The chemical composition of Earth's lower mantle can be constrained by combining seismological observations with mineral physics elasticity measurements. However, the lack of laboratory data for Earth's most abundant mineral, (Mg,Fe,Al)(Al,Fe,Si)O3 bridgmanite (also known as silicate perovskite), has hampered any conclusive result. Here we report single-crystal elasticity data on (Al,Fe)-bearing bridgmanite (Mg0.9Fe0.1Si0.9Al0.1)O3 measured using high-pressure Brillouin spectroscopy and X-ray diffraction. Our measurements show that the elastic behaviour of (Al,Fe)-bearing bridgmanite is markedly different from the behaviour of the MgSiO3 endmember. We use our data to model seismic wave velocities in the top portion of the lower mantle, assuming a pyrolitic mantle composition and accounting for depth-dependent changes in iron partitioning between bridgmanite and ferropericlase. We find excellent agreement between our mineral physics predictions and the seismic Preliminary Reference Earth Model down to at least 1,200 kilometres depth, indicating chemical homogeneity of the upper and shallow lower mantle. A high Fe3+/Fe2+ ratio of about two in shallow-lower-mantle bridgmanite is required to match seismic data, implying the presence of metallic iron in an isochemical mantle. Our calculated velocities are in increasingly poor agreement with those of the lower mantle at depths greater than 1,200 kilometres, indicating either a change in bridgmanite cation ordering or a decrease in the ferric iron content of the lower mantle.