Proof Design Patterns

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
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
“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