Mathematical proof for order of operations

There is no such proof. The order of operations to which we are accustomed is really nothing more than a mathematical convention to which most adhere in order to help eliminate the alternative of ambiguity.

But it never hurts to use parentheses to designate operations to perform first (inner to outer), which is virtually universally understood, thus eliminating our reliance on convention in the hopes that others will know the convention!

The order of operation is defined, but not proved.

There is no such proof, but maybe one can find good motivations. The precedence of multiplication with respect to addition is probably a consequence of natural language:

3 cats and 2 dogs

is something like $3\times c + 2\times d$. This has to do with the fact that multiplication is sometimes expressed by adjectives in the natural language, while summation is expressed by a conjunction.