Understanding the proof of Karlin–Rubin Theorem The Karlin-Rubin Theorem states that for $H_0 \colon \theta \leq \theta_0$, $H_1 \colon \theta >\theta_0$, for distributions that have monotone likelihood property, test given by thresholding the sufficient statistics $T$ is the uniformly most powerful.
I understand that for each $\theta_0 < \theta_1$, the likelihood ratio test is the uniformly most powerful test for $H_0 \colon \theta = \theta_0$ and $H_1 \colon \theta = \theta_1$, which is equivalent to looking at sufficient statistics due to monotone likelihood property.
I also understand our power function will be increasing.
However, I do not see how these properties lead to the proof of Karlin–Rubin Theorem. I'm struggling to see where we use the monotone likelihood ratio property.

$\bullet$ MLR property is required to express the test $\varphi$ in terms of the statistic $T(\mathbf x) $ in that for $\theta_0<\theta_1,$ $$\frac{f_{\theta_1}(\mathbf x) }{f_{\theta_0}(\mathbf x) } \gtreqless k\iff T(\mathbf x) \gtreqless t_0.$$
$\bullet$ The first part of the proof shows the power function is nondecreasing. So, for all $\theta\leq\theta_0,$ $$\mathbb E_\theta\varphi\leq \mathbb E_{\theta_0}\varphi=\alpha.$$ Then the test due to NP lemma is MP at $\theta_1$ among all level $\alpha$ tests for testing $\mathcal H_0: \theta\leq\theta_0$ against $\mathcal H_1: \theta=\theta_1.$
$\bullet$ Now the class of all tests $\varphi$ such that $\mathbb E[\varphi|\theta\leq\theta_0]\leq \alpha$ is a subclass of those satisfying $\mathbb E[\varphi|\theta=\theta_0]\leq \alpha;$ so the MP test of the latter class must be MP for the former subclass too. The test constructed by NP lemma is MP and is not depending on the value of $\theta_1.$ This means it is UMP level $\alpha$ test for testing $\mathcal H_0: \theta\leq \theta_0$ against $\mathcal H_1: \theta> \theta_0.$
