The correct (and simpler) expression is $$P(\text{my card} > \text{your card} | \text{I pick card of value i}) = \frac{(i-1)\times 4}{51}$$

Simply: the numerator is the possible cards that are lower than $i$, and $51$ are the available cards.

AS noted in the comments, this can be expressed as

$$ \dfrac{(i-1) + (i-1)\times 3}{12+13\times 3}$$

where the first terms in each sum correspond to the selected rank. But this does not really help. And of course, the mistake amounts to assuming $\frac{a}{b}+\frac{c}{d}\neq \frac{a+c}{b+d}$ This is already noted by
JMoravitz' comment.