Step |
Hyp |
Ref |
Expression |
1 |
|
etransclem31.s |
|- ( ph -> S e. { RR , CC } ) |
2 |
|
etransclem31.x |
|- ( ph -> X e. ( ( TopOpen ` CCfld ) |`t S ) ) |
3 |
|
etransclem31.p |
|- ( ph -> P e. NN ) |
4 |
|
etransclem31.m |
|- ( ph -> M e. NN0 ) |
5 |
|
etransclem31.f |
|- F = ( x e. X |-> ( ( x ^ ( P - 1 ) ) x. prod_ j e. ( 1 ... M ) ( ( x - j ) ^ P ) ) ) |
6 |
|
etransclem31.n |
|- ( ph -> N e. NN0 ) |
7 |
|
etransclem31.h |
|- H = ( j e. ( 0 ... M ) |-> ( x e. X |-> ( ( x - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) ) ) |
8 |
|
etransclem31.c |
|- C = ( n e. NN0 |-> { c e. ( ( 0 ... n ) ^m ( 0 ... M ) ) | sum_ j e. ( 0 ... M ) ( c ` j ) = n } ) |
9 |
|
etransclem31.y |
|- ( ph -> Y e. X ) |
10 |
1 2 3 4 5 6 7 8
|
etransclem30 |
|- ( ph -> ( ( S Dn F ) ` N ) = ( x e. X |-> sum_ c e. ( C ` N ) ( ( ( ! ` N ) / prod_ j e. ( 0 ... M ) ( ! ` ( c ` j ) ) ) x. prod_ j e. ( 0 ... M ) ( ( ( S Dn ( H ` j ) ) ` ( c ` j ) ) ` x ) ) ) ) |
11 |
|
fveq2 |
|- ( x = Y -> ( ( ( S Dn ( H ` j ) ) ` ( c ` j ) ) ` x ) = ( ( ( S Dn ( H ` j ) ) ` ( c ` j ) ) ` Y ) ) |
12 |
11
|
prodeq2ad |
|- ( x = Y -> prod_ j e. ( 0 ... M ) ( ( ( S Dn ( H ` j ) ) ` ( c ` j ) ) ` x ) = prod_ j e. ( 0 ... M ) ( ( ( S Dn ( H ` j ) ) ` ( c ` j ) ) ` Y ) ) |
13 |
12
|
oveq2d |
|- ( x = Y -> ( ( ( ! ` N ) / prod_ j e. ( 0 ... M ) ( ! ` ( c ` j ) ) ) x. prod_ j e. ( 0 ... M ) ( ( ( S Dn ( H ` j ) ) ` ( c ` j ) ) ` x ) ) = ( ( ( ! ` N ) / prod_ j e. ( 0 ... M ) ( ! ` ( c ` j ) ) ) x. prod_ j e. ( 0 ... M ) ( ( ( S Dn ( H ` j ) ) ` ( c ` j ) ) ` Y ) ) ) |
14 |
13
|
sumeq2sdv |
|- ( x = Y -> sum_ c e. ( C ` N ) ( ( ( ! ` N ) / prod_ j e. ( 0 ... M ) ( ! ` ( c ` j ) ) ) x. prod_ j e. ( 0 ... M ) ( ( ( S Dn ( H ` j ) ) ` ( c ` j ) ) ` x ) ) = sum_ c e. ( C ` N ) ( ( ( ! ` N ) / prod_ j e. ( 0 ... M ) ( ! ` ( c ` j ) ) ) x. prod_ j e. ( 0 ... M ) ( ( ( S Dn ( H ` j ) ) ` ( c ` j ) ) ` Y ) ) ) |
15 |
14
|
adantl |
|- ( ( ph /\ x = Y ) -> sum_ c e. ( C ` N ) ( ( ( ! ` N ) / prod_ j e. ( 0 ... M ) ( ! ` ( c ` j ) ) ) x. prod_ j e. ( 0 ... M ) ( ( ( S Dn ( H ` j ) ) ` ( c ` j ) ) ` x ) ) = sum_ c e. ( C ` N ) ( ( ( ! ` N ) / prod_ j e. ( 0 ... M ) ( ! ` ( c ` j ) ) ) x. prod_ j e. ( 0 ... M ) ( ( ( S Dn ( H ` j ) ) ` ( c ` j ) ) ` Y ) ) ) |
16 |
8 6
|
etransclem16 |
|- ( ph -> ( C ` N ) e. Fin ) |
17 |
6
|
faccld |
|- ( ph -> ( ! ` N ) e. NN ) |
18 |
17
|
nncnd |
|- ( ph -> ( ! ` N ) e. CC ) |
19 |
18
|
adantr |
|- ( ( ph /\ c e. ( C ` N ) ) -> ( ! ` N ) e. CC ) |
20 |
|
fzfid |
|- ( ( ph /\ c e. ( C ` N ) ) -> ( 0 ... M ) e. Fin ) |
21 |
|
fzssnn0 |
|- ( 0 ... N ) C_ NN0 |
22 |
|
ssrab2 |
|- { c e. ( ( 0 ... N ) ^m ( 0 ... M ) ) | sum_ j e. ( 0 ... M ) ( c ` j ) = N } C_ ( ( 0 ... N ) ^m ( 0 ... M ) ) |
23 |
|
simpr |
|- ( ( ph /\ c e. ( C ` N ) ) -> c e. ( C ` N ) ) |
24 |
8 6
|
etransclem12 |
|- ( ph -> ( C ` N ) = { c e. ( ( 0 ... N ) ^m ( 0 ... M ) ) | sum_ j e. ( 0 ... M ) ( c ` j ) = N } ) |
25 |
24
|
adantr |
|- ( ( ph /\ c e. ( C ` N ) ) -> ( C ` N ) = { c e. ( ( 0 ... N ) ^m ( 0 ... M ) ) | sum_ j e. ( 0 ... M ) ( c ` j ) = N } ) |
26 |
23 25
|
eleqtrd |
|- ( ( ph /\ c e. ( C ` N ) ) -> c e. { c e. ( ( 0 ... N ) ^m ( 0 ... M ) ) | sum_ j e. ( 0 ... M ) ( c ` j ) = N } ) |
27 |
22 26
|
sseldi |
|- ( ( ph /\ c e. ( C ` N ) ) -> c e. ( ( 0 ... N ) ^m ( 0 ... M ) ) ) |
28 |
27
|
adantr |
|- ( ( ( ph /\ c e. ( C ` N ) ) /\ j e. ( 0 ... M ) ) -> c e. ( ( 0 ... N ) ^m ( 0 ... M ) ) ) |
29 |
|
elmapi |
|- ( c e. ( ( 0 ... N ) ^m ( 0 ... M ) ) -> c : ( 0 ... M ) --> ( 0 ... N ) ) |
30 |
28 29
|
syl |
|- ( ( ( ph /\ c e. ( C ` N ) ) /\ j e. ( 0 ... M ) ) -> c : ( 0 ... M ) --> ( 0 ... N ) ) |
31 |
|
simpr |
|- ( ( ( ph /\ c e. ( C ` N ) ) /\ j e. ( 0 ... M ) ) -> j e. ( 0 ... M ) ) |
32 |
30 31
|
ffvelrnd |
|- ( ( ( ph /\ c e. ( C ` N ) ) /\ j e. ( 0 ... M ) ) -> ( c ` j ) e. ( 0 ... N ) ) |
33 |
21 32
|
sseldi |
|- ( ( ( ph /\ c e. ( C ` N ) ) /\ j e. ( 0 ... M ) ) -> ( c ` j ) e. NN0 ) |
34 |
33
|
faccld |
|- ( ( ( ph /\ c e. ( C ` N ) ) /\ j e. ( 0 ... M ) ) -> ( ! ` ( c ` j ) ) e. NN ) |
35 |
34
|
nncnd |
|- ( ( ( ph /\ c e. ( C ` N ) ) /\ j e. ( 0 ... M ) ) -> ( ! ` ( c ` j ) ) e. CC ) |
36 |
20 35
|
fprodcl |
|- ( ( ph /\ c e. ( C ` N ) ) -> prod_ j e. ( 0 ... M ) ( ! ` ( c ` j ) ) e. CC ) |
37 |
34
|
nnne0d |
|- ( ( ( ph /\ c e. ( C ` N ) ) /\ j e. ( 0 ... M ) ) -> ( ! ` ( c ` j ) ) =/= 0 ) |
38 |
20 35 37
|
fprodn0 |
|- ( ( ph /\ c e. ( C ` N ) ) -> prod_ j e. ( 0 ... M ) ( ! ` ( c ` j ) ) =/= 0 ) |
39 |
19 36 38
|
divcld |
|- ( ( ph /\ c e. ( C ` N ) ) -> ( ( ! ` N ) / prod_ j e. ( 0 ... M ) ( ! ` ( c ` j ) ) ) e. CC ) |
40 |
1
|
ad2antrr |
|- ( ( ( ph /\ c e. ( C ` N ) ) /\ j e. ( 0 ... M ) ) -> S e. { RR , CC } ) |
41 |
2
|
ad2antrr |
|- ( ( ( ph /\ c e. ( C ` N ) ) /\ j e. ( 0 ... M ) ) -> X e. ( ( TopOpen ` CCfld ) |`t S ) ) |
42 |
3
|
ad2antrr |
|- ( ( ( ph /\ c e. ( C ` N ) ) /\ j e. ( 0 ... M ) ) -> P e. NN ) |
43 |
|
etransclem5 |
|- ( j e. ( 0 ... M ) |-> ( x e. X |-> ( ( x - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) ) ) = ( k e. ( 0 ... M ) |-> ( y e. X |-> ( ( y - k ) ^ if ( k = 0 , ( P - 1 ) , P ) ) ) ) |
44 |
7 43
|
eqtri |
|- H = ( k e. ( 0 ... M ) |-> ( y e. X |-> ( ( y - k ) ^ if ( k = 0 , ( P - 1 ) , P ) ) ) ) |
45 |
40 41 42 44 31 33
|
etransclem20 |
|- ( ( ( ph /\ c e. ( C ` N ) ) /\ j e. ( 0 ... M ) ) -> ( ( S Dn ( H ` j ) ) ` ( c ` j ) ) : X --> CC ) |
46 |
9
|
ad2antrr |
|- ( ( ( ph /\ c e. ( C ` N ) ) /\ j e. ( 0 ... M ) ) -> Y e. X ) |
47 |
45 46
|
ffvelrnd |
|- ( ( ( ph /\ c e. ( C ` N ) ) /\ j e. ( 0 ... M ) ) -> ( ( ( S Dn ( H ` j ) ) ` ( c ` j ) ) ` Y ) e. CC ) |
48 |
20 47
|
fprodcl |
|- ( ( ph /\ c e. ( C ` N ) ) -> prod_ j e. ( 0 ... M ) ( ( ( S Dn ( H ` j ) ) ` ( c ` j ) ) ` Y ) e. CC ) |
49 |
39 48
|
mulcld |
|- ( ( ph /\ c e. ( C ` N ) ) -> ( ( ( ! ` N ) / prod_ j e. ( 0 ... M ) ( ! ` ( c ` j ) ) ) x. prod_ j e. ( 0 ... M ) ( ( ( S Dn ( H ` j ) ) ` ( c ` j ) ) ` Y ) ) e. CC ) |
50 |
16 49
|
fsumcl |
|- ( ph -> sum_ c e. ( C ` N ) ( ( ( ! ` N ) / prod_ j e. ( 0 ... M ) ( ! ` ( c ` j ) ) ) x. prod_ j e. ( 0 ... M ) ( ( ( S Dn ( H ` j ) ) ` ( c ` j ) ) ` Y ) ) e. CC ) |
51 |
10 15 9 50
|
fvmptd |
|- ( ph -> ( ( ( S Dn F ) ` N ) ` Y ) = sum_ c e. ( C ` N ) ( ( ( ! ` N ) / prod_ j e. ( 0 ... M ) ( ! ` ( c ` j ) ) ) x. prod_ j e. ( 0 ... M ) ( ( ( S Dn ( H ` j ) ) ` ( c ` j ) ) ` Y ) ) ) |
52 |
40 41 42 44 31 33 46
|
etransclem21 |
|- ( ( ( ph /\ c e. ( C ` N ) ) /\ j e. ( 0 ... M ) ) -> ( ( ( S Dn ( H ` j ) ) ` ( c ` j ) ) ` Y ) = if ( if ( j = 0 , ( P - 1 ) , P ) < ( c ` j ) , 0 , ( ( ( ! ` if ( j = 0 , ( P - 1 ) , P ) ) / ( ! ` ( if ( j = 0 , ( P - 1 ) , P ) - ( c ` j ) ) ) ) x. ( ( Y - j ) ^ ( if ( j = 0 , ( P - 1 ) , P ) - ( c ` j ) ) ) ) ) ) |
53 |
52
|
prodeq2dv |
|- ( ( ph /\ c e. ( C ` N ) ) -> prod_ j e. ( 0 ... M ) ( ( ( S Dn ( H ` j ) ) ` ( c ` j ) ) ` Y ) = prod_ j e. ( 0 ... M ) if ( if ( j = 0 , ( P - 1 ) , P ) < ( c ` j ) , 0 , ( ( ( ! ` if ( j = 0 , ( P - 1 ) , P ) ) / ( ! ` ( if ( j = 0 , ( P - 1 ) , P ) - ( c ` j ) ) ) ) x. ( ( Y - j ) ^ ( if ( j = 0 , ( P - 1 ) , P ) - ( c ` j ) ) ) ) ) ) |
54 |
|
nn0uz |
|- NN0 = ( ZZ>= ` 0 ) |
55 |
4 54
|
eleqtrdi |
|- ( ph -> M e. ( ZZ>= ` 0 ) ) |
56 |
55
|
adantr |
|- ( ( ph /\ c e. ( C ` N ) ) -> M e. ( ZZ>= ` 0 ) ) |
57 |
52 47
|
eqeltrrd |
|- ( ( ( ph /\ c e. ( C ` N ) ) /\ j e. ( 0 ... M ) ) -> if ( if ( j = 0 , ( P - 1 ) , P ) < ( c ` j ) , 0 , ( ( ( ! ` if ( j = 0 , ( P - 1 ) , P ) ) / ( ! ` ( if ( j = 0 , ( P - 1 ) , P ) - ( c ` j ) ) ) ) x. ( ( Y - j ) ^ ( if ( j = 0 , ( P - 1 ) , P ) - ( c ` j ) ) ) ) ) e. CC ) |
58 |
|
iftrue |
|- ( j = 0 -> if ( j = 0 , ( P - 1 ) , P ) = ( P - 1 ) ) |
59 |
|
fveq2 |
|- ( j = 0 -> ( c ` j ) = ( c ` 0 ) ) |
60 |
58 59
|
breq12d |
|- ( j = 0 -> ( if ( j = 0 , ( P - 1 ) , P ) < ( c ` j ) <-> ( P - 1 ) < ( c ` 0 ) ) ) |
61 |
58
|
fveq2d |
|- ( j = 0 -> ( ! ` if ( j = 0 , ( P - 1 ) , P ) ) = ( ! ` ( P - 1 ) ) ) |
62 |
58 59
|
oveq12d |
|- ( j = 0 -> ( if ( j = 0 , ( P - 1 ) , P ) - ( c ` j ) ) = ( ( P - 1 ) - ( c ` 0 ) ) ) |
63 |
62
|
fveq2d |
|- ( j = 0 -> ( ! ` ( if ( j = 0 , ( P - 1 ) , P ) - ( c ` j ) ) ) = ( ! ` ( ( P - 1 ) - ( c ` 0 ) ) ) ) |
64 |
61 63
|
oveq12d |
|- ( j = 0 -> ( ( ! ` if ( j = 0 , ( P - 1 ) , P ) ) / ( ! ` ( if ( j = 0 , ( P - 1 ) , P ) - ( c ` j ) ) ) ) = ( ( ! ` ( P - 1 ) ) / ( ! ` ( ( P - 1 ) - ( c ` 0 ) ) ) ) ) |
65 |
|
oveq2 |
|- ( j = 0 -> ( Y - j ) = ( Y - 0 ) ) |
66 |
65 62
|
oveq12d |
|- ( j = 0 -> ( ( Y - j ) ^ ( if ( j = 0 , ( P - 1 ) , P ) - ( c ` j ) ) ) = ( ( Y - 0 ) ^ ( ( P - 1 ) - ( c ` 0 ) ) ) ) |
67 |
64 66
|
oveq12d |
|- ( j = 0 -> ( ( ( ! ` if ( j = 0 , ( P - 1 ) , P ) ) / ( ! ` ( if ( j = 0 , ( P - 1 ) , P ) - ( c ` j ) ) ) ) x. ( ( Y - j ) ^ ( if ( j = 0 , ( P - 1 ) , P ) - ( c ` j ) ) ) ) = ( ( ( ! ` ( P - 1 ) ) / ( ! ` ( ( P - 1 ) - ( c ` 0 ) ) ) ) x. ( ( Y - 0 ) ^ ( ( P - 1 ) - ( c ` 0 ) ) ) ) ) |
68 |
60 67
|
ifbieq2d |
|- ( j = 0 -> if ( if ( j = 0 , ( P - 1 ) , P ) < ( c ` j ) , 0 , ( ( ( ! ` if ( j = 0 , ( P - 1 ) , P ) ) / ( ! ` ( if ( j = 0 , ( P - 1 ) , P ) - ( c ` j ) ) ) ) x. ( ( Y - j ) ^ ( if ( j = 0 , ( P - 1 ) , P ) - ( c ` j ) ) ) ) ) = if ( ( P - 1 ) < ( c ` 0 ) , 0 , ( ( ( ! ` ( P - 1 ) ) / ( ! ` ( ( P - 1 ) - ( c ` 0 ) ) ) ) x. ( ( Y - 0 ) ^ ( ( P - 1 ) - ( c ` 0 ) ) ) ) ) ) |
69 |
56 57 68
|
fprod1p |
|- ( ( ph /\ c e. ( C ` N ) ) -> prod_ j e. ( 0 ... M ) if ( if ( j = 0 , ( P - 1 ) , P ) < ( c ` j ) , 0 , ( ( ( ! ` if ( j = 0 , ( P - 1 ) , P ) ) / ( ! ` ( if ( j = 0 , ( P - 1 ) , P ) - ( c ` j ) ) ) ) x. ( ( Y - j ) ^ ( if ( j = 0 , ( P - 1 ) , P ) - ( c ` j ) ) ) ) ) = ( if ( ( P - 1 ) < ( c ` 0 ) , 0 , ( ( ( ! ` ( P - 1 ) ) / ( ! ` ( ( P - 1 ) - ( c ` 0 ) ) ) ) x. ( ( Y - 0 ) ^ ( ( P - 1 ) - ( c ` 0 ) ) ) ) ) x. prod_ j e. ( ( 0 + 1 ) ... M ) if ( if ( j = 0 , ( P - 1 ) , P ) < ( c ` j ) , 0 , ( ( ( ! ` if ( j = 0 , ( P - 1 ) , P ) ) / ( ! ` ( if ( j = 0 , ( P - 1 ) , P ) - ( c ` j ) ) ) ) x. ( ( Y - j ) ^ ( if ( j = 0 , ( P - 1 ) , P ) - ( c ` j ) ) ) ) ) ) ) |
70 |
1 2
|
dvdmsscn |
|- ( ph -> X C_ CC ) |
71 |
70 9
|
sseldd |
|- ( ph -> Y e. CC ) |
72 |
71
|
subid1d |
|- ( ph -> ( Y - 0 ) = Y ) |
73 |
72
|
oveq1d |
|- ( ph -> ( ( Y - 0 ) ^ ( ( P - 1 ) - ( c ` 0 ) ) ) = ( Y ^ ( ( P - 1 ) - ( c ` 0 ) ) ) ) |
74 |
73
|
oveq2d |
|- ( ph -> ( ( ( ! ` ( P - 1 ) ) / ( ! ` ( ( P - 1 ) - ( c ` 0 ) ) ) ) x. ( ( Y - 0 ) ^ ( ( P - 1 ) - ( c ` 0 ) ) ) ) = ( ( ( ! ` ( P - 1 ) ) / ( ! ` ( ( P - 1 ) - ( c ` 0 ) ) ) ) x. ( Y ^ ( ( P - 1 ) - ( c ` 0 ) ) ) ) ) |
75 |
74
|
ifeq2d |
|- ( ph -> if ( ( P - 1 ) < ( c ` 0 ) , 0 , ( ( ( ! ` ( P - 1 ) ) / ( ! ` ( ( P - 1 ) - ( c ` 0 ) ) ) ) x. ( ( Y - 0 ) ^ ( ( P - 1 ) - ( c ` 0 ) ) ) ) ) = if ( ( P - 1 ) < ( c ` 0 ) , 0 , ( ( ( ! ` ( P - 1 ) ) / ( ! ` ( ( P - 1 ) - ( c ` 0 ) ) ) ) x. ( Y ^ ( ( P - 1 ) - ( c ` 0 ) ) ) ) ) ) |
76 |
|
0p1e1 |
|- ( 0 + 1 ) = 1 |
77 |
76
|
oveq1i |
|- ( ( 0 + 1 ) ... M ) = ( 1 ... M ) |
78 |
77
|
prodeq1i |
|- prod_ j e. ( ( 0 + 1 ) ... M ) if ( if ( j = 0 , ( P - 1 ) , P ) < ( c ` j ) , 0 , ( ( ( ! ` if ( j = 0 , ( P - 1 ) , P ) ) / ( ! ` ( if ( j = 0 , ( P - 1 ) , P ) - ( c ` j ) ) ) ) x. ( ( Y - j ) ^ ( if ( j = 0 , ( P - 1 ) , P ) - ( c ` j ) ) ) ) ) = prod_ j e. ( 1 ... M ) if ( if ( j = 0 , ( P - 1 ) , P ) < ( c ` j ) , 0 , ( ( ( ! ` if ( j = 0 , ( P - 1 ) , P ) ) / ( ! ` ( if ( j = 0 , ( P - 1 ) , P ) - ( c ` j ) ) ) ) x. ( ( Y - j ) ^ ( if ( j = 0 , ( P - 1 ) , P ) - ( c ` j ) ) ) ) ) |
79 |
|
0red |
|- ( j e. ( 1 ... M ) -> 0 e. RR ) |
80 |
|
1red |
|- ( j e. ( 1 ... M ) -> 1 e. RR ) |
81 |
|
elfzelz |
|- ( j e. ( 1 ... M ) -> j e. ZZ ) |
82 |
81
|
zred |
|- ( j e. ( 1 ... M ) -> j e. RR ) |
83 |
|
0lt1 |
|- 0 < 1 |
84 |
83
|
a1i |
|- ( j e. ( 1 ... M ) -> 0 < 1 ) |
85 |
|
elfzle1 |
|- ( j e. ( 1 ... M ) -> 1 <_ j ) |
86 |
79 80 82 84 85
|
ltletrd |
|- ( j e. ( 1 ... M ) -> 0 < j ) |
87 |
86
|
gt0ne0d |
|- ( j e. ( 1 ... M ) -> j =/= 0 ) |
88 |
87
|
neneqd |
|- ( j e. ( 1 ... M ) -> -. j = 0 ) |
89 |
88
|
iffalsed |
|- ( j e. ( 1 ... M ) -> if ( j = 0 , ( P - 1 ) , P ) = P ) |
90 |
89
|
breq1d |
|- ( j e. ( 1 ... M ) -> ( if ( j = 0 , ( P - 1 ) , P ) < ( c ` j ) <-> P < ( c ` j ) ) ) |
91 |
89
|
fveq2d |
|- ( j e. ( 1 ... M ) -> ( ! ` if ( j = 0 , ( P - 1 ) , P ) ) = ( ! ` P ) ) |
92 |
89
|
oveq1d |
|- ( j e. ( 1 ... M ) -> ( if ( j = 0 , ( P - 1 ) , P ) - ( c ` j ) ) = ( P - ( c ` j ) ) ) |
93 |
92
|
fveq2d |
|- ( j e. ( 1 ... M ) -> ( ! ` ( if ( j = 0 , ( P - 1 ) , P ) - ( c ` j ) ) ) = ( ! ` ( P - ( c ` j ) ) ) ) |
94 |
91 93
|
oveq12d |
|- ( j e. ( 1 ... M ) -> ( ( ! ` if ( j = 0 , ( P - 1 ) , P ) ) / ( ! ` ( if ( j = 0 , ( P - 1 ) , P ) - ( c ` j ) ) ) ) = ( ( ! ` P ) / ( ! ` ( P - ( c ` j ) ) ) ) ) |
95 |
92
|
oveq2d |
|- ( j e. ( 1 ... M ) -> ( ( Y - j ) ^ ( if ( j = 0 , ( P - 1 ) , P ) - ( c ` j ) ) ) = ( ( Y - j ) ^ ( P - ( c ` j ) ) ) ) |
96 |
94 95
|
oveq12d |
|- ( j e. ( 1 ... M ) -> ( ( ( ! ` if ( j = 0 , ( P - 1 ) , P ) ) / ( ! ` ( if ( j = 0 , ( P - 1 ) , P ) - ( c ` j ) ) ) ) x. ( ( Y - j ) ^ ( if ( j = 0 , ( P - 1 ) , P ) - ( c ` j ) ) ) ) = ( ( ( ! ` P ) / ( ! ` ( P - ( c ` j ) ) ) ) x. ( ( Y - j ) ^ ( P - ( c ` j ) ) ) ) ) |
97 |
90 96
|
ifbieq2d |
|- ( j e. ( 1 ... M ) -> if ( if ( j = 0 , ( P - 1 ) , P ) < ( c ` j ) , 0 , ( ( ( ! ` if ( j = 0 , ( P - 1 ) , P ) ) / ( ! ` ( if ( j = 0 , ( P - 1 ) , P ) - ( c ` j ) ) ) ) x. ( ( Y - j ) ^ ( if ( j = 0 , ( P - 1 ) , P ) - ( c ` j ) ) ) ) ) = if ( P < ( c ` j ) , 0 , ( ( ( ! ` P ) / ( ! ` ( P - ( c ` j ) ) ) ) x. ( ( Y - j ) ^ ( P - ( c ` j ) ) ) ) ) ) |
98 |
97
|
prodeq2i |
|- prod_ j e. ( 1 ... M ) if ( if ( j = 0 , ( P - 1 ) , P ) < ( c ` j ) , 0 , ( ( ( ! ` if ( j = 0 , ( P - 1 ) , P ) ) / ( ! ` ( if ( j = 0 , ( P - 1 ) , P ) - ( c ` j ) ) ) ) x. ( ( Y - j ) ^ ( if ( j = 0 , ( P - 1 ) , P ) - ( c ` j ) ) ) ) ) = prod_ j e. ( 1 ... M ) if ( P < ( c ` j ) , 0 , ( ( ( ! ` P ) / ( ! ` ( P - ( c ` j ) ) ) ) x. ( ( Y - j ) ^ ( P - ( c ` j ) ) ) ) ) |
99 |
78 98
|
eqtri |
|- prod_ j e. ( ( 0 + 1 ) ... M ) if ( if ( j = 0 , ( P - 1 ) , P ) < ( c ` j ) , 0 , ( ( ( ! ` if ( j = 0 , ( P - 1 ) , P ) ) / ( ! ` ( if ( j = 0 , ( P - 1 ) , P ) - ( c ` j ) ) ) ) x. ( ( Y - j ) ^ ( if ( j = 0 , ( P - 1 ) , P ) - ( c ` j ) ) ) ) ) = prod_ j e. ( 1 ... M ) if ( P < ( c ` j ) , 0 , ( ( ( ! ` P ) / ( ! ` ( P - ( c ` j ) ) ) ) x. ( ( Y - j ) ^ ( P - ( c ` j ) ) ) ) ) |
100 |
99
|
a1i |
|- ( ph -> prod_ j e. ( ( 0 + 1 ) ... M ) if ( if ( j = 0 , ( P - 1 ) , P ) < ( c ` j ) , 0 , ( ( ( ! ` if ( j = 0 , ( P - 1 ) , P ) ) / ( ! ` ( if ( j = 0 , ( P - 1 ) , P ) - ( c ` j ) ) ) ) x. ( ( Y - j ) ^ ( if ( j = 0 , ( P - 1 ) , P ) - ( c ` j ) ) ) ) ) = prod_ j e. ( 1 ... M ) if ( P < ( c ` j ) , 0 , ( ( ( ! ` P ) / ( ! ` ( P - ( c ` j ) ) ) ) x. ( ( Y - j ) ^ ( P - ( c ` j ) ) ) ) ) ) |
101 |
75 100
|
oveq12d |
|- ( ph -> ( if ( ( P - 1 ) < ( c ` 0 ) , 0 , ( ( ( ! ` ( P - 1 ) ) / ( ! ` ( ( P - 1 ) - ( c ` 0 ) ) ) ) x. ( ( Y - 0 ) ^ ( ( P - 1 ) - ( c ` 0 ) ) ) ) ) x. prod_ j e. ( ( 0 + 1 ) ... M ) if ( if ( j = 0 , ( P - 1 ) , P ) < ( c ` j ) , 0 , ( ( ( ! ` if ( j = 0 , ( P - 1 ) , P ) ) / ( ! ` ( if ( j = 0 , ( P - 1 ) , P ) - ( c ` j ) ) ) ) x. ( ( Y - j ) ^ ( if ( j = 0 , ( P - 1 ) , P ) - ( c ` j ) ) ) ) ) ) = ( if ( ( P - 1 ) < ( c ` 0 ) , 0 , ( ( ( ! ` ( P - 1 ) ) / ( ! ` ( ( P - 1 ) - ( c ` 0 ) ) ) ) x. ( Y ^ ( ( P - 1 ) - ( c ` 0 ) ) ) ) ) x. prod_ j e. ( 1 ... M ) if ( P < ( c ` j ) , 0 , ( ( ( ! ` P ) / ( ! ` ( P - ( c ` j ) ) ) ) x. ( ( Y - j ) ^ ( P - ( c ` j ) ) ) ) ) ) ) |
102 |
101
|
adantr |
|- ( ( ph /\ c e. ( C ` N ) ) -> ( if ( ( P - 1 ) < ( c ` 0 ) , 0 , ( ( ( ! ` ( P - 1 ) ) / ( ! ` ( ( P - 1 ) - ( c ` 0 ) ) ) ) x. ( ( Y - 0 ) ^ ( ( P - 1 ) - ( c ` 0 ) ) ) ) ) x. prod_ j e. ( ( 0 + 1 ) ... M ) if ( if ( j = 0 , ( P - 1 ) , P ) < ( c ` j ) , 0 , ( ( ( ! ` if ( j = 0 , ( P - 1 ) , P ) ) / ( ! ` ( if ( j = 0 , ( P - 1 ) , P ) - ( c ` j ) ) ) ) x. ( ( Y - j ) ^ ( if ( j = 0 , ( P - 1 ) , P ) - ( c ` j ) ) ) ) ) ) = ( if ( ( P - 1 ) < ( c ` 0 ) , 0 , ( ( ( ! ` ( P - 1 ) ) / ( ! ` ( ( P - 1 ) - ( c ` 0 ) ) ) ) x. ( Y ^ ( ( P - 1 ) - ( c ` 0 ) ) ) ) ) x. prod_ j e. ( 1 ... M ) if ( P < ( c ` j ) , 0 , ( ( ( ! ` P ) / ( ! ` ( P - ( c ` j ) ) ) ) x. ( ( Y - j ) ^ ( P - ( c ` j ) ) ) ) ) ) ) |
103 |
53 69 102
|
3eqtrd |
|- ( ( ph /\ c e. ( C ` N ) ) -> prod_ j e. ( 0 ... M ) ( ( ( S Dn ( H ` j ) ) ` ( c ` j ) ) ` Y ) = ( if ( ( P - 1 ) < ( c ` 0 ) , 0 , ( ( ( ! ` ( P - 1 ) ) / ( ! ` ( ( P - 1 ) - ( c ` 0 ) ) ) ) x. ( Y ^ ( ( P - 1 ) - ( c ` 0 ) ) ) ) ) x. prod_ j e. ( 1 ... M ) if ( P < ( c ` j ) , 0 , ( ( ( ! ` P ) / ( ! ` ( P - ( c ` j ) ) ) ) x. ( ( Y - j ) ^ ( P - ( c ` j ) ) ) ) ) ) ) |
104 |
103
|
oveq2d |
|- ( ( ph /\ c e. ( C ` N ) ) -> ( ( ( ! ` N ) / prod_ j e. ( 0 ... M ) ( ! ` ( c ` j ) ) ) x. prod_ j e. ( 0 ... M ) ( ( ( S Dn ( H ` j ) ) ` ( c ` j ) ) ` Y ) ) = ( ( ( ! ` N ) / prod_ j e. ( 0 ... M ) ( ! ` ( c ` j ) ) ) x. ( if ( ( P - 1 ) < ( c ` 0 ) , 0 , ( ( ( ! ` ( P - 1 ) ) / ( ! ` ( ( P - 1 ) - ( c ` 0 ) ) ) ) x. ( Y ^ ( ( P - 1 ) - ( c ` 0 ) ) ) ) ) x. prod_ j e. ( 1 ... M ) if ( P < ( c ` j ) , 0 , ( ( ( ! ` P ) / ( ! ` ( P - ( c ` j ) ) ) ) x. ( ( Y - j ) ^ ( P - ( c ` j ) ) ) ) ) ) ) ) |
105 |
104
|
sumeq2dv |
|- ( ph -> sum_ c e. ( C ` N ) ( ( ( ! ` N ) / prod_ j e. ( 0 ... M ) ( ! ` ( c ` j ) ) ) x. prod_ j e. ( 0 ... M ) ( ( ( S Dn ( H ` j ) ) ` ( c ` j ) ) ` Y ) ) = sum_ c e. ( C ` N ) ( ( ( ! ` N ) / prod_ j e. ( 0 ... M ) ( ! ` ( c ` j ) ) ) x. ( if ( ( P - 1 ) < ( c ` 0 ) , 0 , ( ( ( ! ` ( P - 1 ) ) / ( ! ` ( ( P - 1 ) - ( c ` 0 ) ) ) ) x. ( Y ^ ( ( P - 1 ) - ( c ` 0 ) ) ) ) ) x. prod_ j e. ( 1 ... M ) if ( P < ( c ` j ) , 0 , ( ( ( ! ` P ) / ( ! ` ( P - ( c ` j ) ) ) ) x. ( ( Y - j ) ^ ( P - ( c ` j ) ) ) ) ) ) ) ) |
106 |
51 105
|
eqtrd |
|- ( ph -> ( ( ( S Dn F ) ` N ) ` Y ) = sum_ c e. ( C ` N ) ( ( ( ! ` N ) / prod_ j e. ( 0 ... M ) ( ! ` ( c ` j ) ) ) x. ( if ( ( P - 1 ) < ( c ` 0 ) , 0 , ( ( ( ! ` ( P - 1 ) ) / ( ! ` ( ( P - 1 ) - ( c ` 0 ) ) ) ) x. ( Y ^ ( ( P - 1 ) - ( c ` 0 ) ) ) ) ) x. prod_ j e. ( 1 ... M ) if ( P < ( c ` j ) , 0 , ( ( ( ! ` P ) / ( ! ` ( P - ( c ` j ) ) ) ) x. ( ( Y - j ) ^ ( P - ( c ` j ) ) ) ) ) ) ) ) |