How did antonio begin to thing himself duke of milan

Prospero used to be the Duke of Milan until his brother, Antonio, betrayed him and stole the dukedom. The king was in on the whole usurping business. Prospero was not on the island then during all the usurping, he was in his library studying magic. Prospero and his daughter Miranda are sailed out to some island where Prosper has been waiting for revenge.

what did antonio do after depriving prospero of his dukedom???