> thing we're trying to prove was one of the preconditions
This entirely depends on what you consider to be a definition of the dot product. There are many possible ways this could be set up. (e.g. if you wanted you could develop this whole theory of vector algebra within the system of Euclid’s axioms. Or you could set it up based on explicit coordinates and concrete arithmetic of numbers with no geometrical basis per se. Or ...)
The hard work leading up to this proof is showing that the algebraic definition a·b = 0 corresponds to the usual notion of perpendicularity in Euclidean space. After that, the algebraic proof of the Pythagorean identity is trivial.
This entirely depends on what you consider to be a definition of the dot product. There are many possible ways this could be set up. (e.g. if you wanted you could develop this whole theory of vector algebra within the system of Euclid’s axioms. Or you could set it up based on explicit coordinates and concrete arithmetic of numbers with no geometrical basis per se. Or ...)
The hard work leading up to this proof is showing that the algebraic definition a·b = 0 corresponds to the usual notion of perpendicularity in Euclidean space. After that, the algebraic proof of the Pythagorean identity is trivial.