Opposite Ring is Ring

Theorem
Let $\struct {R, +, \times}$ be a ring.

Let $\struct {R, +, *}$ be the opposite ring.

Then $\struct {R, +, *}$ is a ring.

Proof
By definition of the opposite ring:
 * $\forall x, y \in R: x * y = y \times x$.

By definition of the ring $R$, $\struct {R, +}$ is an abelian group.

To complete the proof, it remains to be shown that $\struct{R, *}$ is a semigroup.

That is, it remains to show that $\struct{R, *}$ is associative.

Let $a, b, c \in R$.

The result follows.