The premise of the matrix violates the first law of thermodynamics; It's just a plot device.

Except although the movie does say battery. That's not the original plot device that was written. They were originally written as human based cpu computers. But some higher up thought people wouldn't understand "human processors" so just everytime you think about or watch the matrix again. Just think it's a poor mistranslation that made it into the movie, "processor->battery"

Of course battery is just what Morpheus and the others from Zion think is the story.

Considering how wrong they were on other important matters the battery explanation could easily be wrong and the CPU correct.

Wasn't there some deleted dialog about this?

I'm paraphrasing, but Neo says "that's impossible (in reference to the Matrix) since it violates the laws of thermodynamics"

To which Morpheus replies "and where did you learn about the laws of thermodynamics?"

If you're in the Matrix, your knowledge of the Matrix would come from the Matrix itself which would have a bias towards falsifying information to preserve itself.