Step |
Hyp |
Ref |
Expression |
1 |
|
fsumneg.1 |
|- ( ph -> A e. Fin ) |
2 |
|
fsumneg.2 |
|- ( ( ph /\ k e. A ) -> B e. CC ) |
3 |
|
neg1cn |
|- -u 1 e. CC |
4 |
3
|
a1i |
|- ( ph -> -u 1 e. CC ) |
5 |
1 4 2
|
fsummulc2 |
|- ( ph -> ( -u 1 x. sum_ k e. A B ) = sum_ k e. A ( -u 1 x. B ) ) |
6 |
1 2
|
fsumcl |
|- ( ph -> sum_ k e. A B e. CC ) |
7 |
6
|
mulm1d |
|- ( ph -> ( -u 1 x. sum_ k e. A B ) = -u sum_ k e. A B ) |
8 |
2
|
mulm1d |
|- ( ( ph /\ k e. A ) -> ( -u 1 x. B ) = -u B ) |
9 |
8
|
sumeq2dv |
|- ( ph -> sum_ k e. A ( -u 1 x. B ) = sum_ k e. A -u B ) |
10 |
5 7 9
|
3eqtr3rd |
|- ( ph -> sum_ k e. A -u B = -u sum_ k e. A B ) |