1080i vs. 720p

I have a quick question about 1080i vs. 720p. I thought a 1080i signal is 1920 x 1080, meaning it has 1920 pixels of horizontal resolution (1920 vertical scan lines), and since each frame is interlaced and drawn 1/60th of a second, it has 540 pixels of vertical resolution (540 horizontal scan lines). Likewise, a 720p signal has 1280 pixels of horizontal resolution (1280 vertical scan lines) and 720 pixels of vertical resolution (720 horizontal scan lines), so if you convert from 720p to 1080i, you actually gain horizontal resolution, is this correct?

Shawn