let n be Nat; :: thesis: for K being Field
for M1, M2 being Matrix of n,K st n > 0 & M1 is Idempotent & M2 is Idempotent & M1 commutes_with M2 & M1 * M2 = 0. K,n holds
M1 + M2 is Idempotent
let K be Field; :: thesis: for M1, M2 being Matrix of n,K st n > 0 & M1 is Idempotent & M2 is Idempotent & M1 commutes_with M2 & M1 * M2 = 0. K,n holds
M1 + M2 is Idempotent
let M1, M2 be Matrix of n,K; :: thesis: ( n > 0 & M1 is Idempotent & M2 is Idempotent & M1 commutes_with M2 & M1 * M2 = 0. K,n implies M1 + M2 is Idempotent )
assume A1: ( n > 0 & M1 is Idempotent & M2 is Idempotent & M1 commutes_with M2 & M1 * M2 = 0. K,n ) ; :: thesis: M1 + M2 is Idempotent
then A2: ( M1 * M1 = M1 & M2 * M2 = M2 & M1 * M2 = M2 * M1 & M1 * M2 = 0. K,n,n ) by Def1, MATRIX_6:def 1;
(M1 + M2) * (M1 + M2) = (((M1 * M1) + (0. K,n)) + (0. K,n)) + (M2 * M2) by A1, MATRIX_6:36
.= ((M1 * M1) + (0. K,n)) + (M2 * M2) by A2, MATRIX_3:6
.= M1 + M2 by A2, MATRIX_3:6 ;
hence M1 + M2 is Idempotent by Def1; :: thesis: verum