Maybe this will help clarify. Apple supports Audio and Video CONTROL over bluetooth. That is not the same as pushing video from the iOS device to the external display.
http://support.apple.com/kb/HT3647?viewlocale=en_US&locale=en_US
Bluetooth has a video output profile/protocol called VDP, but I do not know of a single device that supports or uses it. Most likely because the bandwidth capacity for BT is ~3mps. You would be severely limited as to the quality of video output, not to mention audio and control data sharing that same connection. BT is also fairly battery intensive. Most likely you would still loose battery charge (even while charging via plug) while pushing video, audio, and remote control over bluetooth.
Apple supports video out over a number of cable centric output options. The wireless method it has chosen is their patented Airplay via WiFi method. Via Airplay you have high quality audio and video, as well as remote control options with "Home Sharing"
Android supports digital video and audio output via cable options (mostly via a mini HDMI to HDMI cable or other ports). It varies from manufacturer to manufacturer (i.e. Samsung vs HTC.) But some support video output such as DLNA.
The overall take away here is DLNA, Airplay, and other wireless video output protocols are very manufacturer specific. HDMI is universal standard that most if not all manufacturers support in some fashion. Therefore the cables will be around in the accessory market for quite a while except for specific accessory manufacturers that want the Apple or Samsung partnerships.
Lastly, if the mobile device is pushing video and audio, its consuming battery. you are going to want it on a charger otherwise you are screwed. With the Apple devices, charging is the same cable as the video.audio output. 