# Right and Left arrow notation in proof.

I'm studying vector spaces and I'm reading a proof where the authour uses the symbols

$$(\Rightarrow)$$

and

$$(\Leftarrow)$$

when proving a theorem. He doesn't use them in context, but rather before starting a part of the proof.

How should I read this? Does it have anything to do with symbols and notation in logic?

Proposition 1.10

Let $V$ be a $K$-vector space, and let $S \subseteq V$. Then $S$ is a subspace of $V$ if and only if:

1. $\mathbf{0} \in S$
2. $\mathbf{v},\mathbf{w} \in S\Rightarrow \mathbf{v}+\mathbf{w}\in S$
3. $\lambda \in K,\mathbf{V}\in S\rightarrow \lambda \cdot \mathbf{v}\in S$

Proof

$(\Rightarrow)$ It is immediate [sigh] to verify that if $S$ is a subspace of $V$ then $1$,$2$ and $3$ hold.

$(\Leftarrow)$ $1.$ ensures $S \neq \emptyset$ $2.$ implies that $+$ is an operation in $S$. $3.$ implies that $\cdot$ is an action.

Asociativity and conmutativity are deduced from their validity in $V$, the neutral element from $1.$, and the inverse additive from $3$. The properties of the action are deduced from their validity in $V$.

He's probably proving a statement of the form "$A$ if and only if $B$", and he's using the first arrow to indicate he's starting the proof of the $A$ implies $B$ part, and the second arrow to indicate he's starting the proof of the $B$ implies $A$ part.