At the core of the standard cosmological model lies the assumption that the redshift of distant galaxies is independent of photon wavelength. This invariance of cosmological redshift with wavelength is routinely found in all galaxy spectra with a precision of Δz~10^−4. The combined use of approximately half a million high-quality galaxy spectra from the Sloan Digital Sky Survey (SDSS) allows us to explore this invariance down to a nominal precision in redshift of one part per million (statistical). Our analysis is performed over the redshift interval 0.02<z<0.25. We use the centroids of spectral lines over the 3700-6800A rest-frame optical window. We do not find any difference in redshift between the blue and red sides down to a precision of 10^−6 at z<0.1 and 10^−5 at 0.1<z<0.25 (i.e. at least an order of magnitude better than with single galaxy spectra). This is the first time the wavelength-independence of the (1+z) redshift law is confirmed over a wide spectral window at this precision level. This result holds independently of the stellar population of the galaxies and their kinematical properties. This result is also robust against wavelength calibration issues. The limited spectral resolution (R~2000) of the SDSS data combined with the asymmetric wavelength sampling of the spectral features in the observed restframe due to the (1+z) stretching of the lines prevent our methodology to achieve a precision higher than 10^−5, at z>0.1. Future attempts to constrain this law will require high quality galaxy spectra at higher resolution (R>10,000).