Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  etransclem44 Structured version   Visualization version   Unicode version

Theorem etransclem44 38255
Description: The given finite sum is nonzero. This is the claim proved after equation (7) in [Juillerat] p. 12 . (Contributed by Glauco Siliprandi, 5-Apr-2020.)
Hypotheses
Ref Expression
etransclem44.a  |-  ( ph  ->  A : NN0 --> ZZ )
etransclem44.a0  |-  ( ph  ->  ( A `  0
)  =/=  0 )
etransclem44.m  |-  ( ph  ->  M  e.  NN0 )
etransclem44.p  |-  ( ph  ->  P  e.  Prime )
etransclem44.ap  |-  ( ph  ->  ( abs `  ( A `  0 )
)  <  P )
etransclem44.mp  |-  ( ph  ->  ( ! `  M
)  <  P )
etransclem44.f  |-  F  =  ( x  e.  RR  |->  ( ( x ^
( P  -  1 ) )  x.  prod_ j  e.  ( 1 ... M ) ( ( x  -  j ) ^ P ) ) )
etransclem44.k  |-  K  =  ( sum_ k  e.  ( ( 0 ... M
)  X.  ( 0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) ) ) ( ( A `  ( 1st `  k ) )  x.  ( ( ( RR  Dn
F ) `  ( 2nd `  k ) ) `
 ( 1st `  k
) ) )  / 
( ! `  ( P  -  1 ) ) )
Assertion
Ref Expression
etransclem44  |-  ( ph  ->  K  =/=  0 )
Distinct variable groups:    A, k    k, F    j, M, k, x    P, j, k, x    ph, j, k, x
Allowed substitution hints:    A( x, j)    F( x, j)    K( x, j, k)

Proof of Theorem etransclem44
Dummy variables  c 
d  n  m  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 etransclem44.k . . . 4  |-  K  =  ( sum_ k  e.  ( ( 0 ... M
)  X.  ( 0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) ) ) ( ( A `  ( 1st `  k ) )  x.  ( ( ( RR  Dn
F ) `  ( 2nd `  k ) ) `
 ( 1st `  k
) ) )  / 
( ! `  ( P  -  1 ) ) )
21a1i 11 . . 3  |-  ( ph  ->  K  =  ( sum_ k  e.  ( (
0 ... M )  X.  ( 0 ... (
( M  x.  P
)  +  ( P  -  1 ) ) ) ) ( ( A `  ( 1st `  k ) )  x.  ( ( ( RR  Dn F ) `
 ( 2nd `  k
) ) `  ( 1st `  k ) ) )  /  ( ! `
 ( P  - 
1 ) ) ) )
3 nfv 1769 . . . . 5  |-  F/ k
ph
4 nfcv 2612 . . . . 5  |-  F/_ k
( ( A ` 
0 )  x.  (
( ( RR  Dn F ) `  ( P  -  1
) ) `  0
) )
5 fzfi 12223 . . . . . . 7  |-  ( 0 ... M )  e. 
Fin
6 fzfi 12223 . . . . . . 7  |-  ( 0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) )  e. 
Fin
7 xpfi 7860 . . . . . . 7  |-  ( ( ( 0 ... M
)  e.  Fin  /\  ( 0 ... (
( M  x.  P
)  +  ( P  -  1 ) ) )  e.  Fin )  ->  ( ( 0 ... M )  X.  (
0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) ) )  e.  Fin )
85, 6, 7mp2an 686 . . . . . 6  |-  ( ( 0 ... M )  X.  ( 0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) ) )  e. 
Fin
98a1i 11 . . . . 5  |-  ( ph  ->  ( ( 0 ... M )  X.  (
0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) ) )  e.  Fin )
10 etransclem44.a . . . . . . . . 9  |-  ( ph  ->  A : NN0 --> ZZ )
1110adantr 472 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ( 0 ... M )  X.  (
0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) ) ) )  ->  A : NN0
--> ZZ )
12 fzssnn0 37627 . . . . . . . . . 10  |-  ( 0 ... M )  C_  NN0
13 xp1st 6842 . . . . . . . . . 10  |-  ( k  e.  ( ( 0 ... M )  X.  ( 0 ... (
( M  x.  P
)  +  ( P  -  1 ) ) ) )  ->  ( 1st `  k )  e.  ( 0 ... M
) )
1412, 13sseldi 3416 . . . . . . . . 9  |-  ( k  e.  ( ( 0 ... M )  X.  ( 0 ... (
( M  x.  P
)  +  ( P  -  1 ) ) ) )  ->  ( 1st `  k )  e. 
NN0 )
1514adantl 473 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ( 0 ... M )  X.  (
0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) ) ) )  ->  ( 1st `  k )  e.  NN0 )
1611, 15ffvelrnd 6038 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ( 0 ... M )  X.  (
0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) ) ) )  ->  ( A `  ( 1st `  k
) )  e.  ZZ )
17 reelprrecn 9649 . . . . . . . . 9  |-  RR  e.  { RR ,  CC }
1817a1i 11 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ( 0 ... M )  X.  (
0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) ) ) )  ->  RR  e.  { RR ,  CC }
)
19 reopn 37591 . . . . . . . . . 10  |-  RR  e.  ( topGen `  ran  (,) )
20 eqid 2471 . . . . . . . . . . 11  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
2120tgioo2 21899 . . . . . . . . . 10  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
2219, 21eleqtri 2547 . . . . . . . . 9  |-  RR  e.  ( ( TopOpen ` fld )t  RR )
2322a1i 11 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ( 0 ... M )  X.  (
0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) ) ) )  ->  RR  e.  ( ( TopOpen ` fld )t  RR ) )
24 etransclem44.p . . . . . . . . . 10  |-  ( ph  ->  P  e.  Prime )
25 prmnn 14704 . . . . . . . . . 10  |-  ( P  e.  Prime  ->  P  e.  NN )
2624, 25syl 17 . . . . . . . . 9  |-  ( ph  ->  P  e.  NN )
2726adantr 472 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ( 0 ... M )  X.  (
0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) ) ) )  ->  P  e.  NN )
28 etransclem44.m . . . . . . . . 9  |-  ( ph  ->  M  e.  NN0 )
2928adantr 472 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ( 0 ... M )  X.  (
0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) ) ) )  ->  M  e.  NN0 )
30 etransclem44.f . . . . . . . 8  |-  F  =  ( x  e.  RR  |->  ( ( x ^
( P  -  1 ) )  x.  prod_ j  e.  ( 1 ... M ) ( ( x  -  j ) ^ P ) ) )
31 xp2nd 6843 . . . . . . . . . 10  |-  ( k  e.  ( ( 0 ... M )  X.  ( 0 ... (
( M  x.  P
)  +  ( P  -  1 ) ) ) )  ->  ( 2nd `  k )  e.  ( 0 ... (
( M  x.  P
)  +  ( P  -  1 ) ) ) )
32 elfznn0 11913 . . . . . . . . . 10  |-  ( ( 2nd `  k )  e.  ( 0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) )  ->  ( 2nd `  k )  e. 
NN0 )
3331, 32syl 17 . . . . . . . . 9  |-  ( k  e.  ( ( 0 ... M )  X.  ( 0 ... (
( M  x.  P
)  +  ( P  -  1 ) ) ) )  ->  ( 2nd `  k )  e. 
NN0 )
3433adantl 473 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ( 0 ... M )  X.  (
0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) ) ) )  ->  ( 2nd `  k )  e.  NN0 )
3515nn0red 10950 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ( 0 ... M )  X.  (
0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) ) ) )  ->  ( 1st `  k )  e.  RR )
3615nn0zd 11061 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ( 0 ... M )  X.  (
0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) ) ) )  ->  ( 1st `  k )  e.  ZZ )
3718, 23, 27, 29, 30, 34, 35, 36etransclem42 38253 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ( 0 ... M )  X.  (
0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) ) ) )  ->  ( (
( RR  Dn
F ) `  ( 2nd `  k ) ) `
 ( 1st `  k
) )  e.  ZZ )
3816, 37zmulcld 11069 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ( 0 ... M )  X.  (
0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) ) ) )  ->  ( ( A `  ( 1st `  k ) )  x.  ( ( ( RR  Dn F ) `
 ( 2nd `  k
) ) `  ( 1st `  k ) ) )  e.  ZZ )
3938zcnd 11064 . . . . 5  |-  ( (
ph  /\  k  e.  ( ( 0 ... M )  X.  (
0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) ) ) )  ->  ( ( A `  ( 1st `  k ) )  x.  ( ( ( RR  Dn F ) `
 ( 2nd `  k
) ) `  ( 1st `  k ) ) )  e.  CC )
40 nn0uz 11217 . . . . . . . 8  |-  NN0  =  ( ZZ>= `  0 )
4128, 40syl6eleq 2559 . . . . . . 7  |-  ( ph  ->  M  e.  ( ZZ>= ` 
0 ) )
42 eluzfz1 11832 . . . . . . 7  |-  ( M  e.  ( ZZ>= `  0
)  ->  0  e.  ( 0 ... M
) )
4341, 42syl 17 . . . . . 6  |-  ( ph  ->  0  e.  ( 0 ... M ) )
44 0zd 10973 . . . . . . . . 9  |-  ( ph  ->  0  e.  ZZ )
4528nn0zd 11061 . . . . . . . . . . 11  |-  ( ph  ->  M  e.  ZZ )
4626nnzd 11062 . . . . . . . . . . 11  |-  ( ph  ->  P  e.  ZZ )
4745, 46zmulcld 11069 . . . . . . . . . 10  |-  ( ph  ->  ( M  x.  P
)  e.  ZZ )
48 nnm1nn0 10935 . . . . . . . . . . . 12  |-  ( P  e.  NN  ->  ( P  -  1 )  e.  NN0 )
4926, 48syl 17 . . . . . . . . . . 11  |-  ( ph  ->  ( P  -  1 )  e.  NN0 )
5049nn0zd 11061 . . . . . . . . . 10  |-  ( ph  ->  ( P  -  1 )  e.  ZZ )
5147, 50zaddcld 11067 . . . . . . . . 9  |-  ( ph  ->  ( ( M  x.  P )  +  ( P  -  1 ) )  e.  ZZ )
5244, 51, 503jca 1210 . . . . . . . 8  |-  ( ph  ->  ( 0  e.  ZZ  /\  ( ( M  x.  P )  +  ( P  -  1 ) )  e.  ZZ  /\  ( P  -  1
)  e.  ZZ ) )
5349nn0ge0d 10952 . . . . . . . 8  |-  ( ph  ->  0  <_  ( P  -  1 ) )
5426nnnn0d 10949 . . . . . . . . . . 11  |-  ( ph  ->  P  e.  NN0 )
5528, 54nn0mulcld 10954 . . . . . . . . . 10  |-  ( ph  ->  ( M  x.  P
)  e.  NN0 )
5655nn0ge0d 10952 . . . . . . . . 9  |-  ( ph  ->  0  <_  ( M  x.  P ) )
5749nn0red 10950 . . . . . . . . . 10  |-  ( ph  ->  ( P  -  1 )  e.  RR )
5847zred 11063 . . . . . . . . . 10  |-  ( ph  ->  ( M  x.  P
)  e.  RR )
5957, 58addge02d 10223 . . . . . . . . 9  |-  ( ph  ->  ( 0  <_  ( M  x.  P )  <->  ( P  -  1 )  <_  ( ( M  x.  P )  +  ( P  -  1 ) ) ) )
6056, 59mpbid 215 . . . . . . . 8  |-  ( ph  ->  ( P  -  1 )  <_  ( ( M  x.  P )  +  ( P  - 
1 ) ) )
6152, 53, 60jca32 544 . . . . . . 7  |-  ( ph  ->  ( ( 0  e.  ZZ  /\  ( ( M  x.  P )  +  ( P  - 
1 ) )  e.  ZZ  /\  ( P  -  1 )  e.  ZZ )  /\  (
0  <_  ( P  -  1 )  /\  ( P  -  1
)  <_  ( ( M  x.  P )  +  ( P  - 
1 ) ) ) ) )
62 elfz2 11817 . . . . . . 7  |-  ( ( P  -  1 )  e.  ( 0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) )  <->  ( (
0  e.  ZZ  /\  ( ( M  x.  P )  +  ( P  -  1 ) )  e.  ZZ  /\  ( P  -  1
)  e.  ZZ )  /\  ( 0  <_ 
( P  -  1 )  /\  ( P  -  1 )  <_ 
( ( M  x.  P )  +  ( P  -  1 ) ) ) ) )
6361, 62sylibr 217 . . . . . 6  |-  ( ph  ->  ( P  -  1 )  e.  ( 0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) ) )
64 opelxp 4869 . . . . . 6  |-  ( <.
0 ,  ( P  -  1 ) >.  e.  ( ( 0 ... M )  X.  (
0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) ) )  <-> 
( 0  e.  ( 0 ... M )  /\  ( P  - 
1 )  e.  ( 0 ... ( ( M  x.  P )  +  ( P  - 
1 ) ) ) ) )
6543, 63, 64sylanbrc 677 . . . . 5  |-  ( ph  -> 
<. 0 ,  ( P  -  1 )
>.  e.  ( ( 0 ... M )  X.  ( 0 ... (
( M  x.  P
)  +  ( P  -  1 ) ) ) ) )
66 fveq2 5879 . . . . . . . 8  |-  ( k  =  <. 0 ,  ( P  -  1 )
>.  ->  ( 1st `  k
)  =  ( 1st `  <. 0 ,  ( P  -  1 )
>. ) )
67 0re 9661 . . . . . . . . 9  |-  0  e.  RR
68 ovex 6336 . . . . . . . . 9  |-  ( P  -  1 )  e. 
_V
69 op1stg 6824 . . . . . . . . 9  |-  ( ( 0  e.  RR  /\  ( P  -  1
)  e.  _V )  ->  ( 1st `  <. 0 ,  ( P  -  1 ) >.
)  =  0 )
7067, 68, 69mp2an 686 . . . . . . . 8  |-  ( 1st `  <. 0 ,  ( P  -  1 )
>. )  =  0
7166, 70syl6eq 2521 . . . . . . 7  |-  ( k  =  <. 0 ,  ( P  -  1 )
>.  ->  ( 1st `  k
)  =  0 )
7271fveq2d 5883 . . . . . 6  |-  ( k  =  <. 0 ,  ( P  -  1 )
>.  ->  ( A `  ( 1st `  k ) )  =  ( A `
 0 ) )
73 fveq2 5879 . . . . . . . . 9  |-  ( k  =  <. 0 ,  ( P  -  1 )
>.  ->  ( 2nd `  k
)  =  ( 2nd `  <. 0 ,  ( P  -  1 )
>. ) )
74 op2ndg 6825 . . . . . . . . . 10  |-  ( ( 0  e.  RR  /\  ( P  -  1
)  e.  _V )  ->  ( 2nd `  <. 0 ,  ( P  -  1 ) >.
)  =  ( P  -  1 ) )
7567, 68, 74mp2an 686 . . . . . . . . 9  |-  ( 2nd `  <. 0 ,  ( P  -  1 )
>. )  =  ( P  -  1 )
7673, 75syl6eq 2521 . . . . . . . 8  |-  ( k  =  <. 0 ,  ( P  -  1 )
>.  ->  ( 2nd `  k
)  =  ( P  -  1 ) )
7776fveq2d 5883 . . . . . . 7  |-  ( k  =  <. 0 ,  ( P  -  1 )
>.  ->  ( ( RR  Dn F ) `
 ( 2nd `  k
) )  =  ( ( RR  Dn
F ) `  ( P  -  1 ) ) )
7877, 71fveq12d 5885 . . . . . 6  |-  ( k  =  <. 0 ,  ( P  -  1 )
>.  ->  ( ( ( RR  Dn F ) `  ( 2nd `  k ) ) `  ( 1st `  k ) )  =  ( ( ( RR  Dn
F ) `  ( P  -  1 ) ) `  0 ) )
7972, 78oveq12d 6326 . . . . 5  |-  ( k  =  <. 0 ,  ( P  -  1 )
>.  ->  ( ( A `
 ( 1st `  k
) )  x.  (
( ( RR  Dn F ) `  ( 2nd `  k ) ) `  ( 1st `  k ) ) )  =  ( ( A `
 0 )  x.  ( ( ( RR  Dn F ) `
 ( P  - 
1 ) ) ` 
0 ) ) )
803, 4, 9, 39, 65, 79fsumsplit1 37747 . . . 4  |-  ( ph  -> 
sum_ k  e.  ( ( 0 ... M
)  X.  ( 0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) ) ) ( ( A `  ( 1st `  k ) )  x.  ( ( ( RR  Dn
F ) `  ( 2nd `  k ) ) `
 ( 1st `  k
) ) )  =  ( ( ( A `
 0 )  x.  ( ( ( RR  Dn F ) `
 ( P  - 
1 ) ) ` 
0 ) )  + 
sum_ k  e.  ( ( ( 0 ... M )  X.  (
0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) ) ) 
\  { <. 0 ,  ( P  - 
1 ) >. } ) ( ( A `  ( 1st `  k ) )  x.  ( ( ( RR  Dn
F ) `  ( 2nd `  k ) ) `
 ( 1st `  k
) ) ) ) )
8180oveq1d 6323 . . 3  |-  ( ph  ->  ( sum_ k  e.  ( ( 0 ... M
)  X.  ( 0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) ) ) ( ( A `  ( 1st `  k ) )  x.  ( ( ( RR  Dn
F ) `  ( 2nd `  k ) ) `
 ( 1st `  k
) ) )  / 
( ! `  ( P  -  1 ) ) )  =  ( ( ( ( A `
 0 )  x.  ( ( ( RR  Dn F ) `
 ( P  - 
1 ) ) ` 
0 ) )  + 
sum_ k  e.  ( ( ( 0 ... M )  X.  (
0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) ) ) 
\  { <. 0 ,  ( P  - 
1 ) >. } ) ( ( A `  ( 1st `  k ) )  x.  ( ( ( RR  Dn
F ) `  ( 2nd `  k ) ) `
 ( 1st `  k
) ) ) )  /  ( ! `  ( P  -  1
) ) ) )
8212, 43sseldi 3416 . . . . . . 7  |-  ( ph  ->  0  e.  NN0 )
8310, 82ffvelrnd 6038 . . . . . 6  |-  ( ph  ->  ( A `  0
)  e.  ZZ )
8417a1i 11 . . . . . . 7  |-  ( ph  ->  RR  e.  { RR ,  CC } )
8522a1i 11 . . . . . . 7  |-  ( ph  ->  RR  e.  ( (
TopOpen ` fld )t  RR ) )
8667a1i 11 . . . . . . 7  |-  ( ph  ->  0  e.  RR )
8784, 85, 26, 28, 30, 49, 86, 44etransclem42 38253 . . . . . 6  |-  ( ph  ->  ( ( ( RR  Dn F ) `
 ( P  - 
1 ) ) ` 
0 )  e.  ZZ )
8883, 87zmulcld 11069 . . . . 5  |-  ( ph  ->  ( ( A ` 
0 )  x.  (
( ( RR  Dn F ) `  ( P  -  1
) ) `  0
) )  e.  ZZ )
8988zcnd 11064 . . . 4  |-  ( ph  ->  ( ( A ` 
0 )  x.  (
( ( RR  Dn F ) `  ( P  -  1
) ) `  0
) )  e.  CC )
90 difss 3549 . . . . . . . 8  |-  ( ( ( 0 ... M
)  X.  ( 0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) ) ) 
\  { <. 0 ,  ( P  - 
1 ) >. } ) 
C_  ( ( 0 ... M )  X.  ( 0 ... (
( M  x.  P
)  +  ( P  -  1 ) ) ) )
91 ssfi 7810 . . . . . . . 8  |-  ( ( ( ( 0 ... M )  X.  (
0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) ) )  e.  Fin  /\  (
( ( 0 ... M )  X.  (
0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) ) ) 
\  { <. 0 ,  ( P  - 
1 ) >. } ) 
C_  ( ( 0 ... M )  X.  ( 0 ... (
( M  x.  P
)  +  ( P  -  1 ) ) ) ) )  -> 
( ( ( 0 ... M )  X.  ( 0 ... (
( M  x.  P
)  +  ( P  -  1 ) ) ) )  \  { <. 0 ,  ( P  -  1 ) >. } )  e.  Fin )
928, 90, 91mp2an 686 . . . . . . 7  |-  ( ( ( 0 ... M
)  X.  ( 0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) ) ) 
\  { <. 0 ,  ( P  - 
1 ) >. } )  e.  Fin
9392a1i 11 . . . . . 6  |-  ( ph  ->  ( ( ( 0 ... M )  X.  ( 0 ... (
( M  x.  P
)  +  ( P  -  1 ) ) ) )  \  { <. 0 ,  ( P  -  1 ) >. } )  e.  Fin )
94 eldifi 3544 . . . . . . 7  |-  ( k  e.  ( ( ( 0 ... M )  X.  ( 0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) ) )  \  { <. 0 ,  ( P  -  1 )
>. } )  ->  k  e.  ( ( 0 ... M )  X.  (
0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) ) ) )
9594, 38sylan2 482 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ( ( 0 ... M )  X.  ( 0 ... (
( M  x.  P
)  +  ( P  -  1 ) ) ) )  \  { <. 0 ,  ( P  -  1 ) >. } ) )  -> 
( ( A `  ( 1st `  k ) )  x.  ( ( ( RR  Dn
F ) `  ( 2nd `  k ) ) `
 ( 1st `  k
) ) )  e.  ZZ )
9693, 95fsumzcl 13878 . . . . 5  |-  ( ph  -> 
sum_ k  e.  ( ( ( 0 ... M )  X.  (
0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) ) ) 
\  { <. 0 ,  ( P  - 
1 ) >. } ) ( ( A `  ( 1st `  k ) )  x.  ( ( ( RR  Dn
F ) `  ( 2nd `  k ) ) `
 ( 1st `  k
) ) )  e.  ZZ )
9796zcnd 11064 . . . 4  |-  ( ph  -> 
sum_ k  e.  ( ( ( 0 ... M )  X.  (
0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) ) ) 
\  { <. 0 ,  ( P  - 
1 ) >. } ) ( ( A `  ( 1st `  k ) )  x.  ( ( ( RR  Dn
F ) `  ( 2nd `  k ) ) `
 ( 1st `  k
) ) )  e.  CC )
9849faccld 37621 . . . . 5  |-  ( ph  ->  ( ! `  ( P  -  1 ) )  e.  NN )
9998nncnd 10647 . . . 4  |-  ( ph  ->  ( ! `  ( P  -  1 ) )  e.  CC )
10098nnne0d 10676 . . . 4  |-  ( ph  ->  ( ! `  ( P  -  1 ) )  =/=  0 )
10189, 97, 99, 100divdird 10443 . . 3  |-  ( ph  ->  ( ( ( ( A `  0 )  x.  ( ( ( RR  Dn F ) `  ( P  -  1 ) ) `
 0 ) )  +  sum_ k  e.  ( ( ( 0 ... M )  X.  (
0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) ) ) 
\  { <. 0 ,  ( P  - 
1 ) >. } ) ( ( A `  ( 1st `  k ) )  x.  ( ( ( RR  Dn
F ) `  ( 2nd `  k ) ) `
 ( 1st `  k
) ) ) )  /  ( ! `  ( P  -  1
) ) )  =  ( ( ( ( A `  0 )  x.  ( ( ( RR  Dn F ) `  ( P  -  1 ) ) `
 0 ) )  /  ( ! `  ( P  -  1
) ) )  +  ( sum_ k  e.  ( ( ( 0 ... M )  X.  (
0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) ) ) 
\  { <. 0 ,  ( P  - 
1 ) >. } ) ( ( A `  ( 1st `  k ) )  x.  ( ( ( RR  Dn
F ) `  ( 2nd `  k ) ) `
 ( 1st `  k
) ) )  / 
( ! `  ( P  -  1 ) ) ) ) )
1022, 81, 1013eqtrd 2509 . 2  |-  ( ph  ->  K  =  ( ( ( ( A ` 
0 )  x.  (
( ( RR  Dn F ) `  ( P  -  1
) ) `  0
) )  /  ( ! `  ( P  -  1 ) ) )  +  ( sum_ k  e.  ( (
( 0 ... M
)  X.  ( 0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) ) ) 
\  { <. 0 ,  ( P  - 
1 ) >. } ) ( ( A `  ( 1st `  k ) )  x.  ( ( ( RR  Dn
F ) `  ( 2nd `  k ) ) `
 ( 1st `  k
) ) )  / 
( ! `  ( P  -  1 ) ) ) ) )
10326nnne0d 10676 . . 3  |-  ( ph  ->  P  =/=  0 )
10483zcnd 11064 . . . . 5  |-  ( ph  ->  ( A `  0
)  e.  CC )
10587zcnd 11064 . . . . 5  |-  ( ph  ->  ( ( ( RR  Dn F ) `
 ( P  - 
1 ) ) ` 
0 )  e.  CC )
106104, 105, 99, 100divassd 10440 . . . 4  |-  ( ph  ->  ( ( ( A `
 0 )  x.  ( ( ( RR  Dn F ) `
 ( P  - 
1 ) ) ` 
0 ) )  / 
( ! `  ( P  -  1 ) ) )  =  ( ( A `  0
)  x.  ( ( ( ( RR  Dn F ) `  ( P  -  1
) ) `  0
)  /  ( ! `
 ( P  - 
1 ) ) ) ) )
107 etransclem5 38216 . . . . . . 7  |-  ( k  e.  ( 0 ... M )  |->  ( y  e.  RR  |->  ( ( y  -  k ) ^ if ( k  =  0 ,  ( P  -  1 ) ,  P ) ) ) )  =  ( j  e.  ( 0 ... M )  |->  ( x  e.  RR  |->  ( ( x  -  j
) ^ if ( j  =  0 ,  ( P  -  1 ) ,  P ) ) ) )
108 etransclem11 38222 . . . . . . 7  |-  ( m  e.  NN0  |->  { d  e.  ( ( 0 ... m )  ^m  ( 0 ... M
) )  |  sum_ k  e.  ( 0 ... M ) ( d `  k )  =  m } )  =  ( n  e. 
NN0  |->  { c  e.  ( ( 0 ... n )  ^m  (
0 ... M ) )  |  sum_ j  e.  ( 0 ... M ) ( c `  j
)  =  n }
)
10984, 85, 26, 28, 30, 49, 107, 108, 43, 86etransclem37 38248 . . . . . 6  |-  ( ph  ->  ( ! `  ( P  -  1 ) )  ||  ( ( ( RR  Dn
F ) `  ( P  -  1 ) ) `  0 ) )
11098nnzd 11062 . . . . . . 7  |-  ( ph  ->  ( ! `  ( P  -  1 ) )  e.  ZZ )
111 dvdsval2 14385 . . . . . . 7  |-  ( ( ( ! `  ( P  -  1 ) )  e.  ZZ  /\  ( ! `  ( P  -  1 ) )  =/=  0  /\  (
( ( RR  Dn F ) `  ( P  -  1
) ) `  0
)  e.  ZZ )  ->  ( ( ! `
 ( P  - 
1 ) )  ||  ( ( ( RR  Dn F ) `
 ( P  - 
1 ) ) ` 
0 )  <->  ( (
( ( RR  Dn F ) `  ( P  -  1
) ) `  0
)  /  ( ! `
 ( P  - 
1 ) ) )  e.  ZZ ) )
112110, 100, 87, 111syl3anc 1292 . . . . . 6  |-  ( ph  ->  ( ( ! `  ( P  -  1
) )  ||  (
( ( RR  Dn F ) `  ( P  -  1
) ) `  0
)  <->  ( ( ( ( RR  Dn
F ) `  ( P  -  1 ) ) `  0 )  /  ( ! `  ( P  -  1
) ) )  e.  ZZ ) )
113109, 112mpbid 215 . . . . 5  |-  ( ph  ->  ( ( ( ( RR  Dn F ) `  ( P  -  1 ) ) `
 0 )  / 
( ! `  ( P  -  1 ) ) )  e.  ZZ )
11483, 113zmulcld 11069 . . . 4  |-  ( ph  ->  ( ( A ` 
0 )  x.  (
( ( ( RR  Dn F ) `
 ( P  - 
1 ) ) ` 
0 )  /  ( ! `  ( P  -  1 ) ) ) )  e.  ZZ )
115106, 114eqeltrd 2549 . . 3  |-  ( ph  ->  ( ( ( A `
 0 )  x.  ( ( ( RR  Dn F ) `
 ( P  - 
1 ) ) ` 
0 ) )  / 
( ! `  ( P  -  1 ) ) )  e.  ZZ )
11694, 39sylan2 482 . . . . 5  |-  ( (
ph  /\  k  e.  ( ( ( 0 ... M )  X.  ( 0 ... (
( M  x.  P
)  +  ( P  -  1 ) ) ) )  \  { <. 0 ,  ( P  -  1 ) >. } ) )  -> 
( ( A `  ( 1st `  k ) )  x.  ( ( ( RR  Dn
F ) `  ( 2nd `  k ) ) `
 ( 1st `  k
) ) )  e.  CC )
11793, 99, 116, 100fsumdivc 13924 . . . 4  |-  ( ph  ->  ( sum_ k  e.  ( ( ( 0 ... M )  X.  (
0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) ) ) 
\  { <. 0 ,  ( P  - 
1 ) >. } ) ( ( A `  ( 1st `  k ) )  x.  ( ( ( RR  Dn
F ) `  ( 2nd `  k ) ) `
 ( 1st `  k
) ) )  / 
( ! `  ( P  -  1 ) ) )  =  sum_ k  e.  ( (
( 0 ... M
)  X.  ( 0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) ) ) 
\  { <. 0 ,  ( P  - 
1 ) >. } ) ( ( ( A `
 ( 1st `  k
) )  x.  (
( ( RR  Dn F ) `  ( 2nd `  k ) ) `  ( 1st `  k ) ) )  /  ( ! `  ( P  -  1
) ) ) )
11816zcnd 11064 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ( 0 ... M )  X.  (
0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) ) ) )  ->  ( A `  ( 1st `  k
) )  e.  CC )
11994, 118sylan2 482 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ( ( 0 ... M )  X.  ( 0 ... (
( M  x.  P
)  +  ( P  -  1 ) ) ) )  \  { <. 0 ,  ( P  -  1 ) >. } ) )  -> 
( A `  ( 1st `  k ) )  e.  CC )
12094, 37sylan2 482 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ( ( 0 ... M )  X.  ( 0 ... (
( M  x.  P
)  +  ( P  -  1 ) ) ) )  \  { <. 0 ,  ( P  -  1 ) >. } ) )  -> 
( ( ( RR  Dn F ) `
 ( 2nd `  k
) ) `  ( 1st `  k ) )  e.  ZZ )
121120zcnd 11064 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ( ( 0 ... M )  X.  ( 0 ... (
( M  x.  P
)  +  ( P  -  1 ) ) ) )  \  { <. 0 ,  ( P  -  1 ) >. } ) )  -> 
( ( ( RR  Dn F ) `
 ( 2nd `  k
) ) `  ( 1st `  k ) )  e.  CC )
12299adantr 472 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ( ( 0 ... M )  X.  ( 0 ... (
( M  x.  P
)  +  ( P  -  1 ) ) ) )  \  { <. 0 ,  ( P  -  1 ) >. } ) )  -> 
( ! `  ( P  -  1 ) )  e.  CC )
123100adantr 472 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ( ( 0 ... M )  X.  ( 0 ... (
( M  x.  P
)  +  ( P  -  1 ) ) ) )  \  { <. 0 ,  ( P  -  1 ) >. } ) )  -> 
( ! `  ( P  -  1 ) )  =/=  0 )
124119, 121, 122, 123divassd 10440 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ( ( 0 ... M )  X.  ( 0 ... (
( M  x.  P
)  +  ( P  -  1 ) ) ) )  \  { <. 0 ,  ( P  -  1 ) >. } ) )  -> 
( ( ( A `
 ( 1st `  k
) )  x.  (
( ( RR  Dn F ) `  ( 2nd `  k ) ) `  ( 1st `  k ) ) )  /  ( ! `  ( P  -  1
) ) )  =  ( ( A `  ( 1st `  k ) )  x.  ( ( ( ( RR  Dn F ) `  ( 2nd `  k ) ) `  ( 1st `  k ) )  / 
( ! `  ( P  -  1 ) ) ) ) )
12594, 16sylan2 482 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ( ( 0 ... M )  X.  ( 0 ... (
( M  x.  P
)  +  ( P  -  1 ) ) ) )  \  { <. 0 ,  ( P  -  1 ) >. } ) )  -> 
( A `  ( 1st `  k ) )  e.  ZZ )
12617a1i 11 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ( ( 0 ... M )  X.  ( 0 ... (
( M  x.  P
)  +  ( P  -  1 ) ) ) )  \  { <. 0 ,  ( P  -  1 ) >. } ) )  ->  RR  e.  { RR ,  CC } )
12722a1i 11 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ( ( 0 ... M )  X.  ( 0 ... (
( M  x.  P
)  +  ( P  -  1 ) ) ) )  \  { <. 0 ,  ( P  -  1 ) >. } ) )  ->  RR  e.  ( ( TopOpen ` fld )t  RR ) )
12826adantr 472 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ( ( 0 ... M )  X.  ( 0 ... (
( M  x.  P
)  +  ( P  -  1 ) ) ) )  \  { <. 0 ,  ( P  -  1 ) >. } ) )  ->  P  e.  NN )
12928adantr 472 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ( ( 0 ... M )  X.  ( 0 ... (
( M  x.  P
)  +  ( P  -  1 ) ) ) )  \  { <. 0 ,  ( P  -  1 ) >. } ) )  ->  M  e.  NN0 )
13094adantl 473 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( ( ( 0 ... M )  X.  ( 0 ... (
( M  x.  P
)  +  ( P  -  1 ) ) ) )  \  { <. 0 ,  ( P  -  1 ) >. } ) )  -> 
k  e.  ( ( 0 ... M )  X.  ( 0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) ) ) )
131130, 33syl 17 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ( ( 0 ... M )  X.  ( 0 ... (
( M  x.  P
)  +  ( P  -  1 ) ) ) )  \  { <. 0 ,  ( P  -  1 ) >. } ) )  -> 
( 2nd `  k
)  e.  NN0 )
132130, 13syl 17 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ( ( 0 ... M )  X.  ( 0 ... (
( M  x.  P
)  +  ( P  -  1 ) ) ) )  \  { <. 0 ,  ( P  -  1 ) >. } ) )  -> 
( 1st `  k
)  e.  ( 0 ... M ) )
13394, 35sylan2 482 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ( ( 0 ... M )  X.  ( 0 ... (
( M  x.  P
)  +  ( P  -  1 ) ) ) )  \  { <. 0 ,  ( P  -  1 ) >. } ) )  -> 
( 1st `  k
)  e.  RR )
134126, 127, 128, 129, 30, 131, 107, 108, 132, 133etransclem37 38248 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ( ( 0 ... M )  X.  ( 0 ... (
( M  x.  P
)  +  ( P  -  1 ) ) ) )  \  { <. 0 ,  ( P  -  1 ) >. } ) )  -> 
( ! `  ( P  -  1 ) )  ||  ( ( ( RR  Dn
F ) `  ( 2nd `  k ) ) `
 ( 1st `  k
) ) )
135110adantr 472 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ( ( 0 ... M )  X.  ( 0 ... (
( M  x.  P
)  +  ( P  -  1 ) ) ) )  \  { <. 0 ,  ( P  -  1 ) >. } ) )  -> 
( ! `  ( P  -  1 ) )  e.  ZZ )
136 dvdsval2 14385 . . . . . . . . 9  |-  ( ( ( ! `  ( P  -  1 ) )  e.  ZZ  /\  ( ! `  ( P  -  1 ) )  =/=  0  /\  (
( ( RR  Dn F ) `  ( 2nd `  k ) ) `  ( 1st `  k ) )  e.  ZZ )  ->  (
( ! `  ( P  -  1 ) )  ||  ( ( ( RR  Dn
F ) `  ( 2nd `  k ) ) `
 ( 1st `  k
) )  <->  ( (
( ( RR  Dn F ) `  ( 2nd `  k ) ) `  ( 1st `  k ) )  / 
( ! `  ( P  -  1 ) ) )  e.  ZZ ) )
137135, 123, 120, 136syl3anc 1292 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ( ( 0 ... M )  X.  ( 0 ... (
( M  x.  P
)  +  ( P  -  1 ) ) ) )  \  { <. 0 ,  ( P  -  1 ) >. } ) )  -> 
( ( ! `  ( P  -  1
) )  ||  (
( ( RR  Dn F ) `  ( 2nd `  k ) ) `  ( 1st `  k ) )  <->  ( (
( ( RR  Dn F ) `  ( 2nd `  k ) ) `  ( 1st `  k ) )  / 
( ! `  ( P  -  1 ) ) )  e.  ZZ ) )
138134, 137mpbid 215 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ( ( 0 ... M )  X.  ( 0 ... (
( M  x.  P
)  +  ( P  -  1 ) ) ) )  \  { <. 0 ,  ( P  -  1 ) >. } ) )  -> 
( ( ( ( RR  Dn F ) `  ( 2nd `  k ) ) `  ( 1st `  k ) )  /  ( ! `
 ( P  - 
1 ) ) )  e.  ZZ )
139125, 138zmulcld 11069 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ( ( 0 ... M )  X.  ( 0 ... (
( M  x.  P
)  +  ( P  -  1 ) ) ) )  \  { <. 0 ,  ( P  -  1 ) >. } ) )  -> 
( ( A `  ( 1st `  k ) )  x.  ( ( ( ( RR  Dn F ) `  ( 2nd `  k ) ) `  ( 1st `  k ) )  / 
( ! `  ( P  -  1 ) ) ) )  e.  ZZ )
140124, 139eqeltrd 2549 . . . . 5  |-  ( (
ph  /\  k  e.  ( ( ( 0 ... M )  X.  ( 0 ... (
( M  x.  P
)  +  ( P  -  1 ) ) ) )  \  { <. 0 ,  ( P  -  1 ) >. } ) )  -> 
( ( ( A `
 ( 1st `  k
) )  x.  (
( ( RR  Dn F ) `  ( 2nd `  k ) ) `  ( 1st `  k ) ) )  /  ( ! `  ( P  -  1
) ) )  e.  ZZ )
14193, 140fsumzcl 13878 . . . 4  |-  ( ph  -> 
sum_ k  e.  ( ( ( 0 ... M )  X.  (
0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) ) ) 
\  { <. 0 ,  ( P  - 
1 ) >. } ) ( ( ( A `
 ( 1st `  k
) )  x.  (
( ( RR  Dn F ) `  ( 2nd `  k ) ) `  ( 1st `  k ) ) )  /  ( ! `  ( P  -  1
) ) )  e.  ZZ )
142117, 141eqeltrd 2549 . . 3  |-  ( ph  ->  ( sum_ k  e.  ( ( ( 0 ... M )  X.  (
0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) ) ) 
\  { <. 0 ,  ( P  - 
1 ) >. } ) ( ( A `  ( 1st `  k ) )  x.  ( ( ( RR  Dn
F ) `  ( 2nd `  k ) ) `
 ( 1st `  k
) ) )  / 
( ! `  ( P  -  1 ) ) )  e.  ZZ )
143 1zzd 10992 . . . . . . . . . . . 12  |-  ( ph  ->  1  e.  ZZ )
144 zabscl 13453 . . . . . . . . . . . . 13  |-  ( ( A `  0 )  e.  ZZ  ->  ( abs `  ( A ` 
0 ) )  e.  ZZ )
14583, 144syl 17 . . . . . . . . . . . 12  |-  ( ph  ->  ( abs `  ( A `  0 )
)  e.  ZZ )
146143, 50, 1453jca 1210 . . . . . . . . . . 11  |-  ( ph  ->  ( 1  e.  ZZ  /\  ( P  -  1 )  e.  ZZ  /\  ( abs `  ( A `
 0 ) )  e.  ZZ ) )
147 nn0abscl 13452 . . . . . . . . . . . . . 14  |-  ( ( A `  0 )  e.  ZZ  ->  ( abs `  ( A ` 
0 ) )  e. 
NN0 )
14883, 147syl 17 . . . . . . . . . . . . 13  |-  ( ph  ->  ( abs `  ( A `  0 )
)  e.  NN0 )
149 etransclem44.a0 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( A `  0
)  =/=  0 )
150104, 149absne0d 13586 . . . . . . . . . . . . 13  |-  ( ph  ->  ( abs `  ( A `  0 )
)  =/=  0 )
151 elnnne0 10907 . . . . . . . . . . . . 13  |-  ( ( abs `  ( A `
 0 ) )  e.  NN  <->  ( ( abs `  ( A ` 
0 ) )  e. 
NN0  /\  ( abs `  ( A `  0
) )  =/=  0
) )
152148, 150, 151sylanbrc 677 . . . . . . . . . . . 12  |-  ( ph  ->  ( abs `  ( A `  0 )
)  e.  NN )
153152nnge1d 10674 . . . . . . . . . . 11  |-  ( ph  ->  1  <_  ( abs `  ( A `  0
) ) )
154 etransclem44.ap . . . . . . . . . . . 12  |-  ( ph  ->  ( abs `  ( A `  0 )
)  <  P )
155 zltlem1 11013 . . . . . . . . . . . . 13  |-  ( ( ( abs `  ( A `  0 )
)  e.  ZZ  /\  P  e.  ZZ )  ->  ( ( abs `  ( A `  0 )
)  <  P  <->  ( abs `  ( A `  0
) )  <_  ( P  -  1 ) ) )
156145, 46, 155syl2anc 673 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( abs `  ( A `  0 )
)  <  P  <->  ( abs `  ( A `  0
) )  <_  ( P  -  1 ) ) )
157154, 156mpbid 215 . . . . . . . . . . 11  |-  ( ph  ->  ( abs `  ( A `  0 )
)  <_  ( P  -  1 ) )
158146, 153, 157jca32 544 . . . . . . . . . 10  |-  ( ph  ->  ( ( 1  e.  ZZ  /\  ( P  -  1 )  e.  ZZ  /\  ( abs `  ( A `  0
) )  e.  ZZ )  /\  ( 1  <_ 
( abs `  ( A `  0 )
)  /\  ( abs `  ( A `  0
) )  <_  ( P  -  1 ) ) ) )
159 elfz2 11817 . . . . . . . . . 10  |-  ( ( abs `  ( A `
 0 ) )  e.  ( 1 ... ( P  -  1 ) )  <->  ( (
1  e.  ZZ  /\  ( P  -  1
)  e.  ZZ  /\  ( abs `  ( A `
 0 ) )  e.  ZZ )  /\  ( 1  <_  ( abs `  ( A ` 
0 ) )  /\  ( abs `  ( A `
 0 ) )  <_  ( P  - 
1 ) ) ) )
160158, 159sylibr 217 . . . . . . . . 9  |-  ( ph  ->  ( abs `  ( A `  0 )
)  e.  ( 1 ... ( P  - 
1 ) ) )
161 fzm1ndvds 14434 . . . . . . . . 9  |-  ( ( P  e.  NN  /\  ( abs `  ( A `
 0 ) )  e.  ( 1 ... ( P  -  1 ) ) )  ->  -.  P  ||  ( abs `  ( A `  0
) ) )
16226, 160, 161syl2anc 673 . . . . . . . 8  |-  ( ph  ->  -.  P  ||  ( abs `  ( A ` 
0 ) ) )
163 dvdsabsb 14399 . . . . . . . . 9  |-  ( ( P  e.  ZZ  /\  ( A `  0 )  e.  ZZ )  -> 
( P  ||  ( A `  0 )  <->  P 
||  ( abs `  ( A `  0 )
) ) )
16446, 83, 163syl2anc 673 . . . . . . . 8  |-  ( ph  ->  ( P  ||  ( A `  0 )  <->  P 
||  ( abs `  ( A `  0 )
) ) )
165162, 164mtbird 308 . . . . . . 7  |-  ( ph  ->  -.  P  ||  ( A `  0 )
)
166 etransclem44.mp . . . . . . . 8  |-  ( ph  ->  ( ! `  M
)  <  P )
16728, 24, 166, 30etransclem41 38252 . . . . . . 7  |-  ( ph  ->  -.  P  ||  (
( ( ( RR  Dn F ) `
 ( P  - 
1 ) ) ` 
0 )  /  ( ! `  ( P  -  1 ) ) ) )
168165, 167jca 541 . . . . . 6  |-  ( ph  ->  ( -.  P  ||  ( A `  0 )  /\  -.  P  ||  ( ( ( ( RR  Dn F ) `  ( P  -  1 ) ) `
 0 )  / 
( ! `  ( P  -  1 ) ) ) ) )
169 pm4.56 503 . . . . . 6  |-  ( ( -.  P  ||  ( A `  0 )  /\  -.  P  ||  (
( ( ( RR  Dn F ) `
 ( P  - 
1 ) ) ` 
0 )  /  ( ! `  ( P  -  1 ) ) ) )  <->  -.  ( P  ||  ( A ` 
0 )  \/  P  ||  ( ( ( ( RR  Dn F ) `  ( P  -  1 ) ) `
 0 )  / 
( ! `  ( P  -  1 ) ) ) ) )
170168, 169sylib 201 . . . . 5  |-  ( ph  ->  -.  ( P  ||  ( A `  0 )  \/  P  ||  (
( ( ( RR  Dn F ) `
 ( P  - 
1 ) ) ` 
0 )  /  ( ! `  ( P  -  1 ) ) ) ) )
171 euclemma 14744 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( A `  0 )  e.  ZZ  /\  ( ( ( ( RR  Dn F ) `  ( P  -  1
) ) `  0
)  /  ( ! `
 ( P  - 
1 ) ) )  e.  ZZ )  -> 
( P  ||  (
( A `  0
)  x.  ( ( ( ( RR  Dn F ) `  ( P  -  1
) ) `  0
)  /  ( ! `
 ( P  - 
1 ) ) ) )  <->  ( P  ||  ( A `  0 )  \/  P  ||  (
( ( ( RR  Dn F ) `
 ( P  - 
1 ) ) ` 
0 )  /  ( ! `  ( P  -  1 ) ) ) ) ) )
17224, 83, 113, 171syl3anc 1292 . . . . 5  |-  ( ph  ->  ( P  ||  (
( A `  0
)  x.  ( ( ( ( RR  Dn F ) `  ( P  -  1
) ) `  0
)  /  ( ! `
 ( P  - 
1 ) ) ) )  <->  ( P  ||  ( A `  0 )  \/  P  ||  (
( ( ( RR  Dn F ) `
 ( P  - 
1 ) ) ` 
0 )  /  ( ! `  ( P  -  1 ) ) ) ) ) )
173170, 172mtbird 308 . . . 4  |-  ( ph  ->  -.  P  ||  (
( A `  0
)  x.  ( ( ( ( RR  Dn F ) `  ( P  -  1
) ) `  0
)  /  ( ! `
 ( P  - 
1 ) ) ) ) )
174106breq2d 4407 . . . 4  |-  ( ph  ->  ( P  ||  (
( ( A ` 
0 )  x.  (
( ( RR  Dn F ) `  ( P  -  1
) ) `  0
) )  /  ( ! `  ( P  -  1 ) ) )  <->  P  ||  ( ( A `  0 )  x.  ( ( ( ( RR  Dn
F ) `  ( P  -  1 ) ) `  0 )  /  ( ! `  ( P  -  1
) ) ) ) ) )
175173, 174mtbird 308 . . 3  |-  ( ph  ->  -.  P  ||  (
( ( A ` 
0 )  x.  (
( ( RR  Dn F ) `  ( P  -  1
) ) `  0
) )  /  ( ! `  ( P  -  1 ) ) ) )
17646adantr 472 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ( ( 0 ... M )  X.  ( 0 ... (
( M  x.  P
)  +  ( P  -  1 ) ) ) )  \  { <. 0 ,  ( P  -  1 ) >. } ) )  ->  P  e.  ZZ )
177176, 125, 1383jca 1210 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ( ( 0 ... M )  X.  ( 0 ... (
( M  x.  P
)  +  ( P  -  1 ) ) ) )  \  { <. 0 ,  ( P  -  1 ) >. } ) )  -> 
( P  e.  ZZ  /\  ( A `  ( 1st `  k ) )  e.  ZZ  /\  (
( ( ( RR  Dn F ) `
 ( 2nd `  k
) ) `  ( 1st `  k ) )  /  ( ! `  ( P  -  1
) ) )  e.  ZZ ) )
178 eldifn 3545 . . . . . . . . . 10  |-  ( k  e.  ( ( ( 0 ... M )  X.  ( 0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) ) )  \  { <. 0 ,  ( P  -  1 )
>. } )  ->  -.  k  e.  { <. 0 ,  ( P  - 
1 ) >. } )
17994adantr 472 . . . . . . . . . . . . 13  |-  ( ( k  e.  ( ( ( 0 ... M
)  X.  ( 0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) ) ) 
\  { <. 0 ,  ( P  - 
1 ) >. } )  /\  ( ( 2nd `  k )  =  ( P  -  1 )  /\  ( 1st `  k
)  =  0 ) )  ->  k  e.  ( ( 0 ... M )  X.  (
0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) ) ) )
180 1st2nd2 6849 . . . . . . . . . . . . 13  |-  ( k  e.  ( ( 0 ... M )  X.  ( 0 ... (
( M  x.  P
)  +  ( P  -  1 ) ) ) )  ->  k  =  <. ( 1st `  k
) ,  ( 2nd `  k ) >. )
181179, 180syl 17 . . . . . . . . . . . 12  |-  ( ( k  e.  ( ( ( 0 ... M
)  X.  ( 0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) ) ) 
\  { <. 0 ,  ( P  - 
1 ) >. } )  /\  ( ( 2nd `  k )  =  ( P  -  1 )  /\  ( 1st `  k
)  =  0 ) )  ->  k  =  <. ( 1st `  k
) ,  ( 2nd `  k ) >. )
182 simpr 468 . . . . . . . . . . . . . 14  |-  ( ( ( 2nd `  k
)  =  ( P  -  1 )  /\  ( 1st `  k )  =  0 )  -> 
( 1st `  k
)  =  0 )
183 simpl 464 . . . . . . . . . . . . . 14  |-  ( ( ( 2nd `  k
)  =  ( P  -  1 )  /\  ( 1st `  k )  =  0 )  -> 
( 2nd `  k
)  =  ( P  -  1 ) )
184182, 183opeq12d 4166 . . . . . . . . . . . . 13  |-  ( ( ( 2nd `  k
)  =  ( P  -  1 )  /\  ( 1st `  k )  =  0 )  ->  <. ( 1st `  k
) ,  ( 2nd `  k ) >.  =  <. 0 ,  ( P  -  1 ) >.
)
185184adantl 473 . . . . . . . . . . . 12  |-  ( ( k  e.  ( ( ( 0 ... M
)  X.  ( 0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) ) ) 
\  { <. 0 ,  ( P  - 
1 ) >. } )  /\  ( ( 2nd `  k )  =  ( P  -  1 )  /\  ( 1st `  k
)  =  0 ) )  ->  <. ( 1st `  k ) ,  ( 2nd `  k )
>.  =  <. 0 ,  ( P  -  1 ) >. )
186181, 185eqtrd 2505 . . . . . . . . . . 11  |-  ( ( k  e.  ( ( ( 0 ... M
)  X.  ( 0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) ) ) 
\  { <. 0 ,  ( P  - 
1 ) >. } )  /\  ( ( 2nd `  k )  =  ( P  -  1 )  /\  ( 1st `  k
)  =  0 ) )  ->  k  =  <. 0 ,  ( P  -  1 ) >.
)
187 elsn 3973 . . . . . . . . . . 11  |-  ( k  e.  { <. 0 ,  ( P  - 
1 ) >. }  <->  k  =  <. 0 ,  ( P  -  1 ) >.
)
188186, 187sylibr 217 . . . . . . . . . 10  |-  ( ( k  e.  ( ( ( 0 ... M
)  X.  ( 0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) ) ) 
\  { <. 0 ,  ( P  - 
1 ) >. } )  /\  ( ( 2nd `  k )  =  ( P  -  1 )  /\  ( 1st `  k
)  =  0 ) )  ->  k  e.  {
<. 0 ,  ( P  -  1 )
>. } )
189178, 188mtand 671 . . . . . . . . 9  |-  ( k  e.  ( ( ( 0 ... M )  X.  ( 0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) ) )  \  { <. 0 ,  ( P  -  1 )
>. } )  ->  -.  ( ( 2nd `  k
)  =  ( P  -  1 )  /\  ( 1st `  k )  =  0 ) )
190189adantl 473 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ( ( 0 ... M )  X.  ( 0 ... (
( M  x.  P
)  +  ( P  -  1 ) ) ) )  \  { <. 0 ,  ( P  -  1 ) >. } ) )  ->  -.  ( ( 2nd `  k
)  =  ( P  -  1 )  /\  ( 1st `  k )  =  0 ) )
191128, 129, 30, 131, 132, 190, 108etransclem38 38249 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ( ( 0 ... M )  X.  ( 0 ... (
( M  x.  P
)  +  ( P  -  1 ) ) ) )  \  { <. 0 ,  ( P  -  1 ) >. } ) )  ->  P  ||  ( ( ( ( RR  Dn
F ) `  ( 2nd `  k ) ) `
 ( 1st `  k
) )  /  ( ! `  ( P  -  1 ) ) ) )
192 dvdsmultr2 14417 . . . . . . 7  |-  ( ( P  e.  ZZ  /\  ( A `  ( 1st `  k ) )  e.  ZZ  /\  ( ( ( ( RR  Dn F ) `  ( 2nd `  k ) ) `  ( 1st `  k ) )  / 
( ! `  ( P  -  1 ) ) )  e.  ZZ )  ->  ( P  ||  ( ( ( ( RR  Dn F ) `  ( 2nd `  k ) ) `  ( 1st `  k ) )  /  ( ! `
 ( P  - 
1 ) ) )  ->  P  ||  (
( A `  ( 1st `  k ) )  x.  ( ( ( ( RR  Dn
F ) `  ( 2nd `  k ) ) `
 ( 1st `  k
) )  /  ( ! `  ( P  -  1 ) ) ) ) ) )
193177, 191, 192sylc 61 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ( ( 0 ... M )  X.  ( 0 ... (
( M  x.  P
)  +  ( P  -  1 ) ) ) )  \  { <. 0 ,  ( P  -  1 ) >. } ) )  ->  P  ||  ( ( A `
 ( 1st `  k
) )  x.  (
( ( ( RR  Dn F ) `
 ( 2nd `  k
) ) `  ( 1st `  k ) )  /  ( ! `  ( P  -  1
) ) ) ) )
194193, 124breqtrrd 4422 . . . . 5  |-  ( (
ph  /\  k  e.  ( ( ( 0 ... M )  X.  ( 0 ... (
( M  x.  P
)  +  ( P  -  1 ) ) ) )  \  { <. 0 ,  ( P  -  1 ) >. } ) )  ->  P  ||  ( ( ( A `  ( 1st `  k ) )  x.  ( ( ( RR  Dn F ) `
 ( 2nd `  k
) ) `  ( 1st `  k ) ) )  /  ( ! `
 ( P  - 
1 ) ) ) )
19593, 46, 140, 194fsumdvds 14425 . . . 4  |-  ( ph  ->  P  ||  sum_ k  e.  ( ( ( 0 ... M )  X.  ( 0 ... (
( M  x.  P
)  +  ( P  -  1 ) ) ) )  \  { <. 0 ,  ( P  -  1 ) >. } ) ( ( ( A `  ( 1st `  k ) )  x.  ( ( ( RR  Dn F ) `  ( 2nd `  k ) ) `  ( 1st `  k ) ) )  /  ( ! `  ( P  -  1 ) ) ) )
196195, 117breqtrrd 4422 . . 3  |-  ( ph  ->  P  ||  ( sum_ k  e.  ( (
( 0 ... M
)  X.  ( 0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) ) ) 
\  { <. 0 ,  ( P  - 
1 ) >. } ) ( ( A `  ( 1st `  k ) )  x.  ( ( ( RR  Dn
F ) `  ( 2nd `  k ) ) `
 ( 1st `  k
) ) )  / 
( ! `  ( P  -  1 ) ) ) )
19746, 103, 115, 142, 175, 196etransclem9 38220 . 2  |-  ( ph  ->  ( ( ( ( A `  0 )  x.  ( ( ( RR  Dn F ) `  ( P  -  1 ) ) `
 0 ) )  /  ( ! `  ( P  -  1
) ) )  +  ( sum_ k  e.  ( ( ( 0 ... M )  X.  (
0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) ) ) 
\  { <. 0 ,  ( P  - 
1 ) >. } ) ( ( A `  ( 1st `  k ) )  x.  ( ( ( RR  Dn
F ) `  ( 2nd `  k ) ) `
 ( 1st `  k
) ) )  / 
( ! `  ( P  -  1 ) ) ) )  =/=  0 )
198102, 197eqnetrd 2710 1  |-  ( ph  ->  K  =/=  0 )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    \/ wo 375    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641   {crab 2760   _Vcvv 3031    \ cdif 3387    C_ wss 3390   ifcif 3872   {csn 3959   {cpr 3961   <.cop 3965   class class class wbr 4395    |-> cmpt 4454    X. cxp 4837   ran crn 4840   -->wf 5585   ` cfv 5589  (class class class)co 6308   1stc1st 6810   2ndc2nd 6811    ^m cmap 7490   Fincfn 7587   CCcc 9555   RRcr 9556   0cc0 9557   1c1 9558    + caddc 9560    x. cmul 9562    < clt 9693    <_ cle 9694    - cmin 9880    / cdiv 10291   NNcn 10631   NN0cn0 10893   ZZcz 10961   ZZ>=cuz 11182   (,)cioo 11660   ...cfz 11810   ^cexp 12310   !cfa 12497   abscabs 13374   sum_csu 13829   prod_cprod 14036    || cdvds 14382   Primecprime 14701   ↾t crest 15397   TopOpenctopn 15398   topGenctg 15414  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-ioo 11664  df-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-fl 12061  df-mod 12130  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-dvds 14383  df-gcd 14548  df-prm 14702  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:  etransclem47  38258
  Copyright terms: Public domain W3C validator