The imamate of the Ahlulbait (peace be upon them) can be proven from the holy Quran by a number of ways.
Amongst them:
- "Your guardian can be only Allah, and His messenger and those who believe, who establish worship and pay the poor due while bowing down (in prayer)." (Quran 5:55) This verse was revealed at the incident where Imam Ali (peace be upon him) gave his ring as charity while he was bowing in prayer. The verse joined his guardianship to the guardianship of the messenger of Allah, which is the leadership of the Muslim nation, i.e. the Imamate.
- "Nor does he speak of (his own) desire" (Quran 53:3). The verse indicates that what the Prophet says is from Almighty Allah. The orders of the Prophet were to follow his household when he said in this commonly known narration: “I leave among you the two weighty things: the book of Allah and my progeny; as long as you are attached to them both, you will never be misguided." The Prophet also ordered us to follow Imam Ali in the famous narration of Ghadeer: "Whosoever’s master I am, Ali is his master." In another narration, the Prophet said: "Ali is your master after my demise," and many more narrations.

This question presents important issues that require an explanation, since this is one of the most important fallacies that the enemies of the Ahlulbait (peace be upon them) present.
We would like to draw your attention to a few important issues in order to clarify the truth:
Firstly: Doubting the Imamate of the Ahlulbait just because it is not mentioned in the Holy Book is in reality doubting the truthfulness of the Messenger of Allah (peace be upon him and his progeny) and belittling his position in being an essential source of religious laws and a speaker of the truth, which in turn raises a big question on the credibility of Islam as a religion and the Holy book as a divine text that “falsehood cannot come at it from before it or from behind it,” as the Quran states clearly. But rather, the Quran itself presents the Messenger of Allah and his conduct as the truth that is not associated with any falsehood, misguidance and forgetfulness. This is explicitly mentioned in the Holy Quran, when the Almighty stated, “Nor does he speak of (his own) desire. It is naught save an inspiration that is inspired,” (Quran 53:3-4) and “whatsoever the Messenger gives you, take it, and whatsoever he forbids, abstain (from it),” (Quran 7:59) and also, “But nay, by thy Lord, they will not have (true) faith until they make thee judge of what is in dispute between them and find within themselves no dislike of that which thou decides, and submit with full submission.”(Quran 65:4)
Based on this, a Muslim should research about Imamate as well as the position of the Sunnah in its regards, just as he looks for the position of the Sunnah with respect to the religious obligations like the prayers, the fast, pilgrimage and the like, as these essential obligations are not mentioned in the Holy Book with detail, but the Sunnah explained the ways to perform them.
It is sufficient to prove the Imamate by referring to the trusted books of narrations of the Sunnis, like the book of al-Bukhari and the book of Muslim and all the authenticated books in which they included narrations of the Prophet (peace be upon him and his progeny) that refer in a clear way to the leadership of the Ahlulbait (peace be upon them) after the Prophet, like when he said: “Whosoever’s master I am, Ali is his master.” Also: “Ali is your wali (leader) after me.” And also, when he said to Ali (peace be upon him): “You are to me like Aaron was to Moses, but there is no prophet after me,” and the like in innumerable narrations. There are other narrations that relate to the religious authority of the Ahlubait (peace be upon them) in general, like: “I am leaving among you two weighty things: the Book of Allah and my household; as long as you are attached to them both, you shall never be misguided.”
Secondly: there are verses of the Holy Quran which, as the narrations indicate, were revealed in respect of Imam Ali (peace be upon him). They are narrated by Sunni Muslims. They also indicate in a clear way the Imamate of Imam Ali (peace be upon them) like when Almighty Allah said, “Your guardian can be only Allah, and His messenger and those who believe, who establish worship and pay the alms, while they bow down (in prayer).” (Quran 55:5) And even the verse, “Obey Allah, and obey the messenger and those of you with authority.” (Quran 59:4) They indicate in a clear way that those in authority whose obedience is necessary are infallible since – based on the verse – one should obey them as one obeys Almighty Allah and likewise the Messenger of Allah, and nobody fulfills this condition except the Imams of the Ahlulbait, whom Almighty Allah purified with a thorough purification as per the verse 33:33. Furthermore, Muslims have unanimously agreed on the righteousness of the Imams of the Ahlulbait, their highness, the strength in their faith, their knowledge in the affairs of religion, as this clearly appears to the individual who looks at their history and biography. Could any Muslim accept that Almighty Allah would order Muslims to obey those in authority by applying this title on anyone that assumes power, whether he does this by force or through inheritance, even if he committed the most heinous of crimes like Yazid bin Mu’awiyah and similar despots? Are the aforementioned verses not sufficient in proving the Imamate of the Ahlulbait from the Quran for the fair-minded searcher for the truth?
Thirdly: What you have mentioned – that if the Holy Quran had named Imam Ali (peace be upon him) clearly as the successor to the Prophet (peace be upon him and his progeny) there would be no conflict between Muslims – is not correct. There are a number of verses that clearly describe certain rulings, and, despite that, Muslims did not agree on them; an example of such a case is the verse of ablution in which the way of performing ablution has been clearly explained, and there is no doubt that the verse is clear in its meaning when it was revealed, and the Messenger of Allah (peace be upon him and his progeny) used to perform wudhu repeatedly in the presence of Muslims. Despite all this, there was a difference in opinion in understanding it. Another example is the verses of divorce which indicate that the three consecutive divorces pronounced in one place and time is not considered as three divorces – by which a man cannot return to the wife – but some Muslim jurists regard this type of divorce as valid and a man cannot return to his wife thereafter. There are many other such examples in jurisprudential issues, although all Muslims base their conclusions on the Holy Book. Furthermore, if one looks in the biography of the Prophet, he could clearly conclude that some of his companions were not obeying him or even respecting him properly. They used to dare to disobey him and not care about his orders; it is sufficient to mention some incidents from the trusted books of narrations of the Sunnis:
(1) Most of the Muslims fled from a number of battles that were led by the Messenger of Allah, like Uhud, Hunayn and Khaybar, whereas the Holy Book states clearly the prohibition to do so and that whoever does so will be punished in Hell for eternity. Almighty Allah said: “Whoso on that day turneth his back to them, unless manoeuvring for battle or intent to join a company, he truly hath incurred wrath from Allah, and his habitation will be hell, a hapless journey's end.”(Quran 16:8)
(2) What took place during the “tragedy of Thursday”: Abdullah bin Abbas – as it is narrated in the Sunni books of narrations like the Musnad of Ahmad – said, weeping, “The day of Thursday! And what day was Thursday!! The Messenger of Allah said to us: “Bring to me a broad bone [to be used to write on] and ink, so I may write for you something that you shall not be misguided after it is written.” Umar said: “The Messenger of Allah is incoherent.” In some of the narrations, they were arguing; some of them were in favor of Umar and some others were saying, “Bring for the Messenger of Allah what he asked for.” The Messenger of Allah said then: “Leave me!! There should not be a fight in my presence.”
From the said incident one can conclude that the refusal of the Prophet to write what he wanted to write was to preserve the dignity of Prophethood, since it is not appropriate for the Prophet to order for something in a clear and direct way and Muslim differ in their opinions whether to obey him or not, since there should be no dispute after the order has been issued.
From this, one can conclude the reason why the name of Imam Ali (peace be upon him) was not mentioned in the Holy Book; Almighty Allah knew that there were some people who wanted to rule and were ready to disobey Allah and His Messenger in an open way, and they had their supporters who were willing to help and encourage them. If the Holy Quran was to mention Imam Ali’s name, then they would doubt the Holy Book in the same way as they have doubted the Messenger of Allah. This would result in the rejection or alteration of the Holy Book.

Some declarations of Imam Ali (peace be upon him) have been reported in the Nahj al-Balaghah in which he presents his view about the Imamate after the Holy Prophet (peace be upon him and his progeny).
For example, he said in sermon number 144, “Where are those who falsely and unjustly claimed that they are deeply versed in knowledge, as against us, although Allah raised us in position and kept them down, bestowed upon us knowledge but deprived them, and entered us (in the fortress of knowledge) but kept them out. With us guidance is to be sought and blindness (of misguidance) is to be changed into brightness. Surely the Imams (divine leaders) will be from the Quraysh. They have been planted in this line through Hashim. It would not suit others, nor would others be suitable as heads of affairs.”
He also said, in sermon number 3, “Beware! By Allah, the son of Abu Quhafah (Abu Bakr) dressed himself with it (the caliphate) and he certainly knew that my position in relation to it was the same as the position of the axis in relation to the hand-mill. The flood water flows down from me and the bird cannot fly upto me…”.

Almighty Allah appointed the Imams to guard the religion and protect it from misguidance and to continue the message of Islam that started with the prophethood of Prophet Mohammad (peace be upon him and his household) for mankind. People did not obey Allah’s orders in this matter due to their desire to possess authority and to be in control. This led to tragedies amongst Muslims for centuries until they separated into different sects and were controlled by the infidels. Almighty Allah said: “And if the people of the townships had believed and kept from evil, surely we should have opened for them blessings from the sky and from the earth. But (unto every messenger) they gave the lie, and so we seized them on account of what they used to earn.” (7:96)

The belief in the imamate is completion of the faith and Islam. many narrations clearly stated that deeds will not be accepted without the belief in the imamate.

We can comment on your question by stating the following:
1. The Imamate of the Imams (peace be upon them) has been legislated by Almighty Allah for people to observe. Every Muslim must obey all that has been proven to be divinely legislated obligations without considering the immediate benefit of the law on his everyday life since believing in a religion is believing in all what is proven to be part of it.
2. Believing in the Wilayat of the Ahlulbait (peace be upon them) is following them and observing their teachings. This has a great effect on the way a believer deals with day-to-day matters based on the teachings and the guidelines that originate from them. We can see that amongst the characteristics of the followers of the Ahlulbait (peace be upon them) by which they are known today is their condemnation of terrorism and tyranny against innocents, especially children, women and old people, and this is not but a reflection of their attachment with the humane teachings of the Ahlulbait (peace be upon them).

What you said is true. Imamate is a divine position that cannot be taken by anyone. The application of this right is represented by the management of the affairs of people with justice according to the divine rules of Almighty Allah, and since Imam Ali (peace be upon him) was not able to exercise this right of authority for some time, we can say that the ruling and management of the affairs of people based on correct religious standards was taken from him.

The omitting of the third testimony during the Adhan does not affect one’s faith, but it is a manifestation of it.

Anybody who submits to the religion of Islam with the two testimonies is a Muslim, provided there is no animosity against any of the Imams (peace be upon them). Iman depends on believing in the Wilayah of the Ahlulbait (peace be upon them).

There are several narrations which indicate that the prophets (peace be upon them) were interceding to Almighty Allah by the five Infallibles (the Final Prophet, Ali, Fatimah, Hasan and Hussain, peace be upon them all). Other texts indicate that their Noor preceded the creation of everything else in glorifying the Almighty Allah. This is an indication of their superiority over the rest of the creation.

This is a big fabrication and cannot be accepted. There is no doubt in the narrations of the Ahlulbait that the Prophet (peace be upon him and his progeny) is better than all mankind and the superiority of the Ahlulbait over mankind is based on their relation to him and being the closest ever to him in their action and faith. It is enough to refer to this narration that Al-Kulayni reported to clear this fallacy:
“A man asked Imam Ali (peace be upon him): Since when has your Lord existed? Imam replied: "When was He ever absent so one could ask: since when did He exist?". The man then asked the Imam: “O Commander of the Believers, are you a prophet?” Imam replied by saying: "Woe to you, I am only one of Mohammad’s slaves."

Shia Muslims prove the infallibility of their Imams (peace be upon them) in various ways; some are logical, and others are from the Holy Quran and the traditions of the Prophet (peace be upon him and his holy progeny). The books of beliefs state these proofs clearly.
Many narrations from Sunni and Shia books of traditions mention that the Imams are twelve, without mentioning their names. Many narrations in the Shia books of traditions mention the names of the Imams (peace be upon them).

There is no dispute amongst the followers of the Ahlul-bait (peace be upon them) that the Prophets and the Imams are sinless, in such a way that they do not commit what opposes the religion. Our scholars have explained in their books of beliefs the evidences to prove this concept. There is no doubt that the Prophet and the Imam are appointed by Almighty Allah. He has ordered people to follow them. So, if it is possible for them to commit sins, they might order people to do so as well and people have to follow them. So their orders would not be trusted to be from Almighty Allah, and He will not order us to follow somebody who can misguide us. Similarly, they are to be taken by their followers as role models. If they were to commit sins, others will be encouraged to do the same.

There is no doubt that the followers of the Ahlulbait (peace be upon them) base the belief in the infallibility of the Imams (peace be upon them) on solid sources, starting from verse 33:33. Other texts indicate this directly and sometimes indirectly. As an example, in Ziyarat al-Jami’ah, which was narrated from Imam Ali Al-Hadi (peace be upon him) with an authentic chain: "Allah has guarded you from lapses and saved you from attractions (of life, Satan, etc)."

The said letter is not different from the rest of the letters found in Nahjul Balagha. The doubt about the authenticity of the letter may come from the fact that this letter includes advises directed to Imam Hassan (peace be upon him) which raises the question about the need of the young Imam to such directions although he is infallible.
Apparently, the said doubt is not valid since this way of presenting advice could be for the sake of reminding the person and repeatedly giving him the advice. The Quran used this way when Almighty Allah said: “And warn, for warning profits believers.” (51:55)
Also, it could be indirect guidance to all Muslims by directing the advice to Imam Hassan (peace be upon him). This way is also mentioned in the Holy book as well. Almighty Allah said: “And cry not unto any other god along with Allah. There is no Allah save Him. Everything will perish save His countenance. His is the command, and unto Him ye will be brought back.” (28:88) In this verse God is addressing the Prophet with guidance, while it is abundantly clear that the Prophet would not worship anyone but Almighty Allah.

The fourteen infallibles are proven to be infallible by a concrete evidence. Some of the children of the imams might have reached a high level of closeness to almighty Allah in such a level that is close to the infallibility.

This belief is invalid, and it is not of the Shia scholars. Shia religious scholars have agreed that the Imams (peace be upon them) are created slaves of Almighty Allah. He created them like He created everybody else. The religious narrations of the Imams themselves confirm this truth. Yes, there is no doubt that the Imams showed miracles, like the prophets (peace be upon them). Almighty Allah said with respect to Jesus (peace be upon him): “I come unto you with a sign from your Lord. Lo! I fashion for you out of clay the likeness of a bird, and I breathe into it and it is a bird, by Allah’s leave. I heal him who was born blind, and the leper, and I raise the dead, by Allah’s leave. And I announce unto you what ye eat and what ye store up in your houses.” (3:49)

The Imams of the Ahlulbait (peace be upon them) are the guardians after the Prophet (peace be upon him), and they have a very high position with Allah, which enables them to perform certain things that normal people do not. They often use this divinely-bequeathed ability to prove their truthfulness.
The Holy Quran states that Prophet Sulayman (peace be upon him) desired to bring the throne of Queen Bilqees to him. Almighty Allah said: "The one with whom was knowledge of the Scripture said: I will bring it thee before thy gaze returneth unto thee. And when he saw it set in his presence, (Solomon) said: This is of the bounty of my Lord..."
This verse clearly indicates that the vicegerent of Prophet Sulayman (Asif bin Barkhiya) brought the throne from Yemen to Jerusalem in a blink of an eye. We believe that the Guardians after our Prophet are more superior in rank then those of other Prophets (peace be upon them all).
There are many narrations that report the various incidents that reflect these miracles.

This phrase refers to seeking intercession of Imam Ali (peace be upon him) to Almighty Allah. Intercession is one of the Islamic concepts founded upon verses of the Holy Book and narrations of the Prophet and the Imams (peace be upon them all).

There is nothing wrong with seeking the mercy directly from Almighty Allah, nor with seeking the intercession of Imam Ali (peace be upon him), since the matter entirely is in hand of Allah. Seeking the intercession of Imam Ali means asking Almighty Allah by the intercession of Imam Ali, so the chances of one’s wishes being answered is greater than directly asking Almighty Allah.