Is prayer valid after a bath taken for refreshment or other purposes, without performing ablution, if water has covered the entire body with the intention of purification?

Intention (niyyah) is a condition for the validity of ablution (wudu'). If the ritual bath (ghusl) is obligatory (like ghusl from janabah), then the ghusl suffices for wudu', and wudu' is not required after it. If the one performing ghusl intends by it the legal purification (tahara shar'iyyah) that removes the state of ritual impurity (hadath) and permits prayer, then his prayer is valid. However, if his intention was merely to cool down or remove dirt, then neither his wudu' nor his prayer is valid. If the ghusl is not obligatory (such as for cooling down), then the intention for wudu' is necessary for the prayer of the one who performed ghusl to be valid.