Complex Numbers under Addition form Monoid

Theorem
The set of complex numbers under addition $\left({\C, +}\right)$ forms a monoid.

Proof
Taking the monoid axioms in turn:

S0: Closure
Complex Addition is Closed.

S1: Associativity
Complex Addition is Associative.

S2: Identity
From Complex Addition Identity is Zero, we have that the identity element of $\left({\C, +}\right)$ is the complex number $0 + 0 i$:
 * $\left({x + i y}\right) + \left({0 + 0 i}\right) = \left({x + 0}\right) + i \left({y + 0}\right) = x + i y$

and similarly for $\left({0 + 0 i}\right) + \left({x + i y}\right)$.

Hence the result.