Canonical Injection on Group Direct Product is Monomorphism/Proof 2

Theorem

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

Let $\struct {G_1 \times G_2, \circ}$ be the group direct product of $\struct {G_1, \circ_1}$ and $\struct {G_2, \circ_2}$

Then the canonical injections:

$\inj_1: \struct {G_1, \circ_1} \to \struct {G_1, \circ_1} \times \struct {G_2, \circ_2}: \forall x \in G_1: \map {\inj_1} x = \tuple {x, e_2}$

$\inj_2: \struct {G_2, \circ_2} \to \struct {G_1, \circ_1} \times \struct {G_2, \circ_2}: \forall x \in G_2: \map {\inj_2} x = \tuple {e_1, x}$

are group monomorphisms.

Proof

A specific instance of Canonical Injection is Monomorphism, where the algebraic structures in question are groups.

$\blacksquare$