let i, j be Nat; :: thesis: for M being Matrix of COMPLEX st [i,j] in Indices M holds
(- M) * (i,j) = - (M * (i,j))
let M be Matrix of COMPLEX; :: thesis: ( [i,j] in Indices M implies (- M) * (i,j) = - (M * (i,j)) )
A1: COMPLEX2Field (- M) = COMPLEX2Field (Field2COMPLEX (- (COMPLEX2Field M))) by MATRIX_5:def 4
.= - (COMPLEX2Field M) by MATRIX_5:6 ;
reconsider m = COMPLEX2Field (- M) as Matrix of COMPLEX by COMPLFLD:def 1;
reconsider m1 = COMPLEX2Field M as Matrix of COMPLEX by COMPLFLD:def 1;
A2: M * (i,j) = m1 * (i,j) by MATRIX_5:def 1
.= (COMPLEX2Field M) * (i,j) by COMPLFLD:def 1 ;
assume [i,j] in Indices M ; :: thesis: (- M) * (i,j) = - (M * (i,j))
then A3: [i,j] in Indices (COMPLEX2Field M) by MATRIX_5:def 1;
(- M) * (i,j) = m * (i,j) by MATRIX_5:def 1
.= (- (COMPLEX2Field M)) * (i,j) by A1, COMPLFLD:def 1
.= - ((COMPLEX2Field M) * (i,j)) by A3, MATRIX_3:def 2 ;
hence (- M) * (i,j) = - (M * (i,j)) by A2, COMPLFLD:2; :: thesis: verum