A multi-user cognitive (secondary) radio system is considered, where the spatial multiplexing mode of operation is implemented amongst the nodes, under the presence of multiple primary transmissions. The secondary receiver carries out minimum mean-squared error (MMSE) detection to effectively decode the secondary data streams, while it performs spectrum sensing at the remaining signal to capture the presence of primary activity or not. New analytical closed-form expressions regarding some important system measures are obtained, namely, the outage and detection probabilities; the transmission power of the secondary nodes; the probability of unexpected interference at the primary nodes; {\color{blue}and the detection efficiency with the aid of the area under the receive operating characteristics curve}. The realistic scenarios of channel fading time variation and channel estimation errors are encountered for the derived results. Finally, the enclosed numerical results verify the accuracy of the proposed framework, while some useful engineering insights are also revealed, such as the key role of the detection accuracy to the overall performance and the impact of transmission power from the secondary nodes to the primary system.