Max and Min Operations on Real Numbers are Isomorphic

Theorem
Let $\R$ denote the set of real numbers.

Let $\vee$ and $\wedge$ denote the max operation and min operation respectively.

Let $\struct {\R, \vee}$ and $\struct {\R, \wedge}$ denote the algebraic structures formed from the above.

Then $\struct {\R, \vee}$ and $\struct {\R, \wedge}$ are isomorphic.

Proof
First we note that from:
 * Min Operation on Toset forms Semigroup

and:
 * Max Operation on Toset forms Semigroup

both $\struct {\R, \vee}$ and $\struct {\R, \wedge}$ are semigroups.

Let $\phi: \R \to \R$ defined as:
 * $\forall x \in \R: \map \phi x = -x$

We have that:
 * $-x = -y \iff x = y$

which demonstrates that $\phi$ is a bijection.

Then we have:

which demonstrates that $\phi$ is a (semigroup) homomorphism.

The result follows by definition of (semigroup) isomorphism.