Step |
Hyp |
Ref |
Expression |
1 |
|
fprodsplit1f.kph |
|- F/ k ph |
2 |
|
fprodsplit1f.fk |
|- ( ph -> F/_ k D ) |
3 |
|
fprodsplit1f.a |
|- ( ph -> A e. Fin ) |
4 |
|
fprodsplit1f.b |
|- ( ( ph /\ k e. A ) -> B e. CC ) |
5 |
|
fprodsplit1f.c |
|- ( ph -> C e. A ) |
6 |
|
fprodsplit1f.d |
|- ( ( ph /\ k = C ) -> B = D ) |
7 |
|
disjdif |
|- ( { C } i^i ( A \ { C } ) ) = (/) |
8 |
7
|
a1i |
|- ( ph -> ( { C } i^i ( A \ { C } ) ) = (/) ) |
9 |
5
|
snssd |
|- ( ph -> { C } C_ A ) |
10 |
|
undif |
|- ( { C } C_ A <-> ( { C } u. ( A \ { C } ) ) = A ) |
11 |
9 10
|
sylib |
|- ( ph -> ( { C } u. ( A \ { C } ) ) = A ) |
12 |
11
|
eqcomd |
|- ( ph -> A = ( { C } u. ( A \ { C } ) ) ) |
13 |
1 8 12 3 4
|
fprodsplitf |
|- ( ph -> prod_ k e. A B = ( prod_ k e. { C } B x. prod_ k e. ( A \ { C } ) B ) ) |
14 |
5
|
ancli |
|- ( ph -> ( ph /\ C e. A ) ) |
15 |
|
nfv |
|- F/ k C e. A |
16 |
1 15
|
nfan |
|- F/ k ( ph /\ C e. A ) |
17 |
|
nfcsb1v |
|- F/_ k [_ C / k ]_ B |
18 |
17
|
nfel1 |
|- F/ k [_ C / k ]_ B e. CC |
19 |
16 18
|
nfim |
|- F/ k ( ( ph /\ C e. A ) -> [_ C / k ]_ B e. CC ) |
20 |
|
eleq1 |
|- ( k = C -> ( k e. A <-> C e. A ) ) |
21 |
20
|
anbi2d |
|- ( k = C -> ( ( ph /\ k e. A ) <-> ( ph /\ C e. A ) ) ) |
22 |
|
csbeq1a |
|- ( k = C -> B = [_ C / k ]_ B ) |
23 |
22
|
eleq1d |
|- ( k = C -> ( B e. CC <-> [_ C / k ]_ B e. CC ) ) |
24 |
21 23
|
imbi12d |
|- ( k = C -> ( ( ( ph /\ k e. A ) -> B e. CC ) <-> ( ( ph /\ C e. A ) -> [_ C / k ]_ B e. CC ) ) ) |
25 |
19 24 4
|
vtoclg1f |
|- ( C e. A -> ( ( ph /\ C e. A ) -> [_ C / k ]_ B e. CC ) ) |
26 |
5 14 25
|
sylc |
|- ( ph -> [_ C / k ]_ B e. CC ) |
27 |
|
prodsns |
|- ( ( C e. A /\ [_ C / k ]_ B e. CC ) -> prod_ k e. { C } B = [_ C / k ]_ B ) |
28 |
5 26 27
|
syl2anc |
|- ( ph -> prod_ k e. { C } B = [_ C / k ]_ B ) |
29 |
1 2 5 6
|
csbiedf |
|- ( ph -> [_ C / k ]_ B = D ) |
30 |
28 29
|
eqtrd |
|- ( ph -> prod_ k e. { C } B = D ) |
31 |
30
|
oveq1d |
|- ( ph -> ( prod_ k e. { C } B x. prod_ k e. ( A \ { C } ) B ) = ( D x. prod_ k e. ( A \ { C } ) B ) ) |
32 |
13 31
|
eqtrd |
|- ( ph -> prod_ k e. A B = ( D x. prod_ k e. ( A \ { C } ) B ) ) |