let K be Field; for M1, M2 being Matrix of K st len M1 = len M2 & width M1 = width M2 holds
- (M1 + M2) = (- M1) + (- M2)
let M1, M2 be Matrix of K; ( len M1 = len M2 & width M1 = width M2 implies - (M1 + M2) = (- M1) + (- M2) )
assume A1:
( len M1 = len M2 & width M1 = width M2 )
; - (M1 + M2) = (- M1) + (- M2)
A2:
width (- M1) = width M1
by MATRIX_3:def 2;
then A3:
width ((- M1) + (- M2)) = width M1
by MATRIX_3:def 3;
A4:
( len (M1 + M2) = len M1 & width (M1 + M2) = width M1 )
by MATRIX_3:def 3;
A5:
len (- M1) = len M1
by MATRIX_3:def 2;
then A6:
len ((- M1) + (- M2)) = len M1
by MATRIX_3:def 3;
A7:
( len (- M2) = len M2 & width (- M2) = width M2 )
by MATRIX_3:def 2;
per cases
( len M1 > 0 or len M1 = 0 )
by NAT_1:3;
suppose A8:
len M1 > 0
;
- (M1 + M2) = (- M1) + (- M2)then A9:
M2 is
Matrix of
len M1,
width M1,
K
by A1, MATRIX_0:20;
A10:
M1 is
Matrix of
len M1,
width M1,
K
by A8, MATRIX_0:20;
(M1 + M2) + ((- M1) + (- M2)) =
(M1 + M2) + ((- M2) + (- M1))
by A1, A5, A2, A7, MATRIX_3:2
.=
((M1 + M2) + (- M2)) + (- M1)
by A1, A7, A4, MATRIX_3:3
.=
(M1 + (M2 + (- M2))) + (- M1)
by A1, MATRIX_3:3
.=
(M1 + (0. (K,(len M1),(width M1)))) + (- M1)
by A9, MATRIX_3:5
.=
M1 + (- M1)
by A10, MATRIX_3:4
.=
0. (
K,
(len M1),
(width M1))
by A10, MATRIX_3:5
;
hence
- (M1 + M2) = (- M1) + (- M2)
by A4, A6, A3, Th8;
verum end;