let a be Real; :: thesis: for n being Nat

for M being Matrix of n,REAL st a <= 0 & M is Nonnegative holds

a * M is Nonpositive

let n be Nat; :: thesis: for M being Matrix of n,REAL st a <= 0 & M is Nonnegative holds

a * M is Nonpositive

let M be Matrix of n,REAL; :: thesis: ( a <= 0 & M is Nonnegative implies a * M is Nonpositive )

assume that

A1: a <= 0 and

A2: M is Nonnegative ; :: thesis: a * M is Nonpositive

A3: Indices (a * M) = Indices M by MATRIXR1:28;

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

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

for M being Matrix of n,REAL st a <= 0 & M is Nonnegative holds

a * M is Nonpositive

let n be Nat; :: thesis: for M being Matrix of n,REAL st a <= 0 & M is Nonnegative holds

a * M is Nonpositive

let M be Matrix of n,REAL; :: thesis: ( a <= 0 & M is Nonnegative implies a * M is Nonpositive )

assume that

A1: a <= 0 and

A2: M is Nonnegative ; :: thesis: a * M is Nonpositive

A3: Indices (a * M) = Indices M by MATRIXR1:28;

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

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

proof

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

assume A4: [i,j] in Indices (a * M) ; :: thesis: (a * M) * (i,j) <= 0

then M * (i,j) >= 0 by A2, A3;

then a * (M * (i,j)) <= 0 by A1;

hence (a * M) * (i,j) <= 0 by A3, A4, Th4; :: thesis: verum

end;assume A4: [i,j] in Indices (a * M) ; :: thesis: (a * M) * (i,j) <= 0

then M * (i,j) >= 0 by A2, A3;

then a * (M * (i,j)) <= 0 by A1;

hence (a * M) * (i,j) <= 0 by A3, A4, Th4; :: thesis: verum