Pointwise Minimum of Simple Functions is Simple/Proof 1

Proof
From Pointwise Sum of Simple Functions is Simple Function:


 * $f + g$ is simple.

From Scalar Multiple of Simple Function is Simple Function:


 * $-g$ is simple.

Then, from Pointwise Sum of Simple Functions is Simple Function we have:


 * $f - g$ is simple.

From Absolute Value of Simple Function is Simple Function:


 * $\size {f - g}$ is simple.

So, from Scalar Multiple of Simple Function is Simple Function, we have:


 * $-\size {f - g}$ is simple.

From Pointwise Sum of Simple Functions is Simple Function, we then have:


 * $\paren {f + g} - \size {f - g}$ is simple.

Finally, from Scalar Multiple of Simple Function is Simple Function, we have:


 * $\dfrac 1 2 \paren {\paren {f + g} - \size {f - g} }$ is simple.

By Minimum Function in terms of Absolute Value, we have:


 * $\ds \min \set {f, g} = \frac 1 2 \paren {\paren {f + g} - \size {f - g} }$

so:


 * $\min \set {f, g}$ is simple.