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.
 A: $\bullet$ 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 $\mathcal H_1: \theta> \theta_0.$
A: Let's take an example with the one sided t-test.
We observe a t-statistic $T$ and the null hypothesis is that $H_0: \mu = 0$ in which case the t-statistic is t-distributed. And the alternative hypothesis is that $H_0: \mu > 0$ in which case the t-statistic is non-central t-distributed.
Below is an image of those distributions and the likelihood ratios (we use $\nu =5$)

The monotone likelihood property means that the likelihood ratio function is either all decreasing or all increasing. This means that the region of maximum likelihood-ratio is for any of the alternative point hypotheses among the composite hypotheses described by an inequality of the same form e.g. "reject if $T_{\text{observed}}>T_c$". The value $T_c$ is eventually chosen based on the distribution under the null hypothesis and the same for every value $\mu_a$ of the alternative hypothesis value.
The monotone property makes that the rejection region will be a region dependent on a single inequality, and independent of the actual value of the true parameter $\theta$ within the range of possibilities of the composite hypothesis.
