Does what occurred during the engagement, in terms of offer, acceptance, and witnesses, make the girl a wife according to Islamic law?

For a marriage to be valid, five conditions are required: identifying the spouses, their consent, the presence of a guardian (wali) as the Prophet, peace and blessings be upon him, said: "There is no marriage except with a guardian," and witnesses to it, as he said: "There is no marriage except with a guardian and two just witnesses," and the absence of impediments to marriage between the spouses, such as kinship, suckling, and difference in religion (except for a Muslim man marrying a chaste woman of the People of the Book). If these conditions are met, the contract is valid, and reciting Al-Fatiha is not a requirement. The woman has the right to withhold herself until the dower is specified and paid.