[ Fish ] Open Question : Need help with aquarium nitrogen cycle (55 gallon)?

[ Fish ] Open Question : Need help with aquarium nitrogen cycle (55 gallon)?

Just a few days ago I obtained a 55 gallon aquarium. I kept the old
gravel, plants, and decor in the tank. Plus there are tons of snails that
I didn't notice until I filled the tank. Yesterday I finally purified the
water and added some different gravel and river stone atop the old gravel.
I also got a new filter (Tetra Whisper EX70). The old one (Tetra Whisper
10i) is in the tank as well but it only has foam in it, no bio-bag. When I
did all of this I thought that I started my nitrogen cycle, however I
forgot to get ammonia to add. So, today before I went out and purchased
some ammonia I tested the water for ammonia, nitrite and nitrate. The
ammonia level was .25, nitrite was 0 and nitrate was 5. The tank is heated
to 80-81 degrees (200 watt heater) and the water is clear. My question is
has my cycle already started ,do I need to get ammonia since there is
already some in the tank (which I assume came from the snails). And, how
is my cycle doing so far. I've had the tank for like 4 days now. When I
got the tank I filled it, but then emptied it out and refilled it so that
I could add more gravel. I'm thinking that because I used so much old
decor and gravel that my cycle is already almost complete,even with no
added ammonia, am I right?