Step |
Hyp |
Ref |
Expression |
1 |
|
etransclem45.p |
|- ( ph -> P e. NN ) |
2 |
|
etransclem45.m |
|- ( ph -> M e. NN0 ) |
3 |
|
etransclem45.f |
|- F = ( x e. RR |-> ( ( x ^ ( P - 1 ) ) x. prod_ j e. ( 1 ... M ) ( ( x - j ) ^ P ) ) ) |
4 |
|
etransclem45.a |
|- ( ph -> A : NN0 --> ZZ ) |
5 |
|
etransclem45.k |
|- K = ( sum_ k e. ( ( 0 ... M ) X. ( 0 ... R ) ) ( ( A ` ( 1st ` k ) ) x. ( ( ( RR Dn F ) ` ( 2nd ` k ) ) ` ( 1st ` k ) ) ) / ( ! ` ( P - 1 ) ) ) |
6 |
|
fzfi |
|- ( 0 ... M ) e. Fin |
7 |
|
fzfi |
|- ( 0 ... R ) e. Fin |
8 |
|
xpfi |
|- ( ( ( 0 ... M ) e. Fin /\ ( 0 ... R ) e. Fin ) -> ( ( 0 ... M ) X. ( 0 ... R ) ) e. Fin ) |
9 |
6 7 8
|
mp2an |
|- ( ( 0 ... M ) X. ( 0 ... R ) ) e. Fin |
10 |
9
|
a1i |
|- ( ph -> ( ( 0 ... M ) X. ( 0 ... R ) ) e. Fin ) |
11 |
|
nnm1nn0 |
|- ( P e. NN -> ( P - 1 ) e. NN0 ) |
12 |
1 11
|
syl |
|- ( ph -> ( P - 1 ) e. NN0 ) |
13 |
12
|
faccld |
|- ( ph -> ( ! ` ( P - 1 ) ) e. NN ) |
14 |
13
|
nncnd |
|- ( ph -> ( ! ` ( P - 1 ) ) e. CC ) |
15 |
4
|
adantr |
|- ( ( ph /\ k e. ( ( 0 ... M ) X. ( 0 ... R ) ) ) -> A : NN0 --> ZZ ) |
16 |
|
xp1st |
|- ( k e. ( ( 0 ... M ) X. ( 0 ... R ) ) -> ( 1st ` k ) e. ( 0 ... M ) ) |
17 |
|
elfznn0 |
|- ( ( 1st ` k ) e. ( 0 ... M ) -> ( 1st ` k ) e. NN0 ) |
18 |
16 17
|
syl |
|- ( k e. ( ( 0 ... M ) X. ( 0 ... R ) ) -> ( 1st ` k ) e. NN0 ) |
19 |
18
|
adantl |
|- ( ( ph /\ k e. ( ( 0 ... M ) X. ( 0 ... R ) ) ) -> ( 1st ` k ) e. NN0 ) |
20 |
15 19
|
ffvelrnd |
|- ( ( ph /\ k e. ( ( 0 ... M ) X. ( 0 ... R ) ) ) -> ( A ` ( 1st ` k ) ) e. ZZ ) |
21 |
20
|
zcnd |
|- ( ( ph /\ k e. ( ( 0 ... M ) X. ( 0 ... R ) ) ) -> ( A ` ( 1st ` k ) ) e. CC ) |
22 |
|
reelprrecn |
|- RR e. { RR , CC } |
23 |
22
|
a1i |
|- ( ( ph /\ k e. ( ( 0 ... M ) X. ( 0 ... R ) ) ) -> RR e. { RR , CC } ) |
24 |
|
reopn |
|- RR e. ( topGen ` ran (,) ) |
25 |
|
eqid |
|- ( TopOpen ` CCfld ) = ( TopOpen ` CCfld ) |
26 |
25
|
tgioo2 |
|- ( topGen ` ran (,) ) = ( ( TopOpen ` CCfld ) |`t RR ) |
27 |
24 26
|
eleqtri |
|- RR e. ( ( TopOpen ` CCfld ) |`t RR ) |
28 |
27
|
a1i |
|- ( ( ph /\ k e. ( ( 0 ... M ) X. ( 0 ... R ) ) ) -> RR e. ( ( TopOpen ` CCfld ) |`t RR ) ) |
29 |
1
|
adantr |
|- ( ( ph /\ k e. ( ( 0 ... M ) X. ( 0 ... R ) ) ) -> P e. NN ) |
30 |
2
|
adantr |
|- ( ( ph /\ k e. ( ( 0 ... M ) X. ( 0 ... R ) ) ) -> M e. NN0 ) |
31 |
|
xp2nd |
|- ( k e. ( ( 0 ... M ) X. ( 0 ... R ) ) -> ( 2nd ` k ) e. ( 0 ... R ) ) |
32 |
|
elfznn0 |
|- ( ( 2nd ` k ) e. ( 0 ... R ) -> ( 2nd ` k ) e. NN0 ) |
33 |
31 32
|
syl |
|- ( k e. ( ( 0 ... M ) X. ( 0 ... R ) ) -> ( 2nd ` k ) e. NN0 ) |
34 |
33
|
adantl |
|- ( ( ph /\ k e. ( ( 0 ... M ) X. ( 0 ... R ) ) ) -> ( 2nd ` k ) e. NN0 ) |
35 |
23 28 29 30 3 34
|
etransclem33 |
|- ( ( ph /\ k e. ( ( 0 ... M ) X. ( 0 ... R ) ) ) -> ( ( RR Dn F ) ` ( 2nd ` k ) ) : RR --> CC ) |
36 |
19
|
nn0red |
|- ( ( ph /\ k e. ( ( 0 ... M ) X. ( 0 ... R ) ) ) -> ( 1st ` k ) e. RR ) |
37 |
35 36
|
ffvelrnd |
|- ( ( ph /\ k e. ( ( 0 ... M ) X. ( 0 ... R ) ) ) -> ( ( ( RR Dn F ) ` ( 2nd ` k ) ) ` ( 1st ` k ) ) e. CC ) |
38 |
21 37
|
mulcld |
|- ( ( ph /\ k e. ( ( 0 ... M ) X. ( 0 ... R ) ) ) -> ( ( A ` ( 1st ` k ) ) x. ( ( ( RR Dn F ) ` ( 2nd ` k ) ) ` ( 1st ` k ) ) ) e. CC ) |
39 |
13
|
nnne0d |
|- ( ph -> ( ! ` ( P - 1 ) ) =/= 0 ) |
40 |
10 14 38 39
|
fsumdivc |
|- ( ph -> ( sum_ k e. ( ( 0 ... M ) X. ( 0 ... R ) ) ( ( A ` ( 1st ` k ) ) x. ( ( ( RR Dn F ) ` ( 2nd ` k ) ) ` ( 1st ` k ) ) ) / ( ! ` ( P - 1 ) ) ) = sum_ k e. ( ( 0 ... M ) X. ( 0 ... R ) ) ( ( ( A ` ( 1st ` k ) ) x. ( ( ( RR Dn F ) ` ( 2nd ` k ) ) ` ( 1st ` k ) ) ) / ( ! ` ( P - 1 ) ) ) ) |
41 |
14
|
adantr |
|- ( ( ph /\ k e. ( ( 0 ... M ) X. ( 0 ... R ) ) ) -> ( ! ` ( P - 1 ) ) e. CC ) |
42 |
39
|
adantr |
|- ( ( ph /\ k e. ( ( 0 ... M ) X. ( 0 ... R ) ) ) -> ( ! ` ( P - 1 ) ) =/= 0 ) |
43 |
21 37 41 42
|
divassd |
|- ( ( ph /\ k e. ( ( 0 ... M ) X. ( 0 ... R ) ) ) -> ( ( ( A ` ( 1st ` k ) ) x. ( ( ( RR Dn F ) ` ( 2nd ` k ) ) ` ( 1st ` k ) ) ) / ( ! ` ( P - 1 ) ) ) = ( ( A ` ( 1st ` k ) ) x. ( ( ( ( RR Dn F ) ` ( 2nd ` k ) ) ` ( 1st ` k ) ) / ( ! ` ( P - 1 ) ) ) ) ) |
44 |
|
etransclem5 |
|- ( k e. ( 0 ... M ) |-> ( y e. RR |-> ( ( y - k ) ^ if ( k = 0 , ( P - 1 ) , P ) ) ) ) = ( j e. ( 0 ... M ) |-> ( x e. RR |-> ( ( x - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) ) ) |
45 |
|
etransclem11 |
|- ( m e. NN0 |-> { d e. ( ( 0 ... m ) ^m ( 0 ... M ) ) | sum_ k e. ( 0 ... M ) ( d ` k ) = m } ) = ( n e. NN0 |-> { c e. ( ( 0 ... n ) ^m ( 0 ... M ) ) | sum_ j e. ( 0 ... M ) ( c ` j ) = n } ) |
46 |
16
|
adantl |
|- ( ( ph /\ k e. ( ( 0 ... M ) X. ( 0 ... R ) ) ) -> ( 1st ` k ) e. ( 0 ... M ) ) |
47 |
23 28 29 30 3 34 44 45 46 36
|
etransclem37 |
|- ( ( ph /\ k e. ( ( 0 ... M ) X. ( 0 ... R ) ) ) -> ( ! ` ( P - 1 ) ) || ( ( ( RR Dn F ) ` ( 2nd ` k ) ) ` ( 1st ` k ) ) ) |
48 |
13
|
nnzd |
|- ( ph -> ( ! ` ( P - 1 ) ) e. ZZ ) |
49 |
48
|
adantr |
|- ( ( ph /\ k e. ( ( 0 ... M ) X. ( 0 ... R ) ) ) -> ( ! ` ( P - 1 ) ) e. ZZ ) |
50 |
19
|
nn0zd |
|- ( ( ph /\ k e. ( ( 0 ... M ) X. ( 0 ... R ) ) ) -> ( 1st ` k ) e. ZZ ) |
51 |
23 28 29 30 3 34 36 50
|
etransclem42 |
|- ( ( ph /\ k e. ( ( 0 ... M ) X. ( 0 ... R ) ) ) -> ( ( ( RR Dn F ) ` ( 2nd ` k ) ) ` ( 1st ` k ) ) e. ZZ ) |
52 |
|
dvdsval2 |
|- ( ( ( ! ` ( P - 1 ) ) e. ZZ /\ ( ! ` ( P - 1 ) ) =/= 0 /\ ( ( ( RR Dn F ) ` ( 2nd ` k ) ) ` ( 1st ` k ) ) e. ZZ ) -> ( ( ! ` ( P - 1 ) ) || ( ( ( RR Dn F ) ` ( 2nd ` k ) ) ` ( 1st ` k ) ) <-> ( ( ( ( RR Dn F ) ` ( 2nd ` k ) ) ` ( 1st ` k ) ) / ( ! ` ( P - 1 ) ) ) e. ZZ ) ) |
53 |
49 42 51 52
|
syl3anc |
|- ( ( ph /\ k e. ( ( 0 ... M ) X. ( 0 ... R ) ) ) -> ( ( ! ` ( P - 1 ) ) || ( ( ( RR Dn F ) ` ( 2nd ` k ) ) ` ( 1st ` k ) ) <-> ( ( ( ( RR Dn F ) ` ( 2nd ` k ) ) ` ( 1st ` k ) ) / ( ! ` ( P - 1 ) ) ) e. ZZ ) ) |
54 |
47 53
|
mpbid |
|- ( ( ph /\ k e. ( ( 0 ... M ) X. ( 0 ... R ) ) ) -> ( ( ( ( RR Dn F ) ` ( 2nd ` k ) ) ` ( 1st ` k ) ) / ( ! ` ( P - 1 ) ) ) e. ZZ ) |
55 |
20 54
|
zmulcld |
|- ( ( ph /\ k e. ( ( 0 ... M ) X. ( 0 ... R ) ) ) -> ( ( A ` ( 1st ` k ) ) x. ( ( ( ( RR Dn F ) ` ( 2nd ` k ) ) ` ( 1st ` k ) ) / ( ! ` ( P - 1 ) ) ) ) e. ZZ ) |
56 |
43 55
|
eqeltrd |
|- ( ( ph /\ k e. ( ( 0 ... M ) X. ( 0 ... R ) ) ) -> ( ( ( A ` ( 1st ` k ) ) x. ( ( ( RR Dn F ) ` ( 2nd ` k ) ) ` ( 1st ` k ) ) ) / ( ! ` ( P - 1 ) ) ) e. ZZ ) |
57 |
10 56
|
fsumzcl |
|- ( ph -> sum_ k e. ( ( 0 ... M ) X. ( 0 ... R ) ) ( ( ( A ` ( 1st ` k ) ) x. ( ( ( RR Dn F ) ` ( 2nd ` k ) ) ` ( 1st ` k ) ) ) / ( ! ` ( P - 1 ) ) ) e. ZZ ) |
58 |
40 57
|
eqeltrd |
|- ( ph -> ( sum_ k e. ( ( 0 ... M ) X. ( 0 ... R ) ) ( ( A ` ( 1st ` k ) ) x. ( ( ( RR Dn F ) ` ( 2nd ` k ) ) ` ( 1st ` k ) ) ) / ( ! ` ( P - 1 ) ) ) e. ZZ ) |
59 |
5 58
|
eqeltrid |
|- ( ph -> K e. ZZ ) |