Focal subgroup of a Sylow subgroup is generated by the commutators with normalizers of non-identity tame intersections

Statement
Suppose $$P$$ is a $$p$$-fact about::Sylow subgroup of a finite group $$G$$ and $$P_0$$ is the focal subgroup of $$P$$ in $$G$$. In particular, by the focal subgroup theorem, $$P_0 = P \cap [G,G]$$.

Then, $$P_0$$ is generated by the subgroups of the form $$[H,N_G(H)]$$, where $$H$$ ranges over all the non-identity fact about::tame Sylow intersections in $$G$$ involving $$P$$ (Note that throwing in the identity adds nothing).

Facts used

 * 1) uses::Focal subgroup theorem
 * 2) uses::Alperin's fusion theorem in terms of tame intersections

Proof
Given: A finite group $$G$$, a $$p$$-Sylow subgroup $$P$$ of $$G$$. $$P_0$$ is the focal subgroup of $$P$$ in $$G$$.

To prove: $$P_0$$ is generated by subgroups $$[H,N_G(H)]$$ where $$H$$ ranges over all non-identity tame $$p$$-Sylow intersections of $$G$$ involving $$P$$.

Proof: Let $$P_1$$ be the subgroup generated by $$[H,N_G(H)]$$ where $$H$$ ranges over all non-identity tame $$p$$-Sylow intersections of $$G$$.

Proof that $$P_1 \le P_0$$

 * 1) $$P_1 \le P$$: Every tame Sylow intersection $$H$$ is by definition contained in $$P$$. Further, since $$H$$ is normal in $$N_G(H)$$, we have $$[H,N_G(H)] \le H$$, so $$[H,N_G(H)] \le P$$ for all $$H$$. Thus, $$P_1 \le P$$.
 * 2) $$P_1 \le G'$$: $$[H,N_G(H)]$$ is, by definition, contained in $$G' = [G,G]$$ for all $$H$$.
 * 3) $$P_1 \le P_0$$: Combining the previous two steps yields $$P_1$$ is contained in $$P \cap G'$$, which is equal to $$P_0$$ by fact (1).

Proof that $$P_0 \le P_1$$
Since $$P_0$$ is generated by elements of the form $$xy^{-1}$$, where $$x,y \in P$$ are conjugate in $$G$$, it suffices to show that elements of this form are in $$P_1$$.

This follows as a consequence of fact (2).