Yes you have yourself a very pretty lyretail male betta.
Doubletail and lyretails are very different tail types but same at the same time.
For example if you look at the betta in my avatar, you could see that he is a real double tail. as for the fish in the below picture, he is a lyretail. The lyretail's caudal fins are a lot more spreaded than the doubletails.
You can easily see the difference in the two. The male in my avatar looks like his caudal can fit in a flower as the one in the picture above does not.