let n be Nat; :: thesis: for R being Ring
for M1, M2 being Matrix of n,R st M1 is invertible & M2 is invertible holds
( M1 * M2 is invertible & (M1 * M2) ~ = (M2 ~) * (M1 ~) )
let R be Ring; :: thesis: for M1, M2 being Matrix of n,R st M1 is invertible & M2 is invertible holds
( M1 * M2 is invertible & (M1 * M2) ~ = (M2 ~) * (M1 ~) )
let M1, M2 be Matrix of n,R; :: thesis: ( M1 is invertible & M2 is invertible implies ( M1 * M2 is invertible & (M1 * M2) ~ = (M2 ~) * (M1 ~) ) )
assume that
A1: M1 is invertible and
A2: M2 is invertible ; :: thesis: ( M1 * M2 is invertible & (M1 * M2) ~ = (M2 ~) * (M1 ~) )
A3: M2 ~ is_reverse_of M2 by A2, Def4;
A4: M1 ~ is_reverse_of M1 by A1, Def4;
A5: len (M2 ~) = n by MATRIX_0:24;
A6: width (M1 ~) = n by MATRIX_0:24;
A7: len M1 = n by MATRIX_0:24;
A8: width M2 = n by MATRIX_0:24;
A9: ( width M1 = n & len M2 = n ) by MATRIX_0:24;
A10: ( width (M2 ~) = n & len (M1 ~) = n ) by MATRIX_0:24;
width (M1 * M2) = n by MATRIX_0:24;
then A11: (M1 * M2) * ((M2 ~) * (M1 ~)) = ((M1 * M2) * (M2 ~)) * (M1 ~) by A5, A10, MATRIX_3:33
.= (M1 * (M2 * (M2 ~))) * (M1 ~) by A9, A8, A5, MATRIX_3:33
.= (M1 * (1. (R,n))) * (M1 ~) by A3
.= M1 * (M1 ~) by MATRIX_3:19
.= 1. (R,n) by A4 ;
width ((M2 ~) * (M1 ~)) = n by MATRIX_0:24;
then ((M2 ~) * (M1 ~)) * (M1 * M2) = (((M2 ~) * (M1 ~)) * M1) * M2 by A9, A7, MATRIX_3:33
.= ((M2 ~) * ((M1 ~) * M1)) * M2 by A7, A6, A10, MATRIX_3:33
.= ((M2 ~) * (1. (R,n))) * M2 by A4
.= (M2 ~) * M2 by MATRIX_3:19
.= 1. (R,n) by A3 ;
then A12: (M2 ~) * (M1 ~) is_reverse_of M1 * M2 by A11;
then M1 * M2 is invertible ;
hence ( M1 * M2 is invertible & (M1 * M2) ~ = (M2 ~) * (M1 ~) ) by A12, Def4; :: thesis: verum