Conditional phase shift for Grover's algorithm

[Francis]

TL;DR

Dear fellas, I am reading a quantum physics tutorial to understand Grover's algorithm and I am stuck with a (very simple) deduction.
I have the pdf attached page 23 at the top.

I am trying to understand the following deduction:
"The conditional phase shift can be represented by the unitary operator 2|0> <0| - I:"

for eq. 4a) I was expecting to be:

[2 |0><0| - I] |0> =
2 |0> <0|0> - I|0> = 2|0> - |0> =
|0>

as for eq. 4b I can't understand it at all. Why does the author considers I|0> = I and I|x> = I? What am I missing?

 

[Haborix]

Just a lot of typos. As to 4b), I think the author just treats the case where $x=0$ separately since it isn't picking up the minus sign.

 

[George Jones]

It seems that this a student project.

It looks like $\left| x \right>$ has has been split into two cases: $\left| x= 0 \right>$ in (4a); $\left| x \neq 0\right>$ with $\left| 0 \right>$ and $\left| x \right>$ orthogonal to each other in (4b),

@Haborix relied with similar comments while I was typing.