Linear Algebra - Prove that 0v = 0

This is a typical proof from a rigorous introduction to Linear Algebra where the goal is to show that 0v = 0 for every vector v in a vector space V.

[LionKimbro]

If you have [0:field][v:vec] = [0:field][v:vec] + [0:field][v:vec], can you show that [0:field][v:vec] must therefore be [0:vec] because there is only one element in [vec] that has the property of adding to an element in [vec] and returning itself, and no other element in [vec] can combine with something to give itself?