And something simpler here for you;
How do we know more CO2 is causing warming?
Climate sensitivity
As the name suggests, climate sensitivity is an estimate of how sensitive the climate is to an increase in a radiative forcing. The climate sensitivity value tells us how much the planet will warm or cool in response to a given radiative forcing change. As you might guess, the temperature change is proportional to the change in the amount of energy reaching the Earth's surface (the radiative forcing), and the climate sensitivity is the coefficient of proportionality:
dT = λ*dF
Where 'dT' is the change in the Earth's average surface temperature, 'λ' is the climate sensitivity, usually with units in Kelvin or degrees Celsius per Watts per square meter (°C/[W/m2]), and 'dF' is the radiative forcing.
So now to calculate the change in temperature, we just need to know the climate sensitivity. Studies have given a possible range of values of 2-4.5°C warming for a doubling of CO2 (IPCC 2007). Using these values it's a simple task to put the climate sensitivity into the units we need, using the formulas above:
λ = dT/dF = dT/(5.35 * ln[2])= [2 to 4.5°C]/3.7 = 0.54 to 1.2°C/(W/m2)
Using this range of possible climate sensitivity values, we can plug λ into the formulas above and calculate the expected temperature change. The atmospheric CO2 concentration as of 2010 is about 390 ppmv. This gives us the value for 'C', and for 'Co' we'll use the pre-industrial value of 280 ppmv.
dT = λ*dF = λ * 5.35 * ln(390/280) = 1.8 * λ
We both know that you don't understand that at all rocks, but the glaring error there is an assumed forcing and an assumed climate sensitivity. We don't know what the climate sensitivity is, therefore your so called proof is nothing more than an unverifiable assumption.
Lets pretend for a second that you actually do understand what you posted. If that is the case, then lets assume that you understand the ideal gas law which is even simpler than what you posted.
PV=nRT where P = the pressure of a gas, V = the volume of a gas, n = the amount of a gas ( the number of molecules), R = the universal gas constant, and T = the temperature of the gas.
Look closely at the equation. Note the = sign. That means that the two sides must remain equal, therefore a change in any one, or more than one of them will result in changing values across the board. P and V are inversely proportional so an increase in pressure will result in a decrease of volume and vise versa.
If you increase either P or V without reducing the other one will require an increase in n, or the gas constant which has a relation to gravity, or T. nRT will then equal PV.
According to the ideal gas law, it isn't possible for PV to not equal nRT and you can't hold P steady while increasing T and V equally but reducing n.
If more greenhouse gasses lead to a higher T, which causes V to rise, then the reduced n must result in a lower V which must result in a lower T.
Explain how to get around that problem within the framework of the greenhouse hypothesis.