Is it not logically wrong to make proofs in one field of mathematics, by using tools belonging to another one, since they start from different axioms?

It is as long as one does not prove the existence of an isomorphism (which between algebra and geometry is provable).

If one postulates the continuity of real numbers and the continuity of points on a line, it can be shown that there is a one-to-one correspondence between numbers and points and that this correspondence preserves distances and proportions.

If one postulates parallelism between the lines of two systems, one can construct a flat grid. And it can be shown that there is a similar correspondence between the points in the plane and those in the grid. That, again, preserves vector distances and geometric similarities.

We can continue in this way, with an arbitrary number of dimensions.

It is the existence of such correspondences that means that whatever happens in one domain has -in the codomain- a corresponding image and vice versa.

Algebra and Geometry are not independent, but are two “languages” that provide two different representations of one and the same thing: the concept of “relationship between entities” of “set theory.”

Like all languages, however, although they have equal “semantic power” (they cover the same problem space) they have different syntactic capacity: there are problems in which the algebraic formulation appears simpler than the geometric one (which can always be shown to exist) and problems for which the opposite is true.

This is the reason why demonstrations done in a different domain are often used to infer properties of the source domain.