Can I use "$\iff$" symbol when I "transform" an expression to another form? I am writing a solution to prove that $\sqrt5$ is not rational. Here is my first half proof:


  Assume $\sqrt{5}$ is a rational number. By the definition of rational number, $\sqrt{5} = \frac{p}{q}$, where $p,q\in\mathbb{Z^+}, q\neq0$, and $gcd(p,q)=1$. We have $5=\frac{p^2}{q^2} \iff 5q^2=p^2$.


Can I use "$\iff$" symbol like this?



How about if a question is asking me to work backward from the desired conclusion and then prove:

If x and y are nonnegative integer, then $\frac{x+y}{2}\geq \sqrt{xy}$. 

Can I do like something like: 


  (first half proof) $\frac{x+y}{2}\geq \sqrt{xy} \iff x+y\geq 2\sqrt{xy} \iff (x+y)^2 \geq 4xy 
\iff x^2+2xy+y^2 \geq 4xy \iff x^2-2xy+y^2 \geq 0 \iff (x-y)^2 \geq 0$. 


Should I use $\iff$, $\Leftarrow$ or $\Rightarrow$?
Yes. "$\iff$" is used to denote logical equivalence, or necessary and sufficient conditions. It is read "if and only if," of which "iff" is a common abbreviation. $p \iff q$ is true if $p$ is true whenever $q$ is true and $q$ is true whenever $p$ is. 


  Assume $\sqrt{5}$ is a rational number. By the definition of rational number, $\sqrt{5} = \frac{p}{q}$, where $p,q\in\mathbb{Z^+}, q\neq0$, and $gcd(p,q)=1$. We have $5=\frac{p^2}{q^2} \iff 5q^2=p^2$.


This is true, since $5=\frac{p^2}{q^2} \Rightarrow 5q^2=p^2$ and $5q^2=p^2 \Rightarrow 5=\frac{p^2}{q^2}$.


  If x and y are nonnegative integer, then $\frac{x+y}{2}\geq \sqrt{xy}$.
  
  Proof: $\frac{x+y}{2}\geq \sqrt{xy} \iff x+y\geq 2\sqrt{xy} \iff (x+y)^2 \geq 4xy 
\iff x^2+2xy+y^2 \geq 4xy \iff x^2-2xy+y^2 \geq 0 \iff (x-y)^2 \geq 0.$ 


This is a fine proof. I would like to see it start where you end, but with $\iff$ used as you do here, it's perfectly valid as is.
You are moving forward in the proof, so you can use $\implies$ (this implies) sign which is more appropriate.