It rests on truth. Any argument that something follows from something else (e.g., that if you are a bachelor, you are not married; if you are a billionaire, you must have at least a million dollars; etc.) rests first on whether or not the inference is valid. Logic is what allows us to conclude that something does or doesn't follow from something else. That's validity.
However, logic is more general. I can determine that something follows from something else even if that something else isn't true.
The great thing about mathematics is that the "world" or "universe" we are dealing with is entirely and explicitly defined (it is "closed" or constitutes a "closed discourse universe"). If I say that grass is green, it is either true or false depending upon not only what the words "green" an "grass" mean but the extent to which these terms combine to make a statement which corresponds to external reality. This isn't true of statements like 1+1=2 or that division by 0 isn't defined or that a negative number multiplied by another negative number yields a positive number. In mathematics, everything is either true by definition or because it follows from (i.e., "is necessarily true") BECAUSE of definitions.
That 0.999...=1 is true because it follows logically (i.e., "is necessarily true") given that numbers are uniquely defined on the real number line (actually, it is true even if we don't assume irrational numbers exist; it follows from the properties of the set of all rational numbers/fractions). Logic doesn't cease to work because of infinities (although it sometimes becomes less intuitive). As an example, and one involving the power of proof by assumption (proof by contradiction), consider a statement that deals only with the whole numbers: "there is no greatest whole number."
How can I prove that? Well, I assume it is false. This implies that there exists some number n such that there is no whole number greater than n. But given any whole number, adding 1 to it yields another whole number. This means that n + 1 is a whole number. The assumption yields a contradiction, so it must be true that there is no greatest whole number.
Likewise, if I assume that 0.999... is not equal to 1, then there must exist some other number it is equal to which differs from 1. Logically, this means that the number 0.999... differs from the number 1 by some fixed amount. In your assertions that this difference is 0.000...1, either this difference really is fixed, in which case I can show it isn't the actual difference, or it must be allowed to become infinitely small. If it is allowed to become infinitely small, than the difference between 0.999...must be smaller than any possible number, which means there is no difference.
The premises (the assumptions) are simply that numbers have the properties they do (and need not even be an assumption that ALL real numbers do, merely the set of rational numbers/fractions).
