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) “complex numbers inherit commutativity via real numbers”

construct then generalize for uniqueness and existence

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

“smallest” 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

- step 1: identify