You cannot work backward by assuming the field axioms (which are unfortunately named because they are not axioms at all but properties) to derive the definition o the field operators.
Of course, when we say they are field axioms we mean those properties hold true for the elements of the field. If you see those properties, they talk about distributivity over the elements of the field and additive inverses of the elements also belong to the field, so the distributivity automatically applies to additive inverses too.
After that with a little algebra, "product of additive inverses of two elements is equal to the product of the two elements" comes out as a result (not a definition).
Of course, by "product" we mean whatever * represents. It is not necessarily the multiplication operator we see in numbers.