Is my iPhone unlocked or not - how do I tell?

I recently bought an iPhone from an Apple store without signing up to any carrier or plan (because I knew I could get a better deal than they were offering). So I believed I was buying an unlocked phone.

Now when I look at settings/General/About, it says the carrier is Optus. But I have just signed up to TPG and am waiting for them to activate my sim. It is a TPG sim in the phone. It is taking a little while to activate and I'm beginning to think there is a problem.

Now I don't know much about this stuff and get easily confused. This is my first ever mobile phone!

I brought my new phone home and plugged it into my Macbook pro and went into iTunes (with the plan of syncing with icloud). I wanted to get all the aps and their current settings that are on my iPod synced over to the phone, so instead of choosing 'set up a new device' I chose 'set up from backup'. I'm a bit worried this was exactly the wrong thing to do. Could this have given some wierd carrier setting to the phone? My ipod was connected via Vodaphone wifi, so I don't understand where the Optus setting could have come from.

How do I check that this is an unlocked iPhone?