let n be Nat; :: thesis: for F being non empty right_complementable Abelian add-associative right_zeroed doubleLoopStr

for A, B being Matrix of n,F holds A + B = B + A

let F be non empty right_complementable Abelian add-associative right_zeroed doubleLoopStr ; :: thesis: for A, B being Matrix of n,F holds A + B = B + A

let A, B be Matrix of n,F; :: thesis: A + B = B + A

A1: Indices A = Indices (A + B) by MATRIX_0:26;

A2: Indices A = Indices B by MATRIX_0:26;

for A, B being Matrix of n,F holds A + B = B + A

let F be non empty right_complementable Abelian add-associative right_zeroed doubleLoopStr ; :: thesis: for A, B being Matrix of n,F holds A + B = B + A

let A, B be Matrix of n,F; :: thesis: A + B = B + A

A1: Indices A = Indices (A + B) by MATRIX_0:26;

A2: Indices A = Indices B by MATRIX_0:26;

now :: thesis: for i, j being Nat st [i,j] in Indices (A + B) holds

(A + B) * (i,j) = (B + A) * (i,j)

hence
A + B = B + A
by MATRIX_0:27; :: thesis: verum(A + B) * (i,j) = (B + A) * (i,j)

let i, j be Nat; :: thesis: ( [i,j] in Indices (A + B) implies (A + B) * (i,j) = (B + A) * (i,j) )

assume A3: [i,j] in Indices (A + B) ; :: thesis: (A + B) * (i,j) = (B + A) * (i,j)

hence (A + B) * (i,j) = (A * (i,j)) + (B * (i,j)) by A1, Def5

.= (B + A) * (i,j) by A2, A1, A3, Def5 ;

:: thesis: verum

end;assume A3: [i,j] in Indices (A + B) ; :: thesis: (A + B) * (i,j) = (B + A) * (i,j)

hence (A + B) * (i,j) = (A * (i,j)) + (B * (i,j)) by A1, Def5

.= (B + A) * (i,j) by A2, A1, A3, Def5 ;

:: thesis: verum