Proof of the no-communication theorem

Let $A, B$ be (finite-dimensional) Hilbert spaces, and $rho$ some mixed state of $A otimes B$. I am trying to show that a measurement performed on the ‘$A$-subsystem’ does not affect $rho^B = text{Tr}_A(rho)$.

I understand a ‘measurement performed on the $A$-subsystem’ as given by some observable $X otimes I$, where $X$ is a self-adjoint operator on $A$ which decomposes as $X = sum m P_m$ (where $P_m$ is the orthogonal projection on the $m$-eigenspace). If this measurement results in outcome $m$, the resulting state should be

$$rho’ = frac{(P_m otimes I) rho (P_m otimes I)}{text{Tr}((P_m otimes I) rho)}$$
and I wish to see that $text{Tr}_A(rho’) = text{Tr}_A(rho)$. Now I can use the cyclicity of trace to see that

$$text{Tr}_A(rho’) = frac{text{Tr}_A((P_m otimes I) rho)}{text{Tr}((P_m otimes I) rho)}$$

but why should this be equal to $text{Tr}_A(rho)$? I did check this is the case if $rho = rho^A otimes rho^B$ decomposes as a product. And I know a general $rho$ will be a linear combination of such cases; but since the equality desired is not linear, it does not seem to follow that it’ll hold (in fact, it seems to indicate it will not hold). Hopefully someone can point to the mistake in my thinking.