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Theorem etransclem44 32300
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 1712 . . . . 5  |-  F/ k
ph
4 nfcv 2616 . . . . 5  |-  F/_ k
( ( A ` 
0 )  x.  (
( ( RR  Dn F ) `  ( P  -  1
) ) `  0
) )
5 fzfi 12064 . . . . . . 7  |-  ( 0 ... M )  e. 
Fin
6 fzfi 12064 . . . . . . 7  |-  ( 0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) )  e. 
Fin
7 xpfi 7783 . . . . . . 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 670 . . . . . 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 463 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ( 0 ... M )  X.  (
0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) ) ) )  ->  A : NN0
--> ZZ )
12 fzssnn0 31761 . . . . . . . . . 10  |-  ( 0 ... M )  C_  NN0
13 xp1st 6803 . . . . . . . . . 10  |-  ( k  e.  ( ( 0 ... M )  X.  ( 0 ... (
( M  x.  P
)  +  ( P  -  1 ) ) ) )  ->  ( 1st `  k )  e.  ( 0 ... M
) )
1412, 13sseldi 3487 . . . . . . . . 9  |-  ( k  e.  ( ( 0 ... M )  X.  ( 0 ... (
( M  x.  P
)  +  ( P  -  1 ) ) ) )  ->  ( 1st `  k )  e. 
NN0 )
1514adantl 464 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ( 0 ... M )  X.  (
0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) ) ) )  ->  ( 1st `  k )  e.  NN0 )
1611, 15ffvelrnd 6008 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ( 0 ... M )  X.  (
0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) ) ) )  ->  ( A `  ( 1st `  k
) )  e.  ZZ )
17 reelprrecn 9573 . . . . . . . . 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 31716 . . . . . . . . . 10  |-  RR  e.  ( topGen `  ran  (,) )
20 eqid 2454 . . . . . . . . . . 11  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
2120tgioo2 21474 . . . . . . . . . 10  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
2219, 21eleqtri 2540 . . . . . . . . 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 14304 . . . . . . . . . 10  |-  ( P  e.  Prime  ->  P  e.  NN )
2624, 25syl 16 . . . . . . . . 9  |-  ( ph  ->  P  e.  NN )
2726adantr 463 . . . . . . . 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 463 . . . . . . . 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 6804 . . . . . . . . . 10  |-  ( k  e.  ( ( 0 ... M )  X.  ( 0 ... (
( M  x.  P
)  +  ( P  -  1 ) ) ) )  ->  ( 2nd `  k )  e.  ( 0 ... (
( M  x.  P
)  +  ( P  -  1 ) ) ) )
32 elfznn0 11775 . . . . . . . . . 10  |-  ( ( 2nd `  k )  e.  ( 0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) )  ->  ( 2nd `  k )  e. 
NN0 )
3331, 32syl 16 . . . . . . . . 9  |-  ( k  e.  ( ( 0 ... M )  X.  ( 0 ... (
( M  x.  P
)  +  ( P  -  1 ) ) ) )  ->  ( 2nd `  k )  e. 
NN0 )
3433adantl 464 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ( 0 ... M )  X.  (
0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) ) ) )  ->  ( 2nd `  k )  e.  NN0 )
3515nn0red 10849 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ( 0 ... M )  X.  (
0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) ) ) )  ->  ( 1st `  k )  e.  RR )
3615nn0zd 10963 . . . . . . . 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 32298 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ( 0 ... M )  X.  (
0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) ) ) )  ->  ( (
( RR  Dn
F ) `  ( 2nd `  k ) ) `
 ( 1st `  k
) )  e.  ZZ )
3816, 37zmulcld 10971 . . . . . 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 10966 . . . . 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 11116 . . . . . . . 8  |-  NN0  =  ( ZZ>= `  0 )
4128, 40syl6eleq 2552 . . . . . . 7  |-  ( ph  ->  M  e.  ( ZZ>= ` 
0 ) )
42 eluzfz1 11696 . . . . . . 7  |-  ( M  e.  ( ZZ>= `  0
)  ->  0  e.  ( 0 ... M
) )
4341, 42syl 16 . . . . . 6  |-  ( ph  ->  0  e.  ( 0 ... M ) )
44 0zd 10872 . . . . . . . . 9  |-  ( ph  ->  0  e.  ZZ )
4528nn0zd 10963 . . . . . . . . . . 11  |-  ( ph  ->  M  e.  ZZ )
4626nnzd 10964 . . . . . . . . . . 11  |-  ( ph  ->  P  e.  ZZ )
4745, 46zmulcld 10971 . . . . . . . . . 10  |-  ( ph  ->  ( M  x.  P
)  e.  ZZ )
48 nnm1nn0 10833 . . . . . . . . . . . 12  |-  ( P  e.  NN  ->  ( P  -  1 )  e.  NN0 )
4926, 48syl 16 . . . . . . . . . . 11  |-  ( ph  ->  ( P  -  1 )  e.  NN0 )
5049nn0zd 10963 . . . . . . . . . 10  |-  ( ph  ->  ( P  -  1 )  e.  ZZ )
5147, 50zaddcld 10969 . . . . . . . . 9  |-  ( ph  ->  ( ( M  x.  P )  +  ( P  -  1 ) )  e.  ZZ )
5244, 51, 503jca 1174 . . . . . . . 8  |-  ( ph  ->  ( 0  e.  ZZ  /\  ( ( M  x.  P )  +  ( P  -  1 ) )  e.  ZZ  /\  ( P  -  1
)  e.  ZZ ) )
5349nn0ge0d 10851 . . . . . . . 8  |-  ( ph  ->  0  <_  ( P  -  1 ) )
5426nnnn0d 10848 . . . . . . . . . . 11  |-  ( ph  ->  P  e.  NN0 )
5528, 54nn0mulcld 10853 . . . . . . . . . 10  |-  ( ph  ->  ( M  x.  P
)  e.  NN0 )
5655nn0ge0d 10851 . . . . . . . . 9  |-  ( ph  ->  0  <_  ( M  x.  P ) )
5749nn0red 10849 . . . . . . . . . 10  |-  ( ph  ->  ( P  -  1 )  e.  RR )
5847zred 10965 . . . . . . . . . 10  |-  ( ph  ->  ( M  x.  P
)  e.  RR )
5957, 58addge02d 10137 . . . . . . . . 9  |-  ( ph  ->  ( 0  <_  ( M  x.  P )  <->  ( P  -  1 )  <_  ( ( M  x.  P )  +  ( P  -  1 ) ) ) )
6056, 59mpbid 210 . . . . . . . 8  |-  ( ph  ->  ( P  -  1 )  <_  ( ( M  x.  P )  +  ( P  - 
1 ) ) )
6152, 53, 60jca32 533 . . . . . . 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 11682 . . . . . . 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 212 . . . . . 6  |-  ( ph  ->  ( P  -  1 )  e.  ( 0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) ) )
64 opelxp 5018 . . . . . 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 662 . . . . 5  |-  ( ph  -> 
<. 0 ,  ( P  -  1 )
>.  e.  ( ( 0 ... M )  X.  ( 0 ... (
( M  x.  P
)  +  ( P  -  1 ) ) ) ) )
66 fveq2 5848 . . . . . . . 8  |-  ( k  =  <. 0 ,  ( P  -  1 )
>.  ->  ( 1st `  k
)  =  ( 1st `  <. 0 ,  ( P  -  1 )
>. ) )
67 0re 9585 . . . . . . . . 9  |-  0  e.  RR
68 ovex 6298 . . . . . . . . 9  |-  ( P  -  1 )  e. 
_V
69 op1stg 6785 . . . . . . . . 9  |-  ( ( 0  e.  RR  /\  ( P  -  1
)  e.  _V )  ->  ( 1st `  <. 0 ,  ( P  -  1 ) >.
)  =  0 )
7067, 68, 69mp2an 670 . . . . . . . 8  |-  ( 1st `  <. 0 ,  ( P  -  1 )
>. )  =  0
7166, 70syl6eq 2511 . . . . . . 7  |-  ( k  =  <. 0 ,  ( P  -  1 )
>.  ->  ( 1st `  k
)  =  0 )
7271fveq2d 5852 . . . . . 6  |-  ( k  =  <. 0 ,  ( P  -  1 )
>.  ->  ( A `  ( 1st `  k ) )  =  ( A `
 0 ) )
73 fveq2 5848 . . . . . . . . 9  |-  ( k  =  <. 0 ,  ( P  -  1 )
>.  ->  ( 2nd `  k
)  =  ( 2nd `  <. 0 ,  ( P  -  1 )
>. ) )
74 op2ndg 6786 . . . . . . . . . 10  |-  ( ( 0  e.  RR  /\  ( P  -  1
)  e.  _V )  ->  ( 2nd `  <. 0 ,  ( P  -  1 ) >.
)  =  ( P  -  1 ) )
7567, 68, 74mp2an 670 . . . . . . . . 9  |-  ( 2nd `  <. 0 ,  ( P  -  1 )
>. )  =  ( P  -  1 )
7673, 75syl6eq 2511 . . . . . . . 8  |-  ( k  =  <. 0 ,  ( P  -  1 )
>.  ->  ( 2nd `  k
)  =  ( P  -  1 ) )
7776fveq2d 5852 . . . . . . 7  |-  ( k  =  <. 0 ,  ( P  -  1 )
>.  ->  ( ( RR  Dn F ) `
 ( 2nd `  k
) )  =  ( ( RR  Dn
F ) `  ( P  -  1 ) ) )
7877, 71fveq12d 5854 . . . . . 6  |-  ( k  =  <. 0 ,  ( P  -  1 )
>.  ->  ( ( ( RR  Dn F ) `  ( 2nd `  k ) ) `  ( 1st `  k ) )  =  ( ( ( RR  Dn
F ) `  ( P  -  1 ) ) `  0 ) )
7972, 78oveq12d 6288 . . . . 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 31812 . . . 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 6285 . . 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 3487 . . . . . . 7  |-  ( ph  ->  0  e.  NN0 )
8310, 82ffvelrnd 6008 . . . . . 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 32298 . . . . . 6  |-  ( ph  ->  ( ( ( RR  Dn F ) `
 ( P  - 
1 ) ) ` 
0 )  e.  ZZ )
8883, 87zmulcld 10971 . . . . 5  |-  ( ph  ->  ( ( A ` 
0 )  x.  (
( ( RR  Dn F ) `  ( P  -  1
) ) `  0
) )  e.  ZZ )
8988zcnd 10966 . . . 4  |-  ( ph  ->  ( ( A ` 
0 )  x.  (
( ( RR  Dn F ) `  ( P  -  1
) ) `  0
) )  e.  CC )
90 difss 3617 . . . . . . . 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 7733 . . . . . . . 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 670 . . . . . . 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 3612 . . . . . . 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 472 . . . . . 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 13639 . . . . 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 10966 . . . 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 31755 . . . . 5  |-  ( ph  ->  ( ! `  ( P  -  1 ) )  e.  NN )
9998nncnd 10547 . . . 4  |-  ( ph  ->  ( ! `  ( P  -  1 ) )  e.  CC )
10098nnne0d 10576 . . . 4  |-  ( ph  ->  ( ! `  ( P  -  1 ) )  =/=  0 )
10189, 97, 99, 100divdird 10354 . . 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 2499 . 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 10576 . . 3  |-  ( ph  ->  P  =/=  0 )
10483zcnd 10966 . . . . 5  |-  ( ph  ->  ( A `  0
)  e.  CC )
10587zcnd 10966 . . . . 5  |-  ( ph  ->  ( ( ( RR  Dn F ) `
 ( P  - 
1 ) ) ` 
0 )  e.  CC )
106104, 105, 99, 100divassd 10351 . . . 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 32261 . . . . . . 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 32267 . . . . . . 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 32293 . . . . . 6  |-  ( ph  ->  ( ! `  ( P  -  1 ) )  ||  ( ( ( RR  Dn
F ) `  ( P  -  1 ) ) `  0 ) )
11098nnzd 10964 . . . . . . 7  |-  ( ph  ->  ( ! `  ( P  -  1 ) )  e.  ZZ )
111 dvdsval2 14073 . . . . . . 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 1226 . . . . . 6  |-  ( ph  ->  ( ( ! `  ( P  -  1
) )  ||  (
( ( RR  Dn F ) `  ( P  -  1
) ) `  0
)  <->  ( ( ( ( RR  Dn
F ) `  ( P  -  1 ) ) `  0 )  /  ( ! `  ( P  -  1
) ) )  e.  ZZ ) )
113109, 112mpbid 210 . . . . 5  |-  ( ph  ->  ( ( ( ( RR  Dn F ) `  ( P  -  1 ) ) `
 0 )  / 
( ! `  ( P  -  1 ) ) )  e.  ZZ )
11483, 113zmulcld 10971 . . . 4  |-  ( ph  ->  ( ( A ` 
0 )  x.  (
( ( ( RR  Dn F ) `
 ( P  - 
1 ) ) ` 
0 )  /  ( ! `  ( P  -  1 ) ) ) )  e.  ZZ )
115106, 114eqeltrd 2542 . . 3  |-  ( ph  ->  ( ( ( A `
 0 )  x.  ( ( ( RR  Dn F ) `
 ( P  - 
1 ) ) ` 
0 ) )  / 
( ! `  ( P  -  1 ) ) )  e.  ZZ )
11694, 39sylan2 472 . . . . 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 13683 . . . 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 10966 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ( 0 ... M )  X.  (
0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) ) ) )  ->  ( A `  ( 1st `  k
) )  e.  CC )
11994, 118sylan2 472 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ( ( 0 ... M )  X.  ( 0 ... (
( M  x.  P
)  +  ( P  -  1 ) ) ) )  \  { <. 0 ,  ( P  -  1 ) >. } ) )  -> 
( A `  ( 1st `  k ) )  e.  CC )
12094, 37sylan2 472 . . . . . . . 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 10966 . . . . . . 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 463 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ( ( 0 ... M )  X.  ( 0 ... (
( M  x.  P
)  +  ( P  -  1 ) ) ) )  \  { <. 0 ,  ( P  -  1 ) >. } ) )  -> 
( ! `  ( P  -  1 ) )  e.  CC )
123100adantr 463 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ( ( 0 ... M )  X.  ( 0 ... (
( M  x.  P
)  +  ( P  -  1 ) ) ) )  \  { <. 0 ,  ( P  -  1 ) >. } ) )  -> 
( ! `  ( P  -  1 ) )  =/=  0 )
124119, 121, 122, 123divassd 10351 . . . . . 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 472 . . . . . . 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 463 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ( ( 0 ... M )  X.  ( 0 ... (
( M  x.  P
)  +  ( P  -  1 ) ) ) )  \  { <. 0 ,  ( P  -  1 ) >. } ) )  ->  P  e.  NN )
12928adantr 463 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ( ( 0 ... M )  X.  ( 0 ... (
( M  x.  P
)  +  ( P  -  1 ) ) ) )  \  { <. 0 ,  ( P  -  1 ) >. } ) )  ->  M  e.  NN0 )
13094adantl 464 . . . . . . . . . 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 16 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ( ( 0 ... M )  X.  ( 0 ... (
( M  x.  P
)  +  ( P  -  1 ) ) ) )  \  { <. 0 ,  ( P  -  1 ) >. } ) )  -> 
( 2nd `  k
)  e.  NN0 )
132130, 13syl 16 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ( ( 0 ... M )  X.  ( 0 ... (
( M  x.  P
)  +  ( P  -  1 ) ) ) )  \  { <. 0 ,  ( P  -  1 ) >. } ) )  -> 
( 1st `  k
)  e.  ( 0 ... M ) )
13394, 35sylan2 472 . . . . . . . . 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 32293 . . . . . . . 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 463 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ( ( 0 ... M )  X.  ( 0 ... (
( M  x.  P
)  +  ( P  -  1 ) ) ) )  \  { <. 0 ,  ( P  -  1 ) >. } ) )  -> 
( ! `  ( P  -  1 ) )  e.  ZZ )
136 dvdsval2 14073 . . . . . . . . 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 1226 . . . . . . . 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 210 . . . . . . 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 10971 . . . . . 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 2542 . . . . 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 13639 . . . 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 2542 . . 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 10891 . . . . . . . . . . . 12  |-  ( ph  ->  1  e.  ZZ )
144 zabscl 13228 . . . . . . . . . . . . 13  |-  ( ( A `  0 )  e.  ZZ  ->  ( abs `  ( A ` 
0 ) )  e.  ZZ )
14583, 144syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  ( abs `  ( A `  0 )
)  e.  ZZ )
146143, 50, 1453jca 1174 . . . . . . . . . . 11  |-  ( ph  ->  ( 1  e.  ZZ  /\  ( P  -  1 )  e.  ZZ  /\  ( abs `  ( A `
 0 ) )  e.  ZZ ) )
147 nn0abscl 13227 . . . . . . . . . . . . . 14  |-  ( ( A `  0 )  e.  ZZ  ->  ( abs `  ( A ` 
0 ) )  e. 
NN0 )
14883, 147syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  ( abs `  ( A `  0 )
)  e.  NN0 )
149 etransclem44.a0 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( A `  0
)  =/=  0 )
150104, 149absne0d 13360 . . . . . . . . . . . . 13  |-  ( ph  ->  ( abs `  ( A `  0 )
)  =/=  0 )
151 elnnne0 10805 . . . . . . . . . . . . 13  |-  ( ( abs `  ( A `
 0 ) )  e.  NN  <->  ( ( abs `  ( A ` 
0 ) )  e. 
NN0  /\  ( abs `  ( A `  0
) )  =/=  0
) )
152148, 150, 151sylanbrc 662 . . . . . . . . . . . 12  |-  ( ph  ->  ( abs `  ( A `  0 )
)  e.  NN )
153152nnge1d 10574 . . . . . . . . . . 11  |-  ( ph  ->  1  <_  ( abs `  ( A `  0
) ) )
154 etransclem44.ap . . . . . . . . . . . 12  |-  ( ph  ->  ( abs `  ( A `  0 )
)  <  P )
155 zltlem1 10912 . . . . . . . . . . . . 13  |-  ( ( ( abs `  ( A `  0 )
)  e.  ZZ  /\  P  e.  ZZ )  ->  ( ( abs `  ( A `  0 )
)  <  P  <->  ( abs `  ( A `  0
) )  <_  ( P  -  1 ) ) )
156145, 46, 155syl2anc 659 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( abs `  ( A `  0 )
)  <  P  <->  ( abs `  ( A `  0
) )  <_  ( P  -  1 ) ) )
157154, 156mpbid 210 . . . . . . . . . . 11  |-  ( ph  ->  ( abs `  ( A `  0 )
)  <_  ( P  -  1 ) )
158146, 153, 157jca32 533 . . . . . . . . . 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 11682 . . . . . . . . . 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 212 . . . . . . . . 9  |-  ( ph  ->  ( abs `  ( A `  0 )
)  e.  ( 1 ... ( P  - 
1 ) ) )
161 fzm1ndvds 14122 . . . . . . . . 9  |-  ( ( P  e.  NN  /\  ( abs `  ( A `
 0 ) )  e.  ( 1 ... ( P  -  1 ) ) )  ->  -.  P  ||  ( abs `  ( A `  0
) ) )
16226, 160, 161syl2anc 659 . . . . . . . 8  |-  ( ph  ->  -.  P  ||  ( abs `  ( A ` 
0 ) ) )
163 dvdsabsb 14087 . . . . . . . . 9  |-  ( ( P  e.  ZZ  /\  ( A `  0 )  e.  ZZ )  -> 
( P  ||  ( A `  0 )  <->  P 
||  ( abs `  ( A `  0 )
) ) )
16446, 83, 163syl2anc 659 . . . . . . . 8  |-  ( ph  ->  ( P  ||  ( A `  0 )  <->  P 
||  ( abs `  ( A `  0 )
) ) )
165162, 164mtbird 299 . . . . . . 7  |-  ( ph  ->  -.  P  ||  ( A `  0 )
)
166 etransclem44.mp . . . . . . . 8  |-  ( ph  ->  ( ! `  M
)  <  P )
16728, 24, 166, 30etransclem41 32297 . . . . . . 7  |-  ( ph  ->  -.  P  ||  (
( ( ( RR  Dn F ) `
 ( P  - 
1 ) ) ` 
0 )  /  ( ! `  ( P  -  1 ) ) ) )
168165, 167jca 530 . . . . . 6  |-  ( ph  ->  ( -.  P  ||  ( A `  0 )  /\  -.  P  ||  ( ( ( ( RR  Dn F ) `  ( P  -  1 ) ) `
 0 )  / 
( ! `  ( P  -  1 ) ) ) ) )
169 pm4.56 493 . . . . . 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 196 . . . . 5  |-  ( ph  ->  -.  ( P  ||  ( A `  0 )  \/  P  ||  (
( ( ( RR  Dn F ) `
 ( P  - 
1 ) ) ` 
0 )  /  ( ! `  ( P  -  1 ) ) ) ) )
171 euclemma 14333 . . . . . 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 1226 . . . . 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 299 . . . 4  |-  ( ph  ->  -.  P  ||  (
( A `  0
)  x.  ( ( ( ( RR  Dn F ) `  ( P  -  1
) ) `  0
)  /  ( ! `
 ( P  - 
1 ) ) ) ) )
174106breq2d 4451 . . . 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 299 . . 3  |-  ( ph  ->  -.  P  ||  (
( ( A ` 
0 )  x.  (
( ( RR  Dn F ) `  ( P  -  1
) ) `  0
) )  /  ( ! `  ( P  -  1 ) ) ) )
17646adantr 463 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ( ( 0 ... M )  X.  ( 0 ... (
( M  x.  P
)  +  ( P  -  1 ) ) ) )  \  { <. 0 ,  ( P  -  1 ) >. } ) )  ->  P  e.  ZZ )
177176, 125, 1383jca 1174 . . . . . . 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 3613 . . . . . . . . . 10  |-  ( k  e.  ( ( ( 0 ... M )  X.  ( 0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) ) )  \  { <. 0 ,  ( P  -  1 )
>. } )  ->  -.  k  e.  { <. 0 ,  ( P  - 
1 ) >. } )
17994adantr 463 . . . . . . . . . . . . 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 6810 . . . . . . . . . . . . 13  |-  ( k  e.  ( ( 0 ... M )  X.  ( 0 ... (
( M  x.  P
)  +  ( P  -  1 ) ) ) )  ->  k  =  <. ( 1st `  k
) ,  ( 2nd `  k ) >. )
181179, 180syl 16 . . . . . . . . . . . 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 459 . . . . . . . . . . . . . 14  |-  ( ( ( 2nd `  k
)  =  ( P  -  1 )  /\  ( 1st `  k )  =  0 )  -> 
( 1st `  k
)  =  0 )
183 simpl 455 . . . . . . . . . . . . . 14  |-  ( ( ( 2nd `  k
)  =  ( P  -  1 )  /\  ( 1st `  k )  =  0 )  -> 
( 2nd `  k
)  =  ( P  -  1 ) )
184182, 183opeq12d 4211 . . . . . . . . . . . . 13  |-  ( ( ( 2nd `  k
)  =  ( P  -  1 )  /\  ( 1st `  k )  =  0 )  ->  <. ( 1st `  k
) ,  ( 2nd `  k ) >.  =  <. 0 ,  ( P  -  1 ) >.
)
185184adantl 464 . . . . . . . . . . . 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 2495 . . . . . . . . . . 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 4030 . . . . . . . . . . 11  |-  ( k  e.  { <. 0 ,  ( P  - 
1 ) >. }  <->  k  =  <. 0 ,  ( P  -  1 ) >.
)
188186, 187sylibr 212 . . . . . . . . . 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 657 . . . . . . . . 9  |-  ( k  e.  ( ( ( 0 ... M )  X.  ( 0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) ) )  \  { <. 0 ,  ( P  -  1 )
>. } )  ->  -.  ( ( 2nd `  k
)  =  ( P  -  1 )  /\  ( 1st `  k )  =  0 ) )
190189adantl 464 . . . . . . . 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 32294 . . . . . . 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 14105 . . . . . . 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 60 . . . . . 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 4465 . . . . 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 14113 . . . 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 4465 . . 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 32265 . 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 2747 1  |-  ( ph  ->  K  =/=  0 )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 366    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   {crab 2808   _Vcvv 3106    \ cdif 3458    C_ wss 3461   ifcif 3929   {csn 4016   {cpr 4018   <.cop 4022   class class class wbr 4439    |-> cmpt 4497    X. cxp 4986   ran crn 4989   -->wf 5566   ` cfv 5570  (class class class)co 6270   1stc1st 6771   2ndc2nd 6772    ^m cmap 7412   Fincfn 7509   CCcc 9479   RRcr 9480   0cc0 9481   1c1 9482    + caddc 9484    x. cmul 9486    < clt 9617    <_ cle 9618    - cmin 9796    / cdiv 10202   NNcn 10531   NN0cn0 10791   ZZcz 10860   ZZ>=cuz 11082   (,)cioo 11532   ...cfz 11675   ^cexp 12148   !cfa 12335   abscabs 13149   sum_csu 13590   prod_cprod 13794    || cdvds 14070   Primecprime 14301   ↾t crest 14910   TopOpenctopn 14911   topGenctg 14927  ℂfldccnfld 18615    Dncdvn 22434
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560  ax-mulf 9561
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-ixp 7463  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fsupp 7822  df-fi 7863  df-sup 7893  df-oi 7927  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10977  df-uz 11083  df-q 11184  df-rp 11222  df-xneg 11321  df-xadd 11322  df-xmul 11323  df-ioo 11536  df-ico 11538  df-icc 11539  df-fz 11676  df-fzo 11800  df-fl 11910  df-mod 11979  df-seq 12090  df-exp 12149  df-fac 12336  df-bc 12363  df-hash 12388  df-cj 13014  df-re 13015  df-im 13016  df-sqrt 13150  df-abs 13151  df-clim 13393  df-sum 13591  df-prod 13795  df-dvds 14071  df-gcd 14229  df-prm 14302  df-struct 14718  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-ress 14723  df-plusg 14797  df-mulr 14798  df-starv 14799  df-sca 14800  df-vsca 14801  df-ip 14802  df-tset 14803  df-ple 14804  df-ds 14806  df-unif 14807  df-hom 14808  df-cco 14809  df-rest 14912  df-topn 14913  df-0g 14931  df-gsum 14932  df-topgen 14933  df-pt 14934  df-prds 14937  df-xrs 14991  df-qtop 14996  df-imas 14997  df-xps 14999  df-mre 15075  df-mrc 15076  df-acs 15078  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-submnd 16166  df-mulg 16259  df-cntz 16554  df-cmn 16999  df-psmet 18606  df-xmet 18607  df-met 18608  df-bl 18609  df-mopn 18610  df-fbas 18611  df-fg 18612  df-cnfld 18616  df-top 19566  df-bases 19568  df-topon 19569  df-topsp 19570  df-cld 19687  df-ntr 19688  df-cls 19689  df-nei 19766  df-lp 19804  df-perf 19805  df-cn 19895  df-cnp 19896  df-haus 19983  df-tx 20229  df-hmeo 20422  df-fil 20513  df-fm 20605  df-flim 20606  df-flf 20607  df-xms 20989  df-ms 20990  df-tms 20991  df-cncf 21548  df-limc 22436  df-dv 22437  df-dvn 22438
This theorem is referenced by:  etransclem47  32303
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