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Theorem etransclem44 38143
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 1761 . . . . 5  |-  F/ k
ph
4 nfcv 2592 . . . . 5  |-  F/_ k
( ( A ` 
0 )  x.  (
( ( RR  Dn F ) `  ( P  -  1
) ) `  0
) )
5 fzfi 12185 . . . . . . 7  |-  ( 0 ... M )  e. 
Fin
6 fzfi 12185 . . . . . . 7  |-  ( 0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) )  e. 
Fin
7 xpfi 7842 . . . . . . 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 678 . . . . . 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 467 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ( 0 ... M )  X.  (
0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) ) ) )  ->  A : NN0
--> ZZ )
12 fzssnn0 37539 . . . . . . . . . 10  |-  ( 0 ... M )  C_  NN0
13 xp1st 6823 . . . . . . . . . 10  |-  ( k  e.  ( ( 0 ... M )  X.  ( 0 ... (
( M  x.  P
)  +  ( P  -  1 ) ) ) )  ->  ( 1st `  k )  e.  ( 0 ... M
) )
1412, 13sseldi 3430 . . . . . . . . 9  |-  ( k  e.  ( ( 0 ... M )  X.  ( 0 ... (
( M  x.  P
)  +  ( P  -  1 ) ) ) )  ->  ( 1st `  k )  e. 
NN0 )
1514adantl 468 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ( 0 ... M )  X.  (
0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) ) ) )  ->  ( 1st `  k )  e.  NN0 )
1611, 15ffvelrnd 6023 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ( 0 ... M )  X.  (
0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) ) ) )  ->  ( A `  ( 1st `  k
) )  e.  ZZ )
17 reelprrecn 9631 . . . . . . . . 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 37502 . . . . . . . . . 10  |-  RR  e.  ( topGen `  ran  (,) )
20 eqid 2451 . . . . . . . . . . 11  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
2120tgioo2 21821 . . . . . . . . . 10  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
2219, 21eleqtri 2527 . . . . . . . . 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 14625 . . . . . . . . . 10  |-  ( P  e.  Prime  ->  P  e.  NN )
2624, 25syl 17 . . . . . . . . 9  |-  ( ph  ->  P  e.  NN )
2726adantr 467 . . . . . . . 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 467 . . . . . . . 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 6824 . . . . . . . . . 10  |-  ( k  e.  ( ( 0 ... M )  X.  ( 0 ... (
( M  x.  P
)  +  ( P  -  1 ) ) ) )  ->  ( 2nd `  k )  e.  ( 0 ... (
( M  x.  P
)  +  ( P  -  1 ) ) ) )
32 elfznn0 11887 . . . . . . . . . 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 468 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ( 0 ... M )  X.  (
0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) ) ) )  ->  ( 2nd `  k )  e.  NN0 )
3515nn0red 10926 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ( 0 ... M )  X.  (
0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) ) ) )  ->  ( 1st `  k )  e.  RR )
3615nn0zd 11038 . . . . . . . 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 38141 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ( 0 ... M )  X.  (
0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) ) ) )  ->  ( (
( RR  Dn
F ) `  ( 2nd `  k ) ) `
 ( 1st `  k
) )  e.  ZZ )
3816, 37zmulcld 11046 . . . . . 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 11041 . . . . 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 11193 . . . . . . . 8  |-  NN0  =  ( ZZ>= `  0 )
4128, 40syl6eleq 2539 . . . . . . 7  |-  ( ph  ->  M  e.  ( ZZ>= ` 
0 ) )
42 eluzfz1 11806 . . . . . . 7  |-  ( M  e.  ( ZZ>= `  0
)  ->  0  e.  ( 0 ... M
) )
4341, 42syl 17 . . . . . 6  |-  ( ph  ->  0  e.  ( 0 ... M ) )
44 0zd 10949 . . . . . . . . 9  |-  ( ph  ->  0  e.  ZZ )
4528nn0zd 11038 . . . . . . . . . . 11  |-  ( ph  ->  M  e.  ZZ )
4626nnzd 11039 . . . . . . . . . . 11  |-  ( ph  ->  P  e.  ZZ )
4745, 46zmulcld 11046 . . . . . . . . . 10  |-  ( ph  ->  ( M  x.  P
)  e.  ZZ )
48 nnm1nn0 10911 . . . . . . . . . . . 12  |-  ( P  e.  NN  ->  ( P  -  1 )  e.  NN0 )
4926, 48syl 17 . . . . . . . . . . 11  |-  ( ph  ->  ( P  -  1 )  e.  NN0 )
5049nn0zd 11038 . . . . . . . . . 10  |-  ( ph  ->  ( P  -  1 )  e.  ZZ )
5147, 50zaddcld 11044 . . . . . . . . 9  |-  ( ph  ->  ( ( M  x.  P )  +  ( P  -  1 ) )  e.  ZZ )
5244, 51, 503jca 1188 . . . . . . . 8  |-  ( ph  ->  ( 0  e.  ZZ  /\  ( ( M  x.  P )  +  ( P  -  1 ) )  e.  ZZ  /\  ( P  -  1
)  e.  ZZ ) )
5349nn0ge0d 10928 . . . . . . . 8  |-  ( ph  ->  0  <_  ( P  -  1 ) )
5426nnnn0d 10925 . . . . . . . . . . 11  |-  ( ph  ->  P  e.  NN0 )
5528, 54nn0mulcld 10930 . . . . . . . . . 10  |-  ( ph  ->  ( M  x.  P
)  e.  NN0 )
5655nn0ge0d 10928 . . . . . . . . 9  |-  ( ph  ->  0  <_  ( M  x.  P ) )
5749nn0red 10926 . . . . . . . . . 10  |-  ( ph  ->  ( P  -  1 )  e.  RR )
5847zred 11040 . . . . . . . . . 10  |-  ( ph  ->  ( M  x.  P
)  e.  RR )
5957, 58addge02d 10202 . . . . . . . . 9  |-  ( ph  ->  ( 0  <_  ( M  x.  P )  <->  ( P  -  1 )  <_  ( ( M  x.  P )  +  ( P  -  1 ) ) ) )
6056, 59mpbid 214 . . . . . . . 8  |-  ( ph  ->  ( P  -  1 )  <_  ( ( M  x.  P )  +  ( P  - 
1 ) ) )
6152, 53, 60jca32 538 . . . . . . 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 11791 . . . . . . 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 216 . . . . . 6  |-  ( ph  ->  ( P  -  1 )  e.  ( 0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) ) )
64 opelxp 4864 . . . . . 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 670 . . . . 5  |-  ( ph  -> 
<. 0 ,  ( P  -  1 )
>.  e.  ( ( 0 ... M )  X.  ( 0 ... (
( M  x.  P
)  +  ( P  -  1 ) ) ) ) )
66 fveq2 5865 . . . . . . . 8  |-  ( k  =  <. 0 ,  ( P  -  1 )
>.  ->  ( 1st `  k
)  =  ( 1st `  <. 0 ,  ( P  -  1 )
>. ) )
67 0re 9643 . . . . . . . . 9  |-  0  e.  RR
68 ovex 6318 . . . . . . . . 9  |-  ( P  -  1 )  e. 
_V
69 op1stg 6805 . . . . . . . . 9  |-  ( ( 0  e.  RR  /\  ( P  -  1
)  e.  _V )  ->  ( 1st `  <. 0 ,  ( P  -  1 ) >.
)  =  0 )
7067, 68, 69mp2an 678 . . . . . . . 8  |-  ( 1st `  <. 0 ,  ( P  -  1 )
>. )  =  0
7166, 70syl6eq 2501 . . . . . . 7  |-  ( k  =  <. 0 ,  ( P  -  1 )
>.  ->  ( 1st `  k
)  =  0 )
7271fveq2d 5869 . . . . . 6  |-  ( k  =  <. 0 ,  ( P  -  1 )
>.  ->  ( A `  ( 1st `  k ) )  =  ( A `
 0 ) )
73 fveq2 5865 . . . . . . . . 9  |-  ( k  =  <. 0 ,  ( P  -  1 )
>.  ->  ( 2nd `  k
)  =  ( 2nd `  <. 0 ,  ( P  -  1 )
>. ) )
74 op2ndg 6806 . . . . . . . . . 10  |-  ( ( 0  e.  RR  /\  ( P  -  1
)  e.  _V )  ->  ( 2nd `  <. 0 ,  ( P  -  1 ) >.
)  =  ( P  -  1 ) )
7567, 68, 74mp2an 678 . . . . . . . . 9  |-  ( 2nd `  <. 0 ,  ( P  -  1 )
>. )  =  ( P  -  1 )
7673, 75syl6eq 2501 . . . . . . . 8  |-  ( k  =  <. 0 ,  ( P  -  1 )
>.  ->  ( 2nd `  k
)  =  ( P  -  1 ) )
7776fveq2d 5869 . . . . . . 7  |-  ( k  =  <. 0 ,  ( P  -  1 )
>.  ->  ( ( RR  Dn F ) `
 ( 2nd `  k
) )  =  ( ( RR  Dn
F ) `  ( P  -  1 ) ) )
7877, 71fveq12d 5871 . . . . . 6  |-  ( k  =  <. 0 ,  ( P  -  1 )
>.  ->  ( ( ( RR  Dn F ) `  ( 2nd `  k ) ) `  ( 1st `  k ) )  =  ( ( ( RR  Dn
F ) `  ( P  -  1 ) ) `  0 ) )
7972, 78oveq12d 6308 . . . . 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 37651 . . . 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 6305 . . 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 3430 . . . . . . 7  |-  ( ph  ->  0  e.  NN0 )
8310, 82ffvelrnd 6023 . . . . . 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 38141 . . . . . 6  |-  ( ph  ->  ( ( ( RR  Dn F ) `
 ( P  - 
1 ) ) ` 
0 )  e.  ZZ )
8883, 87zmulcld 11046 . . . . 5  |-  ( ph  ->  ( ( A ` 
0 )  x.  (
( ( RR  Dn F ) `  ( P  -  1
) ) `  0
) )  e.  ZZ )
8988zcnd 11041 . . . 4  |-  ( ph  ->  ( ( A ` 
0 )  x.  (
( ( RR  Dn F ) `  ( P  -  1
) ) `  0
) )  e.  CC )
90 difss 3560 . . . . . . . 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 7792 . . . . . . . 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 678 . . . . . . 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 3555 . . . . . . 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 477 . . . . . 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 13801 . . . . 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 11041 . . . 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 37533 . . . . 5  |-  ( ph  ->  ( ! `  ( P  -  1 ) )  e.  NN )
9998nncnd 10625 . . . 4  |-  ( ph  ->  ( ! `  ( P  -  1 ) )  e.  CC )
10098nnne0d 10654 . . . 4  |-  ( ph  ->  ( ! `  ( P  -  1 ) )  =/=  0 )
10189, 97, 99, 100divdird 10421 . . 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 2489 . 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 10654 . . 3  |-  ( ph  ->  P  =/=  0 )
10483zcnd 11041 . . . . 5  |-  ( ph  ->  ( A `  0
)  e.  CC )
10587zcnd 11041 . . . . 5  |-  ( ph  ->  ( ( ( RR  Dn F ) `
 ( P  - 
1 ) ) ` 
0 )  e.  CC )
106104, 105, 99, 100divassd 10418 . . . 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 38104 . . . . . . 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 38110 . . . . . . 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 38136 . . . . . 6  |-  ( ph  ->  ( ! `  ( P  -  1 ) )  ||  ( ( ( RR  Dn
F ) `  ( P  -  1 ) ) `  0 ) )
11098nnzd 11039 . . . . . . 7  |-  ( ph  ->  ( ! `  ( P  -  1 ) )  e.  ZZ )
111 dvdsval2 14308 . . . . . . 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 1268 . . . . . 6  |-  ( ph  ->  ( ( ! `  ( P  -  1
) )  ||  (
( ( RR  Dn F ) `  ( P  -  1
) ) `  0
)  <->  ( ( ( ( RR  Dn
F ) `  ( P  -  1 ) ) `  0 )  /  ( ! `  ( P  -  1
) ) )  e.  ZZ ) )
113109, 112mpbid 214 . . . . 5  |-  ( ph  ->  ( ( ( ( RR  Dn F ) `  ( P  -  1 ) ) `
 0 )  / 
( ! `  ( P  -  1 ) ) )  e.  ZZ )
11483, 113zmulcld 11046 . . . 4  |-  ( ph  ->  ( ( A ` 
0 )  x.  (
( ( ( RR  Dn F ) `
 ( P  - 
1 ) ) ` 
0 )  /  ( ! `  ( P  -  1 ) ) ) )  e.  ZZ )
115106, 114eqeltrd 2529 . . 3  |-  ( ph  ->  ( ( ( A `
 0 )  x.  ( ( ( RR  Dn F ) `
 ( P  - 
1 ) ) ` 
0 ) )  / 
( ! `  ( P  -  1 ) ) )  e.  ZZ )
11694, 39sylan2 477 . . . . 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 13847 . . . 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 11041 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ( 0 ... M )  X.  (
0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) ) ) )  ->  ( A `  ( 1st `  k
) )  e.  CC )
11994, 118sylan2 477 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ( ( 0 ... M )  X.  ( 0 ... (
( M  x.  P
)  +  ( P  -  1 ) ) ) )  \  { <. 0 ,  ( P  -  1 ) >. } ) )  -> 
( A `  ( 1st `  k ) )  e.  CC )
12094, 37sylan2 477 . . . . . . . 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 11041 . . . . . . 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 467 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ( ( 0 ... M )  X.  ( 0 ... (
( M  x.  P
)  +  ( P  -  1 ) ) ) )  \  { <. 0 ,  ( P  -  1 ) >. } ) )  -> 
( ! `  ( P  -  1 ) )  e.  CC )
123100adantr 467 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ( ( 0 ... M )  X.  ( 0 ... (
( M  x.  P
)  +  ( P  -  1 ) ) ) )  \  { <. 0 ,  ( P  -  1 ) >. } ) )  -> 
( ! `  ( P  -  1 ) )  =/=  0 )
124119, 121, 122, 123divassd 10418 . . . . . 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 477 . . . . . . 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 467 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ( ( 0 ... M )  X.  ( 0 ... (
( M  x.  P
)  +  ( P  -  1 ) ) ) )  \  { <. 0 ,  ( P  -  1 ) >. } ) )  ->  P  e.  NN )
12928adantr 467 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ( ( 0 ... M )  X.  ( 0 ... (
( M  x.  P
)  +  ( P  -  1 ) ) ) )  \  { <. 0 ,  ( P  -  1 ) >. } ) )  ->  M  e.  NN0 )
13094adantl 468 . . . . . . . . . 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 477 . . . . . . . . 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 38136 . . . . . . . 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 467 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ( ( 0 ... M )  X.  ( 0 ... (
( M  x.  P
)  +  ( P  -  1 ) ) ) )  \  { <. 0 ,  ( P  -  1 ) >. } ) )  -> 
( ! `  ( P  -  1 ) )  e.  ZZ )
136 dvdsval2 14308 . . . . . . . . 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 1268 . . . . . . . 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 214 . . . . . . 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 11046 . . . . . 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 2529 . . . . 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 13801 . . . 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 2529 . . 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 10968 . . . . . . . . . . . 12  |-  ( ph  ->  1  e.  ZZ )
144 zabscl 13376 . . . . . . . . . . . . 13  |-  ( ( A `  0 )  e.  ZZ  ->  ( abs `  ( A ` 
0 ) )  e.  ZZ )
14583, 144syl 17 . . . . . . . . . . . 12  |-  ( ph  ->  ( abs `  ( A `  0 )
)  e.  ZZ )
146143, 50, 1453jca 1188 . . . . . . . . . . 11  |-  ( ph  ->  ( 1  e.  ZZ  /\  ( P  -  1 )  e.  ZZ  /\  ( abs `  ( A `
 0 ) )  e.  ZZ ) )
147 nn0abscl 13375 . . . . . . . . . . . . . 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 13509 . . . . . . . . . . . . 13  |-  ( ph  ->  ( abs `  ( A `  0 )
)  =/=  0 )
151 elnnne0 10883 . . . . . . . . . . . . 13  |-  ( ( abs `  ( A `
 0 ) )  e.  NN  <->  ( ( abs `  ( A ` 
0 ) )  e. 
NN0  /\  ( abs `  ( A `  0
) )  =/=  0
) )
152148, 150, 151sylanbrc 670 . . . . . . . . . . . 12  |-  ( ph  ->  ( abs `  ( A `  0 )
)  e.  NN )
153152nnge1d 10652 . . . . . . . . . . 11  |-  ( ph  ->  1  <_  ( abs `  ( A `  0
) ) )
154 etransclem44.ap . . . . . . . . . . . 12  |-  ( ph  ->  ( abs `  ( A `  0 )
)  <  P )
155 zltlem1 10989 . . . . . . . . . . . . 13  |-  ( ( ( abs `  ( A `  0 )
)  e.  ZZ  /\  P  e.  ZZ )  ->  ( ( abs `  ( A `  0 )
)  <  P  <->  ( abs `  ( A `  0
) )  <_  ( P  -  1 ) ) )
156145, 46, 155syl2anc 667 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( abs `  ( A `  0 )
)  <  P  <->  ( abs `  ( A `  0
) )  <_  ( P  -  1 ) ) )
157154, 156mpbid 214 . . . . . . . . . . 11  |-  ( ph  ->  ( abs `  ( A `  0 )
)  <_  ( P  -  1 ) )
158146, 153, 157jca32 538 . . . . . . . . . 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 11791 . . . . . . . . . 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 216 . . . . . . . . 9  |-  ( ph  ->  ( abs `  ( A `  0 )
)  e.  ( 1 ... ( P  - 
1 ) ) )
161 fzm1ndvds 14357 . . . . . . . . 9  |-  ( ( P  e.  NN  /\  ( abs `  ( A `
 0 ) )  e.  ( 1 ... ( P  -  1 ) ) )  ->  -.  P  ||  ( abs `  ( A `  0
) ) )
16226, 160, 161syl2anc 667 . . . . . . . 8  |-  ( ph  ->  -.  P  ||  ( abs `  ( A ` 
0 ) ) )
163 dvdsabsb 14322 . . . . . . . . 9  |-  ( ( P  e.  ZZ  /\  ( A `  0 )  e.  ZZ )  -> 
( P  ||  ( A `  0 )  <->  P 
||  ( abs `  ( A `  0 )
) ) )
16446, 83, 163syl2anc 667 . . . . . . . 8  |-  ( ph  ->  ( P  ||  ( A `  0 )  <->  P 
||  ( abs `  ( A `  0 )
) ) )
165162, 164mtbird 303 . . . . . . 7  |-  ( ph  ->  -.  P  ||  ( A `  0 )
)
166 etransclem44.mp . . . . . . . 8  |-  ( ph  ->  ( ! `  M
)  <  P )
16728, 24, 166, 30etransclem41 38140 . . . . . . 7  |-  ( ph  ->  -.  P  ||  (
( ( ( RR  Dn F ) `
 ( P  - 
1 ) ) ` 
0 )  /  ( ! `  ( P  -  1 ) ) ) )
168165, 167jca 535 . . . . . 6  |-  ( ph  ->  ( -.  P  ||  ( A `  0 )  /\  -.  P  ||  ( ( ( ( RR  Dn F ) `  ( P  -  1 ) ) `
 0 )  / 
( ! `  ( P  -  1 ) ) ) ) )
169 pm4.56 498 . . . . . 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 200 . . . . 5  |-  ( ph  ->  -.  ( P  ||  ( A `  0 )  \/  P  ||  (
( ( ( RR  Dn F ) `
 ( P  - 
1 ) ) ` 
0 )  /  ( ! `  ( P  -  1 ) ) ) ) )
171 euclemma 14665 . . . . . 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 1268 . . . . 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 303 . . . 4  |-  ( ph  ->  -.  P  ||  (
( A `  0
)  x.  ( ( ( ( RR  Dn F ) `  ( P  -  1
) ) `  0
)  /  ( ! `
 ( P  - 
1 ) ) ) ) )
174106breq2d 4414 . . . 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 303 . . 3  |-  ( ph  ->  -.  P  ||  (
( ( A ` 
0 )  x.  (
( ( RR  Dn F ) `  ( P  -  1
) ) `  0
) )  /  ( ! `  ( P  -  1 ) ) ) )
17646adantr 467 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ( ( 0 ... M )  X.  ( 0 ... (
( M  x.  P
)  +  ( P  -  1 ) ) ) )  \  { <. 0 ,  ( P  -  1 ) >. } ) )  ->  P  e.  ZZ )
177176, 125, 1383jca 1188 . . . . . . 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 3556 . . . . . . . . . 10  |-  ( k  e.  ( ( ( 0 ... M )  X.  ( 0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) ) )  \  { <. 0 ,  ( P  -  1 )
>. } )  ->  -.  k  e.  { <. 0 ,  ( P  - 
1 ) >. } )
17994adantr 467 . . . . . . . . . . . . 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 6830 . . . . . . . . . . . . 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 463 . . . . . . . . . . . . . 14  |-  ( ( ( 2nd `  k
)  =  ( P  -  1 )  /\  ( 1st `  k )  =  0 )  -> 
( 1st `  k
)  =  0 )
183 simpl 459 . . . . . . . . . . . . . 14  |-  ( ( ( 2nd `  k
)  =  ( P  -  1 )  /\  ( 1st `  k )  =  0 )  -> 
( 2nd `  k
)  =  ( P  -  1 ) )
184182, 183opeq12d 4174 . . . . . . . . . . . . 13  |-  ( ( ( 2nd `  k
)  =  ( P  -  1 )  /\  ( 1st `  k )  =  0 )  ->  <. ( 1st `  k
) ,  ( 2nd `  k ) >.  =  <. 0 ,  ( P  -  1 ) >.
)
185184adantl 468 . . . . . . . . . . . 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 2485 . . . . . . . . . . 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 3982 . . . . . . . . . . 11  |-  ( k  e.  { <. 0 ,  ( P  - 
1 ) >. }  <->  k  =  <. 0 ,  ( P  -  1 ) >.
)
188186, 187sylibr 216 . . . . . . . . . 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 665 . . . . . . . . 9  |-  ( k  e.  ( ( ( 0 ... M )  X.  ( 0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) ) )  \  { <. 0 ,  ( P  -  1 )
>. } )  ->  -.  ( ( 2nd `  k
)  =  ( P  -  1 )  /\  ( 1st `  k )  =  0 ) )
190189adantl 468 . . . . . . . 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 38137 . . . . . . 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 14340 . . . . . . 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 62 . . . . . 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 4429 . . . . 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 14348 . . . 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 4429 . . 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 38108 . 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 2691 1  |-  ( ph  ->  K  =/=  0 )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    \/ wo 370    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887    =/= wne 2622   {crab 2741   _Vcvv 3045    \ cdif 3401    C_ wss 3404   ifcif 3881   {csn 3968   {cpr 3970   <.cop 3974   class class class wbr 4402    |-> cmpt 4461    X. cxp 4832   ran crn 4835   -->wf 5578   ` cfv 5582  (class class class)co 6290   1stc1st 6791   2ndc2nd 6792    ^m cmap 7472   Fincfn 7569   CCcc 9537   RRcr 9538   0cc0 9539   1c1 9540    + caddc 9542    x. cmul 9544    < clt 9675    <_ cle 9676    - cmin 9860    / cdiv 10269   NNcn 10609   NN0cn0 10869   ZZcz 10937   ZZ>=cuz 11159   (,)cioo 11635   ...cfz 11784   ^cexp 12272   !cfa 12459   abscabs 13297   sum_csu 13752   prod_cprod 13959    || cdvds 14305   Primecprime 14622   ↾t crest 15319   TopOpenctopn 15320   topGenctg 15336  ℂfldccnfld 18970    Dncdvn 22819
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617  ax-addf 9618  ax-mulf 9619
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-iin 4281  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-of 6531  df-om 6693  df-1st 6793  df-2nd 6794  df-supp 6915  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-2o 7183  df-oadd 7186  df-er 7363  df-map 7474  df-pm 7475  df-ixp 7523  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-fsupp 7884  df-fi 7925  df-sup 7956  df-inf 7957  df-oi 8025  df-card 8373  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-ioo 11639  df-ico 11641  df-icc 11642  df-fz 11785  df-fzo 11916  df-fl 12028  df-mod 12097  df-seq 12214  df-exp 12273  df-fac 12460  df-bc 12488  df-hash 12516  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-clim 13552  df-sum 13753  df-prod 13960  df-dvds 14306  df-gcd 14469  df-prm 14623  df-struct 15123  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-ress 15128  df-plusg 15203  df-mulr 15204  df-starv 15205  df-sca 15206  df-vsca 15207  df-ip 15208  df-tset 15209  df-ple 15210  df-ds 15212  df-unif 15213  df-hom 15214  df-cco 15215  df-rest 15321  df-topn 15322  df-0g 15340  df-gsum 15341  df-topgen 15342  df-pt 15343  df-prds 15346  df-xrs 15400  df-qtop 15406  df-imas 15407  df-xps 15410  df-mre 15492  df-mrc 15493  df-acs 15495  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-submnd 16583  df-mulg 16676  df-cntz 16971  df-cmn 17432  df-psmet 18962  df-xmet 18963  df-met 18964  df-bl 18965  df-mopn 18966  df-fbas 18967  df-fg 18968  df-cnfld 18971  df-top 19921  df-bases 19922  df-topon 19923  df-topsp 19924  df-cld 20034  df-ntr 20035  df-cls 20036  df-nei 20114  df-lp 20152  df-perf 20153  df-cn 20243  df-cnp 20244  df-haus 20331  df-tx 20577  df-hmeo 20770  df-fil 20861  df-fm 20953  df-flim 20954  df-flf 20955  df-xms 21335  df-ms 21336  df-tms 21337  df-cncf 21910  df-limc 22821  df-dv 22822  df-dvn 22823
This theorem is referenced by:  etransclem47  38146
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