Dirac Notation and Magnitude of Bra's Help

[NeedPhysHelp8]

Hi all,

I was diving into my 3rd year quantum assignment and I saw the following which I have to use for the rest of the question to prove the Cauchy-Schwarz inequality:

Homework Statement

$$|| a|x> + b|y> ||^2$$

I only really learned a bit about Dirac notation last year, so please let me know if this simplification is right.

$$( a|x> + b|y> )^{*}( a|x> + b|y> )$$
$$a^{*}a<x|x> + a^{*}b<x|y> + b^{*}a<y|x> + b^{*}b<y|y>$$

Later it says to make substitutions for $$a= -<x|y> , b= <x|x>$$ to prove the inequality, but I want to make sure I didn't screw up the very first part above...which I have a feeling I did.
Thanks

 

[dextercioby]

The multiplications with bra's and kets are correctly done.