It was something like this, in my brain.
Why does your proof show, that it cannot be otherwise? Why cannot long division m/n = irrational number? The axiom of rational numbers must be enforced before we start long division. (I suppose it must be shown or demonstrated that irrational number is impossible to be the quotient of two integers)
Therefore we know that m/n = rational number always. Also m,n belong to domain of integers.
The basic characteristic of rational number is that any rational number in existance can be shown as quotient, or fraction of two integers m,n
also n is unequal to 0
From these assumptions, it follows that m/n is a simple fraction. Therefore (because of integers being domain) Simple Fraction is same as division.
The basic characteristic of simple fraction is that the divisor, must be "any one countable and finite integer", in existance, except zero. Divisor can be as large or small as possible (just like dividend can be as large or small as possible), but it is always "a countable finite integer" (because simple fraction)
Simple fraction m/n can be divided in long division. But, I'm not sure why... we must calculate in long division? What does the calculation in long divison actually prove? We can already say that assuming the axioms about rational numbers being true, then we can simply claim with truthfulness, that long division will not result in irrational number. Sorry for being pessimistic