Step |
Hyp |
Ref |
Expression |
1 |
|
etransclem27.s |
|- ( ph -> S e. { RR , CC } ) |
2 |
|
etransclem27.x |
|- ( ph -> X e. ( ( TopOpen ` CCfld ) |`t S ) ) |
3 |
|
etransclem27.p |
|- ( ph -> P e. NN ) |
4 |
|
etransclem27.h |
|- H = ( j e. ( 0 ... M ) |-> ( x e. X |-> ( ( x - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) ) ) |
5 |
|
etransclem27.cfi |
|- ( ph -> C e. Fin ) |
6 |
|
etransclem27.cf |
|- ( ph -> C : dom C --> ( NN0 ^m ( 0 ... M ) ) ) |
7 |
|
etransclem27.g |
|- G = ( x e. X |-> sum_ l e. dom C prod_ j e. ( 0 ... M ) ( ( ( S Dn ( H ` j ) ) ` ( ( C ` l ) ` j ) ) ` x ) ) |
8 |
|
etransclem27.jx |
|- ( ph -> J e. X ) |
9 |
|
etransclem27.jz |
|- ( ph -> J e. ZZ ) |
10 |
|
fveq2 |
|- ( x = J -> ( ( ( S Dn ( H ` j ) ) ` ( ( C ` l ) ` j ) ) ` x ) = ( ( ( S Dn ( H ` j ) ) ` ( ( C ` l ) ` j ) ) ` J ) ) |
11 |
10
|
prodeq2ad |
|- ( x = J -> prod_ j e. ( 0 ... M ) ( ( ( S Dn ( H ` j ) ) ` ( ( C ` l ) ` j ) ) ` x ) = prod_ j e. ( 0 ... M ) ( ( ( S Dn ( H ` j ) ) ` ( ( C ` l ) ` j ) ) ` J ) ) |
12 |
11
|
sumeq2sdv |
|- ( x = J -> sum_ l e. dom C prod_ j e. ( 0 ... M ) ( ( ( S Dn ( H ` j ) ) ` ( ( C ` l ) ` j ) ) ` x ) = sum_ l e. dom C prod_ j e. ( 0 ... M ) ( ( ( S Dn ( H ` j ) ) ` ( ( C ` l ) ` j ) ) ` J ) ) |
13 |
|
dmfi |
|- ( C e. Fin -> dom C e. Fin ) |
14 |
5 13
|
syl |
|- ( ph -> dom C e. Fin ) |
15 |
|
fzfid |
|- ( ( ph /\ l e. dom C ) -> ( 0 ... M ) e. Fin ) |
16 |
1
|
ad2antrr |
|- ( ( ( ph /\ l e. dom C ) /\ j e. ( 0 ... M ) ) -> S e. { RR , CC } ) |
17 |
2
|
ad2antrr |
|- ( ( ( ph /\ l e. dom C ) /\ j e. ( 0 ... M ) ) -> X e. ( ( TopOpen ` CCfld ) |`t S ) ) |
18 |
3
|
ad2antrr |
|- ( ( ( ph /\ l e. dom C ) /\ j e. ( 0 ... M ) ) -> P e. NN ) |
19 |
|
etransclem5 |
|- ( j e. ( 0 ... M ) |-> ( x e. X |-> ( ( x - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) ) ) = ( z e. ( 0 ... M ) |-> ( y e. X |-> ( ( y - z ) ^ if ( z = 0 , ( P - 1 ) , P ) ) ) ) |
20 |
4 19
|
eqtri |
|- H = ( z e. ( 0 ... M ) |-> ( y e. X |-> ( ( y - z ) ^ if ( z = 0 , ( P - 1 ) , P ) ) ) ) |
21 |
|
simpr |
|- ( ( ( ph /\ l e. dom C ) /\ j e. ( 0 ... M ) ) -> j e. ( 0 ... M ) ) |
22 |
6
|
ffvelrnda |
|- ( ( ph /\ l e. dom C ) -> ( C ` l ) e. ( NN0 ^m ( 0 ... M ) ) ) |
23 |
|
elmapi |
|- ( ( C ` l ) e. ( NN0 ^m ( 0 ... M ) ) -> ( C ` l ) : ( 0 ... M ) --> NN0 ) |
24 |
22 23
|
syl |
|- ( ( ph /\ l e. dom C ) -> ( C ` l ) : ( 0 ... M ) --> NN0 ) |
25 |
24
|
ffvelrnda |
|- ( ( ( ph /\ l e. dom C ) /\ j e. ( 0 ... M ) ) -> ( ( C ` l ) ` j ) e. NN0 ) |
26 |
16 17 18 20 21 25
|
etransclem20 |
|- ( ( ( ph /\ l e. dom C ) /\ j e. ( 0 ... M ) ) -> ( ( S Dn ( H ` j ) ) ` ( ( C ` l ) ` j ) ) : X --> CC ) |
27 |
8
|
ad2antrr |
|- ( ( ( ph /\ l e. dom C ) /\ j e. ( 0 ... M ) ) -> J e. X ) |
28 |
26 27
|
ffvelrnd |
|- ( ( ( ph /\ l e. dom C ) /\ j e. ( 0 ... M ) ) -> ( ( ( S Dn ( H ` j ) ) ` ( ( C ` l ) ` j ) ) ` J ) e. CC ) |
29 |
15 28
|
fprodcl |
|- ( ( ph /\ l e. dom C ) -> prod_ j e. ( 0 ... M ) ( ( ( S Dn ( H ` j ) ) ` ( ( C ` l ) ` j ) ) ` J ) e. CC ) |
30 |
14 29
|
fsumcl |
|- ( ph -> sum_ l e. dom C prod_ j e. ( 0 ... M ) ( ( ( S Dn ( H ` j ) ) ` ( ( C ` l ) ` j ) ) ` J ) e. CC ) |
31 |
7 12 8 30
|
fvmptd3 |
|- ( ph -> ( G ` J ) = sum_ l e. dom C prod_ j e. ( 0 ... M ) ( ( ( S Dn ( H ` j ) ) ` ( ( C ` l ) ` j ) ) ` J ) ) |
32 |
16 17 18 20 21 25 27
|
etransclem21 |
|- ( ( ( ph /\ l e. dom C ) /\ j e. ( 0 ... M ) ) -> ( ( ( S Dn ( H ` j ) ) ` ( ( C ` l ) ` j ) ) ` J ) = if ( if ( j = 0 , ( P - 1 ) , P ) < ( ( C ` l ) ` j ) , 0 , ( ( ( ! ` if ( j = 0 , ( P - 1 ) , P ) ) / ( ! ` ( if ( j = 0 , ( P - 1 ) , P ) - ( ( C ` l ) ` j ) ) ) ) x. ( ( J - j ) ^ ( if ( j = 0 , ( P - 1 ) , P ) - ( ( C ` l ) ` j ) ) ) ) ) ) |
33 |
|
iftrue |
|- ( if ( j = 0 , ( P - 1 ) , P ) < ( ( C ` l ) ` j ) -> if ( if ( j = 0 , ( P - 1 ) , P ) < ( ( C ` l ) ` j ) , 0 , ( ( ( ! ` if ( j = 0 , ( P - 1 ) , P ) ) / ( ! ` ( if ( j = 0 , ( P - 1 ) , P ) - ( ( C ` l ) ` j ) ) ) ) x. ( ( J - j ) ^ ( if ( j = 0 , ( P - 1 ) , P ) - ( ( C ` l ) ` j ) ) ) ) ) = 0 ) |
34 |
|
0zd |
|- ( if ( j = 0 , ( P - 1 ) , P ) < ( ( C ` l ) ` j ) -> 0 e. ZZ ) |
35 |
33 34
|
eqeltrd |
|- ( if ( j = 0 , ( P - 1 ) , P ) < ( ( C ` l ) ` j ) -> if ( if ( j = 0 , ( P - 1 ) , P ) < ( ( C ` l ) ` j ) , 0 , ( ( ( ! ` if ( j = 0 , ( P - 1 ) , P ) ) / ( ! ` ( if ( j = 0 , ( P - 1 ) , P ) - ( ( C ` l ) ` j ) ) ) ) x. ( ( J - j ) ^ ( if ( j = 0 , ( P - 1 ) , P ) - ( ( C ` l ) ` j ) ) ) ) ) e. ZZ ) |
36 |
35
|
adantl |
|- ( ( ( ( ph /\ l e. dom C ) /\ j e. ( 0 ... M ) ) /\ if ( j = 0 , ( P - 1 ) , P ) < ( ( C ` l ) ` j ) ) -> if ( if ( j = 0 , ( P - 1 ) , P ) < ( ( C ` l ) ` j ) , 0 , ( ( ( ! ` if ( j = 0 , ( P - 1 ) , P ) ) / ( ! ` ( if ( j = 0 , ( P - 1 ) , P ) - ( ( C ` l ) ` j ) ) ) ) x. ( ( J - j ) ^ ( if ( j = 0 , ( P - 1 ) , P ) - ( ( C ` l ) ` j ) ) ) ) ) e. ZZ ) |
37 |
|
0zd |
|- ( ( ( ( ph /\ l e. dom C ) /\ j e. ( 0 ... M ) ) /\ -. if ( j = 0 , ( P - 1 ) , P ) < ( ( C ` l ) ` j ) ) -> 0 e. ZZ ) |
38 |
|
nnm1nn0 |
|- ( P e. NN -> ( P - 1 ) e. NN0 ) |
39 |
3 38
|
syl |
|- ( ph -> ( P - 1 ) e. NN0 ) |
40 |
3
|
nnnn0d |
|- ( ph -> P e. NN0 ) |
41 |
39 40
|
ifcld |
|- ( ph -> if ( j = 0 , ( P - 1 ) , P ) e. NN0 ) |
42 |
41
|
nn0zd |
|- ( ph -> if ( j = 0 , ( P - 1 ) , P ) e. ZZ ) |
43 |
42
|
ad3antrrr |
|- ( ( ( ( ph /\ l e. dom C ) /\ j e. ( 0 ... M ) ) /\ -. if ( j = 0 , ( P - 1 ) , P ) < ( ( C ` l ) ` j ) ) -> if ( j = 0 , ( P - 1 ) , P ) e. ZZ ) |
44 |
25
|
nn0zd |
|- ( ( ( ph /\ l e. dom C ) /\ j e. ( 0 ... M ) ) -> ( ( C ` l ) ` j ) e. ZZ ) |
45 |
44
|
adantr |
|- ( ( ( ( ph /\ l e. dom C ) /\ j e. ( 0 ... M ) ) /\ -. if ( j = 0 , ( P - 1 ) , P ) < ( ( C ` l ) ` j ) ) -> ( ( C ` l ) ` j ) e. ZZ ) |
46 |
43 45
|
zsubcld |
|- ( ( ( ( ph /\ l e. dom C ) /\ j e. ( 0 ... M ) ) /\ -. if ( j = 0 , ( P - 1 ) , P ) < ( ( C ` l ) ` j ) ) -> ( if ( j = 0 , ( P - 1 ) , P ) - ( ( C ` l ) ` j ) ) e. ZZ ) |
47 |
45
|
zred |
|- ( ( ( ( ph /\ l e. dom C ) /\ j e. ( 0 ... M ) ) /\ -. if ( j = 0 , ( P - 1 ) , P ) < ( ( C ` l ) ` j ) ) -> ( ( C ` l ) ` j ) e. RR ) |
48 |
43
|
zred |
|- ( ( ( ( ph /\ l e. dom C ) /\ j e. ( 0 ... M ) ) /\ -. if ( j = 0 , ( P - 1 ) , P ) < ( ( C ` l ) ` j ) ) -> if ( j = 0 , ( P - 1 ) , P ) e. RR ) |
49 |
|
simpr |
|- ( ( ( ( ph /\ l e. dom C ) /\ j e. ( 0 ... M ) ) /\ -. if ( j = 0 , ( P - 1 ) , P ) < ( ( C ` l ) ` j ) ) -> -. if ( j = 0 , ( P - 1 ) , P ) < ( ( C ` l ) ` j ) ) |
50 |
47 48 49
|
nltled |
|- ( ( ( ( ph /\ l e. dom C ) /\ j e. ( 0 ... M ) ) /\ -. if ( j = 0 , ( P - 1 ) , P ) < ( ( C ` l ) ` j ) ) -> ( ( C ` l ) ` j ) <_ if ( j = 0 , ( P - 1 ) , P ) ) |
51 |
48 47
|
subge0d |
|- ( ( ( ( ph /\ l e. dom C ) /\ j e. ( 0 ... M ) ) /\ -. if ( j = 0 , ( P - 1 ) , P ) < ( ( C ` l ) ` j ) ) -> ( 0 <_ ( if ( j = 0 , ( P - 1 ) , P ) - ( ( C ` l ) ` j ) ) <-> ( ( C ` l ) ` j ) <_ if ( j = 0 , ( P - 1 ) , P ) ) ) |
52 |
50 51
|
mpbird |
|- ( ( ( ( ph /\ l e. dom C ) /\ j e. ( 0 ... M ) ) /\ -. if ( j = 0 , ( P - 1 ) , P ) < ( ( C ` l ) ` j ) ) -> 0 <_ ( if ( j = 0 , ( P - 1 ) , P ) - ( ( C ` l ) ` j ) ) ) |
53 |
|
0red |
|- ( ( ( ph /\ l e. dom C ) /\ j e. ( 0 ... M ) ) -> 0 e. RR ) |
54 |
25
|
nn0red |
|- ( ( ( ph /\ l e. dom C ) /\ j e. ( 0 ... M ) ) -> ( ( C ` l ) ` j ) e. RR ) |
55 |
41
|
nn0red |
|- ( ph -> if ( j = 0 , ( P - 1 ) , P ) e. RR ) |
56 |
55
|
ad2antrr |
|- ( ( ( ph /\ l e. dom C ) /\ j e. ( 0 ... M ) ) -> if ( j = 0 , ( P - 1 ) , P ) e. RR ) |
57 |
25
|
nn0ge0d |
|- ( ( ( ph /\ l e. dom C ) /\ j e. ( 0 ... M ) ) -> 0 <_ ( ( C ` l ) ` j ) ) |
58 |
53 54 56 57
|
lesub2dd |
|- ( ( ( ph /\ l e. dom C ) /\ j e. ( 0 ... M ) ) -> ( if ( j = 0 , ( P - 1 ) , P ) - ( ( C ` l ) ` j ) ) <_ ( if ( j = 0 , ( P - 1 ) , P ) - 0 ) ) |
59 |
56
|
recnd |
|- ( ( ( ph /\ l e. dom C ) /\ j e. ( 0 ... M ) ) -> if ( j = 0 , ( P - 1 ) , P ) e. CC ) |
60 |
59
|
subid1d |
|- ( ( ( ph /\ l e. dom C ) /\ j e. ( 0 ... M ) ) -> ( if ( j = 0 , ( P - 1 ) , P ) - 0 ) = if ( j = 0 , ( P - 1 ) , P ) ) |
61 |
58 60
|
breqtrd |
|- ( ( ( ph /\ l e. dom C ) /\ j e. ( 0 ... M ) ) -> ( if ( j = 0 , ( P - 1 ) , P ) - ( ( C ` l ) ` j ) ) <_ if ( j = 0 , ( P - 1 ) , P ) ) |
62 |
61
|
adantr |
|- ( ( ( ( ph /\ l e. dom C ) /\ j e. ( 0 ... M ) ) /\ -. if ( j = 0 , ( P - 1 ) , P ) < ( ( C ` l ) ` j ) ) -> ( if ( j = 0 , ( P - 1 ) , P ) - ( ( C ` l ) ` j ) ) <_ if ( j = 0 , ( P - 1 ) , P ) ) |
63 |
37 43 46 52 62
|
elfzd |
|- ( ( ( ( ph /\ l e. dom C ) /\ j e. ( 0 ... M ) ) /\ -. if ( j = 0 , ( P - 1 ) , P ) < ( ( C ` l ) ` j ) ) -> ( if ( j = 0 , ( P - 1 ) , P ) - ( ( C ` l ) ` j ) ) e. ( 0 ... if ( j = 0 , ( P - 1 ) , P ) ) ) |
64 |
|
permnn |
|- ( ( if ( j = 0 , ( P - 1 ) , P ) - ( ( C ` l ) ` j ) ) e. ( 0 ... if ( j = 0 , ( P - 1 ) , P ) ) -> ( ( ! ` if ( j = 0 , ( P - 1 ) , P ) ) / ( ! ` ( if ( j = 0 , ( P - 1 ) , P ) - ( ( C ` l ) ` j ) ) ) ) e. NN ) |
65 |
63 64
|
syl |
|- ( ( ( ( ph /\ l e. dom C ) /\ j e. ( 0 ... M ) ) /\ -. if ( j = 0 , ( P - 1 ) , P ) < ( ( C ` l ) ` j ) ) -> ( ( ! ` if ( j = 0 , ( P - 1 ) , P ) ) / ( ! ` ( if ( j = 0 , ( P - 1 ) , P ) - ( ( C ` l ) ` j ) ) ) ) e. NN ) |
66 |
65
|
nnzd |
|- ( ( ( ( ph /\ l e. dom C ) /\ j e. ( 0 ... M ) ) /\ -. if ( j = 0 , ( P - 1 ) , P ) < ( ( C ` l ) ` j ) ) -> ( ( ! ` if ( j = 0 , ( P - 1 ) , P ) ) / ( ! ` ( if ( j = 0 , ( P - 1 ) , P ) - ( ( C ` l ) ` j ) ) ) ) e. ZZ ) |
67 |
9
|
ad3antrrr |
|- ( ( ( ( ph /\ l e. dom C ) /\ j e. ( 0 ... M ) ) /\ -. if ( j = 0 , ( P - 1 ) , P ) < ( ( C ` l ) ` j ) ) -> J e. ZZ ) |
68 |
|
elfzelz |
|- ( j e. ( 0 ... M ) -> j e. ZZ ) |
69 |
68
|
ad2antlr |
|- ( ( ( ( ph /\ l e. dom C ) /\ j e. ( 0 ... M ) ) /\ -. if ( j = 0 , ( P - 1 ) , P ) < ( ( C ` l ) ` j ) ) -> j e. ZZ ) |
70 |
67 69
|
zsubcld |
|- ( ( ( ( ph /\ l e. dom C ) /\ j e. ( 0 ... M ) ) /\ -. if ( j = 0 , ( P - 1 ) , P ) < ( ( C ` l ) ` j ) ) -> ( J - j ) e. ZZ ) |
71 |
|
elnn0z |
|- ( ( if ( j = 0 , ( P - 1 ) , P ) - ( ( C ` l ) ` j ) ) e. NN0 <-> ( ( if ( j = 0 , ( P - 1 ) , P ) - ( ( C ` l ) ` j ) ) e. ZZ /\ 0 <_ ( if ( j = 0 , ( P - 1 ) , P ) - ( ( C ` l ) ` j ) ) ) ) |
72 |
46 52 71
|
sylanbrc |
|- ( ( ( ( ph /\ l e. dom C ) /\ j e. ( 0 ... M ) ) /\ -. if ( j = 0 , ( P - 1 ) , P ) < ( ( C ` l ) ` j ) ) -> ( if ( j = 0 , ( P - 1 ) , P ) - ( ( C ` l ) ` j ) ) e. NN0 ) |
73 |
|
zexpcl |
|- ( ( ( J - j ) e. ZZ /\ ( if ( j = 0 , ( P - 1 ) , P ) - ( ( C ` l ) ` j ) ) e. NN0 ) -> ( ( J - j ) ^ ( if ( j = 0 , ( P - 1 ) , P ) - ( ( C ` l ) ` j ) ) ) e. ZZ ) |
74 |
70 72 73
|
syl2anc |
|- ( ( ( ( ph /\ l e. dom C ) /\ j e. ( 0 ... M ) ) /\ -. if ( j = 0 , ( P - 1 ) , P ) < ( ( C ` l ) ` j ) ) -> ( ( J - j ) ^ ( if ( j = 0 , ( P - 1 ) , P ) - ( ( C ` l ) ` j ) ) ) e. ZZ ) |
75 |
66 74
|
zmulcld |
|- ( ( ( ( ph /\ l e. dom C ) /\ j e. ( 0 ... M ) ) /\ -. if ( j = 0 , ( P - 1 ) , P ) < ( ( C ` l ) ` j ) ) -> ( ( ( ! ` if ( j = 0 , ( P - 1 ) , P ) ) / ( ! ` ( if ( j = 0 , ( P - 1 ) , P ) - ( ( C ` l ) ` j ) ) ) ) x. ( ( J - j ) ^ ( if ( j = 0 , ( P - 1 ) , P ) - ( ( C ` l ) ` j ) ) ) ) e. ZZ ) |
76 |
37 75
|
ifcld |
|- ( ( ( ( ph /\ l e. dom C ) /\ j e. ( 0 ... M ) ) /\ -. if ( j = 0 , ( P - 1 ) , P ) < ( ( C ` l ) ` j ) ) -> if ( if ( j = 0 , ( P - 1 ) , P ) < ( ( C ` l ) ` j ) , 0 , ( ( ( ! ` if ( j = 0 , ( P - 1 ) , P ) ) / ( ! ` ( if ( j = 0 , ( P - 1 ) , P ) - ( ( C ` l ) ` j ) ) ) ) x. ( ( J - j ) ^ ( if ( j = 0 , ( P - 1 ) , P ) - ( ( C ` l ) ` j ) ) ) ) ) e. ZZ ) |
77 |
36 76
|
pm2.61dan |
|- ( ( ( ph /\ l e. dom C ) /\ j e. ( 0 ... M ) ) -> if ( if ( j = 0 , ( P - 1 ) , P ) < ( ( C ` l ) ` j ) , 0 , ( ( ( ! ` if ( j = 0 , ( P - 1 ) , P ) ) / ( ! ` ( if ( j = 0 , ( P - 1 ) , P ) - ( ( C ` l ) ` j ) ) ) ) x. ( ( J - j ) ^ ( if ( j = 0 , ( P - 1 ) , P ) - ( ( C ` l ) ` j ) ) ) ) ) e. ZZ ) |
78 |
32 77
|
eqeltrd |
|- ( ( ( ph /\ l e. dom C ) /\ j e. ( 0 ... M ) ) -> ( ( ( S Dn ( H ` j ) ) ` ( ( C ` l ) ` j ) ) ` J ) e. ZZ ) |
79 |
15 78
|
fprodzcl |
|- ( ( ph /\ l e. dom C ) -> prod_ j e. ( 0 ... M ) ( ( ( S Dn ( H ` j ) ) ` ( ( C ` l ) ` j ) ) ` J ) e. ZZ ) |
80 |
14 79
|
fsumzcl |
|- ( ph -> sum_ l e. dom C prod_ j e. ( 0 ... M ) ( ( ( S Dn ( H ` j ) ) ` ( ( C ` l ) ` j ) ) ` J ) e. ZZ ) |
81 |
31 80
|
eqeltrd |
|- ( ph -> ( G ` J ) e. ZZ ) |