Lower bound smallest eigenvalue of matrix product

Question

Let $A$ and $B$ be $n \times n$ invertible matrices. Is it true that

$ \lambda_{\min}( [A \, B]'[A \, B]) ) = \lambda_{\min}( [A'A + B'B + A'B + B'A]) ) > \lambda_{\min}(A'A)? $

Here $\lambda_{\min}(M)$ denotes the smallest eigenvalue of matrix $M.$