Conservation of 3-Momentum: Noether’s Theorem

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

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