Step |
Hyp |
Ref |
Expression |
1 |
|
etransclem7.n |
|- ( ph -> P e. NN ) |
2 |
|
etransclem7.c |
|- ( ph -> C : ( 0 ... M ) --> ( 0 ... N ) ) |
3 |
|
etransclem7.j |
|- ( ph -> J e. ( 0 ... M ) ) |
4 |
|
fzfid |
|- ( ph -> ( 1 ... M ) e. Fin ) |
5 |
|
0zd |
|- ( ( ( ph /\ j e. ( 1 ... M ) ) /\ P < ( C ` j ) ) -> 0 e. ZZ ) |
6 |
|
0zd |
|- ( ( ( ph /\ j e. ( 1 ... M ) ) /\ -. P < ( C ` j ) ) -> 0 e. ZZ ) |
7 |
1
|
nnzd |
|- ( ph -> P e. ZZ ) |
8 |
7
|
ad2antrr |
|- ( ( ( ph /\ j e. ( 1 ... M ) ) /\ -. P < ( C ` j ) ) -> P e. ZZ ) |
9 |
7
|
adantr |
|- ( ( ph /\ j e. ( 1 ... M ) ) -> P e. ZZ ) |
10 |
2
|
adantr |
|- ( ( ph /\ j e. ( 1 ... M ) ) -> C : ( 0 ... M ) --> ( 0 ... N ) ) |
11 |
|
0zd |
|- ( j e. ( 1 ... M ) -> 0 e. ZZ ) |
12 |
|
fzp1ss |
|- ( 0 e. ZZ -> ( ( 0 + 1 ) ... M ) C_ ( 0 ... M ) ) |
13 |
11 12
|
syl |
|- ( j e. ( 1 ... M ) -> ( ( 0 + 1 ) ... M ) C_ ( 0 ... M ) ) |
14 |
|
id |
|- ( j e. ( 1 ... M ) -> j e. ( 1 ... M ) ) |
15 |
|
1e0p1 |
|- 1 = ( 0 + 1 ) |
16 |
15
|
oveq1i |
|- ( 1 ... M ) = ( ( 0 + 1 ) ... M ) |
17 |
14 16
|
eleqtrdi |
|- ( j e. ( 1 ... M ) -> j e. ( ( 0 + 1 ) ... M ) ) |
18 |
13 17
|
sseldd |
|- ( j e. ( 1 ... M ) -> j e. ( 0 ... M ) ) |
19 |
18
|
adantl |
|- ( ( ph /\ j e. ( 1 ... M ) ) -> j e. ( 0 ... M ) ) |
20 |
10 19
|
ffvelrnd |
|- ( ( ph /\ j e. ( 1 ... M ) ) -> ( C ` j ) e. ( 0 ... N ) ) |
21 |
20
|
elfzelzd |
|- ( ( ph /\ j e. ( 1 ... M ) ) -> ( C ` j ) e. ZZ ) |
22 |
9 21
|
zsubcld |
|- ( ( ph /\ j e. ( 1 ... M ) ) -> ( P - ( C ` j ) ) e. ZZ ) |
23 |
22
|
adantr |
|- ( ( ( ph /\ j e. ( 1 ... M ) ) /\ -. P < ( C ` j ) ) -> ( P - ( C ` j ) ) e. ZZ ) |
24 |
21
|
zred |
|- ( ( ph /\ j e. ( 1 ... M ) ) -> ( C ` j ) e. RR ) |
25 |
24
|
adantr |
|- ( ( ( ph /\ j e. ( 1 ... M ) ) /\ -. P < ( C ` j ) ) -> ( C ` j ) e. RR ) |
26 |
8
|
zred |
|- ( ( ( ph /\ j e. ( 1 ... M ) ) /\ -. P < ( C ` j ) ) -> P e. RR ) |
27 |
|
simpr |
|- ( ( ( ph /\ j e. ( 1 ... M ) ) /\ -. P < ( C ` j ) ) -> -. P < ( C ` j ) ) |
28 |
25 26 27
|
nltled |
|- ( ( ( ph /\ j e. ( 1 ... M ) ) /\ -. P < ( C ` j ) ) -> ( C ` j ) <_ P ) |
29 |
26 25
|
subge0d |
|- ( ( ( ph /\ j e. ( 1 ... M ) ) /\ -. P < ( C ` j ) ) -> ( 0 <_ ( P - ( C ` j ) ) <-> ( C ` j ) <_ P ) ) |
30 |
28 29
|
mpbird |
|- ( ( ( ph /\ j e. ( 1 ... M ) ) /\ -. P < ( C ` j ) ) -> 0 <_ ( P - ( C ` j ) ) ) |
31 |
|
elfzle1 |
|- ( ( C ` j ) e. ( 0 ... N ) -> 0 <_ ( C ` j ) ) |
32 |
20 31
|
syl |
|- ( ( ph /\ j e. ( 1 ... M ) ) -> 0 <_ ( C ` j ) ) |
33 |
32
|
adantr |
|- ( ( ( ph /\ j e. ( 1 ... M ) ) /\ -. P < ( C ` j ) ) -> 0 <_ ( C ` j ) ) |
34 |
26 25
|
subge02d |
|- ( ( ( ph /\ j e. ( 1 ... M ) ) /\ -. P < ( C ` j ) ) -> ( 0 <_ ( C ` j ) <-> ( P - ( C ` j ) ) <_ P ) ) |
35 |
33 34
|
mpbid |
|- ( ( ( ph /\ j e. ( 1 ... M ) ) /\ -. P < ( C ` j ) ) -> ( P - ( C ` j ) ) <_ P ) |
36 |
6 8 23 30 35
|
elfzd |
|- ( ( ( ph /\ j e. ( 1 ... M ) ) /\ -. P < ( C ` j ) ) -> ( P - ( C ` j ) ) e. ( 0 ... P ) ) |
37 |
|
permnn |
|- ( ( P - ( C ` j ) ) e. ( 0 ... P ) -> ( ( ! ` P ) / ( ! ` ( P - ( C ` j ) ) ) ) e. NN ) |
38 |
36 37
|
syl |
|- ( ( ( ph /\ j e. ( 1 ... M ) ) /\ -. P < ( C ` j ) ) -> ( ( ! ` P ) / ( ! ` ( P - ( C ` j ) ) ) ) e. NN ) |
39 |
38
|
nnzd |
|- ( ( ( ph /\ j e. ( 1 ... M ) ) /\ -. P < ( C ` j ) ) -> ( ( ! ` P ) / ( ! ` ( P - ( C ` j ) ) ) ) e. ZZ ) |
40 |
3
|
elfzelzd |
|- ( ph -> J e. ZZ ) |
41 |
40
|
adantr |
|- ( ( ph /\ j e. ( 1 ... M ) ) -> J e. ZZ ) |
42 |
|
elfzelz |
|- ( j e. ( 1 ... M ) -> j e. ZZ ) |
43 |
42
|
adantl |
|- ( ( ph /\ j e. ( 1 ... M ) ) -> j e. ZZ ) |
44 |
41 43
|
zsubcld |
|- ( ( ph /\ j e. ( 1 ... M ) ) -> ( J - j ) e. ZZ ) |
45 |
44
|
adantr |
|- ( ( ( ph /\ j e. ( 1 ... M ) ) /\ -. P < ( C ` j ) ) -> ( J - j ) e. ZZ ) |
46 |
|
elnn0z |
|- ( ( P - ( C ` j ) ) e. NN0 <-> ( ( P - ( C ` j ) ) e. ZZ /\ 0 <_ ( P - ( C ` j ) ) ) ) |
47 |
23 30 46
|
sylanbrc |
|- ( ( ( ph /\ j e. ( 1 ... M ) ) /\ -. P < ( C ` j ) ) -> ( P - ( C ` j ) ) e. NN0 ) |
48 |
|
zexpcl |
|- ( ( ( J - j ) e. ZZ /\ ( P - ( C ` j ) ) e. NN0 ) -> ( ( J - j ) ^ ( P - ( C ` j ) ) ) e. ZZ ) |
49 |
45 47 48
|
syl2anc |
|- ( ( ( ph /\ j e. ( 1 ... M ) ) /\ -. P < ( C ` j ) ) -> ( ( J - j ) ^ ( P - ( C ` j ) ) ) e. ZZ ) |
50 |
39 49
|
zmulcld |
|- ( ( ( ph /\ j e. ( 1 ... M ) ) /\ -. P < ( C ` j ) ) -> ( ( ( ! ` P ) / ( ! ` ( P - ( C ` j ) ) ) ) x. ( ( J - j ) ^ ( P - ( C ` j ) ) ) ) e. ZZ ) |
51 |
5 50
|
ifclda |
|- ( ( ph /\ j e. ( 1 ... M ) ) -> if ( P < ( C ` j ) , 0 , ( ( ( ! ` P ) / ( ! ` ( P - ( C ` j ) ) ) ) x. ( ( J - j ) ^ ( P - ( C ` j ) ) ) ) ) e. ZZ ) |
52 |
4 51
|
fprodzcl |
|- ( ph -> prod_ j e. ( 1 ... M ) if ( P < ( C ` j ) , 0 , ( ( ( ! ` P ) / ( ! ` ( P - ( C ` j ) ) ) ) x. ( ( J - j ) ^ ( P - ( C ` j ) ) ) ) ) e. ZZ ) |