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

Theorem bpolysum 24860
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 443 . . . 4  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  N  e.  NN0 )
2 nn0uz 10278 . . . 4  |-  NN0  =  ( ZZ>= `  0 )
31, 2syl6eleq 2386 . . 3  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  N  e.  ( ZZ>= ` 
0 ) )
4 elfzelz 10814 . . . . . 6  |-  ( k  e.  ( 0 ... N )  ->  k  e.  ZZ )
5 bccl 11350 . . . . . 6  |-  ( ( N  e.  NN0  /\  k  e.  ZZ )  ->  ( N  _C  k
)  e.  NN0 )
61, 4, 5syl2an 463 . . . . 5  |-  ( ( ( N  e.  NN0  /\  X  e.  CC )  /\  k  e.  ( 0 ... N ) )  ->  ( N  _C  k )  e.  NN0 )
76nn0cnd 10036 . . . 4  |-  ( ( ( N  e.  NN0  /\  X  e.  CC )  /\  k  e.  ( 0 ... N ) )  ->  ( N  _C  k )  e.  CC )
8 elfznn0 10838 . . . . . 6  |-  ( k  e.  ( 0 ... N )  ->  k  e.  NN0 )
9 simpr 447 . . . . . 6  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  X  e.  CC )
10 bpolycl 24859 . . . . . 6  |-  ( ( k  e.  NN0  /\  X  e.  CC )  ->  ( k BernPoly  X )  e.  CC )
118, 9, 10syl2anr 464 . . . . 5  |-  ( ( ( N  e.  NN0  /\  X  e.  CC )  /\  k  e.  ( 0 ... N ) )  ->  ( k BernPoly  X )  e.  CC )
12 fznn0sub 10840 . . . . . . . 8  |-  ( k  e.  ( 0 ... N )  ->  ( N  -  k )  e.  NN0 )
1312adantl 452 . . . . . . 7  |-  ( ( ( N  e.  NN0  /\  X  e.  CC )  /\  k  e.  ( 0 ... N ) )  ->  ( N  -  k )  e. 
NN0 )
14 nn0p1nn 10019 . . . . . . 7  |-  ( ( N  -  k )  e.  NN0  ->  ( ( N  -  k )  +  1 )  e.  NN )
1513, 14syl 15 . . . . . 6  |-  ( ( ( N  e.  NN0  /\  X  e.  CC )  /\  k  e.  ( 0 ... N ) )  ->  ( ( N  -  k )  +  1 )  e.  NN )
1615nncnd 9778 . . . . 5  |-  ( ( ( N  e.  NN0  /\  X  e.  CC )  /\  k  e.  ( 0 ... N ) )  ->  ( ( N  -  k )  +  1 )  e.  CC )
1715nnne0d 9806 . . . . 5  |-  ( ( ( N  e.  NN0  /\  X  e.  CC )  /\  k  e.  ( 0 ... N ) )  ->  ( ( N  -  k )  +  1 )  =/=  0 )
1811, 16, 17divcld 9552 . . . 4  |-  ( ( ( N  e.  NN0  /\  X  e.  CC )  /\  k  e.  ( 0 ... N ) )  ->  ( (
k BernPoly  X )  /  (
( N  -  k
)  +  1 ) )  e.  CC )
197, 18mulcld 8871 . . 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 5882 . . . 4  |-  ( k  =  N  ->  ( N  _C  k )  =  ( N  _C  N
) )
21 oveq1 5881 . . . . 5  |-  ( k  =  N  ->  (
k BernPoly  X )  =  ( N BernPoly  X ) )
22 oveq2 5882 . . . . . 6  |-  ( k  =  N  ->  ( N  -  k )  =  ( N  -  N ) )
2322oveq1d 5889 . . . . 5  |-  ( k  =  N  ->  (
( N  -  k
)  +  1 )  =  ( ( N  -  N )  +  1 ) )
2421, 23oveq12d 5892 . . . 4  |-  ( k  =  N  ->  (
( k BernPoly  X )  /  ( ( N  -  k )  +  1 ) )  =  ( ( N BernPoly  X )  /  ( ( N  -  N )  +  1 ) ) )
2520, 24oveq12d 5892 . . 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 12232 . 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 11340 . . . . . 6  |-  ( N  e.  NN0  ->  ( N  _C  N )  =  1 )
2827adantr 451 . . . . 5  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( N  _C  N
)  =  1 )
29 nn0cn 9991 . . . . . . . . . . 11  |-  ( N  e.  NN0  ->  N  e.  CC )
3029adantr 451 . . . . . . . . . 10  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  N  e.  CC )
3130subidd 9161 . . . . . . . . 9  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( N  -  N
)  =  0 )
3231oveq1d 5889 . . . . . . . 8  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( ( N  -  N )  +  1 )  =  ( 0  +  1 ) )
33 0p1e1 9855 . . . . . . . 8  |-  ( 0  +  1 )  =  1
3432, 33syl6eq 2344 . . . . . . 7  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( ( N  -  N )  +  1 )  =  1 )
3534oveq2d 5890 . . . . . 6  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( ( N BernPoly  X )  /  ( ( N  -  N )  +  1 ) )  =  ( ( N BernPoly  X )  /  1 ) )
36 bpolycl 24859 . . . . . . 7  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( N BernPoly  X )  e.  CC )
3736div1d 9544 . . . . . 6  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( ( N BernPoly  X )  /  1 )  =  ( N BernPoly  X )
)
3835, 37eqtrd 2328 . . . . 5  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( ( N BernPoly  X )  /  ( ( N  -  N )  +  1 ) )  =  ( N BernPoly  X )
)
3928, 38oveq12d 5892 . . . 4  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( ( N  _C  N )  x.  (
( N BernPoly  X )  /  ( ( N  -  N )  +  1 ) ) )  =  ( 1  x.  ( N BernPoly  X )
) )
4036mulid2d 8869 . . . 4  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( 1  x.  ( N BernPoly  X ) )  =  ( N BernPoly  X )
)
4139, 40eqtrd 2328 . . 3  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( ( N  _C  N )  x.  (
( N BernPoly  X )  /  ( ( N  -  N )  +  1 ) ) )  =  ( N BernPoly  X ) )
4241oveq2d 5890 . 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 24856 . . . 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 2301 . . 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 11137 . . . . 5  |-  ( ( X  e.  CC  /\  N  e.  NN0 )  -> 
( X ^ N
)  e.  CC )
4645ancoms 439 . . . 4  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( X ^ N
)  e.  CC )
47 fzfid 11051 . . . . 5  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( 0 ... ( N  -  1 ) )  e.  Fin )
48 fzssp1 10850 . . . . . . . 8  |-  ( 0 ... ( N  - 
1 ) )  C_  ( 0 ... (
( N  -  1 )  +  1 ) )
49 ax-1cn 8811 . . . . . . . . . 10  |-  1  e.  CC
50 npcan 9076 . . . . . . . . . 10  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( ( N  - 
1 )  +  1 )  =  N )
5130, 49, 50sylancl 643 . . . . . . . . 9  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( ( N  - 
1 )  +  1 )  =  N )
5251oveq2d 5890 . . . . . . . 8  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( 0 ... (
( N  -  1 )  +  1 ) )  =  ( 0 ... N ) )
5348, 52syl5sseq 3239 . . . . . . 7  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( 0 ... ( N  -  1 ) )  C_  ( 0 ... N ) )
5453sselda 3193 . . . . . 6  |-  ( ( ( N  e.  NN0  /\  X  e.  CC )  /\  k  e.  ( 0 ... ( N  -  1 ) ) )  ->  k  e.  ( 0 ... N
) )
5554, 19syldan 456 . . . . 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 12222 . . . 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 9191 . . 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 201 . 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 2332 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 358    = wceq 1632    e. wcel 1696   ` cfv 5271  (class class class)co 5874   CCcc 8751   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758    - cmin 9053    / cdiv 9439   NNcn 9762   NN0cn0 9981   ZZcz 10040   ZZ>=cuz 10246   ...cfz 10798   ^cexp 11120    _C cbc 11331   sum_csu 12174   BernPoly cbp 24853
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-fz 10799  df-fzo 10887  df-seq 11063  df-exp 11121  df-fac 11305  df-bc 11332  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-sum 12175  df-pred 24239  df-bpoly 24854
  Copyright terms: Public domain W3C validator