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.