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Theorem etransclem44 37960
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 1751 . . . . 5  |-  F/ k
ph
4 nfcv 2584 . . . . 5  |-  F/_ k
( ( A ` 
0 )  x.  (
( ( RR  Dn F ) `  ( P  -  1
) ) `  0
) )
5 fzfi 12184 . . . . . . 7  |-  ( 0 ... M )  e. 
Fin
6 fzfi 12184 . . . . . . 7  |-  ( 0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) )  e. 
Fin
7 xpfi 7844 . . . . . . 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 676 . . . . . 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 466 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ( 0 ... M )  X.  (
0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) ) ) )  ->  A : NN0
--> ZZ )
12 fzssnn0 37378 . . . . . . . . . 10  |-  ( 0 ... M )  C_  NN0
13 xp1st 6833 . . . . . . . . . 10  |-  ( k  e.  ( ( 0 ... M )  X.  ( 0 ... (
( M  x.  P
)  +  ( P  -  1 ) ) ) )  ->  ( 1st `  k )  e.  ( 0 ... M
) )
1412, 13sseldi 3462 . . . . . . . . 9  |-  ( k  e.  ( ( 0 ... M )  X.  ( 0 ... (
( M  x.  P
)  +  ( P  -  1 ) ) ) )  ->  ( 1st `  k )  e. 
NN0 )
1514adantl 467 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ( 0 ... M )  X.  (
0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) ) ) )  ->  ( 1st `  k )  e.  NN0 )
1611, 15ffvelrnd 6034 . . . . . . 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 37341 . . . . . . . . . 10  |-  RR  e.  ( topGen `  ran  (,) )
20 eqid 2422 . . . . . . . . . . 11  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
2120tgioo2 21805 . . . . . . . . . 10  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
2219, 21eleqtri 2508 . . . . . . . . 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 14610 . . . . . . . . . 10  |-  ( P  e.  Prime  ->  P  e.  NN )
2624, 25syl 17 . . . . . . . . 9  |-  ( ph  ->  P  e.  NN )
2726adantr 466 . . . . . . . 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 466 . . . . . . . 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 6834 . . . . . . . . . 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 467 . . . . . . . 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 37958 . . . . . . 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 2520 . . . . . . 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 1185 . . . . . . . 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 213 . . . . . . . 8  |-  ( ph  ->  ( P  -  1 )  <_  ( ( M  x.  P )  +  ( P  - 
1 ) ) )
6152, 53, 60jca32 537 . . . . . . 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 215 . . . . . 6  |-  ( ph  ->  ( P  -  1 )  e.  ( 0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) ) )
64 opelxp 4879 . . . . . 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 668 . . . . 5  |-  ( ph  -> 
<. 0 ,  ( P  -  1 )
>.  e.  ( ( 0 ... M )  X.  ( 0 ... (
( M  x.  P
)  +  ( P  -  1 ) ) ) ) )
66 fveq2 5877 . . . . . . . 8  |-  ( k  =  <. 0 ,  ( P  -  1 )
>.  ->  ( 1st `  k
)  =  ( 1st `  <. 0 ,  ( P  -  1 )
>. ) )
67 0re 9643 . . . . . . . . 9  |-  0  e.  RR
68 ovex 6329 . . . . . . . . 9  |-  ( P  -  1 )  e. 
_V
69 op1stg 6815 . . . . . . . . 9  |-  ( ( 0  e.  RR  /\  ( P  -  1
)  e.  _V )  ->  ( 1st `  <. 0 ,  ( P  -  1 ) >.
)  =  0 )
7067, 68, 69mp2an 676 . . . . . . . 8  |-  ( 1st `  <. 0 ,  ( P  -  1 )
>. )  =  0
7166, 70syl6eq 2479 . . . . . . 7  |-  ( k  =  <. 0 ,  ( P  -  1 )
>.  ->  ( 1st `  k
)  =  0 )
7271fveq2d 5881 . . . . . 6  |-  ( k  =  <. 0 ,  ( P  -  1 )
>.  ->  ( A `  ( 1st `  k ) )  =  ( A `
 0 ) )
73 fveq2 5877 . . . . . . . . 9  |-  ( k  =  <. 0 ,  ( P  -  1 )
>.  ->  ( 2nd `  k
)  =  ( 2nd `  <. 0 ,  ( P  -  1 )
>. ) )
74 op2ndg 6816 . . . . . . . . . 10  |-  ( ( 0  e.  RR  /\  ( P  -  1
)  e.  _V )  ->  ( 2nd `  <. 0 ,  ( P  -  1 ) >.
)  =  ( P  -  1 ) )
7567, 68, 74mp2an 676 . . . . . . . . 9  |-  ( 2nd `  <. 0 ,  ( P  -  1 )
>. )  =  ( P  -  1 )
7673, 75syl6eq 2479 . . . . . . . 8  |-  ( k  =  <. 0 ,  ( P  -  1 )
>.  ->  ( 2nd `  k
)  =  ( P  -  1 ) )
7776fveq2d 5881 . . . . . . 7  |-  ( k  =  <. 0 ,  ( P  -  1 )
>.  ->  ( ( RR  Dn F ) `
 ( 2nd `  k
) )  =  ( ( RR  Dn
F ) `  ( P  -  1 ) ) )
7877, 71fveq12d 5883 . . . . . 6  |-  ( k  =  <. 0 ,  ( P  -  1 )
>.  ->  ( ( ( RR  Dn F ) `  ( 2nd `  k ) ) `  ( 1st `  k ) )  =  ( ( ( RR  Dn
F ) `  ( P  -  1 ) ) `  0 ) )
7972, 78oveq12d 6319 . . . . 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 37471 . . . 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 6316 . . 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 3462 . . . . . . 7  |-  ( ph  ->  0  e.  NN0 )
8310, 82ffvelrnd 6034 . . . . . 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 37958 . . . . . 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 3592 . . . . . . . 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 7794 . . . . . . . 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 676 . . . . . . 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 3587 . . . . . . 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 476 . . . . . 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 13786 . . . . 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 37372 . . . . 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 2467 . 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 37921 . . . . . . 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 37927 . . . . . . 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 37953 . . . . . 6  |-  ( ph  ->  ( ! `  ( P  -  1 ) )  ||  ( ( ( RR  Dn
F ) `  ( P  -  1 ) ) `  0 ) )
11098nnzd 11039 . . . . . . 7  |-  ( ph  ->  ( ! `  ( P  -  1 ) )  e.  ZZ )
111 dvdsval2 14293 . . . . . . 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 1264 . . . . . 6  |-  ( ph  ->  ( ( ! `  ( P  -  1
) )  ||  (
( ( RR  Dn F ) `  ( P  -  1
) ) `  0
)  <->  ( ( ( ( RR  Dn
F ) `  ( P  -  1 ) ) `  0 )  /  ( ! `  ( P  -  1
) ) )  e.  ZZ ) )
113109, 112mpbid 213 . . . . 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 2510 . . 3  |-  ( ph  ->  ( ( ( A `
 0 )  x.  ( ( ( RR  Dn F ) `
 ( P  - 
1 ) ) ` 
0 ) )  / 
( ! `  ( P  -  1 ) ) )  e.  ZZ )
11694, 39sylan2 476 . . . . 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 13832 . . . 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 476 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ( ( 0 ... M )  X.  ( 0 ... (
( M  x.  P
)  +  ( P  -  1 ) ) ) )  \  { <. 0 ,  ( P  -  1 ) >. } ) )  -> 
( A `  ( 1st `  k ) )  e.  CC )
12094, 37sylan2 476 . . . . . . . 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 466 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ( ( 0 ... M )  X.  ( 0 ... (
( M  x.  P
)  +  ( P  -  1 ) ) ) )  \  { <. 0 ,  ( P  -  1 ) >. } ) )  -> 
( ! `  ( P  -  1 ) )  e.  CC )
123100adantr 466 . . . . . . 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 476 . . . . . . 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 466 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ( ( 0 ... M )  X.  ( 0 ... (
( M  x.  P
)  +  ( P  -  1 ) ) ) )  \  { <. 0 ,  ( P  -  1 ) >. } ) )  ->  P  e.  NN )
12928adantr 466 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ( ( 0 ... M )  X.  ( 0 ... (
( M  x.  P
)  +  ( P  -  1 ) ) ) )  \  { <. 0 ,  ( P  -  1 ) >. } ) )  ->  M  e.  NN0 )
13094adantl 467 . . . . . . . . . 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 476 . . . . . . . . 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 37953 . . . . . . . 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 466 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ( ( 0 ... M )  X.  ( 0 ... (
( M  x.  P
)  +  ( P  -  1 ) ) ) )  \  { <. 0 ,  ( P  -  1 ) >. } ) )  -> 
( ! `  ( P  -  1 ) )  e.  ZZ )
136 dvdsval2 14293 . . . . . . . . 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 1264 . . . . . . . 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 213 . . . . . . 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 2510 . . . . 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 13786 . . . 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 2510 . . 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 13362 . . . . . . . . . . . . 13  |-  ( ( A `  0 )  e.  ZZ  ->  ( abs `  ( A ` 
0 ) )  e.  ZZ )
14583, 144syl 17 . . . . . . . . . . . 12  |-  ( ph  ->  ( abs `  ( A `  0 )
)  e.  ZZ )
146143, 50, 1453jca 1185 . . . . . . . . . . 11  |-  ( ph  ->  ( 1  e.  ZZ  /\  ( P  -  1 )  e.  ZZ  /\  ( abs `  ( A `
 0 ) )  e.  ZZ ) )
147 nn0abscl 13361 . . . . . . . . . . . . . 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 13494 . . . . . . . . . . . . 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 668 . . . . . . . . . . . 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 665 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( abs `  ( A `  0 )
)  <  P  <->  ( abs `  ( A `  0
) )  <_  ( P  -  1 ) ) )
157154, 156mpbid 213 . . . . . . . . . . 11  |-  ( ph  ->  ( abs `  ( A `  0 )
)  <_  ( P  -  1 ) )
158146, 153, 157jca32 537 . . . . . . . . . 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 215 . . . . . . . . 9  |-  ( ph  ->  ( abs `  ( A `  0 )
)  e.  ( 1 ... ( P  - 
1 ) ) )
161 fzm1ndvds 14342 . . . . . . . . 9  |-  ( ( P  e.  NN  /\  ( abs `  ( A `
 0 ) )  e.  ( 1 ... ( P  -  1 ) ) )  ->  -.  P  ||  ( abs `  ( A `  0
) ) )
16226, 160, 161syl2anc 665 . . . . . . . 8  |-  ( ph  ->  -.  P  ||  ( abs `  ( A ` 
0 ) ) )
163 dvdsabsb 14307 . . . . . . . . 9  |-  ( ( P  e.  ZZ  /\  ( A `  0 )  e.  ZZ )  -> 
( P  ||  ( A `  0 )  <->  P 
||  ( abs `  ( A `  0 )
) ) )
16446, 83, 163syl2anc 665 . . . . . . . 8  |-  ( ph  ->  ( P  ||  ( A `  0 )  <->  P 
||  ( abs `  ( A `  0 )
) ) )
165162, 164mtbird 302 . . . . . . 7  |-  ( ph  ->  -.  P  ||  ( A `  0 )
)
166 etransclem44.mp . . . . . . . 8  |-  ( ph  ->  ( ! `  M
)  <  P )
16728, 24, 166, 30etransclem41 37957 . . . . . . 7  |-  ( ph  ->  -.  P  ||  (
( ( ( RR  Dn F ) `
 ( P  - 
1 ) ) ` 
0 )  /  ( ! `  ( P  -  1 ) ) ) )
168165, 167jca 534 . . . . . 6  |-  ( ph  ->  ( -.  P  ||  ( A `  0 )  /\  -.  P  ||  ( ( ( ( RR  Dn F ) `  ( P  -  1 ) ) `
 0 )  / 
( ! `  ( P  -  1 ) ) ) ) )
169 pm4.56 497 . . . . . 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 199 . . . . 5  |-  ( ph  ->  -.  ( P  ||  ( A `  0 )  \/  P  ||  (
( ( ( RR  Dn F ) `
 ( P  - 
1 ) ) ` 
0 )  /  ( ! `  ( P  -  1 ) ) ) ) )
171 euclemma 14650 . . . . . 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 1264 . . . . 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 302 . . . 4  |-  ( ph  ->  -.  P  ||  (
( A `  0
)  x.  ( ( ( ( RR  Dn F ) `  ( P  -  1
) ) `  0
)  /  ( ! `
 ( P  - 
1 ) ) ) ) )
174106breq2d 4432 . . . 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 302 . . 3  |-  ( ph  ->  -.  P  ||  (
( ( A ` 
0 )  x.  (
( ( RR  Dn F ) `  ( P  -  1
) ) `  0
) )  /  ( ! `  ( P  -  1 ) ) ) )
17646adantr 466 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ( ( 0 ... M )  X.  ( 0 ... (
( M  x.  P
)  +  ( P  -  1 ) ) ) )  \  { <. 0 ,  ( P  -  1 ) >. } ) )  ->  P  e.  ZZ )
177176, 125, 1383jca 1185 . . . . . . 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 3588 . . . . . . . . . 10  |-  ( k  e.  ( ( ( 0 ... M )  X.  ( 0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) ) )  \  { <. 0 ,  ( P  -  1 )
>. } )  ->  -.  k  e.  { <. 0 ,  ( P  - 
1 ) >. } )
17994adantr 466 . . . . . . . . . . . . 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 6840 . . . . . . . . . . . . 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 462 . . . . . . . . . . . . . 14  |-  ( ( ( 2nd `  k
)  =  ( P  -  1 )  /\  ( 1st `  k )  =  0 )  -> 
( 1st `  k
)  =  0 )
183 simpl 458 . . . . . . . . . . . . . 14  |-  ( ( ( 2nd `  k
)  =  ( P  -  1 )  /\  ( 1st `  k )  =  0 )  -> 
( 2nd `  k
)  =  ( P  -  1 ) )
184182, 183opeq12d 4192 . . . . . . . . . . . . 13  |-  ( ( ( 2nd `  k
)  =  ( P  -  1 )  /\  ( 1st `  k )  =  0 )  ->  <. ( 1st `  k
) ,  ( 2nd `  k ) >.  =  <. 0 ,  ( P  -  1 ) >.
)
185184adantl 467 . . . . . . . . . . . 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 2463 . . . . . . . . . . 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 4010 . . . . . . . . . . 11  |-  ( k  e.  { <. 0 ,  ( P  - 
1 ) >. }  <->  k  =  <. 0 ,  ( P  -  1 ) >.
)
188186, 187sylibr 215 . . . . . . . . . 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 663 . . . . . . . . 9  |-  ( k  e.  ( ( ( 0 ... M )  X.  ( 0 ... ( ( M  x.  P )  +  ( P  -  1 ) ) ) )  \  { <. 0 ,  ( P  -  1 )
>. } )  ->  -.  ( ( 2nd `  k
)  =  ( P  -  1 )  /\  ( 1st `  k )  =  0 ) )
190189adantl 467 . . . . . . . 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 37954 . . . . . . 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 14325 . . . . . . 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 4447 . . . . 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 14333 . . . 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 4447 . . 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 37925 . 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 2717 1  |-  ( ph  ->  K  =/=  0 )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    \/ wo 369    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1868    =/= wne 2618   {crab 2779   _Vcvv 3081    \ cdif 3433    C_ wss 3436   ifcif 3909   {csn 3996   {cpr 3998   <.cop 4002   class class class wbr 4420    |-> cmpt 4479    X. cxp 4847   ran crn 4850   -->wf 5593   ` cfv 5597  (class class class)co 6301   1stc1st 6801   2ndc2nd 6802    ^m cmap 7476   Fincfn 7573   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 12271   !cfa 12458   abscabs 13283   sum_csu 13737   prod_cprod 13944    || cdvds 14290   Primecprime 14607   ↾t crest 15304   TopOpenctopn 15305   topGenctg 15321  ℂfldccnfld 18955    Dncdvn 22803
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593  ax-inf2 8148  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 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-int 4253  df-iun 4298  df-iin 4299  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4760  df-id 4764  df-po 4770  df-so 4771  df-fr 4808  df-se 4809  df-we 4810  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-pred 5395  df-ord 5441  df-on 5442  df-lim 5443  df-suc 5444  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-f1 5602  df-fo 5603  df-f1o 5604  df-fv 5605  df-isom 5606  df-riota 6263  df-ov 6304  df-oprab 6305  df-mpt2 6306  df-of 6541  df-om 6703  df-1st 6803  df-2nd 6804  df-supp 6922  df-wrecs 7032  df-recs 7094  df-rdg 7132  df-1o 7186  df-2o 7187  df-oadd 7190  df-er 7367  df-map 7478  df-pm 7479  df-ixp 7527  df-en 7574  df-dom 7575  df-sdom 7576  df-fin 7577  df-fsupp 7886  df-fi 7927  df-sup 7958  df-inf 7959  df-oi 8027  df-card 8374  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 12027  df-mod 12096  df-seq 12213  df-exp 12272  df-fac 12459  df-bc 12487  df-hash 12515  df-cj 13148  df-re 13149  df-im 13150  df-sqrt 13284  df-abs 13285  df-clim 13537  df-sum 13738  df-prod 13945  df-dvds 14291  df-gcd 14454  df-prm 14608  df-struct 15108  df-ndx 15109  df-slot 15110  df-base 15111  df-sets 15112  df-ress 15113  df-plusg 15188  df-mulr 15189  df-starv 15190  df-sca 15191  df-vsca 15192  df-ip 15193  df-tset 15194  df-ple 15195  df-ds 15197  df-unif 15198  df-hom 15199  df-cco 15200  df-rest 15306  df-topn 15307  df-0g 15325  df-gsum 15326  df-topgen 15327  df-pt 15328  df-prds 15331  df-xrs 15385  df-qtop 15391  df-imas 15392  df-xps 15395  df-mre 15477  df-mrc 15478  df-acs 15480  df-mgm 16473  df-sgrp 16512  df-mnd 16522  df-submnd 16568  df-mulg 16661  df-cntz 16956  df-cmn 17417  df-psmet 18947  df-xmet 18948  df-met 18949  df-bl 18950  df-mopn 18951  df-fbas 18952  df-fg 18953  df-cnfld 18956  df-top 19905  df-bases 19906  df-topon 19907  df-topsp 19908  df-cld 20018  df-ntr 20019  df-cls 20020  df-nei 20098  df-lp 20136  df-perf 20137  df-cn 20227  df-cnp 20228  df-haus 20315  df-tx 20561  df-hmeo 20754  df-fil 20845  df-fm 20937  df-flim 20938  df-flf 20939  df-xms 21319  df-ms 21320  df-tms 21321  df-cncf 21894  df-limc 22805  df-dv 22806  df-dvn 22807
This theorem is referenced by:  etransclem47  37963
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