Is the prayer valid when one passes over and sits on a mat that was soiled by a shoe with which one entered the bathroom?

The default status of things is purity, including the bathroom floor, unless you are certain of its impurity. If you doubt the impurity of something, do not rule it impure, because doubt does not render a pure object impure. So, if a person walks across a rug with their shoe, the rug is not judged impure unless you are certain that the shoe is impure and wet, in which case only the affected spot becomes impure.

In cases of doubt regarding the transfer of impurity, the default is that it has not transferred. Some scholars have stated that if a person's foot becomes wet and then steps on a dry impure spot, and then steps on a pure spot, it becomes pure by the second step. This opinion is suitable for those afflicted with waswas (definition) (obsessive whispering/doubts).

Shaykh al-Islam Ibn Taymiyyah said: "The default status of the bathroom floor is purity. Any impurity that falls on it, such as urine, is poured over with water to remove it, and it is in a much better state than roads. The default status of roads is purity. Just as one is certain that impurity will inevitably fall on its ground, so too is one certain that water will encompass whatever impurity falls on it. Even if one does not know this, one should not definitively judge a specific spot as impure unless one knows for certain that impurity has occurred on it."