And then I realized that I had forgotten to put in any detergent. Oh well. They should be OK.

Should be just as good as doing it down by the creek.
As someone here noted, many persons put in too much. Very typical irational thinking. If some is good, more has to be
better. Detergent manufacturers themselves warn against this, as it can gradually build up scum in side your washining machine filters and hoses.
Same with using hot water. Unless you need to kill germs, as with underwear or sheets, warm or cold are sufficient.
The most popular and most expensive detergents, such as Tide, are massive overkill for typical washloads.
If you have the option, set up a drying rack indoors or use an outdoor clothesline. Electric dryers have a lint
filter for a reason: they literally shred your clothes with the tumble friction.
It is worth noting that in most places you are washing with recycled sewage water. Not unclean, but very expensive.
Water treatment is a huge tax and environmental burden, and the world is in a drought state right now.
Almost all garments now contain artificial fabrics, which shred off in the washer and end up as microplastics in the ocean and
our food stream.
Hanging up a garment overnight with a window open so it can air out is often more than sufficient to refresh it.