Step |
Hyp |
Ref |
Expression |
1 |
|
fprodsubrecnncnvlem.k |
|- F/ k ph |
2 |
|
fprodsubrecnncnvlem.a |
|- ( ph -> A e. Fin ) |
3 |
|
fprodsubrecnncnvlem.b |
|- ( ( ph /\ k e. A ) -> B e. CC ) |
4 |
|
fprodsubrecnncnvlem.s |
|- S = ( n e. NN |-> prod_ k e. A ( B - ( 1 / n ) ) ) |
5 |
|
fprodsubrecnncnvlem.f |
|- F = ( x e. CC |-> prod_ k e. A ( B - x ) ) |
6 |
|
fprodsubrecnncnvlem.g |
|- G = ( n e. NN |-> ( 1 / n ) ) |
7 |
|
nnuz |
|- NN = ( ZZ>= ` 1 ) |
8 |
|
1zzd |
|- ( ph -> 1 e. ZZ ) |
9 |
1 2 3 5
|
fprodsub2cncf |
|- ( ph -> F e. ( CC -cn-> CC ) ) |
10 |
|
1rp |
|- 1 e. RR+ |
11 |
10
|
a1i |
|- ( n e. NN -> 1 e. RR+ ) |
12 |
|
nnrp |
|- ( n e. NN -> n e. RR+ ) |
13 |
11 12
|
rpdivcld |
|- ( n e. NN -> ( 1 / n ) e. RR+ ) |
14 |
13
|
rpcnd |
|- ( n e. NN -> ( 1 / n ) e. CC ) |
15 |
14
|
adantl |
|- ( ( ph /\ n e. NN ) -> ( 1 / n ) e. CC ) |
16 |
15 6
|
fmptd |
|- ( ph -> G : NN --> CC ) |
17 |
|
1cnd |
|- ( ph -> 1 e. CC ) |
18 |
|
divcnv |
|- ( 1 e. CC -> ( n e. NN |-> ( 1 / n ) ) ~~> 0 ) |
19 |
17 18
|
syl |
|- ( ph -> ( n e. NN |-> ( 1 / n ) ) ~~> 0 ) |
20 |
6
|
a1i |
|- ( ph -> G = ( n e. NN |-> ( 1 / n ) ) ) |
21 |
20
|
breq1d |
|- ( ph -> ( G ~~> 0 <-> ( n e. NN |-> ( 1 / n ) ) ~~> 0 ) ) |
22 |
19 21
|
mpbird |
|- ( ph -> G ~~> 0 ) |
23 |
|
0cnd |
|- ( ph -> 0 e. CC ) |
24 |
7 8 9 16 22 23
|
climcncf |
|- ( ph -> ( F o. G ) ~~> ( F ` 0 ) ) |
25 |
|
nfv |
|- F/ k x e. CC |
26 |
1 25
|
nfan |
|- F/ k ( ph /\ x e. CC ) |
27 |
2
|
adantr |
|- ( ( ph /\ x e. CC ) -> A e. Fin ) |
28 |
3
|
adantlr |
|- ( ( ( ph /\ x e. CC ) /\ k e. A ) -> B e. CC ) |
29 |
|
simplr |
|- ( ( ( ph /\ x e. CC ) /\ k e. A ) -> x e. CC ) |
30 |
28 29
|
subcld |
|- ( ( ( ph /\ x e. CC ) /\ k e. A ) -> ( B - x ) e. CC ) |
31 |
26 27 30
|
fprodclf |
|- ( ( ph /\ x e. CC ) -> prod_ k e. A ( B - x ) e. CC ) |
32 |
31 5
|
fmptd |
|- ( ph -> F : CC --> CC ) |
33 |
|
fcompt |
|- ( ( F : CC --> CC /\ G : NN --> CC ) -> ( F o. G ) = ( n e. NN |-> ( F ` ( G ` n ) ) ) ) |
34 |
32 16 33
|
syl2anc |
|- ( ph -> ( F o. G ) = ( n e. NN |-> ( F ` ( G ` n ) ) ) ) |
35 |
4
|
a1i |
|- ( ph -> S = ( n e. NN |-> prod_ k e. A ( B - ( 1 / n ) ) ) ) |
36 |
|
id |
|- ( n e. NN -> n e. NN ) |
37 |
6
|
fvmpt2 |
|- ( ( n e. NN /\ ( 1 / n ) e. CC ) -> ( G ` n ) = ( 1 / n ) ) |
38 |
36 14 37
|
syl2anc |
|- ( n e. NN -> ( G ` n ) = ( 1 / n ) ) |
39 |
38
|
fveq2d |
|- ( n e. NN -> ( F ` ( G ` n ) ) = ( F ` ( 1 / n ) ) ) |
40 |
39
|
adantl |
|- ( ( ph /\ n e. NN ) -> ( F ` ( G ` n ) ) = ( F ` ( 1 / n ) ) ) |
41 |
|
oveq2 |
|- ( x = ( 1 / n ) -> ( B - x ) = ( B - ( 1 / n ) ) ) |
42 |
41
|
prodeq2ad |
|- ( x = ( 1 / n ) -> prod_ k e. A ( B - x ) = prod_ k e. A ( B - ( 1 / n ) ) ) |
43 |
|
prodex |
|- prod_ k e. A ( B - ( 1 / n ) ) e. _V |
44 |
43
|
a1i |
|- ( ( ph /\ n e. NN ) -> prod_ k e. A ( B - ( 1 / n ) ) e. _V ) |
45 |
5 42 15 44
|
fvmptd3 |
|- ( ( ph /\ n e. NN ) -> ( F ` ( 1 / n ) ) = prod_ k e. A ( B - ( 1 / n ) ) ) |
46 |
40 45
|
eqtr2d |
|- ( ( ph /\ n e. NN ) -> prod_ k e. A ( B - ( 1 / n ) ) = ( F ` ( G ` n ) ) ) |
47 |
46
|
mpteq2dva |
|- ( ph -> ( n e. NN |-> prod_ k e. A ( B - ( 1 / n ) ) ) = ( n e. NN |-> ( F ` ( G ` n ) ) ) ) |
48 |
35 47
|
eqtrd |
|- ( ph -> S = ( n e. NN |-> ( F ` ( G ` n ) ) ) ) |
49 |
34 48
|
eqtr4d |
|- ( ph -> ( F o. G ) = S ) |
50 |
5
|
a1i |
|- ( ph -> F = ( x e. CC |-> prod_ k e. A ( B - x ) ) ) |
51 |
|
nfv |
|- F/ k x = 0 |
52 |
1 51
|
nfan |
|- F/ k ( ph /\ x = 0 ) |
53 |
|
oveq2 |
|- ( x = 0 -> ( B - x ) = ( B - 0 ) ) |
54 |
53
|
ad2antlr |
|- ( ( ( ph /\ x = 0 ) /\ k e. A ) -> ( B - x ) = ( B - 0 ) ) |
55 |
3
|
subid1d |
|- ( ( ph /\ k e. A ) -> ( B - 0 ) = B ) |
56 |
55
|
adantlr |
|- ( ( ( ph /\ x = 0 ) /\ k e. A ) -> ( B - 0 ) = B ) |
57 |
54 56
|
eqtrd |
|- ( ( ( ph /\ x = 0 ) /\ k e. A ) -> ( B - x ) = B ) |
58 |
57
|
ex |
|- ( ( ph /\ x = 0 ) -> ( k e. A -> ( B - x ) = B ) ) |
59 |
52 58
|
ralrimi |
|- ( ( ph /\ x = 0 ) -> A. k e. A ( B - x ) = B ) |
60 |
59
|
prodeq2d |
|- ( ( ph /\ x = 0 ) -> prod_ k e. A ( B - x ) = prod_ k e. A B ) |
61 |
|
prodex |
|- prod_ k e. A B e. _V |
62 |
61
|
a1i |
|- ( ph -> prod_ k e. A B e. _V ) |
63 |
50 60 23 62
|
fvmptd |
|- ( ph -> ( F ` 0 ) = prod_ k e. A B ) |
64 |
49 63
|
breq12d |
|- ( ph -> ( ( F o. G ) ~~> ( F ` 0 ) <-> S ~~> prod_ k e. A B ) ) |
65 |
24 64
|
mpbid |
|- ( ph -> S ~~> prod_ k e. A B ) |