Vector Scaled by Zero is Zero Vector

Theorem

Let $F$ be a field whose zero is $0_F$.

Let $\struct {\mathbf V, +, \circ}_F$ be a vector space over $F$, as defined by the vector space axioms.

Then:

$\forall \mathbf v \in \mathbf V: 0_F \circ \mathbf v = \bszero$

Proof

 $\displaystyle 0_F \circ \mathbf v$ $=$ $\displaystyle \paren {0_F + 0_F} \circ \mathbf v$ Field Axiom $A \, 3$: Identity Element for Addition $\displaystyle$ $=$ $\displaystyle 0_F \circ \mathbf v + 0_F \circ \mathbf v$ Vector Space Axiom $\text V 5$: Distributivity over Scalar Addition $\displaystyle \leadsto \ \$ $\displaystyle 0_F \circ \mathbf v + \paren {-0_F \circ \mathbf v}$ $=$ $\displaystyle \paren {0_F \circ \mathbf v + 0_F \circ \mathbf v} + \paren {-0_F \circ \mathbf v}$ adding $-0_F \circ \mathbf v$ to both sides $\displaystyle$ $=$ $\displaystyle 0_F \circ \mathbf v + \paren {0_F \circ \mathbf v + \paren {-0_F \circ \mathbf v} }$ Vector Space Axiom $\text V 2$: Associativity $\displaystyle \leadsto \ \$ $\displaystyle \bszero$ $=$ $\displaystyle 0_F \circ \mathbf v + \bszero$ Vector Space Axiom $\text V 4$: Inverses $\displaystyle$ $=$ $\displaystyle 0_F \circ \mathbf v$ Vector Space Axiom $\text V 3$: Identity

$\blacksquare$