---
title: "Proof Design Patterns"
source: https://www.jemoka.com/posts/kbhproof_design_patterns-1/
---

Based on the wise words of a crab, I will start writing down some Proof Design Patterns I saw over Axler.
inheriting properties (splitting, doing, merging) &ldquo;complex numbers inherit commutativity via real numbers&rdquo;
construct then generalize for uniqueness and existence
try to remember to go backwards
to prove IFF
zero is cool, and here too!, also \(1-1=0\)
\(0v = 0\) \(1-1 = 0\) \(v-v=0\) a.k.a. \(v+(-v)=0\) \(v+0 = v\) distributivity is epic: it is essentially the only tool to connect scalar multiplication and addition in a vector space
&ldquo;smallest&rdquo; double containement proofs to show set equivalence: prove one way, then prove the converse (\(a \subset b, b\subset a \Rightarrow a=b\))
couple hints
step 1: identify hypothesis (assumptions) desired conclusion (results, trying/to/proof) step 2: define write down precise, mathematical notations
