let n be Nat; :: thesis: for K being Field
for M1, M2 being Matrix of n,K st n > 0 & M1 commutes_with M2 holds
M1 @ commutes_with M2 @
let K be Field; :: thesis: for M1, M2 being Matrix of n,K st n > 0 & M1 commutes_with M2 holds
M1 @ commutes_with M2 @
let M1, M2 be Matrix of n,K; :: thesis: ( n > 0 & M1 commutes_with M2 implies M1 @ commutes_with M2 @ )
A1: ( width M1 = n & width M2 = n & len M1 = n & len M2 = n ) by MATRIX_1:25;
assume A2: ( n > 0 & M1 commutes_with M2 ) ; :: thesis: M1 @ commutes_with M2 @
set M3 = M1 @ ;
set M4 = M2 @ ;
(M1 @ ) * (M2 @ ) = (M2 * M1) @ by A1, A2, MATRIX_3:24
.= (M1 * M2) @ by A2, Def1
.= (M2 @ ) * (M1 @ ) by A1, A2, MATRIX_3:24 ;
hence M1 @ commutes_with M2 @ by Def1; :: thesis: verum