Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bpolysum Structured version   Unicode version

Theorem bpolysum 29392
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 457 . . . 4  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  N  e.  NN0 )
2 nn0uz 11112 . . . 4  |-  NN0  =  ( ZZ>= `  0 )
31, 2syl6eleq 2565 . . 3  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  N  e.  ( ZZ>= ` 
0 ) )
4 elfzelz 11684 . . . . . 6  |-  ( k  e.  ( 0 ... N )  ->  k  e.  ZZ )
5 bccl 12364 . . . . . 6  |-  ( ( N  e.  NN0  /\  k  e.  ZZ )  ->  ( N  _C  k
)  e.  NN0 )
61, 4, 5syl2an 477 . . . . 5  |-  ( ( ( N  e.  NN0  /\  X  e.  CC )  /\  k  e.  ( 0 ... N ) )  ->  ( N  _C  k )  e.  NN0 )
76nn0cnd 10850 . . . 4  |-  ( ( ( N  e.  NN0  /\  X  e.  CC )  /\  k  e.  ( 0 ... N ) )  ->  ( N  _C  k )  e.  CC )
8 elfznn0 11766 . . . . . 6  |-  ( k  e.  ( 0 ... N )  ->  k  e.  NN0 )
9 simpr 461 . . . . . 6  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  X  e.  CC )
10 bpolycl 29391 . . . . . 6  |-  ( ( k  e.  NN0  /\  X  e.  CC )  ->  ( k BernPoly  X )  e.  CC )
118, 9, 10syl2anr 478 . . . . 5  |-  ( ( ( N  e.  NN0  /\  X  e.  CC )  /\  k  e.  ( 0 ... N ) )  ->  ( k BernPoly  X )  e.  CC )
12 fznn0sub 11712 . . . . . . . 8  |-  ( k  e.  ( 0 ... N )  ->  ( N  -  k )  e.  NN0 )
1312adantl 466 . . . . . . 7  |-  ( ( ( N  e.  NN0  /\  X  e.  CC )  /\  k  e.  ( 0 ... N ) )  ->  ( N  -  k )  e. 
NN0 )
14 nn0p1nn 10831 . . . . . . 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 10548 . . . . 5  |-  ( ( ( N  e.  NN0  /\  X  e.  CC )  /\  k  e.  ( 0 ... N ) )  ->  ( ( N  -  k )  +  1 )  e.  CC )
1715nnne0d 10576 . . . . 5  |-  ( ( ( N  e.  NN0  /\  X  e.  CC )  /\  k  e.  ( 0 ... N ) )  ->  ( ( N  -  k )  +  1 )  =/=  0 )
1811, 16, 17divcld 10316 . . . 4  |-  ( ( ( N  e.  NN0  /\  X  e.  CC )  /\  k  e.  ( 0 ... N ) )  ->  ( (
k BernPoly  X )  /  (
( N  -  k
)  +  1 ) )  e.  CC )
197, 18mulcld 9612 . . 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 6290 . . . 4  |-  ( k  =  N  ->  ( N  _C  k )  =  ( N  _C  N
) )
21 oveq1 6289 . . . . 5  |-  ( k  =  N  ->  (
k BernPoly  X )  =  ( N BernPoly  X ) )
22 oveq2 6290 . . . . . 6  |-  ( k  =  N  ->  ( N  -  k )  =  ( N  -  N ) )
2322oveq1d 6297 . . . . 5  |-  ( k  =  N  ->  (
( N  -  k
)  +  1 )  =  ( ( N  -  N )  +  1 ) )
2421, 23oveq12d 6300 . . . 4  |-  ( k  =  N  ->  (
( k BernPoly  X )  /  ( ( N  -  k )  +  1 ) )  =  ( ( N BernPoly  X )  /  ( ( N  -  N )  +  1 ) ) )
2520, 24oveq12d 6300 . . 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 13525 . 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 12354 . . . . . 6  |-  ( N  e.  NN0  ->  ( N  _C  N )  =  1 )
2827adantr 465 . . . . 5  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( N  _C  N
)  =  1 )
29 nn0cn 10801 . . . . . . . . . . 11  |-  ( N  e.  NN0  ->  N  e.  CC )
3029adantr 465 . . . . . . . . . 10  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  N  e.  CC )
3130subidd 9914 . . . . . . . . 9  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( N  -  N
)  =  0 )
3231oveq1d 6297 . . . . . . . 8  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( ( N  -  N )  +  1 )  =  ( 0  +  1 ) )
33 0p1e1 10643 . . . . . . . 8  |-  ( 0  +  1 )  =  1
3432, 33syl6eq 2524 . . . . . . 7  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( ( N  -  N )  +  1 )  =  1 )
3534oveq2d 6298 . . . . . 6  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( ( N BernPoly  X )  /  ( ( N  -  N )  +  1 ) )  =  ( ( N BernPoly  X )  /  1 ) )
36 bpolycl 29391 . . . . . . 7  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( N BernPoly  X )  e.  CC )
3736div1d 10308 . . . . . 6  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( ( N BernPoly  X )  /  1 )  =  ( N BernPoly  X )
)
3835, 37eqtrd 2508 . . . . 5  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( ( N BernPoly  X )  /  ( ( N  -  N )  +  1 ) )  =  ( N BernPoly  X )
)
3928, 38oveq12d 6300 . . . 4  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( ( N  _C  N )  x.  (
( N BernPoly  X )  /  ( ( N  -  N )  +  1 ) ) )  =  ( 1  x.  ( N BernPoly  X )
) )
4036mulid2d 9610 . . . 4  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( 1  x.  ( N BernPoly  X ) )  =  ( N BernPoly  X )
)
4139, 40eqtrd 2508 . . 3  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( ( N  _C  N )  x.  (
( N BernPoly  X )  /  ( ( N  -  N )  +  1 ) ) )  =  ( N BernPoly  X ) )
4241oveq2d 6298 . 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 29388 . . . 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 2475 . . 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 12148 . . . . 5  |-  ( ( X  e.  CC  /\  N  e.  NN0 )  -> 
( X ^ N
)  e.  CC )
4645ancoms 453 . . . 4  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( X ^ N
)  e.  CC )
47 fzfid 12047 . . . . 5  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( 0 ... ( N  -  1 ) )  e.  Fin )
48 fzssp1 11722 . . . . . . . 8  |-  ( 0 ... ( N  - 
1 ) )  C_  ( 0 ... (
( N  -  1 )  +  1 ) )
49 ax-1cn 9546 . . . . . . . . . 10  |-  1  e.  CC
50 npcan 9825 . . . . . . . . . 10  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( ( N  - 
1 )  +  1 )  =  N )
5130, 49, 50sylancl 662 . . . . . . . . 9  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( ( N  - 
1 )  +  1 )  =  N )
5251oveq2d 6298 . . . . . . . 8  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( 0 ... (
( N  -  1 )  +  1 ) )  =  ( 0 ... N ) )
5348, 52syl5sseq 3552 . . . . . . 7  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( 0 ... ( N  -  1 ) )  C_  ( 0 ... N ) )
5453sselda 3504 . . . . . 6  |-  ( ( ( N  e.  NN0  /\  X  e.  CC )  /\  k  e.  ( 0 ... ( N  -  1 ) ) )  ->  k  e.  ( 0 ... N
) )
5554, 19syldan 470 . . . . 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 13514 . . . 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 9944 . . 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 2512 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 1379    e. wcel 1767   ` cfv 5586  (class class class)co 6282   CCcc 9486   0cc0 9488   1c1 9489    + caddc 9491    x. cmul 9493    - cmin 9801    / cdiv 10202   NNcn 10532   NN0cn0 10791   ZZcz 10860   ZZ>=cuz 11078   ...cfz 11668   ^cexp 12130    _C cbc 12344   sum_csu 13467   BernPoly cbp 29385
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-sup 7897  df-oi 7931  df-card 8316  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11079  df-rp 11217  df-fz 11669  df-fzo 11789  df-seq 12072  df-exp 12131  df-fac 12318  df-bc 12345  df-hash 12370  df-cj 12891  df-re 12892  df-im 12893  df-sqrt 13027  df-abs 13028  df-clim 13270  df-sum 13468  df-pred 28821  df-wrecs 28913  df-bpoly 29386
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator