Famous Salafi scholars, Hafiz Mubarakpuri writes in explanation of the part “Part of Me” that this narration tells us that when Holy Prophet (asws) is hurt due to hurting of a person, then it is Haram/prohibited to hurt that person as well, no matter we hurt her more or less, in any case, it is haram. And it is proven without doubt that what hurts Bibi Fatima (asws) hurts Holy Prophet (asws), so this means that anyone who has hurt Bibi Fatima (asws), Has has hurt Holy Prophet . and This Sahih/authentic Hadeeth is witness upon this. His exact words are

فيه تحريم أذى من يتأذى النبي صلى الله عليه وسلم بتأذيه لأن أذى النبي صلى الله عليه وسلم حرام حرام اتفاقًا قليله وكثيره وقد جزم بأنه يؤذيه ما يؤذي فاطمة فكل من وقع منه في حق فاطمة شيء فتأذت به فهو يُؤذي النبي صلى الله عليه وسلم بشهادة هذا الخبر الصحيح.

[Tohfa tul Ahodhi Sharah Jami Tirmidhi, 10/340, Printed by Qadimi Kutb Khana, Karachi]
Translated by Slave of Ahlubait
Research by Brother Ismaeel Deobandi