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 + M2 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 + M2 commutes_with M2
let M1, M2 be Matrix of n,K; :: thesis: ( n > 0 & M1 commutes_with M2 implies M1 + M2 commutes_with M2 )
assume that
A1: n > 0 and
A2: M1 commutes_with M2 ; :: thesis: M1 + M2 commutes_with M2
A3: ( width M2 = n & len M1 = n ) by MATRIX_1:25;
A4: ( width M1 = n & len M2 = n ) by MATRIX_1:25;
then (M1 + M2) * M2 = (M1 * M2) + (M2 * M2) by A1, A3, MATRIX_4:63
.= (M2 * M1) + (M2 * M2) by A2, Def1
.= M2 * (M1 + M2) by A1, A4, A3, MATRIX_4:62 ;
hence M1 + M2 commutes_with M2 by Def1; :: thesis: verum