let M be Matrix of n,K; :: thesis: ( M = M1 + M2 implies M is Anti-subsymmetric )
assume A1: M = M1 + M2 ; :: thesis: M is Anti-subsymmetric
let i, j, k, l be Nat; :: according to MATRIX17:def 2 :: thesis: ( [i,j] in Indices M & k = (n + 1) - j & l = (n + 1) - i implies M * (i,j) = - (M * (k,l)) )
assume that
A2: [i,j] in Indices M and
A3: ( k = (n + 1) - j & l = (n + 1) - i ) ; :: thesis: M * (i,j) = - (M * (k,l))
A4: Indices M = [:(Seg n),(Seg n):] by MATRIX_0:24;
( (n + 1) - j in Seg n & (n + 1) - i in Seg n ) by A2, A4, Lm2;
then A5: [k,l] in [:(Seg n),(Seg n):] by A3, ZFMISC_1:87;
A6: ( Indices M1 = [:(Seg n),(Seg n):] & Indices M2 = [:(Seg n),(Seg n):] ) by MATRIX_0:24;
(M1 + M2) * (i,j) = (M1 * (i,j)) + (M2 * (i,j)) by A2, A4, A6, MATRIX_3:def 3
.= (- (M1 * (k,l))) + (M2 * (i,j)) by A2, A4, A6, A3, Def2
.= (- (M1 * (k,l))) + (- (M2 * (k,l))) by A2, A4, A6, A3, Def2
.= - ((M1 * (k,l)) + (M2 * (k,l))) by RLVECT_1:31
.= - ((M1 + M2) * (k,l)) by A6, A5, MATRIX_3:def 3 ;
hence M * (i,j) = - (M * (k,l)) by A1; :: thesis: verum