let i, j be Nat; for M1, M2 being Matrix of COMPLEX st [i,j] in Indices M1 holds
(M1 + M2) * (i,j) = (M1 * (i,j)) + (M2 * (i,j))
let M1, M2 be Matrix of COMPLEX; ( [i,j] in Indices M1 implies (M1 + M2) * (i,j) = (M1 * (i,j)) + (M2 * (i,j)) )
A1: COMPLEX2Field (M1 + M2) =
COMPLEX2Field (Field2COMPLEX ((COMPLEX2Field M1) + (COMPLEX2Field M2)))
by MATRIX_5:def 3
.=
(COMPLEX2Field M1) + (COMPLEX2Field M2)
by MATRIX_5:6
;
reconsider m1 = COMPLEX2Field M1, m2 = COMPLEX2Field M2 as Matrix of COMPLEX by COMPLFLD:def 1;
set m = COMPLEX2Field (M1 + M2);
reconsider m9 = COMPLEX2Field (M1 + M2) as Matrix of COMPLEX by COMPLFLD:def 1;
A2: M1 * (i,j) =
m1 * (i,j)
by MATRIX_5:def 1
.=
(COMPLEX2Field M1) * (i,j)
by COMPLFLD:def 1
;
assume
[i,j] in Indices M1
; (M1 + M2) * (i,j) = (M1 * (i,j)) + (M2 * (i,j))
then A3:
[i,j] in Indices (COMPLEX2Field M1)
by MATRIX_5:def 1;
A4: M2 * (i,j) =
m2 * (i,j)
by MATRIX_5:def 1
.=
(COMPLEX2Field M2) * (i,j)
by COMPLFLD:def 1
;
(M1 + M2) * (i,j) =
m9 * (i,j)
by MATRIX_5:def 1
.=
(COMPLEX2Field (M1 + M2)) * (i,j)
by COMPLFLD:def 1
.=
((COMPLEX2Field M1) * (i,j)) + ((COMPLEX2Field M2) * (i,j))
by A1, A3, MATRIX_3:def 3
;
hence
(M1 + M2) * (i,j) = (M1 * (i,j)) + (M2 * (i,j))
by A2, A4; verum