Sure. This is a wonderful Question and Request.
The proof is in these verses:
[Quran 6:38] We did not leave anything out of this Book.
[Quran 7:52] We have given them a scripture that is fully detailed, with knowledge, guidance, and mercy for the people who believe.
We Notice, everything is in the Quran. However, as Quran also says, in verse 3:7, some of its verses are Mutishabihat. Mutishabihat are those verses that are not clear, they include allusions, symbols, Metaphors, and also disconnected letters, because these verses are unclear what exactly they mean. However, according to Quran, those who are well grounded and firm in knowledge, who God has given Them the Knowledge, know its interpretation.
It is obvious, Muhammad and the Imams who God kept them pure and give them its knowlege know its interpretations, better than any ordinary believer, regardless if they are scholars even. No one can claim in Islam, they understand Quran better than Muhammad and those who are pure and well grounded in knowledge.
But as regards to your Question, that why we need to present each Hadith is compatible and originated from Quran.
It is because all details are in Quran. Thus, any True Hadith, must have its origin in the Quran. The only difference is, the Hadith makes those details of Quran more clear. But if a Hadith has not originated from Quran, it is a false Hadith, because as Quran says, it has all details.
Let me demonstrate what is meant by the aforementioned discussion, by the following example:
For example here is various translation of
Quran 17:71,:
M. Pickthall:
On the day when We shall summon all men with their record, whoso is given his book in his right hand - such will read their book and they will not be wronged a shred.
Shakir:
(Remember) the day when We will call every people with their Imam; then whoever is given his book in his right hand, these shall read their book; and they shall not be dealt with a whit unjustly.
Yusuf Ali (Saudi Rev. 1985):
One day We shall call together all human beings with their (respective) Imams: those who are given their record in their right hand will read it (with pleasure), and they will not be dealt with unjustly in the least.
Dr. Laleh Bakhtiar :
On a Day when We will call to every clan with their leader, then, whoever was given his book in his right hand, those will recite their book and they will not be wronged in the least.
So, the various Muslims translators based on their imaginations, provided a different translation.
But if they had referred to the interpretations of Imam Sadiq, it would have been clear what this verse is about:
In Al-kafi, ‘Abdallah ibn Sinan narrated, "I asked abu ‘Abdallah (a.s.) about the meaning of , ‘On the day when We will call every nation with their Imam . . ." (17:71) The Imam (a.s.) said, "It refers to the Imam who is with them and he is Qa’im of the people of that time."
So, according to Imam Sadiq that verse is about Imam Qaim. But now we have to have another evidence from Prophet or Quran to validate the Authenticity of the Hadith of Sadiq. Here is a Hadith from Muhammad, as a witness:
With regards to this verse, it is recorded in the Book
Mizanul Hikmat, In a tradition attributed to Prophet Muhammad, that He said, “He shall call all people to the Imam of their age with Book of God, and tradition of their prophet”
My intention here was to demonstrate that, the verses of Quran must be explained and interpreted by aid of recorded Traditions of Those who were Well Grounded in Knowledge. And those Recorded Traditions, in turn must have originated from the verses of Quran, and must have another witness from Quran and Traditions of Muhammad to make sure they are Authentic Hadithes. The Idea of having 'Witness', also originated from teachings of Quran. There are many verses that emphasizes on Having Witness for validity of different things, and here it is also applicable to determine the validity of Hadithes. So, what better Witness than Quran could there be, to demonstrate authenticity of a Hadith? Obviously, for believers, there is none!
