let n be Nat; :: thesis: for M being Matrix of n,REAL st M is Nonnegative holds

- M is Nonpositive

let M be Matrix of n,REAL; :: thesis: ( M is Nonnegative implies - M is Nonpositive )

A1: ( Indices M = [:(Seg n),(Seg n):] & Indices (- M) = [:(Seg n),(Seg n):] ) by MATRIX_0:24;

assume A2: M is Nonnegative ; :: thesis: - M is Nonpositive

for i, j being Nat st [i,j] in Indices (- M) holds

(- M) * (i,j) <= 0

- M is Nonpositive

let M be Matrix of n,REAL; :: thesis: ( M is Nonnegative implies - M is Nonpositive )

A1: ( Indices M = [:(Seg n),(Seg n):] & Indices (- M) = [:(Seg n),(Seg n):] ) by MATRIX_0:24;

assume A2: M is Nonnegative ; :: thesis: - M is Nonpositive

for i, j being Nat st [i,j] in Indices (- M) holds

(- M) * (i,j) <= 0

proof

hence
- M is Nonpositive
; :: thesis: verum
let i, j be Nat; :: thesis: ( [i,j] in Indices (- M) implies (- M) * (i,j) <= 0 )

assume A3: [i,j] in Indices (- M) ; :: thesis: (- M) * (i,j) <= 0

then M * (i,j) >= 0 by A2, A1;

then - (M * (i,j)) <= 0 ;

hence (- M) * (i,j) <= 0 by A1, A3, Th2; :: thesis: verum

end;assume A3: [i,j] in Indices (- M) ; :: thesis: (- M) * (i,j) <= 0

then M * (i,j) >= 0 by A2, A1;

then - (M * (i,j)) <= 0 ;

hence (- M) * (i,j) <= 0 by A1, A3, Th2; :: thesis: verum