Is this proof of the product of determinants in tensor notation correct?

Question

I'll start with the matrix C which is the product of the matrices A and B. $$c^i_k = a^i_jb^j_k$$ The determinant of C is $$\frac{1}{3!}\delta_{ijk}^{rst} c^i_rc^j_sc^k_t $$ by the definition of multiplication plugging in and rearranging: $$\frac{1}{3!}\delta_{ijk}^{rst} a^i_lb^l_ra^j_mb^m_sa^k_nb^n_t = \frac{1}{3!}\delta_{ijk}^{rst} a^i_la^j_ma^k_nb^l_rb^m_sb^n_t $$ now the step I'm unsure of is that I know I can turn this into this on its own: $$\delta_{ijk}^{rst} = \frac{1}{3!} \delta_{ijk}^{lmn} \delta_{lmn}^{rst} $$ But can I plug this into the equation and mix the dummy indices like this so I get: $$ \frac{1}{3!}\frac{1}{3!} \delta_{ijk}^{lmn} \delta_{lmn}^{rst} a^i_la^j_ma^k_nb^l_rb^m_sb^n_t $$ which should be the end of the proof since these are the determinants of A and B multiplied together.