Basis of Vector Space Injects into Generator

Theorem

Let $K$ be a division ring.

Let $V$ be a vector space over $K$.

Let $B$ be a basis of $V$.

Let $G$ be a generator of $V$.

Then there exists an injection from $B$ to $G$.

Proof

By Vector Space has Basis between Linearly Independent Set and Spanning Set, there exists a basis $C \subset G$.

By Bases of Vector Space have Equal Cardinality, there exists a bijection between $B$ and $C$.

By Composite of Injections is Injection, composing this bijection with the inclusion of $C$ in $G$, we obtain an injection from $B$ to $G$.

$\blacksquare$