| Step |
Hyp |
Ref |
Expression |
| 1 |
|
fprodp1s.1 |
|- ( ph -> N e. ( ZZ>= ` M ) ) |
| 2 |
|
fprodp1s.2 |
|- ( ( ph /\ k e. ( M ... ( N + 1 ) ) ) -> A e. CC ) |
| 3 |
2
|
ralrimiva |
|- ( ph -> A. k e. ( M ... ( N + 1 ) ) A e. CC ) |
| 4 |
|
nfcsb1v |
|- F/_ k [_ m / k ]_ A |
| 5 |
4
|
nfel1 |
|- F/ k [_ m / k ]_ A e. CC |
| 6 |
|
csbeq1a |
|- ( k = m -> A = [_ m / k ]_ A ) |
| 7 |
6
|
eleq1d |
|- ( k = m -> ( A e. CC <-> [_ m / k ]_ A e. CC ) ) |
| 8 |
5 7
|
rspc |
|- ( m e. ( M ... ( N + 1 ) ) -> ( A. k e. ( M ... ( N + 1 ) ) A e. CC -> [_ m / k ]_ A e. CC ) ) |
| 9 |
3 8
|
mpan9 |
|- ( ( ph /\ m e. ( M ... ( N + 1 ) ) ) -> [_ m / k ]_ A e. CC ) |
| 10 |
|
csbeq1 |
|- ( m = ( N + 1 ) -> [_ m / k ]_ A = [_ ( N + 1 ) / k ]_ A ) |
| 11 |
1 9 10
|
fprodp1 |
|- ( ph -> prod_ m e. ( M ... ( N + 1 ) ) [_ m / k ]_ A = ( prod_ m e. ( M ... N ) [_ m / k ]_ A x. [_ ( N + 1 ) / k ]_ A ) ) |
| 12 |
|
nfcv |
|- F/_ m A |
| 13 |
12 4 6
|
cbvprodi |
|- prod_ k e. ( M ... ( N + 1 ) ) A = prod_ m e. ( M ... ( N + 1 ) ) [_ m / k ]_ A |
| 14 |
12 4 6
|
cbvprodi |
|- prod_ k e. ( M ... N ) A = prod_ m e. ( M ... N ) [_ m / k ]_ A |
| 15 |
14
|
oveq1i |
|- ( prod_ k e. ( M ... N ) A x. [_ ( N + 1 ) / k ]_ A ) = ( prod_ m e. ( M ... N ) [_ m / k ]_ A x. [_ ( N + 1 ) / k ]_ A ) |
| 16 |
11 13 15
|
3eqtr4g |
|- ( ph -> prod_ k e. ( M ... ( N + 1 ) ) A = ( prod_ k e. ( M ... N ) A x. [_ ( N + 1 ) / k ]_ A ) ) |