It is perfectly legal and you do not need any other proofs, but it all depends on a company HR's attitude and knowledge about immigration issues.
I talked to one company some time ago, and all my attempts to explain what EAD is only triggered suspicions. They knew only two things - H1 or GC (I wish it was as simple as that), and they told me "we don't know about EAD, we have to talk to someone who know, but we do know whom to ask..." It was a nice company, too bad they had a genuine bone-head for the recruiter.