Constant Mapping to Identity is Homomorphism/Groups

Theorem
Let $\struct {G_1, \circ_1}$ and $\struct {G_2, \circ_2}$ be groups with identities $e_1$ and $e_2$ respectively.

Let $\phi_e: \struct {G_1, \circ_1} \to \struct {G_2, \circ_2}$ be the constant mapping defined as:
 * $\forall x \in G_1: \map {\phi_e} x = e_2$

Then $\phi_e$ is a group homomorphism whose image is $\set {e_2}$ and whose kernel is $G_1$.

Proof
Let $x, y \in G_1$.

Then:

So $\phi_e$ is a group homomorphism.

The results about image and kernel follow directly by definition.