vector space axioms imply this?
It is most probably an error. A field action implies some things about the group addition in the vector space and some axioms can be removed, but not this one.
For a complete discussion of vector spaces axioms, see this paper
Independent Axioms for Vector Spaces, J. F. Rigby and James Wiegold. The Mathematical Gazette Vol. 57, No. 399 (Feb., 1973), pp. 56-62.
The first page is freely available and contains all you need, except the proofs. The main point is that the axioms below suffice and are independent, and so form a minimal set:
