Proof of Theorem ICBM

The linear transformation $\ltdefn{I_V}{V}{V}$ is invertible, and its inverse is itself, $I_V$ (check this!). So by Theorem IMR, the matrix $\matrixrep{I_V}{B}{C}=\cbm{B}{C}$ is invertible. Theorem NI says an invertible matrix is nonsingular.

Then \begin{align*} \inverse{\cbm{B}{C}} &=\inverse{\left(\matrixrep{I_V}{B}{C}\right)}&&\knowl{./knowls/definition.CBM.knowl}{\text{Definition CBM}}\\ &=\matrixrep{\ltinverse{I_V}}{C}{B}&&\knowl{./knowls/theorem.IMR.knowl}{\text{Theorem IMR}}\\ &=\matrixrep{I_V}{C}{B}&&\knowl{./knowls/definition.IDLT.knowl}{\text{Definition IDLT}}\\ &=\cbm{C}{B}&&\knowl{./knowls/definition.CBM.knowl}{\text{Definition CBM}}\\ \end{align*}