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Theorem bpolysum 28125
Description: A sum for Bernoulli polynomials. (Contributed by Scott Fenton, 16-May-2014.) (Proof shortened by Mario Carneiro, 22-May-2014.)
Assertion
Ref Expression
bpolysum  |-  ( ( N  e.  NN0  /\  X  e.  CC )  -> 
sum_ k  e.  ( 0 ... N ) ( ( N  _C  k )  x.  (
( k BernPoly  X )  /  ( ( N  -  k )  +  1 ) ) )  =  ( X ^ N ) )
Distinct variable groups:    k, N    k, X

Proof of Theorem bpolysum
StepHypRef Expression
1 simpl 454 . . . 4  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  N  e.  NN0 )
2 nn0uz 10891 . . . 4  |-  NN0  =  ( ZZ>= `  0 )
31, 2syl6eleq 2531 . . 3  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  N  e.  ( ZZ>= ` 
0 ) )
4 elfzelz 11449 . . . . . 6  |-  ( k  e.  ( 0 ... N )  ->  k  e.  ZZ )
5 bccl 12094 . . . . . 6  |-  ( ( N  e.  NN0  /\  k  e.  ZZ )  ->  ( N  _C  k
)  e.  NN0 )
61, 4, 5syl2an 474 . . . . 5  |-  ( ( ( N  e.  NN0  /\  X  e.  CC )  /\  k  e.  ( 0 ... N ) )  ->  ( N  _C  k )  e.  NN0 )
76nn0cnd 10634 . . . 4  |-  ( ( ( N  e.  NN0  /\  X  e.  CC )  /\  k  e.  ( 0 ... N ) )  ->  ( N  _C  k )  e.  CC )
8 elfznn0 11477 . . . . . 6  |-  ( k  e.  ( 0 ... N )  ->  k  e.  NN0 )
9 simpr 458 . . . . . 6  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  X  e.  CC )
10 bpolycl 28124 . . . . . 6  |-  ( ( k  e.  NN0  /\  X  e.  CC )  ->  ( k BernPoly  X )  e.  CC )
118, 9, 10syl2anr 475 . . . . 5  |-  ( ( ( N  e.  NN0  /\  X  e.  CC )  /\  k  e.  ( 0 ... N ) )  ->  ( k BernPoly  X )  e.  CC )
12 fznn0sub 11483 . . . . . . . 8  |-  ( k  e.  ( 0 ... N )  ->  ( N  -  k )  e.  NN0 )
1312adantl 463 . . . . . . 7  |-  ( ( ( N  e.  NN0  /\  X  e.  CC )  /\  k  e.  ( 0 ... N ) )  ->  ( N  -  k )  e. 
NN0 )
14 nn0p1nn 10615 . . . . . . 7  |-  ( ( N  -  k )  e.  NN0  ->  ( ( N  -  k )  +  1 )  e.  NN )
1513, 14syl 16 . . . . . 6  |-  ( ( ( N  e.  NN0  /\  X  e.  CC )  /\  k  e.  ( 0 ... N ) )  ->  ( ( N  -  k )  +  1 )  e.  NN )
1615nncnd 10334 . . . . 5  |-  ( ( ( N  e.  NN0  /\  X  e.  CC )  /\  k  e.  ( 0 ... N ) )  ->  ( ( N  -  k )  +  1 )  e.  CC )
1715nnne0d 10362 . . . . 5  |-  ( ( ( N  e.  NN0  /\  X  e.  CC )  /\  k  e.  ( 0 ... N ) )  ->  ( ( N  -  k )  +  1 )  =/=  0 )
1811, 16, 17divcld 10103 . . . 4  |-  ( ( ( N  e.  NN0  /\  X  e.  CC )  /\  k  e.  ( 0 ... N ) )  ->  ( (
k BernPoly  X )  /  (
( N  -  k
)  +  1 ) )  e.  CC )
197, 18mulcld 9402 . . 3  |-  ( ( ( N  e.  NN0  /\  X  e.  CC )  /\  k  e.  ( 0 ... N ) )  ->  ( ( N  _C  k )  x.  ( ( k BernPoly  X
)  /  ( ( N  -  k )  +  1 ) ) )  e.  CC )
20 oveq2 6098 . . . 4  |-  ( k  =  N  ->  ( N  _C  k )  =  ( N  _C  N
) )
21 oveq1 6097 . . . . 5  |-  ( k  =  N  ->  (
k BernPoly  X )  =  ( N BernPoly  X ) )
22 oveq2 6098 . . . . . 6  |-  ( k  =  N  ->  ( N  -  k )  =  ( N  -  N ) )
2322oveq1d 6105 . . . . 5  |-  ( k  =  N  ->  (
( N  -  k
)  +  1 )  =  ( ( N  -  N )  +  1 ) )
2421, 23oveq12d 6108 . . . 4  |-  ( k  =  N  ->  (
( k BernPoly  X )  /  ( ( N  -  k )  +  1 ) )  =  ( ( N BernPoly  X )  /  ( ( N  -  N )  +  1 ) ) )
2520, 24oveq12d 6108 . . 3  |-  ( k  =  N  ->  (
( N  _C  k
)  x.  ( ( k BernPoly  X )  /  (
( N  -  k
)  +  1 ) ) )  =  ( ( N  _C  N
)  x.  ( ( N BernPoly  X )  /  (
( N  -  N
)  +  1 ) ) ) )
263, 19, 25fsumm1 13216 . 2  |-  ( ( N  e.  NN0  /\  X  e.  CC )  -> 
sum_ k  e.  ( 0 ... N ) ( ( N  _C  k )  x.  (
( k BernPoly  X )  /  ( ( N  -  k )  +  1 ) ) )  =  ( sum_ k  e.  ( 0 ... ( N  -  1 ) ) ( ( N  _C  k )  x.  ( ( k BernPoly  X
)  /  ( ( N  -  k )  +  1 ) ) )  +  ( ( N  _C  N )  x.  ( ( N BernPoly  X )  /  (
( N  -  N
)  +  1 ) ) ) ) )
27 bcnn 12084 . . . . . 6  |-  ( N  e.  NN0  ->  ( N  _C  N )  =  1 )
2827adantr 462 . . . . 5  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( N  _C  N
)  =  1 )
29 nn0cn 10585 . . . . . . . . . . 11  |-  ( N  e.  NN0  ->  N  e.  CC )
3029adantr 462 . . . . . . . . . 10  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  N  e.  CC )
3130subidd 9703 . . . . . . . . 9  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( N  -  N
)  =  0 )
3231oveq1d 6105 . . . . . . . 8  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( ( N  -  N )  +  1 )  =  ( 0  +  1 ) )
33 0p1e1 10429 . . . . . . . 8  |-  ( 0  +  1 )  =  1
3432, 33syl6eq 2489 . . . . . . 7  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( ( N  -  N )  +  1 )  =  1 )
3534oveq2d 6106 . . . . . 6  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( ( N BernPoly  X )  /  ( ( N  -  N )  +  1 ) )  =  ( ( N BernPoly  X )  /  1 ) )
36 bpolycl 28124 . . . . . . 7  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( N BernPoly  X )  e.  CC )
3736div1d 10095 . . . . . 6  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( ( N BernPoly  X )  /  1 )  =  ( N BernPoly  X )
)
3835, 37eqtrd 2473 . . . . 5  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( ( N BernPoly  X )  /  ( ( N  -  N )  +  1 ) )  =  ( N BernPoly  X )
)
3928, 38oveq12d 6108 . . . 4  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( ( N  _C  N )  x.  (
( N BernPoly  X )  /  ( ( N  -  N )  +  1 ) ) )  =  ( 1  x.  ( N BernPoly  X )
) )
4036mulid2d 9400 . . . 4  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( 1  x.  ( N BernPoly  X ) )  =  ( N BernPoly  X )
)
4139, 40eqtrd 2473 . . 3  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( ( N  _C  N )  x.  (
( N BernPoly  X )  /  ( ( N  -  N )  +  1 ) ) )  =  ( N BernPoly  X ) )
4241oveq2d 6106 . 2  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( sum_ k  e.  ( 0 ... ( N  -  1 ) ) ( ( N  _C  k )  x.  (
( k BernPoly  X )  /  ( ( N  -  k )  +  1 ) ) )  +  ( ( N  _C  N )  x.  ( ( N BernPoly  X )  /  ( ( N  -  N )  +  1 ) ) ) )  =  ( sum_ k  e.  ( 0 ... ( N  - 
1 ) ) ( ( N  _C  k
)  x.  ( ( k BernPoly  X )  /  (
( N  -  k
)  +  1 ) ) )  +  ( N BernPoly  X ) ) )
43 bpolyval 28121 . . . 4  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( N BernPoly  X )  =  ( ( X ^ N )  -  sum_ k  e.  ( 0 ... ( N  - 
1 ) ) ( ( N  _C  k
)  x.  ( ( k BernPoly  X )  /  (
( N  -  k
)  +  1 ) ) ) ) )
4443eqcomd 2446 . . 3  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( ( X ^ N )  -  sum_ k  e.  ( 0 ... ( N  - 
1 ) ) ( ( N  _C  k
)  x.  ( ( k BernPoly  X )  /  (
( N  -  k
)  +  1 ) ) ) )  =  ( N BernPoly  X )
)
45 expcl 11879 . . . . 5  |-  ( ( X  e.  CC  /\  N  e.  NN0 )  -> 
( X ^ N
)  e.  CC )
4645ancoms 450 . . . 4  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( X ^ N
)  e.  CC )
47 fzfid 11791 . . . . 5  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( 0 ... ( N  -  1 ) )  e.  Fin )
48 fzssp1 11497 . . . . . . . 8  |-  ( 0 ... ( N  - 
1 ) )  C_  ( 0 ... (
( N  -  1 )  +  1 ) )
49 ax-1cn 9336 . . . . . . . . . 10  |-  1  e.  CC
50 npcan 9615 . . . . . . . . . 10  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( ( N  - 
1 )  +  1 )  =  N )
5130, 49, 50sylancl 657 . . . . . . . . 9  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( ( N  - 
1 )  +  1 )  =  N )
5251oveq2d 6106 . . . . . . . 8  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( 0 ... (
( N  -  1 )  +  1 ) )  =  ( 0 ... N ) )
5348, 52syl5sseq 3401 . . . . . . 7  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( 0 ... ( N  -  1 ) )  C_  ( 0 ... N ) )
5453sselda 3353 . . . . . 6  |-  ( ( ( N  e.  NN0  /\  X  e.  CC )  /\  k  e.  ( 0 ... ( N  -  1 ) ) )  ->  k  e.  ( 0 ... N
) )
5554, 19syldan 467 . . . . 5  |-  ( ( ( N  e.  NN0  /\  X  e.  CC )  /\  k  e.  ( 0 ... ( N  -  1 ) ) )  ->  ( ( N  _C  k )  x.  ( ( k BernPoly  X
)  /  ( ( N  -  k )  +  1 ) ) )  e.  CC )
5647, 55fsumcl 13206 . . . 4  |-  ( ( N  e.  NN0  /\  X  e.  CC )  -> 
sum_ k  e.  ( 0 ... ( N  -  1 ) ) ( ( N  _C  k )  x.  (
( k BernPoly  X )  /  ( ( N  -  k )  +  1 ) ) )  e.  CC )
5746, 56, 36subaddd 9733 . . 3  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( ( ( X ^ N )  -  sum_ k  e.  ( 0 ... ( N  - 
1 ) ) ( ( N  _C  k
)  x.  ( ( k BernPoly  X )  /  (
( N  -  k
)  +  1 ) ) ) )  =  ( N BernPoly  X )  <->  (
sum_ k  e.  ( 0 ... ( N  -  1 ) ) ( ( N  _C  k )  x.  (
( k BernPoly  X )  /  ( ( N  -  k )  +  1 ) ) )  +  ( N BernPoly  X ) )  =  ( X ^ N ) ) )
5844, 57mpbid 210 . 2  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( sum_ k  e.  ( 0 ... ( N  -  1 ) ) ( ( N  _C  k )  x.  (
( k BernPoly  X )  /  ( ( N  -  k )  +  1 ) ) )  +  ( N BernPoly  X ) )  =  ( X ^ N ) )
5926, 42, 583eqtrd 2477 1  |-  ( ( N  e.  NN0  /\  X  e.  CC )  -> 
sum_ k  e.  ( 0 ... N ) ( ( N  _C  k )  x.  (
( k BernPoly  X )  /  ( ( N  -  k )  +  1 ) ) )  =  ( X ^ N ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1364    e. wcel 1761   ` cfv 5415  (class class class)co 6090   CCcc 9276   0cc0 9278   1c1 9279    + caddc 9281    x. cmul 9283    - cmin 9591    / cdiv 9989   NNcn 10318   NN0cn0 10575   ZZcz 10642   ZZ>=cuz 10857   ...cfz 11433   ^cexp 11861    _C cbc 12074   sum_csu 13159   BernPoly cbp 28118
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-sup 7687  df-oi 7720  df-card 8105  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-n0 10576  df-z 10643  df-uz 10858  df-rp 10988  df-fz 11434  df-fzo 11545  df-seq 11803  df-exp 11862  df-fac 12048  df-bc 12075  df-hash 12100  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-clim 12962  df-sum 13160  df-pred 27554  df-wrecs 27646  df-bpoly 28119
This theorem is referenced by: (None)
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