Questing about Vergin Mobile and an Unlocked Samsung Galaxy S3

I hope that this question has not been asked before. I am a noob.

Virgin Mobile uses the Sprint Network as far as my understanding goes. Sprint sells the Samsung Galaxy S3 and it works on the Sprint Network. However, Virgin Mobile does not offer the Samsung Galaxy S3 on there network.

So I called Virgin Mobile customer help line and asked them if I perched a Unlocked Galaxy S3 could I use it on Virgin Mobile's network. They have cheaper unlimited data rates than Sprint, this is why I want to do this.

However, the customer service rep stated that the phone will not work on the Virgin Mobile network, even if the phone is Unlocked because they do not offer the phone.

Is this right? Or could the phone still in theory work on Virgin Mobile's network and they are just not letting that happen? After all if the phone works on Sprint's network and Virgin Mobile is using Sprint's network, then why not? I am confused. Can any one shed some light on this dark matter?