I've surmised that there are four big facts about the relationships between the position and momentum operators, their Hilbert spaces, and their eigenstates in QM. I think I am just about at a point where -- given one of them -- I'm able to derive the other three. I suppose my question is which of them is fundamental? None of the four are explicitly postulates of QM, so I'm wondering which of the four are axiomatic, take-it-on-faith statements. The four facts I'm referring to are:
In position space, momentum eigenstates are plane waves $\psi_p(x)=e^{ikx}$.
Position and momentum wave functions are related by Fourier transforms $\psi(p)=\int_{-\infty}^{\infty}\psi(x)e^{ikx}dx\Rightarrow\psi(x)=\int_{-\infty}^{\infty}\psi(p)e^{-ikx}dp$.
The momentum operator in position space is given as $\hat{p}=-i\hbar\frac{\partial}{\partial{x}}$.
The momentum operator is the generator of translations in position space $e^{i\hat{p}\Delta{x}/\hbar}|x\rangle=|x+\Delta{x}\rangle$.
Obviously (1) is a direct consequence of (2), and by solving a differential equation (3) can easily be derived from (1). I haven't derived for myself yet how (4) follows from any of the others which is what I'm trying to do now and what actually led me to this question. I don't know that (2) can be rigorously derived from (1), but assuming it can, it seems to me that the most likely candidate to be an axiom of QM is (1). Am I correct? Some things I've read have suggested to me that it's actually (4) that is fundamental. Does it even matter in the end?