Yeah, the hairiest part is probably the existence and uniqueness of the antiderivative, followed by the existence of an inverse for exp(1). In fact, I can't quite recall whether the book defined it as a Riemann integral or an antiderivative, but of course it had a statement of the FTC which would connect the two. (It was just a high-school textbook, so it tended to gloss over the finer points of existence and uniqueness.)