I've posted this before, but it bears repeating. Don't use anything but tire lube for mounting tires. A) you don't know what's in the stuff and B) some of that stuff will cause major issues with your tire/wheel down the road.
Dish soap is not a tire lube. Don't use it as such. It is alkaline (the opposite of acid) and will eat your aluminum rim up. The aluminum oxide that forms, expands just like steel does when it rusts.
The expansion will push the tire bead off the rim and leak air. I've experienced exactly this, and had a tire go flat from it in the middle of a long ride. It sucks.
Other stuff that has a petroleum base will degrade the rubber of your tire. You may not be the one to experience the bead failing.
If you are at the last little bit where the bead does not seat, deflate the tire. Push the bead back and lubricate liberally. Then try again.
Making sure the valve core is removed is kind of important. You won't get enough volume of air in the tire, in the short amount of time needed with the core still in.
Pressure and lube are your friend when mounting a tire. Just don't use too much of either. Too much pressure can be dangerous and cause injury or even death if you push that too far. I've seen a rim fail under pressure and it ain't pretty.
Keep pressures relatively low. That doesn't mean 40 psi. That means don't go past 75 psi. If you don't get the bead to seat by then, refer back to where wrote about deflate the tire.
Lube is your friend. Until it isn't. I've had tires spin on the rim due to excessive lube....or too slick a lube. Best stuff I've ever used is made by Murphy's oil soap. It's a tire mounting compound with corrosion inhibitors in the formula.
It looks like grease but is made of soap. It's slick, but not too slick. You can buy it at any Napa auto parts store or on the internetz. I know for certain it is available on Amazon.
It is not prohibitively expensive either. I've seen a 1 gal container for around $20 and that would last you a life time of doing your own tires and your friend's too.