Composite of Group Monomorphisms is Monomorphism

Theorem
Let:
 * $\left({G_1, \circ}\right)$
 * $\left({G_2, *}\right)$
 * $\left({G_3, \oplus}\right)$

be groups.

Let:
 * $\phi: \left({G_1, \circ}\right) \to \left({G_2, *}\right)$
 * $\psi: \left({G_2, *}\right) \to \left({G_3, \oplus}\right)$

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.