Is the Hadith, "Indeed, Fatima is a part of me; Allah is angered by her anger and pleased by her pleasure," authentic?

Al-Bukhari and Muslim narrated that Ali proposed to the daughter of Abu Jahl. Fatimah then came to the Messenger of Allah, peace be upon him, and said, "Your people claim that you do not get angry for your daughters, and this is Ali marrying the daughter of Abu Jahl." So the Prophet, peace be upon him, stood up and said, "Thereafter, I married off Abu al-Aas ibn al-Rabi', and he spoke to me and spoke truthfully. Indeed, Fatimah is a part of me, and I dislike that she should be distressed. By Allah, the daughter of the Messenger of Allah, peace be upon him, and the daughter of the enemy of Allah shall not be combined under one man." So Ali abandoned the proposal. Other narrations in Al-Bukhari state: "A part of me; what troubles her troubles me, and what harms her harms me," and "Whoever angers her has angered me."