A question about proving $\left|-a\right|=\left|a\right|$

Question

I wonder if it is correct to prove this expression in this way.

$\left|a\right|=\sqrt{a^2}$

$\left|-a\right|=\sqrt{(-a)^2}$

$\sqrt{(-a)^2}=\sqrt{a^2}$

Answer 1

Given a variable $x\in\Bbb R$ we define $|x|=\begin{cases} x\iff x\ge 0\\ -x\iff x<0\end{cases}$

So you can see that $x\ge 0\implies |x|=x$ and $|-x|=x$ and in the sane way $x<0\implies |x|=-x$ and $|-x|=-x$