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Theorem fsumkthpow 29745
Description: A closed-form expression for the sum of  K-th powers. (Contributed by Scott Fenton, 16-May-2014.) This is Metamath 100 proof #77. (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 12063 . . . 4  |-  ( ( K  e.  NN0  /\  M  e.  NN0 )  -> 
( 0 ... M
)  e.  Fin )
2 elfzelz 11700 . . . . . 6  |-  ( n  e.  ( 0 ... M )  ->  n  e.  ZZ )
32zcnd 10979 . . . . 5  |-  ( n  e.  ( 0 ... M )  ->  n  e.  CC )
4 simpl 457 . . . . 5  |-  ( ( K  e.  NN0  /\  M  e.  NN0 )  ->  K  e.  NN0 )
5 expcl 12164 . . . . 5  |-  ( ( n  e.  CC  /\  K  e.  NN0 )  -> 
( n ^ K
)  e.  CC )
63, 4, 5syl2anr 478 . . . 4  |-  ( ( ( K  e.  NN0  /\  M  e.  NN0 )  /\  n  e.  (
0 ... M ) )  ->  ( n ^ K )  e.  CC )
71, 6fsumcl 13535 . . 3  |-  ( ( K  e.  NN0  /\  M  e.  NN0 )  ->  sum_ n  e.  ( 0 ... M ) ( n ^ K )  e.  CC )
8 nn0p1nn 10847 . . . . 5  |-  ( K  e.  NN0  ->  ( K  +  1 )  e.  NN )
98adantr 465 . . . 4  |-  ( ( K  e.  NN0  /\  M  e.  NN0 )  -> 
( K  +  1 )  e.  NN )
109nncnd 10564 . . 3  |-  ( ( K  e.  NN0  /\  M  e.  NN0 )  -> 
( K  +  1 )  e.  CC )
119nnne0d 10592 . . 3  |-  ( ( K  e.  NN0  /\  M  e.  NN0 )  -> 
( K  +  1 )  =/=  0 )
127, 10, 11divcan3d 10337 . 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 13579 . . . 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 29744 . . . . . . 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 477 . . . . . 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 10817 . . . . . . . . . 10  |-  ( K  e.  NN0  ->  K  e.  CC )
1716ad2antrr 725 . . . . . . . . 9  |-  ( ( ( K  e.  NN0  /\  M  e.  NN0 )  /\  n  e.  (
0 ... M ) )  ->  K  e.  CC )
18 ax-1cn 9562 . . . . . . . . 9  |-  1  e.  CC
19 pncan 9838 . . . . . . . . 9  |-  ( ( K  e.  CC  /\  1  e.  CC )  ->  ( ( K  + 
1 )  -  1 )  =  K )
2017, 18, 19sylancl 662 . . . . . . . 8  |-  ( ( ( K  e.  NN0  /\  M  e.  NN0 )  /\  n  e.  (
0 ... M ) )  ->  ( ( K  +  1 )  - 
1 )  =  K )
2120oveq2d 6311 . . . . . . 7  |-  ( ( ( K  e.  NN0  /\  M  e.  NN0 )  /\  n  e.  (
0 ... M ) )  ->  ( n ^
( ( K  + 
1 )  -  1 ) )  =  ( n ^ K ) )
2221oveq2d 6311 . . . . . 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 2508 . . . . 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 13505 . . . 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 6303 . . . . 5  |-  ( k  =  n  ->  (
( K  +  1 ) BernPoly  k )  =  ( ( K  + 
1 ) BernPoly  n )
)
26 oveq2 6303 . . . . 5  |-  ( k  =  ( n  + 
1 )  ->  (
( K  +  1 ) BernPoly  k )  =  ( ( K  + 
1 ) BernPoly  ( n  +  1 ) ) )
27 oveq2 6303 . . . . 5  |-  ( k  =  0  ->  (
( K  +  1 ) BernPoly  k )  =  ( ( K  + 
1 ) BernPoly  0 ) )
28 oveq2 6303 . . . . 5  |-  ( k  =  ( M  + 
1 )  ->  (
( K  +  1 ) BernPoly  k )  =  ( ( K  + 
1 ) BernPoly  ( M  +  1 ) ) )
29 nn0z 10899 . . . . . 6  |-  ( M  e.  NN0  ->  M  e.  ZZ )
3029adantl 466 . . . . 5  |-  ( ( K  e.  NN0  /\  M  e.  NN0 )  ->  M  e.  ZZ )
31 peano2nn0 10848 . . . . . . 7  |-  ( M  e.  NN0  ->  ( M  +  1 )  e. 
NN0 )
3231adantl 466 . . . . . 6  |-  ( ( K  e.  NN0  /\  M  e.  NN0 )  -> 
( M  +  1 )  e.  NN0 )
33 nn0uz 11128 . . . . . 6  |-  NN0  =  ( ZZ>= `  0 )
3432, 33syl6eleq 2565 . . . . 5  |-  ( ( K  e.  NN0  /\  M  e.  NN0 )  -> 
( M  +  1 )  e.  ( ZZ>= ` 
0 ) )
35 peano2nn0 10848 . . . . . . 7  |-  ( K  e.  NN0  ->  ( K  +  1 )  e. 
NN0 )
3635ad2antrr 725 . . . . . 6  |-  ( ( ( K  e.  NN0  /\  M  e.  NN0 )  /\  k  e.  (
0 ... ( M  + 
1 ) ) )  ->  ( K  + 
1 )  e.  NN0 )
37 elfznn0 11782 . . . . . . . 8  |-  ( k  e.  ( 0 ... ( M  +  1 ) )  ->  k  e.  NN0 )
3837adantl 466 . . . . . . 7  |-  ( ( ( K  e.  NN0  /\  M  e.  NN0 )  /\  k  e.  (
0 ... ( M  + 
1 ) ) )  ->  k  e.  NN0 )
3938nn0cnd 10866 . . . . . 6  |-  ( ( ( K  e.  NN0  /\  M  e.  NN0 )  /\  k  e.  (
0 ... ( M  + 
1 ) ) )  ->  k  e.  CC )
40 bpolycl 29741 . . . . . 6  |-  ( ( ( K  +  1 )  e.  NN0  /\  k  e.  CC )  ->  ( ( K  + 
1 ) BernPoly  k )  e.  CC )
4136, 39, 40syl2anc 661 . . . . 5  |-  ( ( ( K  e.  NN0  /\  M  e.  NN0 )  /\  k  e.  (
0 ... ( M  + 
1 ) ) )  ->  ( ( K  +  1 ) BernPoly  k
)  e.  CC )
4225, 26, 27, 28, 30, 34, 41telfsum2 13599 . . . 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 2514 . . 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 6310 . 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 2510 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   ` cfv 5594  (class class class)co 6295   CCcc 9502   0cc0 9504   1c1 9505    + caddc 9507    x. cmul 9509    - cmin 9817    / cdiv 10218   NNcn 10548   NN0cn0 10807   ZZcz 10876   ZZ>=cuz 11094   ...cfz 11684   ^cexp 12146   sum_csu 13488   BernPoly cbp 29735
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 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582
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 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-sup 7913  df-oi 7947  df-card 8332  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-n0 10808  df-z 10877  df-uz 11095  df-rp 11233  df-fz 11685  df-fzo 11805  df-seq 12088  df-exp 12147  df-fac 12334  df-bc 12361  df-hash 12386  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-clim 13291  df-sum 13489  df-pred 29171  df-wrecs 29263  df-bpoly 29736
This theorem is referenced by:  fsumcube  29749
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