Canonical Injection on Group Direct Product is Monomorphism/Proof 1

Proof
From Canonical Injection is Injection we have that the canonical injections are in fact injective.

It remains to prove the morphism property.

Let $x, y \in G_1$.

Then:

and the morphism property has been demonstrated to hold for $\inj_1$.

Thus $\inj_1: \struct {G_1, \circ_1} \to \struct {G_1, \circ_1} \times \struct {G_2, \circ_2}$ has been shown to be an injective group homomorphism and therefore a group monomorphism.

The same argument applies to $\inj_2$.