let a, b be Real; for n being Nat
for M1, M2 being Matrix of n,REAL st a <= 0 & b <= a & M1 is Nonpositive & M2 is_less_or_equal_with M1 holds
a * M1 is_less_or_equal_with b * M2
let n be Nat; for M1, M2 being Matrix of n,REAL st a <= 0 & b <= a & M1 is Nonpositive & M2 is_less_or_equal_with M1 holds
a * M1 is_less_or_equal_with b * M2
let M1, M2 be Matrix of n,REAL; ( a <= 0 & b <= a & M1 is Nonpositive & M2 is_less_or_equal_with M1 implies a * M1 is_less_or_equal_with b * M2 )
assume that
A1:
( a <= 0 & b <= a )
and
A2:
( M1 is Nonpositive & M2 is_less_or_equal_with M1 )
; a * M1 is_less_or_equal_with b * M2
A3:
Indices (a * M1) = Indices M1
by MATRIXR1:28;
A4:
( Indices M1 = [:(Seg n),(Seg n):] & Indices M2 = [:(Seg n),(Seg n):] )
by MATRIX_0:24;
for i, j being Nat st [i,j] in Indices (a * M1) holds
(a * M1) * (i,j) <= (b * M2) * (i,j)
proof
let i,
j be
Nat;
( [i,j] in Indices (a * M1) implies (a * M1) * (i,j) <= (b * M2) * (i,j) )
assume A5:
[i,j] in Indices (a * M1)
;
(a * M1) * (i,j) <= (b * M2) * (i,j)
then
(
M1 * (
i,
j)
<= 0 &
M2 * (
i,
j)
<= M1 * (
i,
j) )
by A2, A4, A3;
then
a * (M1 * (i,j)) <= b * (M2 * (i,j))
by A1, XREAL_1:67;
then
(a * M1) * (
i,
j)
<= b * (M2 * (i,j))
by A3, A5, Th4;
hence
(a * M1) * (
i,
j)
<= (b * M2) * (
i,
j)
by A4, A3, A5, Th4;
verum
end;
hence
a * M1 is_less_or_equal_with b * M2
; verum