From the wording of this post it sounds like you made up the term "Definition-Theorem-Proof"? That would be quite amusing, because that's the standard term used for this style of textbooks.
There is a great schism in mathematics between mathematical physicists/applied mathematicians/intuitionists, and pure mathematicians/Bourbaki. The DTP style is strongly characteristic of the latter, and much-bemoaned by the former.
Originally, I did make it up! Lol, my bad. Thanks for letting me know, let me adjust the wording above.
The Definition-Theorem-Proof style is just a way of compressing communication. In reality, heuristic / proof-outline comes first; then, you do some work to fill the technical gaps and match to the existing canon, in order to improve readability and conform to academic standards.
Imho, this is also the proper way of reading maths papers / books: Zoom in on the meat. Once you understood the core argument, it is often unnecessary too read definitions or theorems at all (Definition: Whatever is needed for the core argument to work. Theorem: Whatever the core argument shows). Due to the perennial mismatch between historic definitions and theorems and the specific core arguments this also leaves you with stronger results than are stated in the paper / book, which is quite important: You are standing on the shoulders of giants, but the giants could not foresee where you want to go.
Epistemic status: Half-certain, half-skeptical. This mental model feels a bit too compressed to me to be maximally useful.
I've spent a surprising (=non-zero) number of hours studying mathematics during my summer break, and noticed an interesting pattern in undergraduate textbooks.
With a certain amount of remarks, exercises, and concrete examples thrown in and seasoned to taste, most traditional mathematics textbooks will start out with some rigorous definitions, then state some theorems that use those definitions, and finally prove those theorems. Then the process starts all over again - the "Definition → Theorem → Proof" pipeline, as it's often called by pure mathematicians.*
I think this lens is very obvious when pointed out, but I've also found it useful to structure my thinking around my newfound hobby. Let's see if it pays out.
First, how do we most effectively deal with each step of the core pipeline? Here are some things I've found useful, although YMMV:
One big advantage I see with the DTP pipeline is that, since mathematics is remarkably conservative with its definitions and theorems, someone who's read one book on subject X can often blaze* through significant portions of another book on subject X. The time cost on the duplicating part of the reading is lower than one would expect.
If you've already taken a course on Real Analysis, but you used the easier Understanding Analysis by Abbott instead of Principles of Mathematical Analysis by Rudin, for example, you are still probably going to find Rudin's works much easier to go through because of that background. This, combined with a generally increased ability to read mathematics works thanks to practice, makes the experience more enjoyable, and makes you more likely to stick with it.
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*Axioms are kind of halfway between theorems and definitions. They also show up very infrequently, since they're supposed to be foundational to everything, so I don't include them explicitly.
*Relatively speaking. Blazing through a math textbook is still going to feel like a snail's pace compared to blazing through the HPMOR archives. :)