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:

$\text S 0$: Closure
Complex Addition is Closed.

$\text S 1$: Associativity
Complex Addition is Associative.

$\text S 2$: Identity
From Complex Addition Identity is Zero, we have that the identity element of $\struct {\C, +}$ is the complex number $0 + 0 i$:
 * $\paren {x + i y} + \paren {0 + 0 i} = \paren {x + 0} + i \paren {y + 0} = x + i y$

and similarly for $\paren {0 + 0 i} + \paren {x + i y}$.

Hence the result.