Mathematical proof of Bohr's complementarity principle Complementarity principle, in physics, tenet that a complete knowledge of phenomena on atomic dimensions requires a description of both wave and particle properties. Depending on the experimental arrangement, the behavior of such phenomena as light and electrons is sometimes wavelike and sometimes particle-like; i.e., such things have a wave-particle duality. It is impossible to observe both the wave and particle aspects simultaneously. Together, however, they present a complete description than either of the two taken alone.
It's logically true. But could we mathematically prove it?
 A: It cannot be proven, because "wave-particle duality" is not a mathematical statement. It most definitely is not "logically true". Can you try to make it mathematical?
A mathematical framework
The "complementarity principle" was introduced in order to better understand some features of quantum mechanics in the early days. The problem is that if you consider a classical wave (e.g. a water wave or anything obeying the wave-equation) or a classical particle (e.g. a football, or any extended object with classical trajectories), calculating quantum mechanical answers with only one of these two concepts won't give you the true result. To some degree, this is an experimental fact and beyond mathematical proof.
What you could do is the following: Take a classical framework (such as in Arnold's book "Mathematical Methods of Classical Mechanics"), take the double-slit experiment and/or the photo-effect and try to find a mathematical description of this experiment in the classical framework. You will have to add more assumptions, such as: You have the slit, the screen and the source and nothing else is there - otherwise, you can probably always devise a clever classical theory that does obey the quantum predictions by introducing unobserved waves or particles. Given these assumptions, one should be able to prove that neither classical waves nor classical particles give the corresponding output - which is routinely argued in an introductory QM course.
Does it matter? No, particle-wave duality is obsolete
The best framework for quantum mechanics, quantum field theory, does not need particles and it does not need classical waves. The current "wisdom" in theoretical physics is that all you need is fields governed by noncommutative objects - you never have to evoke any "wave-particle duality". 
Now, fields give rise to "wave-functions", which behave in many ways as "waves" and the representations of the Lorentz group give rise to "particles", which sometimes can be seen as localised particles. In this sense, the wave-particle duality survives. It also survives, because people want to try to visualise quantum mechanical systems - and quantum fields alone cannot be adequately visualised because they are not part of our intuition. It is perfectly fine to think about waves and particles, keeping in mind that this is not the whole picture. Yet, there is no need to do this.
In any case: Introducing classical-like particles in an interacting quantum field theory is problematic. Just saying they are representations makes them often look quite unlike particles. Likewise, the wave function does not fulfil the wave equation, but a wave-type equation (the Schrödinger or Dirac equation). In this sense, we have neither waves, nor particles. Hence "wave-particle duality" as an obsolete semiclassical picture. 
In particular, it's not a mathematical theorem. 
Back to the maths
In a sense, given QFT, you can "prove" the complementarity principle by showing that neither a classical particle picture nor a classical wave picture arises from the standard model. This will be true, because you can prove that you will seldom have well-localised objects (these would be particles), and fields are quantised (this has consequences to "wave"-pictures). However, not much is gained by this "proof".
