Salaato Salaam / Tasleem … Is it quranic?

This very common phrase (AL-SALAT ALA AL-NABI and AL-TASLEEM) derives from the following verse:
1. Allah and His angels send (yusalloona) blessings on the Prophet: O ye that believe! send ye (salloo) blessings on him and (sallimu) salute him with all respect. 33:56
Innallaaha wa malaaikatahu yu salloo na alan nabi yaa aiyuhal lazeena aamanu salloo alaihi wa  salli  mu   tasleema
Immediately after the completion of this verse the Muslims say this:
SAHIH BUKHARI 8:368  & 4:589     Abdur-Rahman bin Abi Laila
Ka’b bin ‘Ujra met me and said, “Shall I give you a present? Once the Prophet came to us and we said, ‘O Allah’s Apostle! We know how to greet you; but how to send Salat upon you? He said, ‘Say: Allahumma Salli ala Muhammadin wa ‘ala Ali Muhammadin, kama sal-laita ‘ala all Ibrahima innaka Hamidun Majid. Allahumma barik ‘ala Muhammadin wa ‘ala all Muhammadin, kama baarakta ‘ala all Ibrahima, innaka Hamidun Majid.’ “
Ka’b bin Ujrah met me and said, “Shall I not give you a present I got from the Prophet?” ‘Abdur-Rahman said, “Yes, give it to me.” I said, “We asked Allah’s Apostle saying, ‘O Allah’s Apostle! How should one (ask Allah to) send blessings on you, the members of the family, for Allah has taught us how to salute you (in the prayer)?’ He said, ‘Say: O Allah! Send Your Mercy on Muhammad and on the family of Muhammad, as You sent Your Mercy on Abraham and on the family of Abraham, for You are the Most Praiseworthy, the Most Glorious. O Allah! Send Your Blessings on Muhammad and the family of Muhammad, as You sent your Blessings on Abraham and on the family of Abraham, for You are the Most Praiseworthy ............. read more>>