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Theorem bpolysum 26003
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 444 . . . 4  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  N  e.  NN0 )
2 nn0uz 10476 . . . 4  |-  NN0  =  ( ZZ>= `  0 )
31, 2syl6eleq 2494 . . 3  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  N  e.  ( ZZ>= ` 
0 ) )
4 elfzelz 11015 . . . . . 6  |-  ( k  e.  ( 0 ... N )  ->  k  e.  ZZ )
5 bccl 11568 . . . . . 6  |-  ( ( N  e.  NN0  /\  k  e.  ZZ )  ->  ( N  _C  k
)  e.  NN0 )
61, 4, 5syl2an 464 . . . . 5  |-  ( ( ( N  e.  NN0  /\  X  e.  CC )  /\  k  e.  ( 0 ... N ) )  ->  ( N  _C  k )  e.  NN0 )
76nn0cnd 10232 . . . 4  |-  ( ( ( N  e.  NN0  /\  X  e.  CC )  /\  k  e.  ( 0 ... N ) )  ->  ( N  _C  k )  e.  CC )
8 elfznn0 11039 . . . . . 6  |-  ( k  e.  ( 0 ... N )  ->  k  e.  NN0 )
9 simpr 448 . . . . . 6  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  X  e.  CC )
10 bpolycl 26002 . . . . . 6  |-  ( ( k  e.  NN0  /\  X  e.  CC )  ->  ( k BernPoly  X )  e.  CC )
118, 9, 10syl2anr 465 . . . . 5  |-  ( ( ( N  e.  NN0  /\  X  e.  CC )  /\  k  e.  ( 0 ... N ) )  ->  ( k BernPoly  X )  e.  CC )
12 fznn0sub 11041 . . . . . . . 8  |-  ( k  e.  ( 0 ... N )  ->  ( N  -  k )  e.  NN0 )
1312adantl 453 . . . . . . 7  |-  ( ( ( N  e.  NN0  /\  X  e.  CC )  /\  k  e.  ( 0 ... N ) )  ->  ( N  -  k )  e. 
NN0 )
14 nn0p1nn 10215 . . . . . . 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 9972 . . . . 5  |-  ( ( ( N  e.  NN0  /\  X  e.  CC )  /\  k  e.  ( 0 ... N ) )  ->  ( ( N  -  k )  +  1 )  e.  CC )
1715nnne0d 10000 . . . . 5  |-  ( ( ( N  e.  NN0  /\  X  e.  CC )  /\  k  e.  ( 0 ... N ) )  ->  ( ( N  -  k )  +  1 )  =/=  0 )
1811, 16, 17divcld 9746 . . . 4  |-  ( ( ( N  e.  NN0  /\  X  e.  CC )  /\  k  e.  ( 0 ... N ) )  ->  ( (
k BernPoly  X )  /  (
( N  -  k
)  +  1 ) )  e.  CC )
197, 18mulcld 9064 . . 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 6048 . . . 4  |-  ( k  =  N  ->  ( N  _C  k )  =  ( N  _C  N
) )
21 oveq1 6047 . . . . 5  |-  ( k  =  N  ->  (
k BernPoly  X )  =  ( N BernPoly  X ) )
22 oveq2 6048 . . . . . 6  |-  ( k  =  N  ->  ( N  -  k )  =  ( N  -  N ) )
2322oveq1d 6055 . . . . 5  |-  ( k  =  N  ->  (
( N  -  k
)  +  1 )  =  ( ( N  -  N )  +  1 ) )
2421, 23oveq12d 6058 . . . 4  |-  ( k  =  N  ->  (
( k BernPoly  X )  /  ( ( N  -  k )  +  1 ) )  =  ( ( N BernPoly  X )  /  ( ( N  -  N )  +  1 ) ) )
2520, 24oveq12d 6058 . . 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 12492 . 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 11558 . . . . . 6  |-  ( N  e.  NN0  ->  ( N  _C  N )  =  1 )
2827adantr 452 . . . . 5  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( N  _C  N
)  =  1 )
29 nn0cn 10187 . . . . . . . . . . 11  |-  ( N  e.  NN0  ->  N  e.  CC )
3029adantr 452 . . . . . . . . . 10  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  N  e.  CC )
3130subidd 9355 . . . . . . . . 9  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( N  -  N
)  =  0 )
3231oveq1d 6055 . . . . . . . 8  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( ( N  -  N )  +  1 )  =  ( 0  +  1 ) )
33 0p1e1 10049 . . . . . . . 8  |-  ( 0  +  1 )  =  1
3432, 33syl6eq 2452 . . . . . . 7  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( ( N  -  N )  +  1 )  =  1 )
3534oveq2d 6056 . . . . . 6  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( ( N BernPoly  X )  /  ( ( N  -  N )  +  1 ) )  =  ( ( N BernPoly  X )  /  1 ) )
36 bpolycl 26002 . . . . . . 7  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( N BernPoly  X )  e.  CC )
3736div1d 9738 . . . . . 6  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( ( N BernPoly  X )  /  1 )  =  ( N BernPoly  X )
)
3835, 37eqtrd 2436 . . . . 5  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( ( N BernPoly  X )  /  ( ( N  -  N )  +  1 ) )  =  ( N BernPoly  X )
)
3928, 38oveq12d 6058 . . . 4  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( ( N  _C  N )  x.  (
( N BernPoly  X )  /  ( ( N  -  N )  +  1 ) ) )  =  ( 1  x.  ( N BernPoly  X )
) )
4036mulid2d 9062 . . . 4  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( 1  x.  ( N BernPoly  X ) )  =  ( N BernPoly  X )
)
4139, 40eqtrd 2436 . . 3  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( ( N  _C  N )  x.  (
( N BernPoly  X )  /  ( ( N  -  N )  +  1 ) ) )  =  ( N BernPoly  X ) )
4241oveq2d 6056 . 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 25999 . . . 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 2409 . . 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 11354 . . . . 5  |-  ( ( X  e.  CC  /\  N  e.  NN0 )  -> 
( X ^ N
)  e.  CC )
4645ancoms 440 . . . 4  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( X ^ N
)  e.  CC )
47 fzfid 11267 . . . . 5  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( 0 ... ( N  -  1 ) )  e.  Fin )
48 fzssp1 11051 . . . . . . . 8  |-  ( 0 ... ( N  - 
1 ) )  C_  ( 0 ... (
( N  -  1 )  +  1 ) )
49 ax-1cn 9004 . . . . . . . . . 10  |-  1  e.  CC
50 npcan 9270 . . . . . . . . . 10  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( ( N  - 
1 )  +  1 )  =  N )
5130, 49, 50sylancl 644 . . . . . . . . 9  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( ( N  - 
1 )  +  1 )  =  N )
5251oveq2d 6056 . . . . . . . 8  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( 0 ... (
( N  -  1 )  +  1 ) )  =  ( 0 ... N ) )
5348, 52syl5sseq 3356 . . . . . . 7  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( 0 ... ( N  -  1 ) )  C_  ( 0 ... N ) )
5453sselda 3308 . . . . . 6  |-  ( ( ( N  e.  NN0  /\  X  e.  CC )  /\  k  e.  ( 0 ... ( N  -  1 ) ) )  ->  k  e.  ( 0 ... N
) )
5554, 19syldan 457 . . . . 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 12482 . . . 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 9385 . . 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 202 . 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 2440 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 set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   ` cfv 5413  (class class class)co 6040   CCcc 8944   0cc0 8946   1c1 8947    + caddc 8949    x. cmul 8951    - cmin 9247    / cdiv 9633   NNcn 9956   NN0cn0 10177   ZZcz 10238   ZZ>=cuz 10444   ...cfz 10999   ^cexp 11337    _C cbc 11548   sum_csu 12434   BernPoly cbp 25996
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-oi 7435  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-fz 11000  df-fzo 11091  df-seq 11279  df-exp 11338  df-fac 11522  df-bc 11549  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237  df-sum 12435  df-pred 25382  df-bpoly 25997
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