Whether or not partner shows 6 spades (partnership agreement), you'll have trouble catching up if you don't show support now. This hand is too good not to give you and pard the chance to explore for slam. The hardest question here, imo, is what 3NT shows over 3S. For some it's a slam try, for others it's an offer to play. There are merits to either POV, it's just key that the partnership has a clear rule. (Fwiw, my preference is that 3NT is non-serious/mild slam try when we have a major suit fit and 2NT was not bid naturally.)