# “Smaller” math proofs discovered as a means to prove a more substantial theorem

I know the title may not be fully descriptive, but please bear with me.

When I do math, whether I have formed a conjecture and seek to prove it or am simply proving nontrivial theorems for homework assignments, I will in some cases look for "intermediate" steps, so to speak. For example, if I know that $A \Rightarrow B$, $B \Rightarrow C$, and $C \Rightarrow D$, then proving $A \Rightarrow D$ simply amounts to wisely choosing these intermediate steps to get the result.

What I am interested in is the opposite. For example, if a mathematician was attempting to prove $A \Rightarrow D$, and he/she also knew that $A \Rightarrow B$ and $C \Rightarrow D$, that person may attempt to prove that $B \Rightarrow C$ to complete the proof. Are there any historical examples of a situation like this where $B \Rightarrow C$ had never previously been rigorously proven, but was eventually shown to be true as a result of someone's attempt to prove $A \Rightarrow D$?

I apologize for the soft question, but I was just thinking about how I do math one day, and in almost every case I can think of, I am using these "smaller" results as building blocks to a larger proof, which seems generally logical. Even in textbooks, almost every result is presented this way because it makes the most sense for students understand the pieces of a proof before they can understand the whole thing. However, it would not surprise me if researchers grappling with open questions have taken the approach I described here, so I am curious if there are any noteworthy results that were discovered this way.

• Someone correct me if I'm wrong, but wasn't this the case with FLT? Didn't Wiles prove something that implied that the theorem holds? – Andrew Tawfeek Aug 8 '17 at 17:46
• @AndrewTawfeek Not quite, it was something like $A\implies B$ and $B\implies C$. I did not quite understand the details, but the prove contained essentially of two steps (of course both steps were extremely difficult) – Peter Aug 8 '17 at 17:49
• I'm pretty sure this happens a lot. Ramsey's theorem was originally just a minor lemma for a paper where Ramsey was trying to prove something else. – Michael Biro Aug 8 '17 at 17:50
• I would guess that solving the "small steps" in order to prove the big theorem is almost always how math is done. – Kaynex Aug 8 '17 at 17:51
• I think it would be harder to find an example of something where all the significant small steps existed before the major result. – DanielV Aug 8 '17 at 17:52