whats the proof for PV=nRT?

The ideal gas equation is just the combination of three gas laws - Boyle's law, Charles's law, and Avogadro's principle.
Boyle's law states that pressure and volume are inversely proportional to each other;
P α 1/V
therefore, PV = constant.
Charles's law states that volume and temperature are directly proportional to each other;
V α T
therefore, (V/T) = constant.
Avogadro's principle states that volume and number of moles (n) are directly proportional to each other;
V α n
therefore, (V/n) = constant.
Combining these three equations, we get
PV / nT = constant
P =The gas constant is R; therefore,
PV / nT = R
We can rewrite the equation to this form:
PV = nRT

A gas that would obey Boyle’s , Charle’s law  and Avogadro’s law under the conditions of temperature and pressure is called an ideal gas.  

Here, we combine four measurable variables P, V, T and n to give a single equation.  

V ∝ n [P, T constant] 

Avogadro’s law  

V ∝ T [n, P constant]  

Charle’s law

V ∝ 1/P [n, T constant] 

Boyle’s law 

The combined gas law can be written as 

V ∝ nT / p

or pV ∝ nT  

 pV = nRT  

This is called ideal gas equation  

where R is the constant of proportionality or universal gas constant    

The value of R was found out to be  

R = 8.314 J mol–1 K–1  

R = 0.0821 litre atm K–1 mol–1  

R = 2 cal K–1 mol–1