Canonical Injection of Real Number Line into Complex Plane

Theorem
Let $\struct {\C, +}$ be the additive group of complex numbers.

Let $\struct {\R, +}$ be the additive group of real numbers.

Let $f: \R \to \C$ be the mapping from the real numbers to the complex numbers defined as:
 * $\forall x \in \R: \map f z = x + 0 y$

Then $f: \struct {\R, +} \to \struct {\C, +}$ is a monomorphism.

Proof
Consider the mapping $g: \C \to \R$ defined as:
 * $\forall z \in \C: \map f z = \map \Re z$

where $\map \Re z$ denotes the real part of $z$.

From Real Part as Mapping is Endomorphism for Complex Addition, this is a projection from $\C$ to $\R$.

The result follows from Canonical Injection is Right Inverse of Projection.