Step |
Hyp |
Ref |
Expression |
1 |
|
etransclem34.s |
|- ( ph -> S e. { RR , CC } ) |
2 |
|
etransclem34.a |
|- ( ph -> X e. ( ( TopOpen ` CCfld ) |`t S ) ) |
3 |
|
etransclem34.p |
|- ( ph -> P e. NN ) |
4 |
|
etransclem34.m |
|- ( ph -> M e. NN0 ) |
5 |
|
etransclem34.f |
|- F = ( x e. X |-> ( ( x ^ ( P - 1 ) ) x. prod_ k e. ( 1 ... M ) ( ( x - k ) ^ P ) ) ) |
6 |
|
etransclem34.n |
|- ( ph -> N e. NN0 ) |
7 |
|
etransclem34.h |
|- H = ( k e. ( 0 ... M ) |-> ( x e. X |-> ( ( x - k ) ^ if ( k = 0 , ( P - 1 ) , P ) ) ) ) |
8 |
|
etransclem34.c |
|- C = ( n e. NN0 |-> { c e. ( ( 0 ... n ) ^m ( 0 ... M ) ) | sum_ k e. ( 0 ... M ) ( c ` k ) = n } ) |
9 |
1 2 3 4 5 6 7 8
|
etransclem30 |
|- ( ph -> ( ( S Dn F ) ` N ) = ( x e. X |-> sum_ c e. ( C ` N ) ( ( ( ! ` N ) / prod_ k e. ( 0 ... M ) ( ! ` ( c ` k ) ) ) x. prod_ k e. ( 0 ... M ) ( ( ( S Dn ( H ` k ) ) ` ( c ` k ) ) ` x ) ) ) ) |
10 |
1 2
|
dvdmsscn |
|- ( ph -> X C_ CC ) |
11 |
8 6
|
etransclem16 |
|- ( ph -> ( C ` N ) e. Fin ) |
12 |
10
|
adantr |
|- ( ( ph /\ c e. ( C ` N ) ) -> X C_ CC ) |
13 |
6
|
faccld |
|- ( ph -> ( ! ` N ) e. NN ) |
14 |
13
|
nncnd |
|- ( ph -> ( ! ` N ) e. CC ) |
15 |
14
|
adantr |
|- ( ( ph /\ c e. ( C ` N ) ) -> ( ! ` N ) e. CC ) |
16 |
|
fzfid |
|- ( ( ph /\ c e. ( C ` N ) ) -> ( 0 ... M ) e. Fin ) |
17 |
|
fzssnn0 |
|- ( 0 ... N ) C_ NN0 |
18 |
|
ssrab2 |
|- { c e. ( ( 0 ... N ) ^m ( 0 ... M ) ) | sum_ k e. ( 0 ... M ) ( c ` k ) = N } C_ ( ( 0 ... N ) ^m ( 0 ... M ) ) |
19 |
|
simpr |
|- ( ( ph /\ c e. ( C ` N ) ) -> c e. ( C ` N ) ) |
20 |
8 6
|
etransclem12 |
|- ( ph -> ( C ` N ) = { c e. ( ( 0 ... N ) ^m ( 0 ... M ) ) | sum_ k e. ( 0 ... M ) ( c ` k ) = N } ) |
21 |
20
|
adantr |
|- ( ( ph /\ c e. ( C ` N ) ) -> ( C ` N ) = { c e. ( ( 0 ... N ) ^m ( 0 ... M ) ) | sum_ k e. ( 0 ... M ) ( c ` k ) = N } ) |
22 |
19 21
|
eleqtrd |
|- ( ( ph /\ c e. ( C ` N ) ) -> c e. { c e. ( ( 0 ... N ) ^m ( 0 ... M ) ) | sum_ k e. ( 0 ... M ) ( c ` k ) = N } ) |
23 |
18 22
|
sseldi |
|- ( ( ph /\ c e. ( C ` N ) ) -> c e. ( ( 0 ... N ) ^m ( 0 ... M ) ) ) |
24 |
|
elmapi |
|- ( c e. ( ( 0 ... N ) ^m ( 0 ... M ) ) -> c : ( 0 ... M ) --> ( 0 ... N ) ) |
25 |
23 24
|
syl |
|- ( ( ph /\ c e. ( C ` N ) ) -> c : ( 0 ... M ) --> ( 0 ... N ) ) |
26 |
25
|
ffvelrnda |
|- ( ( ( ph /\ c e. ( C ` N ) ) /\ k e. ( 0 ... M ) ) -> ( c ` k ) e. ( 0 ... N ) ) |
27 |
17 26
|
sseldi |
|- ( ( ( ph /\ c e. ( C ` N ) ) /\ k e. ( 0 ... M ) ) -> ( c ` k ) e. NN0 ) |
28 |
27
|
faccld |
|- ( ( ( ph /\ c e. ( C ` N ) ) /\ k e. ( 0 ... M ) ) -> ( ! ` ( c ` k ) ) e. NN ) |
29 |
28
|
nncnd |
|- ( ( ( ph /\ c e. ( C ` N ) ) /\ k e. ( 0 ... M ) ) -> ( ! ` ( c ` k ) ) e. CC ) |
30 |
16 29
|
fprodcl |
|- ( ( ph /\ c e. ( C ` N ) ) -> prod_ k e. ( 0 ... M ) ( ! ` ( c ` k ) ) e. CC ) |
31 |
28
|
nnne0d |
|- ( ( ( ph /\ c e. ( C ` N ) ) /\ k e. ( 0 ... M ) ) -> ( ! ` ( c ` k ) ) =/= 0 ) |
32 |
16 29 31
|
fprodn0 |
|- ( ( ph /\ c e. ( C ` N ) ) -> prod_ k e. ( 0 ... M ) ( ! ` ( c ` k ) ) =/= 0 ) |
33 |
15 30 32
|
divcld |
|- ( ( ph /\ c e. ( C ` N ) ) -> ( ( ! ` N ) / prod_ k e. ( 0 ... M ) ( ! ` ( c ` k ) ) ) e. CC ) |
34 |
|
ssid |
|- CC C_ CC |
35 |
34
|
a1i |
|- ( ( ph /\ c e. ( C ` N ) ) -> CC C_ CC ) |
36 |
12 33 35
|
constcncfg |
|- ( ( ph /\ c e. ( C ` N ) ) -> ( x e. X |-> ( ( ! ` N ) / prod_ k e. ( 0 ... M ) ( ! ` ( c ` k ) ) ) ) e. ( X -cn-> CC ) ) |
37 |
1
|
ad2antrr |
|- ( ( ( ph /\ c e. ( C ` N ) ) /\ k e. ( 0 ... M ) ) -> S e. { RR , CC } ) |
38 |
2
|
ad2antrr |
|- ( ( ( ph /\ c e. ( C ` N ) ) /\ k e. ( 0 ... M ) ) -> X e. ( ( TopOpen ` CCfld ) |`t S ) ) |
39 |
3
|
ad2antrr |
|- ( ( ( ph /\ c e. ( C ` N ) ) /\ k e. ( 0 ... M ) ) -> P e. NN ) |
40 |
|
etransclem5 |
|- ( k e. ( 0 ... M ) |-> ( x e. X |-> ( ( x - k ) ^ if ( k = 0 , ( P - 1 ) , P ) ) ) ) = ( j e. ( 0 ... M ) |-> ( y e. X |-> ( ( y - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) ) ) |
41 |
7 40
|
eqtri |
|- H = ( j e. ( 0 ... M ) |-> ( y e. X |-> ( ( y - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) ) ) |
42 |
|
simpr |
|- ( ( ( ph /\ c e. ( C ` N ) ) /\ k e. ( 0 ... M ) ) -> k e. ( 0 ... M ) ) |
43 |
37 38 39 41 42 27
|
etransclem20 |
|- ( ( ( ph /\ c e. ( C ` N ) ) /\ k e. ( 0 ... M ) ) -> ( ( S Dn ( H ` k ) ) ` ( c ` k ) ) : X --> CC ) |
44 |
43
|
3adant2 |
|- ( ( ( ph /\ c e. ( C ` N ) ) /\ x e. X /\ k e. ( 0 ... M ) ) -> ( ( S Dn ( H ` k ) ) ` ( c ` k ) ) : X --> CC ) |
45 |
|
simp2 |
|- ( ( ( ph /\ c e. ( C ` N ) ) /\ x e. X /\ k e. ( 0 ... M ) ) -> x e. X ) |
46 |
44 45
|
ffvelrnd |
|- ( ( ( ph /\ c e. ( C ` N ) ) /\ x e. X /\ k e. ( 0 ... M ) ) -> ( ( ( S Dn ( H ` k ) ) ` ( c ` k ) ) ` x ) e. CC ) |
47 |
43
|
feqmptd |
|- ( ( ( ph /\ c e. ( C ` N ) ) /\ k e. ( 0 ... M ) ) -> ( ( S Dn ( H ` k ) ) ` ( c ` k ) ) = ( x e. X |-> ( ( ( S Dn ( H ` k ) ) ` ( c ` k ) ) ` x ) ) ) |
48 |
37 38 39 41 42 27
|
etransclem22 |
|- ( ( ( ph /\ c e. ( C ` N ) ) /\ k e. ( 0 ... M ) ) -> ( ( S Dn ( H ` k ) ) ` ( c ` k ) ) e. ( X -cn-> CC ) ) |
49 |
47 48
|
eqeltrrd |
|- ( ( ( ph /\ c e. ( C ` N ) ) /\ k e. ( 0 ... M ) ) -> ( x e. X |-> ( ( ( S Dn ( H ` k ) ) ` ( c ` k ) ) ` x ) ) e. ( X -cn-> CC ) ) |
50 |
12 16 46 49
|
fprodcncf |
|- ( ( ph /\ c e. ( C ` N ) ) -> ( x e. X |-> prod_ k e. ( 0 ... M ) ( ( ( S Dn ( H ` k ) ) ` ( c ` k ) ) ` x ) ) e. ( X -cn-> CC ) ) |
51 |
36 50
|
mulcncf |
|- ( ( ph /\ c e. ( C ` N ) ) -> ( x e. X |-> ( ( ( ! ` N ) / prod_ k e. ( 0 ... M ) ( ! ` ( c ` k ) ) ) x. prod_ k e. ( 0 ... M ) ( ( ( S Dn ( H ` k ) ) ` ( c ` k ) ) ` x ) ) ) e. ( X -cn-> CC ) ) |
52 |
10 11 51
|
fsumcncf |
|- ( ph -> ( x e. X |-> sum_ c e. ( C ` N ) ( ( ( ! ` N ) / prod_ k e. ( 0 ... M ) ( ! ` ( c ` k ) ) ) x. prod_ k e. ( 0 ... M ) ( ( ( S Dn ( H ` k ) ) ` ( c ` k ) ) ` x ) ) ) e. ( X -cn-> CC ) ) |
53 |
9 52
|
eqeltrd |
|- ( ph -> ( ( S Dn F ) ` N ) e. ( X -cn-> CC ) ) |