let n be Nat; :: thesis: for K being Field

for M1, M2 being Matrix of n,K st M1 is Orthogonal & M1 commutes_with M2 holds

M1 @ commutes_with M2

let K be Field; :: thesis: for M1, M2 being Matrix of n,K st M1 is Orthogonal & M1 commutes_with M2 holds

M1 @ commutes_with M2

let M1, M2 be Matrix of n,K; :: thesis: ( M1 is Orthogonal & M1 commutes_with M2 implies M1 @ commutes_with M2 )

set M3 = M1 @ ;

assume that

A1: M1 is Orthogonal and

A2: M1 commutes_with M2 ; :: thesis: M1 @ commutes_with M2

M1 is invertible by A1;

then A3: M1 ~ is_reverse_of M1 by Def4;

A4: width M2 = n by MATRIX_0:24;

A5: width M1 = n by MATRIX_0:24;

A6: ( len M2 = n & width (M1 ~) = n ) by MATRIX_0:24;

A7: ( len (M1 ~) = n & width ((M1 ~) * M2) = n ) by MATRIX_0:24;

A8: len M1 = n by MATRIX_0:24;

M2 * (M1 @) = ((1. (K,n)) * M2) * (M1 @) by MATRIX_3:18

.= (((M1 ~) * M1) * M2) * (M1 @) by A3

.= ((M1 ~) * (M1 * M2)) * (M1 @) by A5, A8, A6, MATRIX_3:33

.= ((M1 ~) * (M2 * M1)) * (M1 @) by A2

.= ((M1 ~) * (M2 * M1)) * (M1 ~) by A1

.= (((M1 ~) * M2) * M1) * (M1 ~) by A4, A8, A6, MATRIX_3:33

.= ((M1 ~) * M2) * (M1 * (M1 ~)) by A5, A8, A7, MATRIX_3:33

.= ((M1 ~) * M2) * (1. (K,n)) by A3

.= (M1 ~) * M2 by MATRIX_3:19

.= (M1 @) * M2 by A1 ;

hence M1 @ commutes_with M2 ; :: thesis: verum

for M1, M2 being Matrix of n,K st M1 is Orthogonal & M1 commutes_with M2 holds

M1 @ commutes_with M2

let K be Field; :: thesis: for M1, M2 being Matrix of n,K st M1 is Orthogonal & M1 commutes_with M2 holds

M1 @ commutes_with M2

let M1, M2 be Matrix of n,K; :: thesis: ( M1 is Orthogonal & M1 commutes_with M2 implies M1 @ commutes_with M2 )

set M3 = M1 @ ;

assume that

A1: M1 is Orthogonal and

A2: M1 commutes_with M2 ; :: thesis: M1 @ commutes_with M2

M1 is invertible by A1;

then A3: M1 ~ is_reverse_of M1 by Def4;

A4: width M2 = n by MATRIX_0:24;

A5: width M1 = n by MATRIX_0:24;

A6: ( len M2 = n & width (M1 ~) = n ) by MATRIX_0:24;

A7: ( len (M1 ~) = n & width ((M1 ~) * M2) = n ) by MATRIX_0:24;

A8: len M1 = n by MATRIX_0:24;

M2 * (M1 @) = ((1. (K,n)) * M2) * (M1 @) by MATRIX_3:18

.= (((M1 ~) * M1) * M2) * (M1 @) by A3

.= ((M1 ~) * (M1 * M2)) * (M1 @) by A5, A8, A6, MATRIX_3:33

.= ((M1 ~) * (M2 * M1)) * (M1 @) by A2

.= ((M1 ~) * (M2 * M1)) * (M1 ~) by A1

.= (((M1 ~) * M2) * M1) * (M1 ~) by A4, A8, A6, MATRIX_3:33

.= ((M1 ~) * M2) * (M1 * (M1 ~)) by A5, A8, A7, MATRIX_3:33

.= ((M1 ~) * M2) * (1. (K,n)) by A3

.= (M1 ~) * M2 by MATRIX_3:19

.= (M1 @) * M2 by A1 ;

hence M1 @ commutes_with M2 ; :: thesis: verum