= Vanishing component theorem for a Lorentz four-vector
If a fixed component of a <Lorentz four-vector> vanishes in every <Lorentz frame>, the vector is zero. For its time component, arbitrary infinitesimal boosts force every spatial component to vanish; for a spatial component, first use rotations and then a boost. Applied to the incoming-minus-outgoing total <four-momentum>, this shows that energy balance in every <inertial frame> implies full <four-momentum conservation>. The statement fails for general <tensors>: a nonzero antisymmetric tensor has zero diagonal components in all frames.
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