Light cannot be at rest. Can this be proven mathematically? So, we all know the world famous equations
$$E^2=\left(mc^2\right)^2+(pc)^2\tag{1}$$
$$E=hf={hc\over{\lambda}}{\rm\ (for\ light)}\tag{2}$$
$$p={h\over{\lambda}}\tag{3}$$
We can realise in eq.2 and 3. The $hc\over{\lambda}$ can be replaced by $pc$
And the $E=pc$ can be used in eq.1 to obtain $(mc^2)^2=0$ or the rest energy is $0$.
This can conclude that light if at rest would reach $0~\rm K$. This is a major contradiction in our non-utopian world. And thus light cannot be at rest.
Is this reasoning correct?
 A: 
Light cannot be at rest. Can this be proven mathematically?

Given Maxwell's equations based on four observational laws, yes.That the velocity of light should be c in vacuum is proven.
Then you go on:

$$E=hf={hc\over{\lambda}}{\rm\ (for\ light)}\tag{2}$$

That is wrong, this is not for light, it is for photons. Photons build up the electromagnetic field in a superposition of their wavefunctions.. The classical electromagnetic field, light, emerges from zillions of photons.
Photons are quantum mechanical particles, not bits of light. A building is made up of bricks. Bricks are not a building.
So the question you are proving morphs :

Photons cannot be at rest. Can this be proven mathematically?

You derive :

or the rest energy is 0.

Look again at your functions.There is the alternative or $m=0$
So you just reached an algebraic relationship. 

This can conclude that light if at rest would reach $0K$ . 

If the mass is zero,  a photon  cannot be at rest from the mathematics of Lorenz transformations. If it has a mass, it will have the energy of the mass which together with the uncertainty principle will not allow $0K$ . 
So this is no proof. 
That the photons have zero mass is an assumption that makes consistent macroscopic data and microscopic data.
Again I stress , do not confuse light with photons. The velocity of light comes out of the Maxwell equations in a clean proof. 
I should also stress that physics tests the frontiers, and so there are people checking on the mass of the photon, example Photon and Graviton Mass Limits. So any proof needs feedback with experiment and observations.
