Not understanding this division problem $\frac{52}{x}=13$

It says to next step

$\frac{52}{13}=x$

Ok, I can do future problems like this, but is there a rule that explains this? What just happened to both sides of the equal sign?
There are two steps here:
starting with
$$\frac{52}{x}=13$$
we multiply both sides by $x$ to get
$$\frac{52}{x}\cdot x=13\cdot x$$
$$52=13\cdot x$$
and then dividing both sides of that by $13$ to get
$$\frac{52}{13}=\frac{13\cdot x}{13}$$
$$\frac{52}{13}=x$$
I'd count several steps here.  Anyways...

We start with $$\left(\frac{52}{x}\right)=13$$.  Multiplying both sides by x we have $$\left(\left(\frac{52}{x}\right)*{x}\right)=(13*{x}).$$  Since $$\left(\frac{x}{1}\right)=x$$ we have  $$\left(\left(\frac{52}{x}\right)*\left(\frac{x}{1}\right)\right)=(13*{x}).$$  Applying the product rule for quotients, we have $$\left(\frac{(52*x)}{(x*1)}\right)=(13*{x}).$$  Since multiplication commutes, we have $$\left(\frac{(x*52)}{(x*1)}\right)=(13*{x}).$$  Applying the product rule for quotients again, we have  $$\left(\left(\frac{x}{x}\right)*\left(\frac{52}{1}\right)\right)=(13*{x}).$$  Since $$\left(\frac{x}{x}\right) =\left(\frac{1}{1}\right)$$ we obtain $$\left(\left(\frac{1}{1}\right)*\left(\frac{52}{1}\right)\right)=(13*{x}).$$  Applying the product rule for quotients once again we have  $$\left(\frac{(1*52)}{(1*1)}\right)=(13*{x}).$$  Since $(1*1)=1$, and $(1*52)=52$ we obtain  $$\left(\frac{52}{1}\right)=(13*{x}).$$  Now we multiply both sides by $$\left(\frac{1}{13}\right)$$ and obtain $$\left(\left(\frac{1}{13}\right)*\left(\frac{52}{1}\right)\right)=\left(\left(\frac{1}{13}\right)*\left(13*{x}\right)\right).$$  Since 13 equals $$\left(\frac{13}{1}\right)$$ we obtain $$\left(\left(\frac{1}{13}\right)*\left(\frac{52}{1}\right)\right)=\left(\frac{1}{13}\right)*\left(\left(\frac{13}{1})*{x}\right)\right).$$  Since * associates we have $$\left(\left(\frac{1}{13}\right)*\left(\frac{52}{1}\right)\right)=\left(\left(\left(\frac{1}{13}\right)*\left(\frac{13}{1}\right)\right)*{x}\right).$$  Applying the product rule for quotients on both sides we obtain $$\left(\frac{(1*52)}{(13*1)}\right)=\left(\left(\frac{(1*13)}{(13*1)}\right)*{x}\right).$$  Since $(1*52)=52$, $(13*1)=13$, and $(1*13)=13$, we obtain $$\left(\frac{52}{13}\right)=\left(\left(\frac{13}{13}\right)*{x}\right).$$  Since $$\left(\frac{13}{13}\right)=1$$ we obtain $$\left(\frac{52}{13}\right)=(1*x).$$  Since $(1*x)=x$, we obtain $$\left(\frac{52}{13}\right)=x$$

Even reasoning in this way steps have gotten left out, such as how we can "multiply both sides by x."