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

let M be Matrix of n,REAL; :: thesis: |:M:| is Nonnegative

for i, j being Nat st [i,j] in Indices |:M:| holds

|:M:| * (i,j) >= 0

let M be Matrix of n,REAL; :: thesis: |:M:| is Nonnegative

for i, j being Nat st [i,j] in Indices |:M:| holds

|:M:| * (i,j) >= 0

proof

hence
|:M:| is Nonnegative
; :: thesis: verum
let i, j be Nat; :: thesis: ( [i,j] in Indices |:M:| implies |:M:| * (i,j) >= 0 )

assume A1: [i,j] in Indices |:M:| ; :: thesis: |:M:| * (i,j) >= 0

Indices |:M:| = Indices M by Th5;

then |:M:| * (i,j) = |.(M * (i,j)).| by A1, Def7;

hence |:M:| * (i,j) >= 0 by COMPLEX1:46; :: thesis: verum

end;assume A1: [i,j] in Indices |:M:| ; :: thesis: |:M:| * (i,j) >= 0

Indices |:M:| = Indices M by Th5;

then |:M:| * (i,j) = |.(M * (i,j)).| by A1, Def7;

hence |:M:| * (i,j) >= 0 by COMPLEX1:46; :: thesis: verum