I've been reading through a derivation of the LSZ reduction formula (http://www2.ph.ed.ac.uk/~egardi/MQFT_2013/MQFT_2013_lecture_2.pdf , c.f. pages 2-3) and I'm slightly confused about the arguments made about the assumptions: $$\langle\Omega\vert\phi(x)\vert\Omega\rangle =0\\ \langle\mathbf{k}\vert\phi(x)\vert\Omega\rangle =e^{ik\cdot x}$$ For both assumptions the author first relates $\phi(x)$ to $\phi(0)$ by using the 4-momentum operator $P^{\mu}$, i.e. $$\phi(x)=e^{iP\cdot x}\phi(0)e^{-iP\cdot x}$$ such that, in the case of the first assumption, one has $$\langle\Omega\vert\phi(x)\vert\Omega\rangle =\langle\Omega\vert e^{iP\cdot x}\phi(0)e^{-iP\cdot x}\vert\Omega\rangle =\langle\Omega\vert\phi(0)\vert\Omega\rangle$$ where we have used that the vacuum state satisfies $P^{\mu}\lvert\Omega\rangle =0$, such that $e^{-iP\cdot x}\vert\Omega\rangle = \vert\Omega\rangle$.

What I don't understand is, why do we need to do this in the first place? Both $\langle\Omega\vert\phi(x)\vert\Omega\rangle$ and $\langle\Omega\vert\phi(0)\vert\Omega\rangle$ are Lorentz invariant.

Is it simply because, by showing that for any $x^{\mu}$, $\langle\Omega\vert\phi(x)\vert\Omega\rangle$ is equal to the Lorentz invariant number, $v\equiv\langle\Omega\vert\phi(0)\vert\Omega\rangle$, we can then simply shift the field $\phi(x)\rightarrow \phi(x)-v$, such that the condition $\langle\Omega\vert\phi(x)\vert\Omega\rangle=0$ is satisfied?! (If this is the case, then I'm guessing the argument is similar for the second condition.)