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I understand your iPad mini shows that it is connected, but you aren't able to access the internet with it. The resource below provides some troubleshooting steps. One of the most effective is to turn off the iPad mini, unplug the power to your modem and/or router, wait a couple of minutes, then plug the power cord back into the modem/router. Restart your iPad mini to refresh the connection to the internet.
If your iPhone, iPad, or iPod touch won’t connect to a Wi-Fi network