Extended Real Numbers under Multiplication form Monoid

Theorem
Denote with $\overline{\R}$ the extended real numbers.

Denote with $\cdot_{\overline{\R}}$ the extended real multiplication.

The algebraic structure $\left({\overline{\R}, \cdot_{\overline{\R}}}\right)$ is a monoid.

Proof
Checking the axioms for a monoid in turn:

Closure
Immediate as $\cdot_{\overline{\R}}: \overline{\R} \times \overline{\R} \to \overline{\R}$ is a mapping.

Associativity
Proved on Extended Real Multiplication is Associative.

Identity
For all $x \in \R$, it holds that $1 \cdot_{\overline{\R}} x = x \cdot_{\overline{\R}} 1 = x$ by definition of $\cdot_{\overline{\R}}$ on $\R$.

Furthermore, by definition, $1 \cdot_{\overline{\R}} \left({+\infty}\right) = \left({+\infty}\right) \cdot_{\overline{\R}} 1 = \left({+\infty}\right)$.

Lastly $1 \cdot_{\overline{\R}} \left({-\infty}\right) = \left({-\infty}\right) \cdot_{\overline{\R}} 1 = \left({-\infty}\right)$.

That is, $1 \in \overline{\R}$ is an identity for $\cdot_{\overline{\R}}$.

Hence, satisfying all the axioms, $\left({\overline{\R}, \cdot_{\overline{\R}}}\right)$ is a monoid.