let n be Nat; :: thesis: for R being commutative Ring
for M1, M2 being Matrix of n,R st n > 0 & M1 commutes_with M2 holds
M1 @ commutes_with M2 @
let R be commutative Ring; :: thesis: for M1, M2 being Matrix of n,R st n > 0 & M1 commutes_with M2 holds
M1 @ commutes_with M2 @
let M1, M2 be Matrix of n,R; :: thesis: ( n > 0 & M1 commutes_with M2 implies M1 @ commutes_with M2 @ )
A1: ( width M1 = n & width M2 = n ) by MATRIX_0:24;
set M3 = M1 @ ;
set M4 = M2 @ ;
A2: len M2 = n by MATRIX_0:24;
assume that
A3: n > 0 and
A4: M1 commutes_with M2 ; :: thesis: M1 @ commutes_with M2 @
len M1 = n by MATRIX_0:24;
then (M1 @) * (M2 @) = (M2 * M1) @ by A1, A3, MATRIX_3:22
.= (M1 * M2) @ by A4
.= (M2 @) * (M1 @) by A1, A2, A3, MATRIX_3:22 ;
hence M1 @ commutes_with M2 @ ; :: thesis: verum