I'm currently reading John Lee's Introduction to Smooth Manifolds, and this is a topic that draws heavily from various fields in math, including real analysis, topology, and even some algebra. Now, while I do have some basic experience in these fields (and a fairly decent amount sufficient to read this book), I don't know how to prove every fact from these fields that Lee mentions in the book, nor do I know every fact from these fields that you need for some of the exercises. As I go along learning these new facts from topology and algebra (I'm fine with the analysis stuff mostly), should I try to prove everything I learn or take some of the facts from outside for granted? While it is definitely good to know how to prove things you're using, a lot of the facts from other fields are non-trivial and take time to prove (and possibly even learn the machinery that goes behind proving them), and in this sense just take away significantly from the time I'm spending on learning about manifolds, which is what the book is about.
In this sense, is it fine to take some facts "for granted" as I learn of them?