Voltage and Current in transformers

There is a well known transformation law for the effective load seen through a transformer.

Let $R_o$ be the load in the output circuit.

$V_o = I_o R_o$

Assuming all power is transferred into the output circuit,

$V_o I_o = V_i I_i$

It then follows simply that

$V_i / I_i = (V_i / V_o)^2 R_o$

This is the effective load seen by the input circuit.

as pointed out by @Georg, $V=IR$ is a different thing and conservation of energy accross the transformers a different!!

you can validate the Ohm's Law $V=IR$ in case where you have current flowing through a resistive medium, which as a matter of fact applies everywhere expect the less abundant superconductors...

If you want to apply Ohm's law in transformer then you can do that provided you do that in primary and secondary independently since primary and secondary in a transformer is not electrically coupled rather they are magnetically connected. So applying Ohm's law between primary and secondary is not correct.

In fact when you actually see the conservation of energy though primary and secondary you will notice that they are not balanced i.e. when current is halved, theoretically voltage should double; but practically this doesn't happen since resistance takes its share and reduces the voltage slightly below its expected value.