0+x = x

need a proof, that 0+x = x
while x ∈ V and (V, +, •) is a vector space over F

>>  0x = (1-1)x = x -x = 0
This is sort of legal, but it uses -1 * x = (-x), which is not an axiom.  By (-x) I mean the additive inverse of x.  The proof of -1 * x = (-x) is easy using 0*x = 0 though.

>>is 0+x = x an axiom then? Oo
Yep, because V is a (abelian) group with composition denoted by "+".