Some of that stuff may be partially true for irrational numbers, but most would fall in the set of non-real(imaginary) numbers where you can divide by zero and have numbers like sqrt(-1); we haven't really discovered enough about that set to make practical use of them. Also, some infinities are larger than others and you can't treat them like variables and do operations on them.
Here is the problem- you really can't try to make proofs like this without defining the domain. Also, every line where an operation is performed to produce an equivalent statement should be justified with an established law. Some of them might seem obvious (most can be done with basic properties of arithmetic), but it would stop most bullshit proofs after 2 or 3 lines and get you really good grades in discrete math and most other math classes as well.