Composite of Group Monomorphisms is Monomorphism

Theorem
Let:
 * $\struct {G_1, \circ}$
 * $\struct {G_2, *}$
 * $\struct {G_3, \oplus}$

be groups.

Let:
 * $\phi: \struct {G_1, \circ} \to \struct {G_2, *}$
 * $\psi: \struct {G_2, *} \to \struct {G_3, \oplus}$

be monomorphisms.

Then the composite of $\phi$ and $\psi$ is also a monomorphism.

Proof
A monomorphism is a homomorphism which is also an injection.

From Composite of Group Homomorphisms is Homomorphism, $\psi \circ \phi$ is a homomorphism.

From Composite of Injections is Injection, $\psi \circ \phi$ is an injection.