It's YOUR wedding so you and your partner choose who to invite.
If your Dad chooses not to come if his wife isn't there, that is his choice.
I was confused by the original post but the follow up helped me understand what's going on.
I can assume your Dad and his "mistress" (which does rather make it sound like she's some evil woman who lured him away when we all know it takes two to tango) will both be hurt and possibly angry about the snub but I get the understandable impression that it's more important to you that your mum is comfortable and content at your wedding.
You could just post out invitations. You could include a letter with your father's invitation which explains your decision and what your priorities are. e.g.
"Dad, you will notice that your invitation is just for you and that Valerie is not included in the invitation. The main reason for this is that it's of the utmost importance to me that my mum is present at my wedding day and I want her to feel comfortable. I think Valerie's presence will be upsetting to her and a distraction for the whole day. I understand that you won't be happy with this decision and you are welcome to choose to come to your child's wedding or stay at home with your wife.
"As I stand on the precipice of my own marriage, I am embarking on a life long commitment to Sam, the person that I love with all my heart and want to spend the rest of my life with. I want Sam to be embraced and welcomed into our family. I am still incredibly hurt by your choices and actions with regards to your affair with Valerie and the devastating impact that your betrayal had on my mum, my family, and my sense of trust and stability. To have her present would do a disservice to the sanctity of marriage and taint the vibe of positivity, love and celebration that I want my wedding to have.
"If you feel that you cannot bring yourself in your role as my Dad to be present on your own, I don't want you to be there. I love you and want you to be there but please understand that if you are to prioritise Valerie's happiness over mine on what should be one of the most important days of my life, you will do further irreparable damage to our relationship and I don't want that drama.
"You are an adult and it is your choice to make. Sam and I are paying for the wedding day and we are inviting the people that we love and trust to share this special day with us. I hope that you can make it."
Or something like that. I don't know.