I'm terrible at these proofs, so I was wondering if someone could please check this proof for me:
Question: Suppose a and b are non-zero elements of a field F. Using only the field axioms, prove that is a multiplicative inverse of ab.
By the axiom that every non-zero real number has a multiplicative inverse.
By the axiom that 1 is the multiplicative identity.
Since the multiplicative inverse of an element of a field is unique, this implies is the multiplicative inverse of ab.
How does it look? Do I have to mention at the end that the multiplicative inverse of an element of a field is unique, or would it be good enough if I left that out?