the rigorous part is the proof that 0 times any number is 0
as found in the link that I provided
it is based on the properties of Real numbers that are accepted as true
namely a*1/a=1 ; a=/=0
that is why you can't do Algebra with 1/0
in fact the Algebraic definition of division clearly states that a/b = a*1/b; b=/=0.