Need help understanding the conclusion of the proof here
How does $-g(t)=-log|X+tV|$ (where $\textbf X, X+tV$ are positive definite) being convex prove that $-f(Z)=-log|Z|$ is convex, Z positive definite nxn matrix?
How do they get from (I wrote this, but I think it's the correct definition of concavity)
$$\begin{split}-log|X+(\lambda t_1+(1-\lambda)t_2)V|&\le-\lambda log|X+t_1V|-(1-\lambda)log|X+t_2V| \text { to}\\ -log|\lambda Z_1+(1-\lambda)Z_2|&\le-\lambda log|Z_1|-(1-\lambda)log |Z_2|\end{split}$$
Steps:
$$\begin{split}-log|X+\lambda t_1 V+(1-\lambda)t_2V|&\le-\lambda log|X+t_1V|-(1-\lambda)log|X+t_2V|\\ -log|X+Z|&\le-\lambda log|X+Z_1|-(1-\lambda)log|X+Z_2|\\\end{split}$$
