Let $f: [a,b] \to \mathbb{R}$ be bounded. Show that $f$ is Riemann intergrable iff $\bar{\int_{a}^{b}} f = -\left[\bar{\int_{a}^{b}} -f\right]$
My attempt is as follows.
"$\Leftarrow$" $\bar{\int_{a}^{b}} -f= \inf\{U(-f;P)\}=\inf\{-L(f;P)\}$
So,
$-[\bar{\int_{a}^{b}} -f = -[-\sup\{L(f;P)\}]=\sup\{f;P\}$
I'm stuck on making sure I'm pushing definitions through properly and the $\to$ direction of the proof.