Isometric surjections on rank one idempotents

:Since the structure of idempotent element sets is more complex than that of projection sets, and idempotent element sets of any rank are unbounded sets, which thus has posed many obstacles to the generalization of Wigner's theorem on idempotent element sets. First, prove the fact that if φ is an isometric mapping on the metaset and there exist two mutually orthogonal rank 1 projections P、Q which make φ(P)、φ(Q) mutually their own rank 1 orthogonal projections, then φ makes that all rank 1 projections are mapped into rank 1 projections. According to the classic Wigner theorem, a specific characterizations can be done for the constraints on the set of rank-1 projections.