[impendia]

As a working mathematician, I would say "obvious" is most often used for omitted arguments whose analogues are widely established elsewhere.

As an elementary example, in an analytic number theory paper one might need to know that for some constant C we have

5x + x^3 + ln(x) < Ce^x

for all positive x. (In practice, one would not need to determine such a C, just know that one exists.)

Such a claim would usually be described as "obvious" or else stated without any justification. I doubt you could find precisely this claim in the literature, but one can find plenty of very similar arguments in textbooks, and anyone doing research in analytic number theory would find this "obvious".