Step |
Hyp |
Ref |
Expression |
1 |
|
fprodcnlem.1 |
|- F/ k ph |
2 |
|
fprodcnlem.k |
|- K = ( TopOpen ` CCfld ) |
3 |
|
fprodcnlem.j |
|- ( ph -> J e. ( TopOn ` X ) ) |
4 |
|
fprodcnlem.a |
|- ( ph -> A e. Fin ) |
5 |
|
fprodcnlem.b |
|- ( ( ph /\ k e. A ) -> ( x e. X |-> B ) e. ( J Cn K ) ) |
6 |
|
fprodcnlem.z |
|- ( ph -> Z C_ A ) |
7 |
|
fprodcnlem.w |
|- ( ph -> W e. ( A \ Z ) ) |
8 |
|
fprodcnlem.p |
|- ( ph -> ( x e. X |-> prod_ k e. Z B ) e. ( J Cn K ) ) |
9 |
|
nfv |
|- F/ k x e. X |
10 |
1 9
|
nfan |
|- F/ k ( ph /\ x e. X ) |
11 |
|
nfcsb1v |
|- F/_ k [_ W / k ]_ B |
12 |
4 6
|
ssfid |
|- ( ph -> Z e. Fin ) |
13 |
12
|
adantr |
|- ( ( ph /\ x e. X ) -> Z e. Fin ) |
14 |
7
|
adantr |
|- ( ( ph /\ x e. X ) -> W e. ( A \ Z ) ) |
15 |
14
|
eldifbd |
|- ( ( ph /\ x e. X ) -> -. W e. Z ) |
16 |
6
|
sselda |
|- ( ( ph /\ k e. Z ) -> k e. A ) |
17 |
16
|
adantlr |
|- ( ( ( ph /\ x e. X ) /\ k e. Z ) -> k e. A ) |
18 |
3
|
adantr |
|- ( ( ph /\ k e. A ) -> J e. ( TopOn ` X ) ) |
19 |
2
|
cnfldtopon |
|- K e. ( TopOn ` CC ) |
20 |
19
|
a1i |
|- ( ( ph /\ k e. A ) -> K e. ( TopOn ` CC ) ) |
21 |
|
cnf2 |
|- ( ( J e. ( TopOn ` X ) /\ K e. ( TopOn ` CC ) /\ ( x e. X |-> B ) e. ( J Cn K ) ) -> ( x e. X |-> B ) : X --> CC ) |
22 |
18 20 5 21
|
syl3anc |
|- ( ( ph /\ k e. A ) -> ( x e. X |-> B ) : X --> CC ) |
23 |
|
eqid |
|- ( x e. X |-> B ) = ( x e. X |-> B ) |
24 |
23
|
fmpt |
|- ( A. x e. X B e. CC <-> ( x e. X |-> B ) : X --> CC ) |
25 |
22 24
|
sylibr |
|- ( ( ph /\ k e. A ) -> A. x e. X B e. CC ) |
26 |
25
|
adantlr |
|- ( ( ( ph /\ x e. X ) /\ k e. A ) -> A. x e. X B e. CC ) |
27 |
|
simplr |
|- ( ( ( ph /\ x e. X ) /\ k e. A ) -> x e. X ) |
28 |
|
rspa |
|- ( ( A. x e. X B e. CC /\ x e. X ) -> B e. CC ) |
29 |
26 27 28
|
syl2anc |
|- ( ( ( ph /\ x e. X ) /\ k e. A ) -> B e. CC ) |
30 |
17 29
|
syldan |
|- ( ( ( ph /\ x e. X ) /\ k e. Z ) -> B e. CC ) |
31 |
|
csbeq1a |
|- ( k = W -> B = [_ W / k ]_ B ) |
32 |
14
|
eldifad |
|- ( ( ph /\ x e. X ) -> W e. A ) |
33 |
|
nfv |
|- F/ k W e. A |
34 |
10 33
|
nfan |
|- F/ k ( ( ph /\ x e. X ) /\ W e. A ) |
35 |
11
|
nfel1 |
|- F/ k [_ W / k ]_ B e. CC |
36 |
34 35
|
nfim |
|- F/ k ( ( ( ph /\ x e. X ) /\ W e. A ) -> [_ W / k ]_ B e. CC ) |
37 |
|
eleq1 |
|- ( k = W -> ( k e. A <-> W e. A ) ) |
38 |
37
|
anbi2d |
|- ( k = W -> ( ( ( ph /\ x e. X ) /\ k e. A ) <-> ( ( ph /\ x e. X ) /\ W e. A ) ) ) |
39 |
31
|
eleq1d |
|- ( k = W -> ( B e. CC <-> [_ W / k ]_ B e. CC ) ) |
40 |
38 39
|
imbi12d |
|- ( k = W -> ( ( ( ( ph /\ x e. X ) /\ k e. A ) -> B e. CC ) <-> ( ( ( ph /\ x e. X ) /\ W e. A ) -> [_ W / k ]_ B e. CC ) ) ) |
41 |
36 40 29
|
vtoclg1f |
|- ( W e. A -> ( ( ( ph /\ x e. X ) /\ W e. A ) -> [_ W / k ]_ B e. CC ) ) |
42 |
41
|
anabsi7 |
|- ( ( ( ph /\ x e. X ) /\ W e. A ) -> [_ W / k ]_ B e. CC ) |
43 |
32 42
|
mpdan |
|- ( ( ph /\ x e. X ) -> [_ W / k ]_ B e. CC ) |
44 |
10 11 13 14 15 30 31 43
|
fprodsplitsn |
|- ( ( ph /\ x e. X ) -> prod_ k e. ( Z u. { W } ) B = ( prod_ k e. Z B x. [_ W / k ]_ B ) ) |
45 |
44
|
mpteq2dva |
|- ( ph -> ( x e. X |-> prod_ k e. ( Z u. { W } ) B ) = ( x e. X |-> ( prod_ k e. Z B x. [_ W / k ]_ B ) ) ) |
46 |
7
|
eldifad |
|- ( ph -> W e. A ) |
47 |
1 33
|
nfan |
|- F/ k ( ph /\ W e. A ) |
48 |
|
nfcv |
|- F/_ k X |
49 |
48 11
|
nfmpt |
|- F/_ k ( x e. X |-> [_ W / k ]_ B ) |
50 |
49
|
nfel1 |
|- F/ k ( x e. X |-> [_ W / k ]_ B ) e. ( J Cn K ) |
51 |
47 50
|
nfim |
|- F/ k ( ( ph /\ W e. A ) -> ( x e. X |-> [_ W / k ]_ B ) e. ( J Cn K ) ) |
52 |
37
|
anbi2d |
|- ( k = W -> ( ( ph /\ k e. A ) <-> ( ph /\ W e. A ) ) ) |
53 |
31
|
mpteq2dv |
|- ( k = W -> ( x e. X |-> B ) = ( x e. X |-> [_ W / k ]_ B ) ) |
54 |
53
|
eleq1d |
|- ( k = W -> ( ( x e. X |-> B ) e. ( J Cn K ) <-> ( x e. X |-> [_ W / k ]_ B ) e. ( J Cn K ) ) ) |
55 |
52 54
|
imbi12d |
|- ( k = W -> ( ( ( ph /\ k e. A ) -> ( x e. X |-> B ) e. ( J Cn K ) ) <-> ( ( ph /\ W e. A ) -> ( x e. X |-> [_ W / k ]_ B ) e. ( J Cn K ) ) ) ) |
56 |
51 55 5
|
vtoclg1f |
|- ( W e. A -> ( ( ph /\ W e. A ) -> ( x e. X |-> [_ W / k ]_ B ) e. ( J Cn K ) ) ) |
57 |
56
|
anabsi7 |
|- ( ( ph /\ W e. A ) -> ( x e. X |-> [_ W / k ]_ B ) e. ( J Cn K ) ) |
58 |
46 57
|
mpdan |
|- ( ph -> ( x e. X |-> [_ W / k ]_ B ) e. ( J Cn K ) ) |
59 |
19
|
a1i |
|- ( ph -> K e. ( TopOn ` CC ) ) |
60 |
2
|
mpomulcn |
|- ( u e. CC , v e. CC |-> ( u x. v ) ) e. ( ( K tX K ) Cn K ) |
61 |
60
|
a1i |
|- ( ph -> ( u e. CC , v e. CC |-> ( u x. v ) ) e. ( ( K tX K ) Cn K ) ) |
62 |
|
oveq12 |
|- ( ( u = prod_ k e. Z B /\ v = [_ W / k ]_ B ) -> ( u x. v ) = ( prod_ k e. Z B x. [_ W / k ]_ B ) ) |
63 |
3 8 58 59 59 61 62
|
cnmpt12 |
|- ( ph -> ( x e. X |-> ( prod_ k e. Z B x. [_ W / k ]_ B ) ) e. ( J Cn K ) ) |
64 |
45 63
|
eqeltrd |
|- ( ph -> ( x e. X |-> prod_ k e. ( Z u. { W } ) B ) e. ( J Cn K ) ) |