Question Regarding The Ideal Gas Law I have been studying thermodynamics and heat transfers and there I met with the ideal gas law, in fact I have used it a lot of times until now, but when i see the proof of this law, I saw that it has been derived by combining Boyle's law, Charles' law, and the third one is Avogadro's law I think, so now i am confused because Boyle's law assumes that Temperature of the gas is constant, Charles' law assumes that the Pressure of the gas is constant and Avogadro's law says that temperature and pressure both must be constant, so how can we combine all these proportional pieces into one single cake, when all these pieces have different dependencies ? 
 A: The easiest way to handle this is to postulate the ideal gas equation (IGE), $pV=nRT$, and then to show that it is consistent with the various laws. 
The only completely straightforward case is Boyle's Law. "For a fixed mass of gas at constant temperature…" immediately renders constant the right hand side of the IGE, so $pV$ is constant, in accordance with Boyle's law, which is a purely experimental law.
Charles's law deals with a fixed mass of gas (therefore fixed n) at constant p, so the IGE gives V proportional to T in accordance with the law. [What is not wholly straightforward here is that one has to be sure that the temperature is measured on – or converted to a value on – the thermodynamic temperature scale.]
Avogadro's Law considers equal volumes of (different) gases at the same temperature and pressure. Re-arranging the IGE we have $nR=\frac{pV}{T}$, so nR, and hence (if R is a universal constant) n is the same for all gases. There are two caveats: (1) Avogadro's law started life as a hypothesis, not an experimental law, (2) It doesn't guarantee that n appears in the IGE as a simple multiplied factor. An equation of the form $pV=f(n)\ RT$ would still be consistent with Avogadro, as, according to this equation $f(n)$ and hence n would be the same for equal volumes of gas at the same T and p.
So a somewhat stronger law than Avogadro's would be needed in order to justify the IGE experimentally; for example that the pressure exerted by equal volumes of gas at the same temperature was proportional to the number of moles. As far as I know, this isn't a named experimental law, but nobody worries. 
Another issue, but perhaps not your main cause of concern, is that the various so-called gas laws are not quite correct for real gases, and we have to extrapolate the observed behaviour back to very low gas densities, when all real gases approach ideal behaviour.
A: Phillip Wood provided a very useful summary, mentioning various necessary ideas.  However, I recall myself as a student of thermodynamics having similar questions, and these answers still never completely satisfied me and I perceive that it still might not completely answer the concern.  Really, the question already states that the various laws have already been outlined and brought together in the IDE, so I'm not convinced that even the best summary like Phillip's really answer the question.
The steps of a scientific method are taught in school, but that method is hardly ever revealed in explaining how many scientific laws were obtained. I think an understanding of the historical process would be enlightening and fill in gaps.  Regrettably I'm not the historian to share the actual details.  But it would show a process in which the Ideal Gas Law could not be postulated first, one in which the concept of especially temperature was still being explored and not fully understood, one in which the idea of separate particles (i.e. atoms and particles) was not yet taken for granted... and was sometimes taken as bogus.  We'd see how various experimental results and incomplete formulas were scratched on various sheets and in notebooks.  Results would be shared in letters and publish in journals, but at far slower speed than we realize today.  Technology had to improve along with the science, otherwise there was no good way to measure and/or maintain constant temperatures... or to keep a gas sealed without leaking out of suitable containers, etc.
Regardless of what intuitive ideas we have developed today, some of what we consider simple relationships could not be taken for granted in the past.  Using rather basic measurement devices, the scientists would necessarily choose to keep one or many things constant.  It would have been nigh impossible to do experiments and learn anything if all aspects were wildly fluctuating.  And then it took a long time in sharing results and for others to piece all the experimental results together, probably with a lot of incorrect or incomplete formulas along the way.
So if one excludes the possibility of postulating the ideal gas, the answer to 

... how can we combine all these proportional pieces into one single cake, when all these pieces have different dependencies?

is that first of all it is absolutely necessary to perform separate experiments, keeping at least one of the dependencies constant in each.  Then one-by-one recognize similarities, do some mathematical manipulations and relate constants, make some guesses (i.e. hypotheses)--including wrong ones, then finally deduce the correct relationship and re-test. 
Pardon me if this sounds like a stupid answer, but I recall that nearly all thermodynamic explanations always started with some sort of postulate / axiom that was plucked from air and nobody (in my realm) really elucidated how or where such axioms came from.  I've realized it is because there is no neat outline or short answer, none that anyone could explain easily and so they didn't really try.
A: The easiest way to explain this is to say that for a gas that is confined, pressure, temperature and volume are all related. This relation carries over to any final state after a state change of any or all variables from an initial state. You mentioned the particular relation in your question: they are proportional. When you have two or more proportional variables, they are related according to this proportionality. The change of one variable will cause a change in another variable.
If you multiply the values P1 * V1 * T1, you end up with a number representing a state of all variables. This is considered the initial state of the gas. This state is directly related to a final state when any of the variables change:
P1 * V1 * T1 = P2 * V2 * T2
This will work when you are only dealing with a single instance of a gas. If one or more of the variables changes due to the introduction of a different gas (for example, to change volume or pressure), this will change the gaseous composition, and you would have to account for that.
The volume of one gas may differ from the volume of another gas at a certain temperature. Carbon dioxide could be used as an example. It becomes a solid at -56.6 Celsius. If changing temperature below this (considering pressure), it could go through a change in phase where it is no longer a gas. The relation will no longer apply to it, and the proportionality will be broken with respect to it.
