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The video portion of HDMI is simply DVI, down to using nearly every DVI pin in some position.

Almost, but not quite. There are minor differences that can mostly be explained by saying that HDMI is a newer standard and they're evolving things. Among other things, HDMI can transmit YCrCb data, while DVI is RGB-only. Also, HDMI implies HDCP, while DVI does not necessarily imply HDCP.