If our ahadith (Shia Twelver) has any legitimacy, then it's established that Fatima is better then all previous Prophets except Mohammad (s), by sheer amount of reports, salawat, ziyarats, and Du'as that testify to this from Ahlulbayt (a) including how they are blessed in Saheefa Sajjadiya in the Du'a Arafa. But my approach is I don't believe anything in ahadith unless I find it in Quran.
As for the proof in Quran, it's in the name Mohamad and Ahmad explaining one another. Mohammad means Ahmad and Ahmad means Mohammad per Quran, and Ahmad is what Angels call Mohammad and humans call who is called Ahmad by Angels, to be Mohammad per ahadith.
In Quran, however, it's quoting Isa (a) who speaks a different language, and so this is a way of God explaining what Mohammad (s) means.
Ahmad means most praised. But is that he just most praised in this world or all worlds, or does it mean something higher. If we look Mohamad implying that name, it can only be if it means praised without anything of the opposite.
This means there is no neglect in praise at each stage with Mohammad (s). There is no way to find any defect as in you could've done better at any stage, Mohammad is fully praised to the extent of utmost praised (Ahmad).
And so when he is Ahmad, he is not just the best, but the best possible way a creation of God can be.
Now what does this have to do with Fatima and Twelve Imams?
You should know, that any neglect from them, any of them, at any stage, would make them fall SO BEHIND Mohammad (s), they couldn't be the "Twelve Captains with him" and they won't be his family. This is because of the vast difference between his rank and other Prophets who swerved time to time and God has to distinguish them again, and choose them again.
The Ahlulbayts are such that they ascend with another. Some of our ahadith say that God hid who was better Yahya or Jesus, and they themselves didn't know who had more Taqwa. This is because the Ahlulbayts ascend to one another and they are blessed together.
When you say "peace be upon Ibrahim" per Quran, this is same as saying "Peace be upon the family of Ibrahim". They are all blessed together.
While the Ahlulbayts of the past was such they would ascend and God would keep secret who was better of them with respect to each other, it happens to be, with Mohammad (s), to be with him, they too can't have any neglect and so with Ahlulbayt of Mohammad (s), we have to see that they are ALL EQUAL.
And so while other Prophets had swerved and fell off the path, only to return, God blessed them all together with their family and they ascend together with their Ahlulbayt, it happens to our Imams to be with Mohammad and blessed together with him, can't have neglect in praise as well.
Four our Ahlulbayt with addition to what I said, have been named Mohammad, and so this shows they have this station of utmost submission to God.
Also according to verse 33:33 Mohammad (s), Ali (a) are of one stage of purity always, we have to see them same stage in purity, and there are stages of purity on top of purity, per Saheefa Sajjadiya about Angels, "blessing that increases their purity on top of their purity", and Angels are completely pure, but there is no end to purity, as God's purity and holiness is absolutely more then all, and so this means Ahlulbayt (a) are to be seen ascending constantly together, blessed together.
Finally, this affirmed in "God and his Angels bless the Prophet" and what we say in Salah, and the verse "Peace be upon the family of Yaseen" which the verses that follow show the reward of Mohammad his family share in it, and it's one peace level they are in, just Harun is with Musa, and Elyas is part of the family of Aaron, and Ishmael and Isaac with Ibrahim, so the level of peace Mohammad (s) is in, his family share.
As for the proof in Quran, it's in the name Mohamad and Ahmad explaining one another. Mohammad means Ahmad and Ahmad means Mohammad per Quran, and Ahmad is what Angels call Mohammad and humans call who is called Ahmad by Angels, to be Mohammad per ahadith.
In Quran, however, it's quoting Isa (a) who speaks a different language, and so this is a way of God explaining what Mohammad (s) means.
Ahmad means most praised. But is that he just most praised in this world or all worlds, or does it mean something higher. If we look Mohamad implying that name, it can only be if it means praised without anything of the opposite.
This means there is no neglect in praise at each stage with Mohammad (s). There is no way to find any defect as in you could've done better at any stage, Mohammad is fully praised to the extent of utmost praised (Ahmad).
And so when he is Ahmad, he is not just the best, but the best possible way a creation of God can be.
Now what does this have to do with Fatima and Twelve Imams?
You should know, that any neglect from them, any of them, at any stage, would make them fall SO BEHIND Mohammad (s), they couldn't be the "Twelve Captains with him" and they won't be his family. This is because of the vast difference between his rank and other Prophets who swerved time to time and God has to distinguish them again, and choose them again.
The Ahlulbayts are such that they ascend with another. Some of our ahadith say that God hid who was better Yahya or Jesus, and they themselves didn't know who had more Taqwa. This is because the Ahlulbayts ascend to one another and they are blessed together.
When you say "peace be upon Ibrahim" per Quran, this is same as saying "Peace be upon the family of Ibrahim". They are all blessed together.
While the Ahlulbayts of the past was such they would ascend and God would keep secret who was better of them with respect to each other, it happens to be, with Mohammad (s), to be with him, they too can't have any neglect and so with Ahlulbayt of Mohammad (s), we have to see that they are ALL EQUAL.
And so while other Prophets had swerved and fell off the path, only to return, God blessed them all together with their family and they ascend together with their Ahlulbayt, it happens to our Imams to be with Mohammad and blessed together with him, can't have neglect in praise as well.
Four our Ahlulbayt with addition to what I said, have been named Mohammad, and so this shows they have this station of utmost submission to God.
Also according to verse 33:33 Mohammad (s), Ali (a) are of one stage of purity always, we have to see them same stage in purity, and there are stages of purity on top of purity, per Saheefa Sajjadiya about Angels, "blessing that increases their purity on top of their purity", and Angels are completely pure, but there is no end to purity, as God's purity and holiness is absolutely more then all, and so this means Ahlulbayt (a) are to be seen ascending constantly together, blessed together.
Finally, this affirmed in "God and his Angels bless the Prophet" and what we say in Salah, and the verse "Peace be upon the family of Yaseen" which the verses that follow show the reward of Mohammad his family share in it, and it's one peace level they are in, just Harun is with Musa, and Elyas is part of the family of Aaron, and Ishmael and Isaac with Ibrahim, so the level of peace Mohammad (s) is in, his family share.