The failure of the "proof" is that it depends on how you define "God", which in your defintion is rather narrow (ie has to have infinite powers). Of course our mathematical reality many not be "God's" for one thing.

Math proofs have the advantage of not being cluttered by semantics. Some examples of faulty/coercive logic (that I'm providing as a public service):

https://www.inc.com/geoffrey-james/7-illogical-statements-that-smart-people-never-make.html#:~:text=Classic%20Example%3A%20%22We%20cannot%20let,whether%20it's%20true%20or%20false.

Not saying "God" exists or doesn't, but if "God" is defined broadly (ie the power of creation, the big bang, evolution etc) then "God" exists.