关于停时布朗运动$B_{t\wedge 1}$满足性质$(P)$的证明检查及简化解法问询

Hey folks,

I've been working through this probability theory exercise and I think I've got a solution, but it feels really weird and overly complicated. I'd really appreciate it if someone could take a look over my work—especially the fourth equality I mention later on—and/or let me know if there's a much simpler way to prove this.

The Exercise #

Let $B_t$ be a $\mathscr{F}t$ Brownian motion started from zero. I want to show $B{t\wedge 1}$ satisfies "Property $(P)$." That is, for any fixed $p\geq 1$, there exists a constant $C$ (depending on $p$) such that for any $\mathscr{F}_t$ stopping time $T\leq 1$, we have
$$E\left[_{(B_1-B_T)^p}| \mathscr{F}_T\right]\leq C.$$

备注：内容来源于stack exchange，提问作者famous mortimer