Suppose $X$ is a set and $\mathbb{K}$ is a field, with sum operation $(x,y)\in X\times X\mapsto x+y\in X$ and multiplication operation $(\lambda, x)\in \mathbb{K}\times X\mapsto \lambda\cdot x\in X$. But we don't know any properties about these operations (for instance, we don't even know if the sum is associative).
Now, let $E$ be a $\mathbb{K}$-vector space and suppose $\phi: E\to X$ is a linear isomorphism (i.e. $\phi$ is a bijection, $\phi(u+v) = \phi(u)+\phi(v)$ and $\phi(\lambda\cdot u) = \lambda\cdot\phi(u)$, for all $u,v\in E$ and all $\lambda\in\mathbb{K}$).
My question: is this enough to assure $X$ is a $\mathbb{K}$-vector space?