Proof
The proof is a routine verification. Here are the different parts.
(R × R', *× *',(1,1'))
is a monoid.
(R × R', + × +', (0,0')) is a commutative group.
Left distributivity.
Right distributivity.
Application of the direct product
construction for groups.
(a,a') * (b,b')
=
(a * b, a' * b')
=
(b * a, b' * a') =
(b,b') * (a,a') .
Observe that the direct product of two commutative groups is again commutative:
(a,a') * ((b,b') + (c,c'))
|
= (Cartesian addition) |
(a,a') * (b + c,b' + c')
| = (Cartesian multiplication) |
(a * (b + c),a' * (b' + c'))
| = (left distributivity for R and R') |
(a * b + a * c,a' * b' + a' * c')
| = (Cartesian addition) |
(a * b,a' * b') +
(a * c + a' * c')
| = (Cartesian multiplication) |
(a,a') * (b,b' ) +
(a,a') * (c,c'). |
Just like left distributivity.