Extended Real Numbers under Multiplication form Commutative Monoid

Theorem

Denote with $\overline \R$ the extended real numbers.

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

The algebraic structure $\struct {\overline \R, \cdot_{\overline \R} }$ is a commutative monoid.

Proof

By Extended Real Numbers under Multiplication form Monoid, $\struct {\overline \R, \cdot_{\overline \R} }$ is a monoid.

By Extended Real Multiplication is Commutative, $\cdot_{\overline \R}$ is commutative.

Hence $\struct {\overline \R, \cdot_{\overline \R} }$ is a commutative monoid.

$\blacksquare$