if you have an ovary removed, will you go through menopause sooner?

I understand my question doesn't deal directly with breast cancer, but I have found lump in my breast and with one of the risk factors being menopause, I was concerned. 

I had one of my ovaries removed when I was 23.  I am now 36 and seem to be having signs of menopause.  Hot flashes, no period or if I do have one it may be after not having one for 5-6 months and then will be more like spotting. (I have also had my tubes tied so I am definately not pregnant).  I also went through the LUPRON treatment for endometriosis which put me in a state of menopause for 6 months when I was 23.

 I was wondering if the removal of the ovary will mean that I would go through menopause sooner than I would have if I had both my ovaries? I had read that you have all the eggs you will ever have when you are born so it seems logical that if one of those ovaries are removed, your eggs are reduced to half, thus "running out of viable eggs" sooner.  I this true?

Thanks for your time.