Indeed the Cook-Levin theorem
> https://en.wikipedia.org/wiki/Cook%E2%80%93Levin_theorem
which actually proves that there exist NP-complete problems in NP is not a trivial result (and was at that time a highly surprising one).
So, indeed, "one half of the proof" has been done; the other half (which seems to be much harder) of P vs NP is still open.
Indeed the Cook-Levin theorem
> https://en.wikipedia.org/wiki/Cook%E2%80%93Levin_theorem
which actually proves that there exist NP-complete problems in NP is not a trivial result (and was at that time a highly surprising one).
So, indeed, "one half of the proof" has been done; the other half (which seems to be much harder) of P vs NP is still open.