Is there an adapter to allow video output from lightning to 30 pin adapter?

The treadmills at the gym I go to have cable tv with a 30 pin cable for hooking up a phone with audio and video outputting to the treadmill tv. I have an IPhone 5 so need an adapter that will output video in addition to audio. Any idea if this exists?

Lightning to 30-pin Adapter

Lightning to 30-pin Adapter

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