Step |
Hyp |
Ref |
Expression |
1 |
|
fprodcllemf.ph |
|- F/ k ph |
2 |
|
fprodcllemf.s |
|- ( ph -> S C_ CC ) |
3 |
|
fprodcllemf.xy |
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x x. y ) e. S ) |
4 |
|
fprodcllemf.a |
|- ( ph -> A e. Fin ) |
5 |
|
fprodcllemf.b |
|- ( ( ph /\ k e. A ) -> B e. S ) |
6 |
|
fprodcllemf.1 |
|- ( ph -> 1 e. S ) |
7 |
|
nfcv |
|- F/_ j B |
8 |
|
nfcsb1v |
|- F/_ k [_ j / k ]_ B |
9 |
|
csbeq1a |
|- ( k = j -> B = [_ j / k ]_ B ) |
10 |
7 8 9
|
cbvprodi |
|- prod_ k e. A B = prod_ j e. A [_ j / k ]_ B |
11 |
5
|
ex |
|- ( ph -> ( k e. A -> B e. S ) ) |
12 |
1 11
|
ralrimi |
|- ( ph -> A. k e. A B e. S ) |
13 |
|
rspsbc |
|- ( j e. A -> ( A. k e. A B e. S -> [. j / k ]. B e. S ) ) |
14 |
12 13
|
mpan9 |
|- ( ( ph /\ j e. A ) -> [. j / k ]. B e. S ) |
15 |
|
sbcel1g |
|- ( j e. _V -> ( [. j / k ]. B e. S <-> [_ j / k ]_ B e. S ) ) |
16 |
15
|
elv |
|- ( [. j / k ]. B e. S <-> [_ j / k ]_ B e. S ) |
17 |
14 16
|
sylib |
|- ( ( ph /\ j e. A ) -> [_ j / k ]_ B e. S ) |
18 |
2 3 4 17 6
|
fprodcllem |
|- ( ph -> prod_ j e. A [_ j / k ]_ B e. S ) |
19 |
10 18
|
eqeltrid |
|- ( ph -> prod_ k e. A B e. S ) |