let n be Nat; :: thesis: for M1, M2 being Matrix of n,REAL st M1 is Nonpositive & M2 is Nonpositive holds

M1 + M2 is Nonpositive

let M1, M2 be Matrix of n,REAL; :: thesis: ( M1 is Nonpositive & M2 is Nonpositive implies M1 + M2 is Nonpositive )

assume that

A1: M1 is Nonpositive and

A2: M2 is Nonpositive ; :: thesis: M1 + M2 is Nonpositive

A3: Indices M1 = [:(Seg n),(Seg n):] by MATRIX_0:24;

A4: Indices (M1 + M2) = [:(Seg n),(Seg n):] by MATRIX_0:24;

A5: Indices M2 = [:(Seg n),(Seg n):] by MATRIX_0:24;

for i, j being Nat st [i,j] in Indices (M1 + M2) holds

(M1 + M2) * (i,j) <= 0

M1 + M2 is Nonpositive

let M1, M2 be Matrix of n,REAL; :: thesis: ( M1 is Nonpositive & M2 is Nonpositive implies M1 + M2 is Nonpositive )

assume that

A1: M1 is Nonpositive and

A2: M2 is Nonpositive ; :: thesis: M1 + M2 is Nonpositive

A3: Indices M1 = [:(Seg n),(Seg n):] by MATRIX_0:24;

A4: Indices (M1 + M2) = [:(Seg n),(Seg n):] by MATRIX_0:24;

A5: Indices M2 = [:(Seg n),(Seg n):] by MATRIX_0:24;

for i, j being Nat st [i,j] in Indices (M1 + M2) holds

(M1 + M2) * (i,j) <= 0

proof

hence
M1 + M2 is Nonpositive
; :: thesis: verum
let i, j be Nat; :: thesis: ( [i,j] in Indices (M1 + M2) implies (M1 + M2) * (i,j) <= 0 )

assume A6: [i,j] in Indices (M1 + M2) ; :: thesis: (M1 + M2) * (i,j) <= 0

then M1 * (i,j) <= 0 by A1, A3, A4;

then A7: (M1 * (i,j)) + (M2 * (i,j)) <= M2 * (i,j) by XREAL_1:32;

M2 * (i,j) <= 0 by A2, A5, A4, A6;

hence (M1 + M2) * (i,j) <= 0 by A3, A4, A6, A7, MATRIXR1:25; :: thesis: verum

end;assume A6: [i,j] in Indices (M1 + M2) ; :: thesis: (M1 + M2) * (i,j) <= 0

then M1 * (i,j) <= 0 by A1, A3, A4;

then A7: (M1 * (i,j)) + (M2 * (i,j)) <= M2 * (i,j) by XREAL_1:32;

M2 * (i,j) <= 0 by A2, A5, A4, A6;

hence (M1 + M2) * (i,j) <= 0 by A3, A4, A6, A7, MATRIXR1:25; :: thesis: verum