I'm teaching myself grammar rules from the book "Teach Yourself Russian Grammar" and finding it extremely helpful. However, I've hit an exception I can't quite deconstruct, no doubt one of you can guide me through. Please correct me if any of my thinking below is wrong!

A single shoe is Туфля, so a pair of shoes is Туфли (nominative plural).

На женщине туфли - On woman shoes. "Woman" is prepositional singular because we're using "На".

На мужчние нет туфель - On man (prep sing) not shoes.

Here's where I'm stuck - I have no idea what case "shoes" is, so I don't understand the construction of туфли / туфель. I'm guessing the case is different in the two examples because of "нет", but that still doesn't get me anywhere.

Thanks,
Lindsay