As an economist, I could not agree more. If the minimumwage is higher than the equilibrium price, that will cause unemployment.
However, have you considered the possibility that the equilibrium price rises if wages rise? If I have a shop in LA, yes I will have to pay a higher wage to the guy helping me stock the shelves. But if at the same time, the people who frequent my shop buy more stuff, I may still need him to prevent my customers from seeing empty shelves and moving to a competitor down the street. So in effect, raising the minimum wage has increased the equilibrium price of having a guy help me stock shelves as I stand to lose more if customers go elsewhere in that circumstance.
Now that still doesn't tell me whether THIS increase in the minimum wage increases unemployment or not. For example because what we also don't know is what the current equilibrium price is. You seem to assume that is lies somewhere near the current minimumwage. Why? If I look at profits for corporations, I might assume that the equilibrium wage is considerably higher than the current minimumwage.
What if the equilibrium price for most labor in LA is already 14 dollars an hour or more? And the fact that millions are paid less than 14 dollars an hour just means more money in the pocket for a few business owners 2000 miles away? In that case I would expect the increase of the minimumwage to have a beneficial effect on employment. After all, it would mean a huge increase in wages for many people who live and spend locally, at the detriment of some business-owners who may live thousands of miles away, and even if they live in the area certainly will not consume every bit of their earnings locally. The increase in wages could even very well push the equilibrium price to a level higher than 15$ an hour.