Commutative Linear Transformation is G-Module Homomorphism

Theorem
Let $\rho:G\to\operatorname{GL}\left(V\right)$ be a representation, and $h\in G$. Then:

If $\rho(g)\rho(h)=\rho(h)\rho(g)$ for all $g\in G$, $\rho(h):V\to V$ is a $G$-module homomorphism.

Proof
Assume that $\rho(g)\rho(h)=\rho(h)\rho(g)$ for all $g\in G$.

Let $v$ be a vector $v\in V$.

Then $\rho(g)\rho(h)(v)=\rho(h)\rho(g)(v)$.

Using the properties from Equivalence of Representation Definitions there exists $\left(V,\phi\right)$, a $G$-module associated with $\rho$.

Now using that $\phi(g,v)=\rho(g)(v)$ for $g\in G$ and $v\in V$.

It is concluded that $\rho(g)\rho(h)(v)=\phi(g,\rho(h)(v))=\rho(h)(\phi(g,v))=\rho(h)\rho(g)(v)$.

Thus $\rho(h):V\to V$ is a $G$-module homomorphism by definition.