What is the ruling concerning prayers performed by one who experienced a wet dream and did not perform ghusl, but rather performed tayammum and prayed due to the lack of sufficient water and intense cold, then continued to pray with tayammum due to the unavailability of a place for ghusl until his return home?

If washing with warm water is available without harm, then neglecting it is a dereliction of duty regarding the purity required for the prayer to be valid, based on the Almighty's saying: "And if you are in a state of janabah (definition), then purify yourselves." However, if the use of water by any means causes harm, then the action is valid in that case.