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Theorem fsumkthpow 26006
Description: A closed-form expression for the sum of  K-th powers. (Contributed by Scott Fenton, 16-May-2014.) (Revised by Mario Carneiro, 16-Jun-2014.)
Assertion
Ref Expression
fsumkthpow  |-  ( ( K  e.  NN0  /\  M  e.  NN0 )  ->  sum_ n  e.  ( 0 ... M ) ( n ^ K )  =  ( ( ( ( K  +  1 ) BernPoly  ( M  + 
1 ) )  -  ( ( K  + 
1 ) BernPoly  0 ) )  /  ( K  +  1 ) ) )
Distinct variable groups:    n, K    n, M

Proof of Theorem fsumkthpow
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 fzfid 11267 . . . 4  |-  ( ( K  e.  NN0  /\  M  e.  NN0 )  -> 
( 0 ... M
)  e.  Fin )
2 elfzelz 11015 . . . . . 6  |-  ( n  e.  ( 0 ... M )  ->  n  e.  ZZ )
32zcnd 10332 . . . . 5  |-  ( n  e.  ( 0 ... M )  ->  n  e.  CC )
4 simpl 444 . . . . 5  |-  ( ( K  e.  NN0  /\  M  e.  NN0 )  ->  K  e.  NN0 )
5 expcl 11354 . . . . 5  |-  ( ( n  e.  CC  /\  K  e.  NN0 )  -> 
( n ^ K
)  e.  CC )
63, 4, 5syl2anr 465 . . . 4  |-  ( ( ( K  e.  NN0  /\  M  e.  NN0 )  /\  n  e.  (
0 ... M ) )  ->  ( n ^ K )  e.  CC )
71, 6fsumcl 12482 . . 3  |-  ( ( K  e.  NN0  /\  M  e.  NN0 )  ->  sum_ n  e.  ( 0 ... M ) ( n ^ K )  e.  CC )
8 nn0p1nn 10215 . . . . 5  |-  ( K  e.  NN0  ->  ( K  +  1 )  e.  NN )
98adantr 452 . . . 4  |-  ( ( K  e.  NN0  /\  M  e.  NN0 )  -> 
( K  +  1 )  e.  NN )
109nncnd 9972 . . 3  |-  ( ( K  e.  NN0  /\  M  e.  NN0 )  -> 
( K  +  1 )  e.  CC )
119nnne0d 10000 . . 3  |-  ( ( K  e.  NN0  /\  M  e.  NN0 )  -> 
( K  +  1 )  =/=  0 )
127, 10, 11divcan3d 9751 . 2  |-  ( ( K  e.  NN0  /\  M  e.  NN0 )  -> 
( ( ( K  +  1 )  x. 
sum_ n  e.  (
0 ... M ) ( n ^ K ) )  /  ( K  +  1 ) )  =  sum_ n  e.  ( 0 ... M ) ( n ^ K
) )
131, 10, 6fsummulc2 12522 . . . 4  |-  ( ( K  e.  NN0  /\  M  e.  NN0 )  -> 
( ( K  + 
1 )  x.  sum_ n  e.  ( 0 ... M ) ( n ^ K ) )  =  sum_ n  e.  ( 0 ... M ) ( ( K  + 
1 )  x.  (
n ^ K ) ) )
14 bpolydif 26005 . . . . . . 7  |-  ( ( ( K  +  1 )  e.  NN  /\  n  e.  CC )  ->  ( ( ( K  +  1 ) BernPoly  (
n  +  1 ) )  -  ( ( K  +  1 ) BernPoly  n ) )  =  ( ( K  + 
1 )  x.  (
n ^ ( ( K  +  1 )  -  1 ) ) ) )
159, 3, 14syl2an 464 . . . . . 6  |-  ( ( ( K  e.  NN0  /\  M  e.  NN0 )  /\  n  e.  (
0 ... M ) )  ->  ( ( ( K  +  1 ) BernPoly  ( n  +  1
) )  -  (
( K  +  1 ) BernPoly  n ) )  =  ( ( K  + 
1 )  x.  (
n ^ ( ( K  +  1 )  -  1 ) ) ) )
16 nn0cn 10187 . . . . . . . . . 10  |-  ( K  e.  NN0  ->  K  e.  CC )
1716ad2antrr 707 . . . . . . . . 9  |-  ( ( ( K  e.  NN0  /\  M  e.  NN0 )  /\  n  e.  (
0 ... M ) )  ->  K  e.  CC )
18 ax-1cn 9004 . . . . . . . . 9  |-  1  e.  CC
19 pncan 9267 . . . . . . . . 9  |-  ( ( K  e.  CC  /\  1  e.  CC )  ->  ( ( K  + 
1 )  -  1 )  =  K )
2017, 18, 19sylancl 644 . . . . . . . 8  |-  ( ( ( K  e.  NN0  /\  M  e.  NN0 )  /\  n  e.  (
0 ... M ) )  ->  ( ( K  +  1 )  - 
1 )  =  K )
2120oveq2d 6056 . . . . . . 7  |-  ( ( ( K  e.  NN0  /\  M  e.  NN0 )  /\  n  e.  (
0 ... M ) )  ->  ( n ^
( ( K  + 
1 )  -  1 ) )  =  ( n ^ K ) )
2221oveq2d 6056 . . . . . 6  |-  ( ( ( K  e.  NN0  /\  M  e.  NN0 )  /\  n  e.  (
0 ... M ) )  ->  ( ( K  +  1 )  x.  ( n ^ (
( K  +  1 )  -  1 ) ) )  =  ( ( K  +  1 )  x.  ( n ^ K ) ) )
2315, 22eqtrd 2436 . . . . 5  |-  ( ( ( K  e.  NN0  /\  M  e.  NN0 )  /\  n  e.  (
0 ... M ) )  ->  ( ( ( K  +  1 ) BernPoly  ( n  +  1
) )  -  (
( K  +  1 ) BernPoly  n ) )  =  ( ( K  + 
1 )  x.  (
n ^ K ) ) )
2423sumeq2dv 12452 . . . 4  |-  ( ( K  e.  NN0  /\  M  e.  NN0 )  ->  sum_ n  e.  ( 0 ... M ) ( ( ( K  + 
1 ) BernPoly  ( n  +  1 ) )  -  ( ( K  +  1 ) BernPoly  n
) )  =  sum_ n  e.  ( 0 ... M ) ( ( K  +  1 )  x.  ( n ^ K ) ) )
25 oveq2 6048 . . . . 5  |-  ( k  =  n  ->  (
( K  +  1 ) BernPoly  k )  =  ( ( K  + 
1 ) BernPoly  n )
)
26 oveq2 6048 . . . . 5  |-  ( k  =  ( n  + 
1 )  ->  (
( K  +  1 ) BernPoly  k )  =  ( ( K  + 
1 ) BernPoly  ( n  +  1 ) ) )
27 oveq2 6048 . . . . 5  |-  ( k  =  0  ->  (
( K  +  1 ) BernPoly  k )  =  ( ( K  + 
1 ) BernPoly  0 ) )
28 oveq2 6048 . . . . 5  |-  ( k  =  ( M  + 
1 )  ->  (
( K  +  1 ) BernPoly  k )  =  ( ( K  + 
1 ) BernPoly  ( M  +  1 ) ) )
29 nn0z 10260 . . . . . 6  |-  ( M  e.  NN0  ->  M  e.  ZZ )
3029adantl 453 . . . . 5  |-  ( ( K  e.  NN0  /\  M  e.  NN0 )  ->  M  e.  ZZ )
31 peano2nn0 10216 . . . . . . 7  |-  ( M  e.  NN0  ->  ( M  +  1 )  e. 
NN0 )
3231adantl 453 . . . . . 6  |-  ( ( K  e.  NN0  /\  M  e.  NN0 )  -> 
( M  +  1 )  e.  NN0 )
33 nn0uz 10476 . . . . . 6  |-  NN0  =  ( ZZ>= `  0 )
3432, 33syl6eleq 2494 . . . . 5  |-  ( ( K  e.  NN0  /\  M  e.  NN0 )  -> 
( M  +  1 )  e.  ( ZZ>= ` 
0 ) )
35 peano2nn0 10216 . . . . . . 7  |-  ( K  e.  NN0  ->  ( K  +  1 )  e. 
NN0 )
3635ad2antrr 707 . . . . . 6  |-  ( ( ( K  e.  NN0  /\  M  e.  NN0 )  /\  k  e.  (
0 ... ( M  + 
1 ) ) )  ->  ( K  + 
1 )  e.  NN0 )
37 elfznn0 11039 . . . . . . . 8  |-  ( k  e.  ( 0 ... ( M  +  1 ) )  ->  k  e.  NN0 )
3837adantl 453 . . . . . . 7  |-  ( ( ( K  e.  NN0  /\  M  e.  NN0 )  /\  k  e.  (
0 ... ( M  + 
1 ) ) )  ->  k  e.  NN0 )
3938nn0cnd 10232 . . . . . 6  |-  ( ( ( K  e.  NN0  /\  M  e.  NN0 )  /\  k  e.  (
0 ... ( M  + 
1 ) ) )  ->  k  e.  CC )
40 bpolycl 26002 . . . . . 6  |-  ( ( ( K  +  1 )  e.  NN0  /\  k  e.  CC )  ->  ( ( K  + 
1 ) BernPoly  k )  e.  CC )
4136, 39, 40syl2anc 643 . . . . 5  |-  ( ( ( K  e.  NN0  /\  M  e.  NN0 )  /\  k  e.  (
0 ... ( M  + 
1 ) ) )  ->  ( ( K  +  1 ) BernPoly  k
)  e.  CC )
4225, 26, 27, 28, 30, 34, 41fsumtscop2 12539 . . . 4  |-  ( ( K  e.  NN0  /\  M  e.  NN0 )  ->  sum_ n  e.  ( 0 ... M ) ( ( ( K  + 
1 ) BernPoly  ( n  +  1 ) )  -  ( ( K  +  1 ) BernPoly  n
) )  =  ( ( ( K  + 
1 ) BernPoly  ( M  +  1 ) )  -  ( ( K  +  1 ) BernPoly  0
) ) )
4313, 24, 423eqtr2d 2442 . . 3  |-  ( ( K  e.  NN0  /\  M  e.  NN0 )  -> 
( ( K  + 
1 )  x.  sum_ n  e.  ( 0 ... M ) ( n ^ K ) )  =  ( ( ( K  +  1 ) BernPoly  ( M  +  1
) )  -  (
( K  +  1 ) BernPoly  0 ) ) )
4443oveq1d 6055 . 2  |-  ( ( K  e.  NN0  /\  M  e.  NN0 )  -> 
( ( ( K  +  1 )  x. 
sum_ n  e.  (
0 ... M ) ( n ^ K ) )  /  ( K  +  1 ) )  =  ( ( ( ( K  +  1 ) BernPoly  ( M  + 
1 ) )  -  ( ( K  + 
1 ) BernPoly  0 ) )  /  ( K  +  1 ) ) )
4512, 44eqtr3d 2438 1  |-  ( ( K  e.  NN0  /\  M  e.  NN0 )  ->  sum_ n  e.  ( 0 ... M ) ( n ^ K )  =  ( ( ( ( K  +  1 ) BernPoly  ( M  + 
1 ) )  -  ( ( K  + 
1 ) BernPoly  0 ) )  /  ( K  +  1 ) ) )
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   sum_csu 12434   BernPoly cbp 25996
This theorem is referenced by:  fsumcube  26010
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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