let n be Nat; for M1, M2, M3 being Matrix of n,REAL st M1 is_less_than M2 holds
M1 + M3 is_less_than M2 + M3
let M1, M2, M3 be Matrix of n,REAL; ( M1 is_less_than M2 implies M1 + M3 is_less_than M2 + M3 )
A1:
Indices M1 = [:(Seg n),(Seg n):]
by MATRIX_0:24;
A2:
Indices M2 = [:(Seg n),(Seg n):]
by MATRIX_0:24;
A3:
Indices (M1 + M3) = [:(Seg n),(Seg n):]
by MATRIX_0:24;
assume A4:
M1 is_less_than M2
; M1 + M3 is_less_than M2 + M3
for i, j being Nat st [i,j] in Indices (M1 + M3) holds
(M1 + M3) * (i,j) < (M2 + M3) * (i,j)
proof
let i,
j be
Nat;
( [i,j] in Indices (M1 + M3) implies (M1 + M3) * (i,j) < (M2 + M3) * (i,j) )
assume A5:
[i,j] in Indices (M1 + M3)
;
(M1 + M3) * (i,j) < (M2 + M3) * (i,j)
then
M1 * (
i,
j)
< M2 * (
i,
j)
by A4, A1, A3;
then
(M1 * (i,j)) + (M3 * (i,j)) < (M2 * (i,j)) + (M3 * (i,j))
by XREAL_1:8;
then
(M1 + M3) * (
i,
j)
< (M2 * (i,j)) + (M3 * (i,j))
by A1, A3, A5, MATRIXR1:25;
hence
(M1 + M3) * (
i,
j)
< (M2 + M3) * (
i,
j)
by A2, A3, A5, MATRIXR1:25;
verum
end;
hence
M1 + M3 is_less_than M2 + M3
; verum