Did you buy two separate tickets? A roudtrip ticket DUB-LHR-DUB and the a complete separate ticket LHR-LAX-LHR? This would have needed separate transactions with two separate charges to your credit card. You would have booked one ticket and then booked the second ticket.
If you did this in one step with one charge then you do not have two separate tickets. You have one ticket on two airlines.