let i, j be Nat; for M1, M2 being Matrix of COMPLEX st len M1 = len M2 & width M1 = width M2 & [i,j] in Indices M1 holds
(M1 - M2) * (i,j) = (M1 * (i,j)) - (M2 * (i,j))
let M1, M2 be Matrix of COMPLEX; ( len M1 = len M2 & width M1 = width M2 & [i,j] in Indices M1 implies (M1 - M2) * (i,j) = (M1 * (i,j)) - (M2 * (i,j)) )
assume that
A1:
len M1 = len M2
and
A2:
width M1 = width M2
and
A3:
[i,j] in Indices M1
; (M1 - M2) * (i,j) = (M1 * (i,j)) - (M2 * (i,j))
A4:
j <= width M2
by A2, A3, Th1;
A5:
1 <= j
by A3, Th1;
A6:
1 <= i
by A3, Th1;
i <= len M2
by A1, A3, Th1;
then
[i,j] in Indices M2
by A6, A5, A4, Th1;
then A7:
[i,j] in Indices (COMPLEX2Field M2)
by MATRIX_5:def 1;
reconsider m2 = COMPLEX2Field M2 as Matrix of COMPLEX by COMPLFLD:def 1;
reconsider m1 = COMPLEX2Field M1 as Matrix of COMPLEX by COMPLFLD:def 1;
set m = COMPLEX2Field (M1 - M2);
A8: COMPLEX2Field (M1 - M2) =
COMPLEX2Field (Field2COMPLEX ((COMPLEX2Field M1) - (COMPLEX2Field M2)))
by MATRIX_5:def 5
.=
(COMPLEX2Field M1) - (COMPLEX2Field M2)
by MATRIX_5:6
;
reconsider m9 = COMPLEX2Field (M1 - M2) as Matrix of COMPLEX by COMPLFLD:def 1;
A9: M1 * (i,j) =
m1 * (i,j)
by MATRIX_5:def 1
.=
(COMPLEX2Field M1) * (i,j)
by COMPLFLD:def 1
;
A10:
[i,j] in Indices (COMPLEX2Field M1)
by A3, MATRIX_5:def 1;
M2 * (i,j) =
m2 * (i,j)
by MATRIX_5:def 1
.=
(COMPLEX2Field M2) * (i,j)
by COMPLFLD:def 1
;
then A11:
- (M2 * (i,j)) = - ((COMPLEX2Field M2) * (i,j))
by COMPLFLD:2;
(M1 - M2) * (i,j) =
m9 * (i,j)
by MATRIX_5:def 1
.=
(COMPLEX2Field (M1 - M2)) * (i,j)
by COMPLFLD:def 1
.=
((COMPLEX2Field M1) + (- (COMPLEX2Field M2))) * (i,j)
by A8, MATRIX_4:def 1
.=
((COMPLEX2Field M1) * (i,j)) + ((- (COMPLEX2Field M2)) * (i,j))
by A10, MATRIX_3:def 3
.=
((COMPLEX2Field M1) * (i,j)) + (- ((COMPLEX2Field M2) * (i,j)))
by A7, MATRIX_3:def 2
;
hence
(M1 - M2) * (i,j) = (M1 * (i,j)) - (M2 * (i,j))
by A9, A11; verum