I always thought pH was equivalent to [H+] = 10-pH? 
I've attached their solution btw.
This is from Thushan. The values are different but it's the same principle

You forget that the system re-equilibriates. There is a reversible reaction going on:
2H2O <--> H3O+ + OH-
When the solution is pH = 10, [OH-] = 10^-4 M and [H+] = 10^-10 M, so there is a lot more OH- than H+ originally.
When you dilute the solution, [OH-] initially decreases 100fold to 10^-6 M, and so does [H+] to 10^-12 M.
However, the system has to reach equlibrium, with K = 10^-14 and the reaction quotient at that time Q = 10^-18 M.
Hence, the system will shift to the right (also consistent with Le Chatelier's principle, given we have a dilution and the system favours the reaction with more aqueous particles).
However, since there is 1000000 times more OH- than there is H+, the amount of OH- formed will be negligible compared to its initial diluted concentration 10^-6 M.
Hence, we can say that [H+] would increase from 10^-12 to about 10^-8 M. This is the same as the amount of OH- formed, which would be much smaller than 10^-6 M. So,
Final:
[OH-] = 10^-6 M
[H+] = 10^-8 M