This doesn't answer the question. You just assume that the situation is the same elsewhere - but the question asks for prove of this assumption.

@asdfex I didn't assume that the situation is the same elsewhere. I proved it, assuming only Hubble's law. The second equation follows from the first.

And where is the proof of Hubble's law? It's also only valid to describe observations made by us, unless you add assumptions.

@asdfex The point I have made is that if Hubble's law is valid for observations made by us, that mathematically implies that Hubble's law is also valid for observations made by anyone else.

@asdfex And, what do you mean by "where is the proof of Hubble's law"? You can't prove it from first principles unless you assume homogeneity, which is begging the question. I'm just taking it to be observational fact.

A mathematical equation designed to describe observations made from Earth can't be used to prove that the same law is valid elsewhere. And exactly this is what is asked @armand (Unless I misunderstand the question). I could equally well write an equation describing the flatness of my desk and extend it to assume that everything is flat.

@asdfex Yes, it can. That's exactly what I proved. At least, for observers that we can ourselves observe (is that your objection?)

@asdfex but again, this follows from the particular form of Hubble's law and would not be true for other velocity-distance relationships. I'm not saying what's true for us is necessarily true for other observers. I'm specifically using the form of Hubble's law to prove it, for Hubble's law only.

@Sten But if space legitimately isn't expanding isotropically, though, but radially outwards from us, then wouldn't that mean that their relative measurement of the velocity of an object at separation x⃗ −x⃗' wouldn't simply be v⃗ −v⃗′?

@JustinHilyard Sorry, are you referring to anisotropic but homogeneous expansion (different in different directions but the same in every place), or inhomogeneous expansion (different in different places)? In general, as long as $\vec v=\mathbf{H}\vec x$, where $\mathbf{H}$ is a linear operator (a matrix) and not necessarily just a number, the expansion is homogeneous but may not be isotropic (and rotation is possible too). But if you break the linearity and go with something like $\vec v\propto\vec x|\vec x|$, then expansion could appear isotropic from your position but would be inhomogeneous

It's in the latter case that we could say we are the center of expansion.

I meant the latter, yeah, since the question is asking about a model where everything is expanding away from us and we do in fact have such a privileged position. Could we not have a situation where, say, the universe is finite and bounded, and expanding from Earth as a center with a strictly linear relationship between distance and velocity? Where, say, all points on a sphere of distance x from us have velocity Hx directly away from us.

...Ah, wait, no, now that I think about it more, the issue there is that you'd have shells "running into" each other in some sense, isn't it?

@JustinHilyard In principle, the expansion could follow $\vec v=H\vec x$, but there could be an edge to the mass distribution (and only empty space beyond), and that distribution could be a sphere centered on us, which is larger than the observable universe. Then, in some sense, we could be the center of expansion. But we couldn't know it, and neither could other observers (unless they were far enough to see the edge).

I had to draw it on paper to visualize it but it actually makes sense. If the scaling is homothetic and the plane infinite then it looks the same from every point and it makes no sense to speak of a center. Remains the problem of how we know the universe is infinite, but that's another question.

@asdfex If you did actually write an equation describing the flatness of your desk, you should quickly observe that there are many things (starting with yourself) that do not conform to it.

@ARaybould These observations regarding my desk translate into traveling a billion light years and measure the movement of galaxies from there. An option I have because my desk is slightly smaller, but not with Hubble. Without the option we can only state that the formula as-is describes observations from Earth, and that there are good reasons to assume it is valid elsewhere. But "because the formula says so" is not a proof that the universe actually obeys its results in every place.

@armand To suppose that we are at the center of the universe's expansion, simply proposing that the universe is finite is not sufficient - you would have to suppose, in addition, that it has a particular geometry putting us in its center, and we have no evidence justifying any such premise. Without that, any other galaxy in the observable universe would be just as good a candidate as ours for being closest to the center of this supposedly finite universe.

@asdfex You do not have to explain to me why your most recent (so far) attempt to disprove Sten's answer fails!

@ARaybould you misunderstood my comment. If the universe is infinite and expands in every direction according to Hubble's law then this expansion looks the same from every point, I.e. every point looks to be the center, l.e. It doesn't make sense to talk about said center. Which is what Sten's answer made me realize.

@armand If this is what your question was about, I suggest to rephrase it. As it is now you asked for reasons to assume that Hubbles law is true. But it seems you wanted to ask why Hubbles law implies that expansion looks the same from everywhere.

@armand My apologies for misunderstanding you, but since you raised the issue of how we know the universe is infinite, I would like to make it clear (for the benefit of third parties) that disputing that it is infinite would not, by itself, justify putting us at the center of its expansion.

@asdfex A quick check of the question reveals that it is not about either of the options you propose! Nevertheless, your questions here are not entirely without merit. The interpretation of the observed red shift as a uniform expansion of the universe depends on a number of assumptions, and the one closest to your questions seems to be this: the geometry of spacetime is such that, given the velocities of two distant objects relative to us, we can derive the velocity of each one relative to the other. Einsteinian spacetime is one such geometry (though others work - e.g. the Galilean one...)

... At this point in our knowledge, I do not think there is any alternative that both implies the Earth is the center of an expanding universe and is consistent with all the observations we have made of the natural world. You could still say that this does not prove anything, but if you apply this level of skepticism consistently, you end up knowing almost nothing at all - to be consistent, you should also, for example, doubt that stellar parallax is evidence for the earth orbiting the sun!

@asdfex I think you missed the implication that an observer at another point would also be in motion. If you subtract their motion away from us from a third location's motion away from us the result is always going to be equation 1, so if the equation is correct relative to us it is correct relative to the observer moving according to it.