Min Operation is Associative

Theorem
The Min operation is associative:


 * $\map \min {\map \min {x, y}, z} = \map \min {x, \map \min {y, z}}$

Thus we are justified in writing $\map \min {x, y, z}$.

Proof
To simplify our notation:
 * Let $\map \min {x, y}$ be (temporarily) denoted $x \underline \vee y$.

There are the following cases to consider:
 * $(1): \quad x \le y \le z$
 * $(2): \quad x \le z \le y$
 * $(3): \quad y \le x \le z$
 * $(4): \quad y \le z \le x$
 * $(5): \quad z \le x \le y$
 * $(6): \quad z \le y \le x$

Taking each one in turn:


 * $(1): \quad$ Let $x \le y \le z$. Then:


 * $(2): \quad$ Let $x \le z \le y$. Then:


 * $(3): \quad$ Let $y \le x \le z$. Then:


 * $(4): \quad$ Let $y \le z \le x$. Then:


 * $(5): \quad$ Let $z \le x \le y$. Then:


 * $(6): \quad$ Let $z \le y \le x$. Then:

Thus in all cases it can be seen that the result holds.

Also see

 * Max Operation is Associative