<p style='text-indent:20px;'>We consider an attraction-repulsion chemotaxis model coupled with the Navier-Stokes system. This model describes the interaction between a type of cells (e.g., bacteria), which proliferate following a logistic law, and two chemical signals produced by the cells themselves that degraded at a constant rate. Also, it is considered that the chemoattractant is consumed with a rate proportional to the amount of organisms. The cells and chemical substances are transported by a viscous incompressible fluid under the influence of a force due to the aggregation of cells. We prove the existence of global mild solutions in bounded domains of <inline-formula><tex-math id="M1">\begin{document}$\mathbb{R}^N,$\end{document}</tex-math></inline-formula> <inline-formula><tex-math id="M2">\begin{document}$N = 2, 3,$\end{document}</tex-math></inline-formula> for small initial data in <inline-formula><tex-math id="M3">\begin{document}$L^p$\end{document}</tex-math></inline-formula>-spaces.