What are the joists made out of? How far apart are they?
"Looks really solid" isn't a very scientific assessment. ;-)
If the joists are wood, what kind are they? Are they suitable for wet outdoor conditions? Or is the deck going to be sealed so that the joists never get wet?
You want to start by determining what kind of wood the joists are made out of to determine if they're even suitable for the deck you have in mind.
Then you should look at some span tables to see if they're going to be strong enough to support the load you want to put on the deck.
Here are a couple of examples.
EDIT after a few comments: I'd probably still talk to an engineer, just to be on the safe side both with the 15 foot (?) span and the 28" centers (was that centers? Or distance between the beams?), and to make sure the 4x4 columns are adequate. Let's say 15 people weigh 3,000 pounds, plus whatever furniture you put on the deck, plus the weight of whatever additional treatment you use to create the floor of the deck.
You're more or less talking about parking a car on top of that thing.
What are those slats that make up the flat roof? I think for the 2 foot span between those beams, 2x4's (on edge, not flat) would be the minimum size if you were using joists (on 16" centers). Since those existing slats are continuous rather than joists, that might make a difference. But I'd get a little local advice.
Playing around with the calculator here, for Douglas Fir, it seems like the minimum beams for the main span might need to be doubled 2x10's. You're pretty close to that, but I do not think it's overbuilt for what you want to do.
Are the 4x4 columns pressure-treated? If not, they're subject to rot especially at the base and your deck could collapse, which could be catastrophic if people are on it or under it.
EDIT 2: As @fungku mentioned in his answer, it's super important to confirm that you have a solid ledger connection to the house. I've seen decks where the ledger was just nailed to the house (presumable some of the nails hit studs) where the deck was literally pulling away and falling off the house. This really requires lag screws at the very least, preferably lag bolts running clear through with washers and nuts on the back side, and/or joist hangers designed for decks (lateral load hangers).