Proof of P-Hardness by reduction
By definition, a language $L$ is P-complete if for every language $M \in P$ there is a logspace reduction $f_M$ such that $x \in M$ iff $f_M(x) \in L$. Now suppose that $L$ is P-complete and that for some language $L' \in P$ there exists a logspace reduction $g$ such that $x \in L$ iff $g(x) \in L'$. Then for every language $M \in P$, $x \in M$ iff $g(f_M(x)) \in L'$, and so $L'$ is P-hard since the composition of logspace reductions is a logspace reduction.

Generalizing, this sort of argument always works as long as the composition of two legal reductions is a legal reduction. This is the case for both polynomial time reductions and logspace reductions.