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Theorem fsumkthpow 25809
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 11232 . . . 4  |-  ( ( K  e.  NN0  /\  M  e.  NN0 )  -> 
( 0 ... M
)  e.  Fin )
2 elfzelz 10984 . . . . . 6  |-  ( n  e.  ( 0 ... M )  ->  n  e.  ZZ )
32zcnd 10301 . . . . 5  |-  ( n  e.  ( 0 ... M )  ->  n  e.  CC )
4 simpl 444 . . . . 5  |-  ( ( K  e.  NN0  /\  M  e.  NN0 )  ->  K  e.  NN0 )
5 expcl 11319 . . . . 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 12447 . . 3  |-  ( ( K  e.  NN0  /\  M  e.  NN0 )  ->  sum_ n  e.  ( 0 ... M ) ( n ^ K )  e.  CC )
8 nn0p1nn 10184 . . . . 5  |-  ( K  e.  NN0  ->  ( K  +  1 )  e.  NN )
98adantr 452 . . . 4  |-  ( ( K  e.  NN0  /\  M  e.  NN0 )  -> 
( K  +  1 )  e.  NN )
109nncnd 9941 . . 3  |-  ( ( K  e.  NN0  /\  M  e.  NN0 )  -> 
( K  +  1 )  e.  CC )
119nnne0d 9969 . . 3  |-  ( ( K  e.  NN0  /\  M  e.  NN0 )  -> 
( K  +  1 )  =/=  0 )
127, 10, 11divcan3d 9720 . 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 12487 . . . 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 25808 . . . . . . 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 10156 . . . . . . . . . 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 8974 . . . . . . . . 9  |-  1  e.  CC
19 pncan 9236 . . . . . . . . 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 6029 . . . . . . 7  |-  ( ( ( K  e.  NN0  /\  M  e.  NN0 )  /\  n  e.  (
0 ... M ) )  ->  ( n ^
( ( K  + 
1 )  -  1 ) )  =  ( n ^ K ) )
2221oveq2d 6029 . . . . . 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 2412 . . . . 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 12417 . . . 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 6021 . . . . 5  |-  ( k  =  n  ->  (
( K  +  1 ) BernPoly  k )  =  ( ( K  + 
1 ) BernPoly  n )
)
26 oveq2 6021 . . . . 5  |-  ( k  =  ( n  + 
1 )  ->  (
( K  +  1 ) BernPoly  k )  =  ( ( K  + 
1 ) BernPoly  ( n  +  1 ) ) )
27 oveq2 6021 . . . . 5  |-  ( k  =  0  ->  (
( K  +  1 ) BernPoly  k )  =  ( ( K  + 
1 ) BernPoly  0 ) )
28 oveq2 6021 . . . . 5  |-  ( k  =  ( M  + 
1 )  ->  (
( K  +  1 ) BernPoly  k )  =  ( ( K  + 
1 ) BernPoly  ( M  +  1 ) ) )
29 nn0z 10229 . . . . . 6  |-  ( M  e.  NN0  ->  M  e.  ZZ )
3029adantl 453 . . . . 5  |-  ( ( K  e.  NN0  /\  M  e.  NN0 )  ->  M  e.  ZZ )
31 peano2nn0 10185 . . . . . . 7  |-  ( M  e.  NN0  ->  ( M  +  1 )  e. 
NN0 )
3231adantl 453 . . . . . 6  |-  ( ( K  e.  NN0  /\  M  e.  NN0 )  -> 
( M  +  1 )  e.  NN0 )
33 nn0uz 10445 . . . . . 6  |-  NN0  =  ( ZZ>= `  0 )
3432, 33syl6eleq 2470 . . . . 5  |-  ( ( K  e.  NN0  /\  M  e.  NN0 )  -> 
( M  +  1 )  e.  ( ZZ>= ` 
0 ) )
35 peano2nn0 10185 . . . . . . 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 11008 . . . . . . . 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 10201 . . . . . 6  |-  ( ( ( K  e.  NN0  /\  M  e.  NN0 )  /\  k  e.  (
0 ... ( M  + 
1 ) ) )  ->  k  e.  CC )
40 bpolycl 25805 . . . . . 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 12504 . . . 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 2418 . . 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 6028 . 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 2414 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 1717   ` cfv 5387  (class class class)co 6013   CCcc 8914   0cc0 8916   1c1 8917    + caddc 8919    x. cmul 8921    - cmin 9216    / cdiv 9602   NNcn 9925   NN0cn0 10146   ZZcz 10207   ZZ>=cuz 10413   ...cfz 10968   ^cexp 11302   sum_csu 12399   BernPoly cbp 25799
This theorem is referenced by:  fsumcube  25813
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-inf2 7522  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993  ax-pre-sup 8994
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-se 4476  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-isom 5396  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-riota 6478  df-recs 6562  df-rdg 6597  df-1o 6653  df-oadd 6657  df-er 6834  df-en 7039  df-dom 7040  df-sdom 7041  df-fin 7042  df-sup 7374  df-oi 7405  df-card 7752  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-div 9603  df-nn 9926  df-2 9983  df-3 9984  df-n0 10147  df-z 10208  df-uz 10414  df-rp 10538  df-fz 10969  df-fzo 11059  df-seq 11244  df-exp 11303  df-fac 11487  df-bc 11514  df-hash 11539  df-cj 11824  df-re 11825  df-im 11826  df-sqr 11960  df-abs 11961  df-clim 12202  df-sum 12400  df-pred 25185  df-bpoly 25800
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