Proving Coherent States are Eigenfunctions of Annihilation Operators

[aaaa202]

Look at the following attached picture, where they prove the coherent states are eigenfunctions of the annihiliation operators by simply proving aexp(φa^†)l0> = φexp(φa^†)l0>. I understand the proof but does that also prove that:
a_iexp(Σφ_ia_i^†)l0> = φ_iexp(Σφ_ia_i^†)l0> ?
I can see that it would if you can use that:
exp(A+B+...) = exp(A)exp(B)exp(C)...
but does that identity hold for operators and how do you see that?
Because if you just taylor expand the operator sum you get cross terms between i and j and I'm not sure what to do with these.
Edit: the picture might be a bit too small, so you can also just look at p158-159 of http://nanotheoryou.wikispaces.com/file/view/Atland+And+Simons.pdf

 

[DrDu]

If the operators A, B, C commute, you can reorder them like ordinary numbers and hence the factorization of the exponential holds.