Proofs from Subspaces

If $S$ is a subspace, the three properties are part of the vector-space axioms. Conversely, the inherited operations already satisfy associativity, commutativity and distributivity because they do so in $V$. Choose $u\in S$ using non-emptiness. Scalar closure gives

and, for every $u\in S$,

Thus $S$ has the zero vector and additive inverses, while the three listed properties provide the remaining requirements.