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Theorem bpolysum 24788
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 10262 . . . 4  |-  NN0  =  ( ZZ>= `  0 )
31, 2syl6eleq 2373 . . 3  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  N  e.  ( ZZ>= ` 
0 ) )
4 elfzelz 10798 . . . . . 6  |-  ( k  e.  ( 0 ... N )  ->  k  e.  ZZ )
5 bccl 11334 . . . . . 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 10020 . . . 4  |-  ( ( ( N  e.  NN0  /\  X  e.  CC )  /\  k  e.  ( 0 ... N ) )  ->  ( N  _C  k )  e.  CC )
8 elfznn0 10822 . . . . . 6  |-  ( k  e.  ( 0 ... N )  ->  k  e.  NN0 )
9 simpr 447 . . . . . 6  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  X  e.  CC )
10 bpolycl 24787 . . . . . 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 10824 . . . . . . . 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 10003 . . . . . . 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 9762 . . . . 5  |-  ( ( ( N  e.  NN0  /\  X  e.  CC )  /\  k  e.  ( 0 ... N ) )  ->  ( ( N  -  k )  +  1 )  e.  CC )
1715nnne0d 9790 . . . . 5  |-  ( ( ( N  e.  NN0  /\  X  e.  CC )  /\  k  e.  ( 0 ... N ) )  ->  ( ( N  -  k )  +  1 )  =/=  0 )
1811, 16, 17divcld 9536 . . . 4  |-  ( ( ( N  e.  NN0  /\  X  e.  CC )  /\  k  e.  ( 0 ... N ) )  ->  ( (
k BernPoly  X )  /  (
( N  -  k
)  +  1 ) )  e.  CC )
197, 18mulcld 8855 . . 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 5866 . . . 4  |-  ( k  =  N  ->  ( N  _C  k )  =  ( N  _C  N
) )
21 oveq1 5865 . . . . 5  |-  ( k  =  N  ->  (
k BernPoly  X )  =  ( N BernPoly  X ) )
22 oveq2 5866 . . . . . 6  |-  ( k  =  N  ->  ( N  -  k )  =  ( N  -  N ) )
2322oveq1d 5873 . . . . 5  |-  ( k  =  N  ->  (
( N  -  k
)  +  1 )  =  ( ( N  -  N )  +  1 ) )
2421, 23oveq12d 5876 . . . 4  |-  ( k  =  N  ->  (
( k BernPoly  X )  /  ( ( N  -  k )  +  1 ) )  =  ( ( N BernPoly  X )  /  ( ( N  -  N )  +  1 ) ) )
2520, 24oveq12d 5876 . . 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 12216 . 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 11324 . . . . . 6  |-  ( N  e.  NN0  ->  ( N  _C  N )  =  1 )
2827adantr 451 . . . . 5  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( N  _C  N
)  =  1 )
29 nn0cn 9975 . . . . . . . . . . 11  |-  ( N  e.  NN0  ->  N  e.  CC )
3029adantr 451 . . . . . . . . . 10  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  N  e.  CC )
3130subidd 9145 . . . . . . . . 9  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( N  -  N
)  =  0 )
3231oveq1d 5873 . . . . . . . 8  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( ( N  -  N )  +  1 )  =  ( 0  +  1 ) )
33 0p1e1 9839 . . . . . . . 8  |-  ( 0  +  1 )  =  1
3432, 33syl6eq 2331 . . . . . . 7  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( ( N  -  N )  +  1 )  =  1 )
3534oveq2d 5874 . . . . . 6  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( ( N BernPoly  X )  /  ( ( N  -  N )  +  1 ) )  =  ( ( N BernPoly  X )  /  1 ) )
36 bpolycl 24787 . . . . . . 7  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( N BernPoly  X )  e.  CC )
3736div1d 9528 . . . . . 6  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( ( N BernPoly  X )  /  1 )  =  ( N BernPoly  X )
)
3835, 37eqtrd 2315 . . . . 5  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( ( N BernPoly  X )  /  ( ( N  -  N )  +  1 ) )  =  ( N BernPoly  X )
)
3928, 38oveq12d 5876 . . . 4  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( ( N  _C  N )  x.  (
( N BernPoly  X )  /  ( ( N  -  N )  +  1 ) ) )  =  ( 1  x.  ( N BernPoly  X )
) )
4036mulid2d 8853 . . . 4  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( 1  x.  ( N BernPoly  X ) )  =  ( N BernPoly  X )
)
4139, 40eqtrd 2315 . . 3  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( ( N  _C  N )  x.  (
( N BernPoly  X )  /  ( ( N  -  N )  +  1 ) ) )  =  ( N BernPoly  X ) )
4241oveq2d 5874 . 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 24784 . . . 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 2288 . . 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 11121 . . . . 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 11035 . . . . 5  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( 0 ... ( N  -  1 ) )  e.  Fin )
48 fzssp1 10834 . . . . . . . 8  |-  ( 0 ... ( N  - 
1 ) )  C_  ( 0 ... (
( N  -  1 )  +  1 ) )
49 ax-1cn 8795 . . . . . . . . . 10  |-  1  e.  CC
50 npcan 9060 . . . . . . . . . 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 5874 . . . . . . . 8  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( 0 ... (
( N  -  1 )  +  1 ) )  =  ( 0 ... N ) )
5348, 52syl5sseq 3226 . . . . . . 7  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( 0 ... ( N  -  1 ) )  C_  ( 0 ... N ) )
5453sselda 3180 . . . . . 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 12206 . . . 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 9175 . . 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 2319 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 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858   CCcc 8735   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742    - cmin 9037    / cdiv 9423   NNcn 9746   NN0cn0 9965   ZZcz 10024   ZZ>=cuz 10230   ...cfz 10782   ^cexp 11104    _C cbc 11315   sum_csu 12158   BernPoly cbp 24781
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fz 10783  df-fzo 10871  df-seq 11047  df-exp 11105  df-fac 11289  df-bc 11316  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-sum 12159  df-pred 24168  df-bpoly 24782
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