I do not think she cheated on you . You moved away to a different location followed by the fight. Technically speaking, it was a break up.It might have been a rebound sex with a friend available, when she was emotionally vulnerable.It seems you having sex with a hooker was pre-planned,you had every intention of getting laid even if you had to pay  .I say,You should focus more on resolving the issues that led to fights with each other rather than who is more guilty.