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It's not correct; how do you justify the step \(\displaystyle (nN\cap nM)m=m(nN\cap nM)\)?

Let \(\displaystyle C=m^{-1}n^{-1}mn\). Then \(\displaystyle C = (m^{-1}n^{-1}m)n \in N\), and \(\displaystyle C=m^{-1}(n^{-1}mn) \in M\). Therefore \(\displaystyle C=1\), i.e. \(\displaystyle mn=nm\).