Conservation of 3-Momentum: Noether’s Theorem

[dipole]

I'm blanking on this at the moment, but a quick application of Noether's theorem would imply that, aside from the momentum-four vector, $p_{\mu} = (E/c,-\vec{p})$ being conserved, that each component of $\vec{p}$ is also conserved due to invariance of the Lagrangian under translations - this is certaintly true in classical mechanics, but it also applies to special relativity, correct?

 

[Ben Niehoff]

A vector being preserved is the same as all of its components being preserved.