamic "laws". Of course one realizes that laws were originally discovered in the classical frameworks, and applied to the mathematical theories. When the quantum mechanical framework was discovered it was found that the law of conservation of energy applied there too. In special relativity the definition of energy was expanded so that the laws would still hold, and relativistic quantum mechanics still describes data and predicts new set ups, as has been seen at the LHC the last few years.

General Relativity, which applies to a framework of large dimensions and masses is another story. It does not emerge from the quantum mechanical framework and theorists are searching for a unifying theory , but this is at the frontier of physics research.