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Theorem etransclem27 38238
Description: The  N-th derivative of  F applied to  J is an integer. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
Hypotheses
Ref Expression
etransclem27.s  |-  ( ph  ->  S  e.  { RR ,  CC } )
etransclem27.x  |-  ( ph  ->  X  e.  ( (
TopOpen ` fld )t  S ) )
etransclem27.p  |-  ( ph  ->  P  e.  NN )
etransclem27.h  |-  H  =  ( j  e.  ( 0 ... M ) 
|->  ( x  e.  X  |->  ( ( x  -  j ) ^ if ( j  =  0 ,  ( P  - 
1 ) ,  P
) ) ) )
etransclem27.cfi  |-  ( ph  ->  C  e.  Fin )
etransclem27.cf  |-  ( ph  ->  C : dom  C --> ( NN0  ^m  ( 0 ... M ) ) )
etransclem27.g  |-  G  =  ( x  e.  X  |-> 
sum_ l  e.  dom  C
prod_ j  e.  (
0 ... M ) ( ( ( S  Dn ( H `  j ) ) `  ( ( C `  l ) `  j
) ) `  x
) )
etransclem27.jx  |-  ( ph  ->  J  e.  X )
etransclem27.jz  |-  ( ph  ->  J  e.  ZZ )
Assertion
Ref Expression
etransclem27  |-  ( ph  ->  ( G `  J
)  e.  ZZ )
Distinct variable groups:    C, j,
l, x    x, H    j, J, l, x    j, M, x    P, j, x   
x, S    j, X, x    ph, j, l, x
Allowed substitution hints:    P( l)    S( j, l)    G( x, j, l)    H( j, l)    M( l)    X( l)

Proof of Theorem etransclem27
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 etransclem27.g . . . 4  |-  G  =  ( x  e.  X  |-> 
sum_ l  e.  dom  C
prod_ j  e.  (
0 ... M ) ( ( ( S  Dn ( H `  j ) ) `  ( ( C `  l ) `  j
) ) `  x
) )
21a1i 11 . . 3  |-  ( ph  ->  G  =  ( x  e.  X  |->  sum_ l  e.  dom  C prod_ j  e.  ( 0 ... M
) ( ( ( S  Dn ( H `  j ) ) `  ( ( C `  l ) `
 j ) ) `
 x ) ) )
3 fveq2 5879 . . . . . 6  |-  ( x  =  J  ->  (
( ( S  Dn ( H `  j ) ) `  ( ( C `  l ) `  j
) ) `  x
)  =  ( ( ( S  Dn
( H `  j
) ) `  (
( C `  l
) `  j )
) `  J )
)
43prodeq2ad 37769 . . . . 5  |-  ( 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 ) )
54sumeq2ad 37740 . . . 4  |-  ( 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
) )
65adantl 473 . . 3  |-  ( (
ph  /\  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
) )
7 etransclem27.jx . . 3  |-  ( ph  ->  J  e.  X )
8 etransclem27.cfi . . . . 5  |-  ( ph  ->  C  e.  Fin )
9 dmfi 7872 . . . . 5  |-  ( C  e.  Fin  ->  dom  C  e.  Fin )
108, 9syl 17 . . . 4  |-  ( ph  ->  dom  C  e.  Fin )
11 fzfid 12224 . . . . 5  |-  ( (
ph  /\  l  e.  dom  C )  ->  (
0 ... M )  e. 
Fin )
12 etransclem27.s . . . . . . . 8  |-  ( ph  ->  S  e.  { RR ,  CC } )
1312ad2antrr 740 . . . . . . 7  |-  ( ( ( ph  /\  l  e.  dom  C )  /\  j  e.  ( 0 ... M ) )  ->  S  e.  { RR ,  CC } )
14 etransclem27.x . . . . . . . 8  |-  ( ph  ->  X  e.  ( (
TopOpen ` fld )t  S ) )
1514ad2antrr 740 . . . . . . 7  |-  ( ( ( ph  /\  l  e.  dom  C )  /\  j  e.  ( 0 ... M ) )  ->  X  e.  ( ( TopOpen ` fld )t  S ) )
16 etransclem27.p . . . . . . . 8  |-  ( ph  ->  P  e.  NN )
1716ad2antrr 740 . . . . . . 7  |-  ( ( ( ph  /\  l  e.  dom  C )  /\  j  e.  ( 0 ... M ) )  ->  P  e.  NN )
18 etransclem27.h . . . . . . . 8  |-  H  =  ( j  e.  ( 0 ... M ) 
|->  ( x  e.  X  |->  ( ( x  -  j ) ^ if ( j  =  0 ,  ( P  - 
1 ) ,  P
) ) ) )
19 etransclem5 38216 . . . . . . . 8  |-  ( 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 ) ) ) )
2018, 19eqtri 2493 . . . . . . 7  |-  H  =  ( z  e.  ( 0 ... M ) 
|->  ( y  e.  X  |->  ( ( y  -  z ) ^ if ( z  =  0 ,  ( P  - 
1 ) ,  P
) ) ) )
21 simpr 468 . . . . . . 7  |-  ( ( ( ph  /\  l  e.  dom  C )  /\  j  e.  ( 0 ... M ) )  ->  j  e.  ( 0 ... M ) )
22 etransclem27.cf . . . . . . . . . 10  |-  ( ph  ->  C : dom  C --> ( NN0  ^m  ( 0 ... M ) ) )
2322ffvelrnda 6037 . . . . . . . . 9  |-  ( (
ph  /\  l  e.  dom  C )  ->  ( C `  l )  e.  ( NN0  ^m  (
0 ... M ) ) )
24 elmapi 7511 . . . . . . . . 9  |-  ( ( C `  l )  e.  ( NN0  ^m  ( 0 ... M
) )  ->  ( C `  l ) : ( 0 ... M ) --> NN0 )
2523, 24syl 17 . . . . . . . 8  |-  ( (
ph  /\  l  e.  dom  C )  ->  ( C `  l ) : ( 0 ... M ) --> NN0 )
2625ffvelrnda 6037 . . . . . . 7  |-  ( ( ( ph  /\  l  e.  dom  C )  /\  j  e.  ( 0 ... M ) )  ->  ( ( C `
 l ) `  j )  e.  NN0 )
2713, 15, 17, 20, 21, 26etransclem20 38231 . . . . . 6  |-  ( ( ( ph  /\  l  e.  dom  C )  /\  j  e.  ( 0 ... M ) )  ->  ( ( S  Dn ( H `
 j ) ) `
 ( ( C `
 l ) `  j ) ) : X --> CC )
287ad2antrr 740 . . . . . 6  |-  ( ( ( ph  /\  l  e.  dom  C )  /\  j  e.  ( 0 ... M ) )  ->  J  e.  X
)
2927, 28ffvelrnd 6038 . . . . 5  |-  ( ( ( ph  /\  l  e.  dom  C )  /\  j  e.  ( 0 ... M ) )  ->  ( ( ( S  Dn ( H `  j ) ) `  ( ( C `  l ) `
 j ) ) `
 J )  e.  CC )
3011, 29fprodcl 14083 . . . 4  |-  ( (
ph  /\  l  e.  dom  C )  ->  prod_ j  e.  ( 0 ... M ) ( ( ( S  Dn
( H `  j
) ) `  (
( C `  l
) `  j )
) `  J )  e.  CC )
3110, 30fsumcl 13876 . . 3  |-  ( ph  -> 
sum_ l  e.  dom  C
prod_ j  e.  (
0 ... M ) ( ( ( S  Dn ( H `  j ) ) `  ( ( C `  l ) `  j
) ) `  J
)  e.  CC )
322, 6, 7, 31fvmptd 5969 . 2  |-  ( ph  ->  ( G `  J
)  =  sum_ l  e.  dom  C prod_ j  e.  ( 0 ... M
) ( ( ( S  Dn ( H `  j ) ) `  ( ( C `  l ) `
 j ) ) `
 J ) )
3313, 15, 17, 20, 21, 26, 28etransclem21 38232 . . . . 5  |-  ( ( ( 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 ) ) ) ) ) )
34 iftrue 3878 . . . . . . . 8  |-  ( 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 )
35 0zd 10973 . . . . . . . 8  |-  ( if ( j  =  0 ,  ( P  - 
1 ) ,  P
)  <  ( ( C `  l ) `  j )  ->  0  e.  ZZ )
3634, 35eqeltrd 2549 . . . . . . 7  |-  ( 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 )
3736adantl 473 . . . . . 6  |-  ( ( ( ( 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 )
38 0zd 10973 . . . . . . 7  |-  ( ( ( ( ph  /\  l  e.  dom  C )  /\  j  e.  ( 0 ... M ) )  /\  -.  if ( j  =  0 ,  ( P  - 
1 ) ,  P
)  <  ( ( C `  l ) `  j ) )  -> 
0  e.  ZZ )
39 nnm1nn0 10935 . . . . . . . . . . . . . . . . 17  |-  ( P  e.  NN  ->  ( P  -  1 )  e.  NN0 )
4016, 39syl 17 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( P  -  1 )  e.  NN0 )
4116nnnn0d 10949 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  P  e.  NN0 )
4240, 41ifcld 3915 . . . . . . . . . . . . . . 15  |-  ( ph  ->  if ( j  =  0 ,  ( P  -  1 ) ,  P )  e.  NN0 )
4342nn0zd 11061 . . . . . . . . . . . . . 14  |-  ( ph  ->  if ( j  =  0 ,  ( P  -  1 ) ,  P )  e.  ZZ )
4443ad3antrrr 744 . . . . . . . . . . . . 13  |-  ( ( ( ( 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 )
4526nn0zd 11061 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  l  e.  dom  C )  /\  j  e.  ( 0 ... M ) )  ->  ( ( C `
 l ) `  j )  e.  ZZ )
4645adantr 472 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  l  e.  dom  C )  /\  j  e.  ( 0 ... M ) )  /\  -.  if ( j  =  0 ,  ( P  - 
1 ) ,  P
)  <  ( ( C `  l ) `  j ) )  -> 
( ( C `  l ) `  j
)  e.  ZZ )
4744, 46zsubcld 11068 . . . . . . . . . . . . 13  |-  ( ( ( ( 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 )
4838, 44, 473jca 1210 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  l  e.  dom  C )  /\  j  e.  ( 0 ... M ) )  /\  -.  if ( j  =  0 ,  ( P  - 
1 ) ,  P
)  <  ( ( C `  l ) `  j ) )  -> 
( 0  e.  ZZ  /\  if ( j  =  0 ,  ( P  -  1 ) ,  P )  e.  ZZ  /\  ( if ( j  =  0 ,  ( P  -  1 ) ,  P )  -  ( ( C `  l ) `  j
) )  e.  ZZ ) )
4946zred 11063 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  l  e.  dom  C )  /\  j  e.  ( 0 ... M ) )  /\  -.  if ( j  =  0 ,  ( P  - 
1 ) ,  P
)  <  ( ( C `  l ) `  j ) )  -> 
( ( C `  l ) `  j
)  e.  RR )
5044zred 11063 . . . . . . . . . . . . . 14  |-  ( ( ( ( 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 )
51 simpr 468 . . . . . . . . . . . . . 14  |-  ( ( ( ( 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 )
)
5249, 50, 51nltled 9802 . . . . . . . . . . . . 13  |-  ( ( ( ( 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 ) )
5350, 49subge0d 10224 . . . . . . . . . . . . 13  |-  ( ( ( ( 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 ) ) )
5452, 53mpbird 240 . . . . . . . . . . . 12  |-  ( ( ( ( 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 ) ) )
55 0red 9662 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  l  e.  dom  C )  /\  j  e.  ( 0 ... M ) )  ->  0  e.  RR )
5626nn0red 10950 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  l  e.  dom  C )  /\  j  e.  ( 0 ... M ) )  ->  ( ( C `
 l ) `  j )  e.  RR )
5742nn0red 10950 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  if ( j  =  0 ,  ( P  -  1 ) ,  P )  e.  RR )
5857ad2antrr 740 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  l  e.  dom  C )  /\  j  e.  ( 0 ... M ) )  ->  if ( j  =  0 ,  ( P  -  1 ) ,  P )  e.  RR )
5926nn0ge0d 10952 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  l  e.  dom  C )  /\  j  e.  ( 0 ... M ) )  ->  0  <_  (
( C `  l
) `  j )
)
6055, 56, 58, 59lesub2dd 10251 . . . . . . . . . . . . . 14  |-  ( ( ( 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 ) )
6158recnd 9687 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  l  e.  dom  C )  /\  j  e.  ( 0 ... M ) )  ->  if ( j  =  0 ,  ( P  -  1 ) ,  P )  e.  CC )
6261subid1d 9994 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  l  e.  dom  C )  /\  j  e.  ( 0 ... M ) )  ->  ( if ( j  =  0 ,  ( P  -  1 ) ,  P )  -  0 )  =  if ( j  =  0 ,  ( P  -  1 ) ,  P ) )
6360, 62breqtrd 4420 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  l  e.  dom  C )  /\  j  e.  ( 0 ... M ) )  ->  ( if ( j  =  0 ,  ( P  -  1 ) ,  P )  -  ( ( C `
 l ) `  j ) )  <_  if ( j  =  0 ,  ( P  - 
1 ) ,  P
) )
6463adantr 472 . . . . . . . . . . . 12  |-  ( ( ( ( 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
) )
6548, 54, 64jca32 544 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  l  e.  dom  C )  /\  j  e.  ( 0 ... M ) )  /\  -.  if ( j  =  0 ,  ( P  - 
1 ) ,  P
)  <  ( ( C `  l ) `  j ) )  -> 
( ( 0  e.  ZZ  /\  if ( j  =  0 ,  ( P  -  1 ) ,  P )  e.  ZZ  /\  ( if ( j  =  0 ,  ( P  - 
1 ) ,  P
)  -  ( ( C `  l ) `
 j ) )  e.  ZZ )  /\  ( 0  <_  ( if ( j  =  0 ,  ( P  - 
1 ) ,  P
)  -  ( ( C `  l ) `
 j ) )  /\  ( if ( j  =  0 ,  ( P  -  1 ) ,  P )  -  ( ( C `
 l ) `  j ) )  <_  if ( j  =  0 ,  ( P  - 
1 ) ,  P
) ) ) )
66 elfz2 11817 . . . . . . . . . . 11  |-  ( ( if ( j  =  0 ,  ( P  -  1 ) ,  P )  -  (
( C `  l
) `  j )
)  e.  ( 0 ... if ( j  =  0 ,  ( P  -  1 ) ,  P ) )  <-> 
( ( 0  e.  ZZ  /\  if ( j  =  0 ,  ( P  -  1 ) ,  P )  e.  ZZ  /\  ( if ( j  =  0 ,  ( P  - 
1 ) ,  P
)  -  ( ( C `  l ) `
 j ) )  e.  ZZ )  /\  ( 0  <_  ( if ( j  =  0 ,  ( P  - 
1 ) ,  P
)  -  ( ( C `  l ) `
 j ) )  /\  ( if ( j  =  0 ,  ( P  -  1 ) ,  P )  -  ( ( C `
 l ) `  j ) )  <_  if ( j  =  0 ,  ( P  - 
1 ) ,  P
) ) ) )
6765, 66sylibr 217 . . . . . . . . . 10  |-  ( ( ( ( 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 ) ) )
68 permnn 12549 . . . . . . . . . 10  |-  ( ( 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 )
6967, 68syl 17 . . . . . . . . 9  |-  ( ( ( ( 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 )
7069nnzd 11062 . . . . . . . 8  |-  ( ( ( ( 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 )
71 etransclem27.jz . . . . . . . . . . 11  |-  ( ph  ->  J  e.  ZZ )
7271ad3antrrr 744 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  l  e.  dom  C )  /\  j  e.  ( 0 ... M ) )  /\  -.  if ( j  =  0 ,  ( P  - 
1 ) ,  P
)  <  ( ( C `  l ) `  j ) )  ->  J  e.  ZZ )
73 elfzelz 11826 . . . . . . . . . . 11  |-  ( j  e.  ( 0 ... M )  ->  j  e.  ZZ )
7473ad2antlr 741 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  l  e.  dom  C )  /\  j  e.  ( 0 ... M ) )  /\  -.  if ( j  =  0 ,  ( P  - 
1 ) ,  P
)  <  ( ( C `  l ) `  j ) )  -> 
j  e.  ZZ )
7572, 74zsubcld 11068 . . . . . . . . 9  |-  ( ( ( ( ph  /\  l  e.  dom  C )  /\  j  e.  ( 0 ... M ) )  /\  -.  if ( j  =  0 ,  ( P  - 
1 ) ,  P
)  <  ( ( C `  l ) `  j ) )  -> 
( J  -  j
)  e.  ZZ )
76 elnn0z 10974 . . . . . . . . . 10  |-  ( ( 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
) ) ) )
7747, 54, 76sylanbrc 677 . . . . . . . . 9  |-  ( ( ( ( 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 )
78 zexpcl 12325 . . . . . . . . 9  |-  ( ( ( 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 )
7975, 77, 78syl2anc 673 . . . . . . . 8  |-  ( ( ( ( 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 )
8070, 79zmulcld 11069 . . . . . . 7  |-  ( ( ( ( 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 )
8138, 80ifcld 3915 . . . . . 6  |-  ( ( ( ( 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 )
8237, 81pm2.61dan 808 . . . . 5  |-  ( ( ( 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 )
8333, 82eqeltrd 2549 . . . 4  |-  ( ( ( ph  /\  l  e.  dom  C )  /\  j  e.  ( 0 ... M ) )  ->  ( ( ( S  Dn ( H `  j ) ) `  ( ( C `  l ) `
 j ) ) `
 J )  e.  ZZ )
8411, 83fprodzcl 14085 . . 3  |-  ( (
ph  /\  l  e.  dom  C )  ->  prod_ j  e.  ( 0 ... M ) ( ( ( S  Dn
( H `  j
) ) `  (
( C `  l
) `  j )
) `  J )  e.  ZZ )
8510, 84fsumzcl 13878 . 2  |-  ( ph  -> 
sum_ l  e.  dom  C
prod_ j  e.  (
0 ... M ) ( ( ( S  Dn ( H `  j ) ) `  ( ( C `  l ) `  j
) ) `  J
)  e.  ZZ )
8632, 85eqeltrd 2549 1  |-  ( ph  ->  ( G `  J
)  e.  ZZ )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904   ifcif 3872   {cpr 3961   class class class wbr 4395    |-> cmpt 4454   dom cdm 4839   -->wf 5585   ` cfv 5589  (class class class)co 6308    ^m cmap 7490   Fincfn 7587   CCcc 9555   RRcr 9556   0cc0 9557   1c1 9558    x. cmul 9562    < clt 9693    <_ cle 9694    - cmin 9880    / cdiv 10291   NNcn 10631   NN0cn0 10893   ZZcz 10961   ...cfz 11810   ^cexp 12310   !cfa 12497   sum_csu 13829   prod_cprod 14036   ↾t crest 15397   TopOpenctopn 15398  ℂfldccnfld 19047    Dncdvn 22898
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-addf 9636  ax-mulf 9637
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-fi 7943  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-icc 11667  df-fz 11811  df-fzo 11943  df-seq 12252  df-exp 12311  df-fac 12498  df-bc 12526  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-clim 13629  df-sum 13830  df-prod 14037  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-starv 15283  df-sca 15284  df-vsca 15285  df-ip 15286  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-hom 15292  df-cco 15293  df-rest 15399  df-topn 15400  df-0g 15418  df-gsum 15419  df-topgen 15420  df-pt 15421  df-prds 15424  df-xrs 15478  df-qtop 15484  df-imas 15485  df-xps 15488  df-mre 15570  df-mrc 15571  df-acs 15573  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-mulg 16754  df-cntz 17049  df-cmn 17510  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-fbas 19044  df-fg 19045  df-cnfld 19048  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-cld 20111  df-ntr 20112  df-cls 20113  df-nei 20191  df-lp 20229  df-perf 20230  df-cn 20320  df-cnp 20321  df-haus 20408  df-tx 20654  df-hmeo 20847  df-fil 20939  df-fm 21031  df-flim 21032  df-flf 21033  df-xms 21413  df-ms 21414  df-tms 21415  df-cncf 21988  df-limc 22900  df-dv 22901  df-dvn 22902
This theorem is referenced by: (None)
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