Learn Russian: Ты не прав, но это нельзя доказать.

When нельзя is followed by a perfective verb, it means the action is physically or logically impossible to successfully complete. Here, no matter how hard someone tries, the result of proving it simply cannot be achieved.

What would it mean if we used the imperfective verb доказывать with нельзя instead?

If you used the imperfective доказывать, the meaning would change from 'impossible' to 'forbidden'. Нельзя доказывать would mean 'it is not allowed to prove' or 'one must not prove it'. Since the speaker means the proof just can't be done, the perfective is required.

Why is прав used instead of the full adjective правый?

прав is a short-form adjective. Russian uses short adjectives for certain conditions, and for a person 'being right', the short forms (прав, права, правы) are the standard way to express it. The full form правый is mostly used to mean 'right' as a physical direction (like the right hand).

Is there a single Russian word for 'wrong' that I could use instead of не прав?