Yes, it's common to use theorems in mathematical proofs.
You might set up a lot of minor theorems first, easy to prove and then a big theorem at the end.
If it is a minor result, you can also call it a lemma - usually used for mini theorems that are not especially interesting in their own right, but are useful for the theorems you can deduce from them.
Though sometimes lemmas end up being as important as theorems in a branch of maths :). Lemma (mathematics)
All of this has nothing to do with whether the maths is interesting or not or valuable or breakthrough maths or not.
You can also just set it out with numbered statements and no theorem at all, just say - statement 12 is derived from statement 3, and statement 107 is derived from statement 7 combined with statement 100 and statement 55 etc. You could do an entire paper like that and never have a single theorem in it anywhere, not even the conclusion. It would work logically but is hard for even mathematicians to follow easily.
So the theorems and lemmas are really a kindness to the reader - and to your self - and a way to reduce the chance of errors by "chunking" the maths into mini results that can be proved separately and understood separately making it easier to spot errors and to understand what it is all about.
This may be helpful for anyone who needs help with how to structure mathematical proof: An Introduction to Proofs and the Mathematical Vernacular - gives lots of examples to work through, including many example proofs, as well as several what he calls "spoofs" that look like proofs but aren't. And discussion of how mathematical language works.
You might set up a lot of minor theorems first, easy to prove and then a big theorem at the end.
If it is a minor result, you can also call it a lemma - usually used for mini theorems that are not especially interesting in their own right, but are useful for the theorems you can deduce from them.
Though sometimes lemmas end up being as important as theorems in a branch of maths :). Lemma (mathematics)
All of this has nothing to do with whether the maths is interesting or not or valuable or breakthrough maths or not.
You can also just set it out with numbered statements and no theorem at all, just say - statement 12 is derived from statement 3, and statement 107 is derived from statement 7 combined with statement 100 and statement 55 etc. You could do an entire paper like that and never have a single theorem in it anywhere, not even the conclusion. It would work logically but is hard for even mathematicians to follow easily.
So the theorems and lemmas are really a kindness to the reader - and to your self - and a way to reduce the chance of errors by "chunking" the maths into mini results that can be proved separately and understood separately making it easier to spot errors and to understand what it is all about.
This may be helpful for anyone who needs help with how to structure mathematical proof: An Introduction to Proofs and the Mathematical Vernacular - gives lots of examples to work through, including many example proofs, as well as several what he calls "spoofs" that look like proofs but aren't. And discussion of how mathematical language works.
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Robert Walker
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