Learn French: Vous n'avez rien prouvé. - You didn't prove anything.

In the passé composé, the negative word rien (nothing/anything) goes between the helper verb (avez) and the past participle (prouvé). This is different from personne (nobody), which always goes after the past participle at the very end of the phrase.

Why isn't there a pas in this sentence?

The word rien acts as the second half of the negative sandwich, replacing pas. You use ne ... rien just like you use ne ... pas. Saying n'avez pas rien would be a double negative and is incorrect.

The translation says "didn't prove anything". Does this literally mean "You have proved nothing"?

Exactly! "You have proved nothing" is exactly how the French sentence is built. English speakers usually say "didn't prove anything" to sound more natural, but French uses ne ... rien to cover both phrases.

How do we know to use avez here instead of êtes?

The verb prouver (to prove) uses avoir as its helping verb, just like most French verbs do in the passé composé. You only use être for verbs of motion, changes of state, or reflexive verbs.