I am confused how the operation $\cdot : F\times V\rightarrow V$ is defined for a vector space over a field. Is $\cdot : F\times V\rightarrow V$ simultaneously defined as $\cdot : F\times F\rightarrow F$ in regards to the field, or should there be symbols distinguishing the axioms below? Any help would be greatly appreciated. If I was to guess the operations are simultaneously defined, but I could be wrong.
Definition 1: A vector space over a field F is defined to be a set $V$. This set is said to have two operations $+: V\times V\rightarrow V$ and $\cdot: F\times V\rightarrow V$. Also, $V$ is defined to have the following conditions:
$(C1)$ $V, +$ forms a commutative group with an identity element $=0$,
$(C2)$ For all $a, b\in F$, for all $v\in V$, $(a⋅ b)⋅ v=a⋅ (b⋅ v)$,
$(C3)$ For all $v\in V$, $1⋅v=v$,
$(C4)$ For all $a\in F$, for all $u, v\in V$, $a⋅(u+v)=a⋅u+a⋅v$,
$(C5)$ For all $a, b\in F$, for all $v\in F$, $(a+b)⋅v=a⋅v+b⋅v$.
Definition 2: A $\textbf{field}$, $F$, is a set with two operations, $+$ and $\cdot$ which satisfies the following conditions:
$(C1)$ $F, +$ forms a commutative group with an identity element $= 0$,
$(C2)$ $F^{\neq 0}, \cdot$ forms a commutative group with an identity element $= 1$,
$(C3)$ For every $a, b, c\in F$, $a\cdot (b+c)=a\cdot b+a\cdot c$.