Although it is some years since the discovery of high-T_c cuprate superconductors, the nature and doping dependence of the condensate in such materials is still uncertain. There is no consensus as to whether the supercurrent is carried by a macroscopic quantum condensate of electron pairs or hole pairs. Here the authors show that the experimentally measured muon spin relaxation rate, which is proportional to the condensate density in high-T_c cuprate superconductors, closely follows the upper bound given by C.N. Yang's theorem for condensation within a single band. This is significant because Yang's well known exposition of superconductivity as off-diagonal long-range order is very widely accepted. The authors infer that the condensate transports negative charge in electron-doped and hole-doped cuprate layers, which rationalizes the experimental observation that the London moments of conventional superconducting metals and superconducting ceramics have the same sign. The models that reproduce this behaviour and give a short coherence length involve real-space pairing on a lattice, possibly arising from repulsive correlations or some other pairing mechanisms.