let n be Nat; for M1, M2 being Matrix of n,REAL st M1 is Positive & M2 is Negative & |:M1:| is_less_than |:M2:| holds
M1 + M2 is Negative
let M1, M2 be Matrix of n,REAL; ( M1 is Positive & M2 is Negative & |:M1:| is_less_than |:M2:| implies M1 + M2 is Negative )
assume that
A1:
M1 is Positive
and
A2:
M2 is Negative
and
A3:
|:M1:| is_less_than |:M2:|
; M1 + M2 is Negative
A4:
Indices M1 = [:(Seg n),(Seg n):]
by MATRIX_0:24;
A5:
Indices (M1 + M2) = [:(Seg n),(Seg n):]
by MATRIX_0:24;
A6:
Indices M2 = [:(Seg n),(Seg n):]
by MATRIX_0:24;
for i, j being Nat st [i,j] in Indices (M1 + M2) holds
(M1 + M2) * (i,j) < 0
proof
let i,
j be
Nat;
( [i,j] in Indices (M1 + M2) implies (M1 + M2) * (i,j) < 0 )
assume A7:
[i,j] in Indices (M1 + M2)
;
(M1 + M2) * (i,j) < 0
then
[i,j] in Indices |:M1:|
by A4, A5, Th5;
then
|:M1:| * (
i,
j)
< |:M2:| * (
i,
j)
by A3;
then
|.(M1 * (i,j)).| < |:M2:| * (
i,
j)
by A4, A5, A7, Def7;
then
|.(M1 * (i,j)).| < |.(M2 * (i,j)).|
by A6, A5, A7, Def7;
then A8:
|.(M1 * (i,j)).| - |.(M2 * (i,j)).| < |.(M2 * (i,j)).| - |.(M2 * (i,j)).|
by XREAL_1:9;
M2 * (
i,
j)
< 0
by A2, A6, A5, A7;
then A9:
- (M2 * (i,j)) = |.(M2 * (i,j)).|
by ABSVALUE:def 1;
M1 * (
i,
j)
> 0
by A1, A4, A5, A7;
then
|.(M1 * (i,j)).| = M1 * (
i,
j)
by ABSVALUE:def 1;
then
(M1 * (i,j)) + (M2 * (i,j)) = |.(M1 * (i,j)).| - |.(M2 * (i,j)).|
by A9;
hence
(M1 + M2) * (
i,
j)
< 0
by A4, A5, A7, A8, MATRIXR1:25;
verum
end;
hence
M1 + M2 is Negative
; verum