Is the word provei the same in Brazilian and European Portuguese?

Yes

The word provei is identical in meaning, spelling, and grammatical function in both Brazilian and Continental Portuguese. The only difference lies in pronunciation. In Brazilian Portuguese, the vowels are generally more "open" and the rhythm of speech is more melodic (syllable-timed). In Continental Portuguese, vowels—especially unstressed ones—are much more "closed" or even reduced/elided, and the rhythm of speech is more stress-timed.

Brazilian Portuguese Examples

1. Eu provei o bolo e achei muito bom. (I tasted the cake and thought it was very good.)
2. Eu provei minha inocência durante o processo. (I proved my innocence during the trial.)
3. Eu provei essa camisa no shopping. (I tried on this shirt at the mall.)
4. Eu provei a sopa, mas achei salgada demais. (I tasted the soup, but I found it too salty.)
5. Eu provei que ele estava mentindo para mim. (I proved that he was lying to me.)

Continental Portuguese Examples

1. Eu provei o bolo e achei muito bom. (I tasted the cake and thought it was very good.)
2. Eu provei a minha inocência durante o processo. (I proved my innocence during the trial.)
3. Eu provei esta camisa no centro comercial. (I tried on this shirt at the shopping center.)
4. Eu provei a sopa, mas achei demasiado salgada. (I tasted the soup, but I found it too salty.)
5. Eu provei que ele estava a mentir-me. (I proved that he was lying to me.)