The evidence on how to perform prayer is after the qur'an the sunnah of the prophet () where he himself said:
... and offer your prayers in the way you saw me offering my prayers ...
(Sahih al-Bukhari)
The evidence for that is that the sahabah() have quoted examples of what the prophet () have been reading/reciting of the qur'an.
You may find some examples in my answer on Why do some Muslim pray Zuhur/Asr silently while Allah in [Al-Quran:17:110] says to pray in a moderate tone?.
Here are some:
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Narrated Jubair bin Mut 'im:
My father said, "I heard Allah's Messenger (ﷺ) reciting "at-Tur" (52) in the Maghrib prayer."
(Sahih al-Bukhari and Sahih Muslim and elsewhere)
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Narrated Al-Bara:
The Prophet (ﷺ) was on a journey and recited in one of the first two rak'at of the 'Isha' prayer "Wa t-teeni wa z-zaitun." (95)
(Sahih al-Bukhari and Sahih Muslim and elsewhere)
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Narrated Ibn 'Abbas:
The Prophet (ﷺ) set out with the intention of going to Suq 'Ukaz (market of 'Ukaz) along with some of his companions. At the same time, a barrier was put between the devils and the news of heaven. Fire commenced to be thrown at them. The Devils went to their people, who asked them, "What is wrong with you?" They said, "A barrier has been placed between us and the news of heaven. And fire has been thrown at us." They said, "The thing which has put a barrier between you and the news of heaven must be something which has happened recently. Go eastward and westward and see what has put a barrier between you and the news of heaven." Those who went towards Tuhama came across the Prophet at a place called Nakhla and it was on the way to Suq 'Ukaz and the Prophet (ﷺ) was offering the Fajr prayer with his companions. When they heard the Qur'an they listened to it and said, "By Allah, this is the thing which has put a barrier between us and the news of heaven." They went to their people and said, "O our people; verily we have heard a wonderful recital (Qur'an) which shows the true path; we believed in it and would not ascribe partners to our Lord." Allah revealed the following verses to his Prophet (Sura 'Jinn') (72): "Say: It has been revealed to me." And what was revealed to him was the conversation of the Jinns.
(Sahih al-Bukhari and Sahih Muslim and elsewhere)
One can also find in the sunnah the doing of the sahabah:
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Yahya related to me from Malik from Hisham ibn Urwa from his father that Abu Bakr as-Siddiq prayed subh and recited surat al-Baqara in the two rak'as.
(Al-Muwatta', subh is a synonym to fajr)
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Narrated Abu Huraira:
The Qur'an is recited in every prayer and in those prayers in which Allah's Messenger (ﷺ) recited aloud for us, we recite aloud in the same prayers for you; and the prayers in which the Prophet (ﷺ) recited quietly, we recite quietly. If you recite "Al-Fatiha" only it is sufficient but if you recite something else in addition, it is better.
(Sahih al-Bukhari and Sahih Muslim and elsewhere -with slight differences in the wording)
On the other hand, the silent prayers were identified in other ahadith:
Narrated Abu Ma'mar:
We asked Khabbab whether Allah's Messenger (ﷺ) used to recite (the Qur'an) in the Zuhr and the 'Asr prayers. He replied in the affirmative. We said, "How did you come to know about it?" He said, "By the movement of his beard."
(Sahih al-Bukhari)
You may also read in fiqh books like in imam an-Nawawi's al-Majmo':
(In the following I'm translating from Arabic with the help of google translate, as these translations are mainly of my own take them with the necessary care!)
فالسنة الجهر في ركعتي الصبح والمغرب والعشاء وفي صلاة الجمعة والإسرار في الظهر والعصر وثالثة المغرب والثالة والرابعة من العشاء وهذا كله بإجماع المسلمين مع الأحاديث الصحيحة المتظاهرة على ذلك
The Sunnah is to recite loudly (with audible voice) in the fajr, Maghrib, and 'Isha's rak’ahs, and in the Friday prayer, and recite silently in the dhohr and 'asr prayers, and the third rak’ah of Maghrib, and the third and fourth rak’ahs of the 'Ishaa' prayer, and all of this is according to the consensus of the Muslims, with the authentic hadiths that demonstrate this.