if P=NP, then is E=NE?

Question

If the computational complexity class P equals to NP, does the complexity class E equal to the class NE?

E is defined as $DTIME(O(2^{O(n)}))$ NE is defined as $NTIME(O(2^{O(n)}))$

Thank you very much.

Answer 1

Yes, by using the padding argument. The Wikipedia article on the padding argument contains a proof of the implication P=NP ⇒ EXP=NEXP, and the implication P=NP ⇒ E=NE can be proved in the same way. I leave the details as an exercise.