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Theorem etransclem27 38120
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 5863 . . . . . 6  |-  ( x  =  J  ->  (
( ( S  Dn ( H `  j ) ) `  ( ( C `  l ) `  j
) ) `  x
)  =  ( ( ( S  Dn
( H `  j
) ) `  (
( C `  l
) `  j )
) `  J )
)
43prodeq2ad 37666 . . . . 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 37638 . . . 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 468 . . 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 7851 . . . . 5  |-  ( C  e.  Fin  ->  dom  C  e.  Fin )
108, 9syl 17 . . . 4  |-  ( ph  ->  dom  C  e.  Fin )
11 fzfid 12183 . . . . 5  |-  ( (
ph  /\  l  e.  dom  C )  ->  (
0 ... M )  e. 
Fin )
12 etransclem27.s . . . . . . . 8  |-  ( ph  ->  S  e.  { RR ,  CC } )
1312ad2antrr 731 . . . . . . 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 731 . . . . . . 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 731 . . . . . . 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 38098 . . . . . . . 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 2472 . . . . . . 7  |-  H  =  ( z  e.  ( 0 ... M ) 
|->  ( y  e.  X  |->  ( ( y  -  z ) ^ if ( z  =  0 ,  ( P  - 
1 ) ,  P
) ) ) )
21 simpr 463 . . . . . . 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 6020 . . . . . . . . 9  |-  ( (
ph  /\  l  e.  dom  C )  ->  ( C `  l )  e.  ( NN0  ^m  (
0 ... M ) ) )
24 elmapi 7490 . . . . . . . . 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 6020 . . . . . . 7  |-  ( ( ( ph  /\  l  e.  dom  C )  /\  j  e.  ( 0 ... M ) )  ->  ( ( C `
 l ) `  j )  e.  NN0 )
2713, 15, 17, 20, 21, 26etransclem20 38113 . . . . . 6  |-  ( ( ( ph  /\  l  e.  dom  C )  /\  j  e.  ( 0 ... M ) )  ->  ( ( S  Dn ( H `
 j ) ) `
 ( ( C `
 l ) `  j ) ) : X --> CC )
287ad2antrr 731 . . . . . 6  |-  ( ( ( ph  /\  l  e.  dom  C )  /\  j  e.  ( 0 ... M ) )  ->  J  e.  X
)
2927, 28ffvelrnd 6021 . . . . 5  |-  ( ( ( ph  /\  l  e.  dom  C )  /\  j  e.  ( 0 ... M ) )  ->  ( ( ( S  Dn ( H `  j ) ) `  ( ( C `  l ) `
 j ) ) `
 J )  e.  CC )
3011, 29fprodcl 13999 . . . 4  |-  ( (
ph  /\  l  e.  dom  C )  ->  prod_ j  e.  ( 0 ... M ) ( ( ( S  Dn
( H `  j
) ) `  (
( C `  l
) `  j )
) `  J )  e.  CC )
3110, 30fsumcl 13792 . . 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 5952 . 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 38114 . . . . 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 3886 . . . . . . . 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 10946 . . . . . . . 8  |-  ( if ( j  =  0 ,  ( P  - 
1 ) ,  P
)  <  ( ( C `  l ) `  j )  ->  0  e.  ZZ )
3634, 35eqeltrd 2528 . . . . . . 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 468 . . . . . 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 10946 . . . . . . 7  |-  ( ( ( ( ph  /\  l  e.  dom  C )  /\  j  e.  ( 0 ... M ) )  /\  -.  if ( j  =  0 ,  ( P  - 
1 ) ,  P
)  <  ( ( C `  l ) `  j ) )  -> 
0  e.  ZZ )
39 nnm1nn0 10908 . . . . . . . . . . . . . . . . 17  |-  ( P  e.  NN  ->  ( P  -  1 )  e.  NN0 )
4016, 39syl 17 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( P  -  1 )  e.  NN0 )
4116nnnn0d 10922 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  P  e.  NN0 )
4240, 41ifcld 3923 . . . . . . . . . . . . . . 15  |-  ( ph  ->  if ( j  =  0 ,  ( P  -  1 ) ,  P )  e.  NN0 )
4342nn0zd 11035 . . . . . . . . . . . . . 14  |-  ( ph  ->  if ( j  =  0 ,  ( P  -  1 ) ,  P )  e.  ZZ )
4443ad3antrrr 735 . . . . . . . . . . . . 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 11035 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  l  e.  dom  C )  /\  j  e.  ( 0 ... M ) )  ->  ( ( C `
 l ) `  j )  e.  ZZ )
4645adantr 467 . . . . . . . . . . . . . 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 11042 . . . . . . . . . . . . 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 1187 . . . . . . . . . . . 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 11037 . . . . . . . . . . . . . 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 11037 . . . . . . . . . . . . . 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 463 . . . . . . . . . . . . . 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 9782 . . . . . . . . . . . . 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 10200 . . . . . . . . . . . . 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 236 . . . . . . . . . . . 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 9641 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  l  e.  dom  C )  /\  j  e.  ( 0 ... M ) )  ->  0  e.  RR )
5626nn0red 10923 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  l  e.  dom  C )  /\  j  e.  ( 0 ... M ) )  ->  ( ( C `
 l ) `  j )  e.  RR )
5742nn0red 10923 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  if ( j  =  0 ,  ( P  -  1 ) ,  P )  e.  RR )
5857ad2antrr 731 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  l  e.  dom  C )  /\  j  e.  ( 0 ... M ) )  ->  if ( j  =  0 ,  ( P  -  1 ) ,  P )  e.  RR )
5926nn0ge0d 10925 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  l  e.  dom  C )  /\  j  e.  ( 0 ... M ) )  ->  0  <_  (
( C `  l
) `  j )
)
6055, 56, 58, 59lesub2dd 10227 . . . . . . . . . . . . . 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 9666 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  l  e.  dom  C )  /\  j  e.  ( 0 ... M ) )  ->  if ( j  =  0 ,  ( P  -  1 ) ,  P )  e.  CC )
6261subid1d 9972 . . . . . . . . . . . . . 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 4426 . . . . . . . . . . . . 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 467 . . . . . . . . . . . 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 538 . . . . . . . . . . 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 11788 . . . . . . . . . . 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 216 . . . . . . . . . 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 12508 . . . . . . . . . 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 11036 . . . . . . . 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 735 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  l  e.  dom  C )  /\  j  e.  ( 0 ... M ) )  /\  -.  if ( j  =  0 ,  ( P  - 
1 ) ,  P
)  <  ( ( C `  l ) `  j ) )  ->  J  e.  ZZ )
73 elfzelz 11797 . . . . . . . . . . 11  |-  ( j  e.  ( 0 ... M )  ->  j  e.  ZZ )
7473ad2antlr 732 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  l  e.  dom  C )  /\  j  e.  ( 0 ... M ) )  /\  -.  if ( j  =  0 ,  ( P  - 
1 ) ,  P
)  <  ( ( C `  l ) `  j ) )  -> 
j  e.  ZZ )
7572, 74zsubcld 11042 . . . . . . . . 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 10947 . . . . . . . . . 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 669 . . . . . . . . 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 12284 . . . . . . . . 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 666 . . . . . . . 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 11043 . . . . . . 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 3923 . . . . . 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 799 . . . . 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 2528 . . . 4  |-  ( ( ( ph  /\  l  e.  dom  C )  /\  j  e.  ( 0 ... M ) )  ->  ( ( ( S  Dn ( H `  j ) ) `  ( ( C `  l ) `
 j ) ) `
 J )  e.  ZZ )
8411, 83fprodzcl 14001 . . 3  |-  ( (
ph  /\  l  e.  dom  C )  ->  prod_ j  e.  ( 0 ... M ) ( ( ( S  Dn
( H `  j
) ) `  (
( C `  l
) `  j )
) `  J )  e.  ZZ )
8510, 84fsumzcl 13794 . 2  |-  ( ph  -> 
sum_ l  e.  dom  C
prod_ j  e.  (
0 ... M ) ( ( ( S  Dn ( H `  j ) ) `  ( ( C `  l ) `  j
) ) `  J
)  e.  ZZ )
8632, 85eqeltrd 2528 1  |-  ( ph  ->  ( G `  J
)  e.  ZZ )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 371    /\ w3a 984    = wceq 1443    e. wcel 1886   ifcif 3880   {cpr 3969   class class class wbr 4401    |-> cmpt 4460   dom cdm 4833   -->wf 5577   ` cfv 5581  (class class class)co 6288    ^m cmap 7469   Fincfn 7566   CCcc 9534   RRcr 9535   0cc0 9536   1c1 9537    x. cmul 9541    < clt 9672    <_ cle 9673    - cmin 9857    / cdiv 10266   NNcn 10606   NN0cn0 10866   ZZcz 10934   ...cfz 11781   ^cexp 12269   !cfa 12456   sum_csu 13745   prod_cprod 13952   ↾t crest 15312   TopOpenctopn 15313  ℂfldccnfld 18963    Dncdvn 22812
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-inf2 8143  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613  ax-pre-sup 9614  ax-addf 9615  ax-mulf 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-fal 1449  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-iin 4280  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-se 4793  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-of 6528  df-om 6690  df-1st 6790  df-2nd 6791  df-supp 6912  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-1o 7179  df-2o 7180  df-oadd 7183  df-er 7360  df-map 7471  df-pm 7472  df-ixp 7520  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  df-fsupp 7881  df-fi 7922  df-sup 7953  df-inf 7954  df-oi 8022  df-card 8370  df-cda 8595  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-div 10267  df-nn 10607  df-2 10665  df-3 10666  df-4 10667  df-5 10668  df-6 10669  df-7 10670  df-8 10671  df-9 10672  df-10 10673  df-n0 10867  df-z 10935  df-dec 11049  df-uz 11157  df-q 11262  df-rp 11300  df-xneg 11406  df-xadd 11407  df-xmul 11408  df-icc 11639  df-fz 11782  df-fzo 11913  df-seq 12211  df-exp 12270  df-fac 12457  df-bc 12485  df-hash 12513  df-cj 13155  df-re 13156  df-im 13157  df-sqrt 13291  df-abs 13292  df-clim 13545  df-sum 13746  df-prod 13953  df-struct 15116  df-ndx 15117  df-slot 15118  df-base 15119  df-sets 15120  df-ress 15121  df-plusg 15196  df-mulr 15197  df-starv 15198  df-sca 15199  df-vsca 15200  df-ip 15201  df-tset 15202  df-ple 15203  df-ds 15205  df-unif 15206  df-hom 15207  df-cco 15208  df-rest 15314  df-topn 15315  df-0g 15333  df-gsum 15334  df-topgen 15335  df-pt 15336  df-prds 15339  df-xrs 15393  df-qtop 15399  df-imas 15400  df-xps 15403  df-mre 15485  df-mrc 15486  df-acs 15488  df-mgm 16481  df-sgrp 16520  df-mnd 16530  df-submnd 16576  df-mulg 16669  df-cntz 16964  df-cmn 17425  df-psmet 18955  df-xmet 18956  df-met 18957  df-bl 18958  df-mopn 18959  df-fbas 18960  df-fg 18961  df-cnfld 18964  df-top 19914  df-bases 19915  df-topon 19916  df-topsp 19917  df-cld 20027  df-ntr 20028  df-cls 20029  df-nei 20107  df-lp 20145  df-perf 20146  df-cn 20236  df-cnp 20237  df-haus 20324  df-tx 20570  df-hmeo 20763  df-fil 20854  df-fm 20946  df-flim 20947  df-flf 20948  df-xms 21328  df-ms 21329  df-tms 21330  df-cncf 21903  df-limc 22814  df-dv 22815  df-dvn 22816
This theorem is referenced by: (None)
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