Is it true that wudu (ablution), tayammum (dry ablution), and prayer with movements such as prostration existed before Islam in Manichaean and Zoroastrian religions, as some Christians claim?

The Sharia evidence indicates that prayer, ablution, fasting, and Zakat were legislated acts of worship in previous divine laws. This Ummah, however, was uniquely granted tayammum (dry ablution), as indicated by the Prophet's (peace be upon him) saying: "And the entire earth has been made for me and my Ummah a place of prayer and a means of purification." It is not surprising to find prayer or ablution among the Manichaeans, as they could have adopted these practices from the law of Isa (Jesus), peace be upon him. As for the Christians' claim that these practices existed among previous nations to disparage Islam, it is false, because the Prophet (peace be upon him) could neither read nor write.