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Theorem etransclem45 37964
Description:  K is an integer. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
Hypotheses
Ref Expression
etransclem45.p  |-  ( ph  ->  P  e.  NN )
etransclem45.m  |-  ( ph  ->  M  e.  NN0 )
etransclem45.f  |-  F  =  ( x  e.  RR  |->  ( ( x ^
( P  -  1 ) )  x.  prod_ j  e.  ( 1 ... M ) ( ( x  -  j ) ^ P ) ) )
etransclem45.a  |-  ( ph  ->  A : NN0 --> ZZ )
etransclem45.k  |-  K  =  ( sum_ k  e.  ( ( 0 ... M
)  X.  ( 0 ... R ) ) ( ( A `  ( 1st `  k ) )  x.  ( ( ( RR  Dn
F ) `  ( 2nd `  k ) ) `
 ( 1st `  k
) ) )  / 
( ! `  ( P  -  1 ) ) )
Assertion
Ref Expression
etransclem45  |-  ( ph  ->  K  e.  ZZ )
Distinct variable groups:    j, M, k, x    P, j, k, x    R, j, k, x    ph, j, k, x
Allowed substitution hints:    A( x, j, k)    F( x, j, k)    K( x, j, k)

Proof of Theorem etransclem45
Dummy variables  c 
d  n  m  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 etransclem45.k . 2  |-  K  =  ( sum_ k  e.  ( ( 0 ... M
)  X.  ( 0 ... R ) ) ( ( A `  ( 1st `  k ) )  x.  ( ( ( RR  Dn
F ) `  ( 2nd `  k ) ) `
 ( 1st `  k
) ) )  / 
( ! `  ( P  -  1 ) ) )
2 fzfi 12185 . . . . . 6  |-  ( 0 ... M )  e. 
Fin
3 fzfi 12185 . . . . . 6  |-  ( 0 ... R )  e. 
Fin
4 xpfi 7845 . . . . . 6  |-  ( ( ( 0 ... M
)  e.  Fin  /\  ( 0 ... R
)  e.  Fin )  ->  ( ( 0 ... M )  X.  (
0 ... R ) )  e.  Fin )
52, 3, 4mp2an 676 . . . . 5  |-  ( ( 0 ... M )  X.  ( 0 ... R ) )  e. 
Fin
65a1i 11 . . . 4  |-  ( ph  ->  ( ( 0 ... M )  X.  (
0 ... R ) )  e.  Fin )
7 etransclem45.p . . . . . . 7  |-  ( ph  ->  P  e.  NN )
8 nnm1nn0 10912 . . . . . . 7  |-  ( P  e.  NN  ->  ( P  -  1 )  e.  NN0 )
97, 8syl 17 . . . . . 6  |-  ( ph  ->  ( P  -  1 )  e.  NN0 )
109faccld 37375 . . . . 5  |-  ( ph  ->  ( ! `  ( P  -  1 ) )  e.  NN )
1110nncnd 10626 . . . 4  |-  ( ph  ->  ( ! `  ( P  -  1 ) )  e.  CC )
12 etransclem45.a . . . . . . . 8  |-  ( ph  ->  A : NN0 --> ZZ )
1312adantr 466 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ( 0 ... M )  X.  (
0 ... R ) ) )  ->  A : NN0
--> ZZ )
14 xp1st 6834 . . . . . . . . 9  |-  ( k  e.  ( ( 0 ... M )  X.  ( 0 ... R
) )  ->  ( 1st `  k )  e.  ( 0 ... M
) )
15 elfznn0 11888 . . . . . . . . 9  |-  ( ( 1st `  k )  e.  ( 0 ... M )  ->  ( 1st `  k )  e. 
NN0 )
1614, 15syl 17 . . . . . . . 8  |-  ( k  e.  ( ( 0 ... M )  X.  ( 0 ... R
) )  ->  ( 1st `  k )  e. 
NN0 )
1716adantl 467 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ( 0 ... M )  X.  (
0 ... R ) ) )  ->  ( 1st `  k )  e.  NN0 )
1813, 17ffvelrnd 6035 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ( 0 ... M )  X.  (
0 ... R ) ) )  ->  ( A `  ( 1st `  k
) )  e.  ZZ )
1918zcnd 11042 . . . . 5  |-  ( (
ph  /\  k  e.  ( ( 0 ... M )  X.  (
0 ... R ) ) )  ->  ( A `  ( 1st `  k
) )  e.  CC )
20 reelprrecn 9632 . . . . . . . 8  |-  RR  e.  { RR ,  CC }
2120a1i 11 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ( 0 ... M )  X.  (
0 ... R ) ) )  ->  RR  e.  { RR ,  CC }
)
22 reopn 37344 . . . . . . . . 9  |-  RR  e.  ( topGen `  ran  (,) )
23 eqid 2422 . . . . . . . . . 10  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
2423tgioo2 21808 . . . . . . . . 9  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
2522, 24eleqtri 2508 . . . . . . . 8  |-  RR  e.  ( ( TopOpen ` fld )t  RR )
2625a1i 11 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ( 0 ... M )  X.  (
0 ... R ) ) )  ->  RR  e.  ( ( TopOpen ` fld )t  RR ) )
277adantr 466 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ( 0 ... M )  X.  (
0 ... R ) ) )  ->  P  e.  NN )
28 etransclem45.m . . . . . . . 8  |-  ( ph  ->  M  e.  NN0 )
2928adantr 466 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ( 0 ... M )  X.  (
0 ... R ) ) )  ->  M  e.  NN0 )
30 etransclem45.f . . . . . . 7  |-  F  =  ( x  e.  RR  |->  ( ( x ^
( P  -  1 ) )  x.  prod_ j  e.  ( 1 ... M ) ( ( x  -  j ) ^ P ) ) )
31 xp2nd 6835 . . . . . . . . 9  |-  ( k  e.  ( ( 0 ... M )  X.  ( 0 ... R
) )  ->  ( 2nd `  k )  e.  ( 0 ... R
) )
32 elfznn0 11888 . . . . . . . . 9  |-  ( ( 2nd `  k )  e.  ( 0 ... R )  ->  ( 2nd `  k )  e. 
NN0 )
3331, 32syl 17 . . . . . . . 8  |-  ( k  e.  ( ( 0 ... M )  X.  ( 0 ... R
) )  ->  ( 2nd `  k )  e. 
NN0 )
3433adantl 467 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ( 0 ... M )  X.  (
0 ... R ) ) )  ->  ( 2nd `  k )  e.  NN0 )
3521, 26, 27, 29, 30, 34etransclem33 37952 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ( 0 ... M )  X.  (
0 ... R ) ) )  ->  ( ( RR  Dn F ) `
 ( 2nd `  k
) ) : RR --> CC )
3617nn0red 10927 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ( 0 ... M )  X.  (
0 ... R ) ) )  ->  ( 1st `  k )  e.  RR )
3735, 36ffvelrnd 6035 . . . . 5  |-  ( (
ph  /\  k  e.  ( ( 0 ... M )  X.  (
0 ... R ) ) )  ->  ( (
( RR  Dn
F ) `  ( 2nd `  k ) ) `
 ( 1st `  k
) )  e.  CC )
3819, 37mulcld 9664 . . . 4  |-  ( (
ph  /\  k  e.  ( ( 0 ... M )  X.  (
0 ... R ) ) )  ->  ( ( A `  ( 1st `  k ) )  x.  ( ( ( RR  Dn F ) `
 ( 2nd `  k
) ) `  ( 1st `  k ) ) )  e.  CC )
3910nnne0d 10655 . . . 4  |-  ( ph  ->  ( ! `  ( P  -  1 ) )  =/=  0 )
406, 11, 38, 39fsumdivc 13835 . . 3  |-  ( ph  ->  ( sum_ k  e.  ( ( 0 ... M
)  X.  ( 0 ... R ) ) ( ( A `  ( 1st `  k ) )  x.  ( ( ( RR  Dn
F ) `  ( 2nd `  k ) ) `
 ( 1st `  k
) ) )  / 
( ! `  ( P  -  1 ) ) )  =  sum_ k  e.  ( (
0 ... M )  X.  ( 0 ... R
) ) ( ( ( A `  ( 1st `  k ) )  x.  ( ( ( RR  Dn F ) `  ( 2nd `  k ) ) `  ( 1st `  k ) ) )  /  ( ! `  ( P  -  1 ) ) ) )
4111adantr 466 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ( 0 ... M )  X.  (
0 ... R ) ) )  ->  ( ! `  ( P  -  1 ) )  e.  CC )
4239adantr 466 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ( 0 ... M )  X.  (
0 ... R ) ) )  ->  ( ! `  ( P  -  1 ) )  =/=  0
)
4319, 37, 41, 42divassd 10419 . . . . 5  |-  ( (
ph  /\  k  e.  ( ( 0 ... M )  X.  (
0 ... R ) ) )  ->  ( (
( 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 ) ) ) ) )
44 etransclem5 37924 . . . . . . . 8  |-  ( 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 ) ) ) )
45 etransclem11 37930 . . . . . . . 8  |-  ( 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 }
)
4614adantl 467 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ( 0 ... M )  X.  (
0 ... R ) ) )  ->  ( 1st `  k )  e.  ( 0 ... M ) )
4721, 26, 27, 29, 30, 34, 44, 45, 46, 36etransclem37 37956 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ( 0 ... M )  X.  (
0 ... R ) ) )  ->  ( ! `  ( P  -  1 ) )  ||  (
( ( RR  Dn F ) `  ( 2nd `  k ) ) `  ( 1st `  k ) ) )
4810nnzd 11040 . . . . . . . . 9  |-  ( ph  ->  ( ! `  ( P  -  1 ) )  e.  ZZ )
4948adantr 466 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ( 0 ... M )  X.  (
0 ... R ) ) )  ->  ( ! `  ( P  -  1 ) )  e.  ZZ )
5017nn0zd 11039 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ( 0 ... M )  X.  (
0 ... R ) ) )  ->  ( 1st `  k )  e.  ZZ )
5121, 26, 27, 29, 30, 34, 36, 50etransclem42 37961 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ( 0 ... M )  X.  (
0 ... R ) ) )  ->  ( (
( RR  Dn
F ) `  ( 2nd `  k ) ) `
 ( 1st `  k
) )  e.  ZZ )
52 dvdsval2 14296 . . . . . . . 8  |-  ( ( ( ! `  ( 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 ) )
5349, 42, 51, 52syl3anc 1264 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ( 0 ... M )  X.  (
0 ... R ) ) )  ->  ( ( ! `  ( P  -  1 ) ) 
||  ( ( ( RR  Dn F ) `  ( 2nd `  k ) ) `  ( 1st `  k ) )  <->  ( ( ( ( RR  Dn
F ) `  ( 2nd `  k ) ) `
 ( 1st `  k
) )  /  ( ! `  ( P  -  1 ) ) )  e.  ZZ ) )
5447, 53mpbid 213 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ( 0 ... M )  X.  (
0 ... R ) ) )  ->  ( (
( ( RR  Dn F ) `  ( 2nd `  k ) ) `  ( 1st `  k ) )  / 
( ! `  ( P  -  1 ) ) )  e.  ZZ )
5518, 54zmulcld 11047 . . . . 5  |-  ( (
ph  /\  k  e.  ( ( 0 ... M )  X.  (
0 ... R ) ) )  ->  ( ( A `  ( 1st `  k ) )  x.  ( ( ( ( RR  Dn F ) `  ( 2nd `  k ) ) `  ( 1st `  k ) )  /  ( ! `
 ( P  - 
1 ) ) ) )  e.  ZZ )
5643, 55eqeltrd 2510 . . . 4  |-  ( (
ph  /\  k  e.  ( ( 0 ... M )  X.  (
0 ... R ) ) )  ->  ( (
( A `  ( 1st `  k ) )  x.  ( ( ( RR  Dn F ) `  ( 2nd `  k ) ) `  ( 1st `  k ) ) )  /  ( ! `  ( P  -  1 ) ) )  e.  ZZ )
576, 56fsumzcl 13789 . . 3  |-  ( ph  -> 
sum_ k  e.  ( ( 0 ... M
)  X.  ( 0 ... R ) ) ( ( ( A `
 ( 1st `  k
) )  x.  (
( ( RR  Dn F ) `  ( 2nd `  k ) ) `  ( 1st `  k ) ) )  /  ( ! `  ( P  -  1
) ) )  e.  ZZ )
5840, 57eqeltrd 2510 . 2  |-  ( ph  ->  ( sum_ k  e.  ( ( 0 ... M
)  X.  ( 0 ... R ) ) ( ( A `  ( 1st `  k ) )  x.  ( ( ( RR  Dn
F ) `  ( 2nd `  k ) ) `
 ( 1st `  k
) ) )  / 
( ! `  ( P  -  1 ) ) )  e.  ZZ )
591, 58syl5eqel 2514 1  |-  ( ph  ->  K  e.  ZZ )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1868    =/= wne 2618   {crab 2779   ifcif 3909   {cpr 3998   class class class wbr 4420    |-> cmpt 4479    X. cxp 4848   ran crn 4851   -->wf 5594   ` cfv 5598  (class class class)co 6302   1stc1st 6802   2ndc2nd 6803    ^m cmap 7477   Fincfn 7574   CCcc 9538   RRcr 9539   0cc0 9540   1c1 9541    x. cmul 9545    - cmin 9861    / cdiv 10270   NNcn 10610   NN0cn0 10870   ZZcz 10938   (,)cioo 11636   ...cfz 11785   ^cexp 12272   !cfa 12459   sum_csu 13740   prod_cprod 13947    || cdvds 14293   ↾t crest 15307   TopOpenctopn 15308   topGenctg 15324  ℂfldccnfld 18958    Dncdvn 22806
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 4552  ax-pow 4599  ax-pr 4657  ax-un 6594  ax-inf2 8149  ax-cnex 9596  ax-resscn 9597  ax-1cn 9598  ax-icn 9599  ax-addcl 9600  ax-addrcl 9601  ax-mulcl 9602  ax-mulrcl 9603  ax-mulcom 9604  ax-addass 9605  ax-mulass 9606  ax-distr 9607  ax-i2m1 9608  ax-1ne0 9609  ax-1rid 9610  ax-rnegex 9611  ax-rrecex 9612  ax-cnre 9613  ax-pre-lttri 9614  ax-pre-lttrn 9615  ax-pre-ltadd 9616  ax-pre-mulgt0 9617  ax-pre-sup 9618  ax-addf 9619  ax-mulf 9620
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 4761  df-id 4765  df-po 4771  df-so 4772  df-fr 4809  df-se 4810  df-we 4811  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863  df-pred 5396  df-ord 5442  df-on 5443  df-lim 5444  df-suc 5445  df-iota 5562  df-fun 5600  df-fn 5601  df-f 5602  df-f1 5603  df-fo 5604  df-f1o 5605  df-fv 5606  df-isom 5607  df-riota 6264  df-ov 6305  df-oprab 6306  df-mpt2 6307  df-of 6542  df-om 6704  df-1st 6804  df-2nd 6805  df-supp 6923  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-2o 7188  df-oadd 7191  df-er 7368  df-map 7479  df-pm 7480  df-ixp 7528  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-fsupp 7887  df-fi 7928  df-sup 7959  df-inf 7960  df-oi 8028  df-card 8375  df-cda 8599  df-pnf 9678  df-mnf 9679  df-xr 9680  df-ltxr 9681  df-le 9682  df-sub 9863  df-neg 9864  df-div 10271  df-nn 10611  df-2 10669  df-3 10670  df-4 10671  df-5 10672  df-6 10673  df-7 10674  df-8 10675  df-9 10676  df-10 10677  df-n0 10871  df-z 10939  df-dec 11053  df-uz 11161  df-q 11266  df-rp 11304  df-xneg 11410  df-xadd 11411  df-xmul 11412  df-ioo 11640  df-ico 11642  df-icc 11643  df-fz 11786  df-fzo 11917  df-seq 12214  df-exp 12273  df-fac 12460  df-bc 12488  df-hash 12516  df-cj 13151  df-re 13152  df-im 13153  df-sqrt 13287  df-abs 13288  df-clim 13540  df-sum 13741  df-prod 13948  df-dvds 14294  df-struct 15111  df-ndx 15112  df-slot 15113  df-base 15114  df-sets 15115  df-ress 15116  df-plusg 15191  df-mulr 15192  df-starv 15193  df-sca 15194  df-vsca 15195  df-ip 15196  df-tset 15197  df-ple 15198  df-ds 15200  df-unif 15201  df-hom 15202  df-cco 15203  df-rest 15309  df-topn 15310  df-0g 15328  df-gsum 15329  df-topgen 15330  df-pt 15331  df-prds 15334  df-xrs 15388  df-qtop 15394  df-imas 15395  df-xps 15398  df-mre 15480  df-mrc 15481  df-acs 15483  df-mgm 16476  df-sgrp 16515  df-mnd 16525  df-submnd 16571  df-mulg 16664  df-cntz 16959  df-cmn 17420  df-psmet 18950  df-xmet 18951  df-met 18952  df-bl 18953  df-mopn 18954  df-fbas 18955  df-fg 18956  df-cnfld 18959  df-top 19908  df-bases 19909  df-topon 19910  df-topsp 19911  df-cld 20021  df-ntr 20022  df-cls 20023  df-nei 20101  df-lp 20139  df-perf 20140  df-cn 20230  df-cnp 20231  df-haus 20318  df-tx 20564  df-hmeo 20757  df-fil 20848  df-fm 20940  df-flim 20941  df-flf 20942  df-xms 21322  df-ms 21323  df-tms 21324  df-cncf 21897  df-limc 22808  df-dv 22809  df-dvn 22810
This theorem is referenced by:  etransclem47  37966
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