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Theorem fsumkthpow 26104
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 11314 . . . 4  |-  ( ( K  e.  NN0  /\  M  e.  NN0 )  -> 
( 0 ... M
)  e.  Fin )
2 elfzelz 11061 . . . . . 6  |-  ( n  e.  ( 0 ... M )  ->  n  e.  ZZ )
32zcnd 10378 . . . . 5  |-  ( n  e.  ( 0 ... M )  ->  n  e.  CC )
4 simpl 445 . . . . 5  |-  ( ( K  e.  NN0  /\  M  e.  NN0 )  ->  K  e.  NN0 )
5 expcl 11401 . . . . 5  |-  ( ( n  e.  CC  /\  K  e.  NN0 )  -> 
( n ^ K
)  e.  CC )
63, 4, 5syl2anr 466 . . . 4  |-  ( ( ( K  e.  NN0  /\  M  e.  NN0 )  /\  n  e.  (
0 ... M ) )  ->  ( n ^ K )  e.  CC )
71, 6fsumcl 12529 . . 3  |-  ( ( K  e.  NN0  /\  M  e.  NN0 )  ->  sum_ n  e.  ( 0 ... M ) ( n ^ K )  e.  CC )
8 nn0p1nn 10261 . . . . 5  |-  ( K  e.  NN0  ->  ( K  +  1 )  e.  NN )
98adantr 453 . . . 4  |-  ( ( K  e.  NN0  /\  M  e.  NN0 )  -> 
( K  +  1 )  e.  NN )
109nncnd 10018 . . 3  |-  ( ( K  e.  NN0  /\  M  e.  NN0 )  -> 
( K  +  1 )  e.  CC )
119nnne0d 10046 . . 3  |-  ( ( K  e.  NN0  /\  M  e.  NN0 )  -> 
( K  +  1 )  =/=  0 )
127, 10, 11divcan3d 9797 . 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 12569 . . . 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 26103 . . . . . . 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 465 . . . . . 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 10233 . . . . . . . . . 10  |-  ( K  e.  NN0  ->  K  e.  CC )
1716ad2antrr 708 . . . . . . . . 9  |-  ( ( ( K  e.  NN0  /\  M  e.  NN0 )  /\  n  e.  (
0 ... M ) )  ->  K  e.  CC )
18 ax-1cn 9050 . . . . . . . . 9  |-  1  e.  CC
19 pncan 9313 . . . . . . . . 9  |-  ( ( K  e.  CC  /\  1  e.  CC )  ->  ( ( K  + 
1 )  -  1 )  =  K )
2017, 18, 19sylancl 645 . . . . . . . 8  |-  ( ( ( K  e.  NN0  /\  M  e.  NN0 )  /\  n  e.  (
0 ... M ) )  ->  ( ( K  +  1 )  - 
1 )  =  K )
2120oveq2d 6099 . . . . . . 7  |-  ( ( ( K  e.  NN0  /\  M  e.  NN0 )  /\  n  e.  (
0 ... M ) )  ->  ( n ^
( ( K  + 
1 )  -  1 ) )  =  ( n ^ K ) )
2221oveq2d 6099 . . . . . 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 2470 . . . . 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 12499 . . . 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 6091 . . . . 5  |-  ( k  =  n  ->  (
( K  +  1 ) BernPoly  k )  =  ( ( K  + 
1 ) BernPoly  n )
)
26 oveq2 6091 . . . . 5  |-  ( k  =  ( n  + 
1 )  ->  (
( K  +  1 ) BernPoly  k )  =  ( ( K  + 
1 ) BernPoly  ( n  +  1 ) ) )
27 oveq2 6091 . . . . 5  |-  ( k  =  0  ->  (
( K  +  1 ) BernPoly  k )  =  ( ( K  + 
1 ) BernPoly  0 ) )
28 oveq2 6091 . . . . 5  |-  ( k  =  ( M  + 
1 )  ->  (
( K  +  1 ) BernPoly  k )  =  ( ( K  + 
1 ) BernPoly  ( M  +  1 ) ) )
29 nn0z 10306 . . . . . 6  |-  ( M  e.  NN0  ->  M  e.  ZZ )
3029adantl 454 . . . . 5  |-  ( ( K  e.  NN0  /\  M  e.  NN0 )  ->  M  e.  ZZ )
31 peano2nn0 10262 . . . . . . 7  |-  ( M  e.  NN0  ->  ( M  +  1 )  e. 
NN0 )
3231adantl 454 . . . . . 6  |-  ( ( K  e.  NN0  /\  M  e.  NN0 )  -> 
( M  +  1 )  e.  NN0 )
33 nn0uz 10522 . . . . . 6  |-  NN0  =  ( ZZ>= `  0 )
3432, 33syl6eleq 2528 . . . . 5  |-  ( ( K  e.  NN0  /\  M  e.  NN0 )  -> 
( M  +  1 )  e.  ( ZZ>= ` 
0 ) )
35 peano2nn0 10262 . . . . . . 7  |-  ( K  e.  NN0  ->  ( K  +  1 )  e. 
NN0 )
3635ad2antrr 708 . . . . . 6  |-  ( ( ( K  e.  NN0  /\  M  e.  NN0 )  /\  k  e.  (
0 ... ( M  + 
1 ) ) )  ->  ( K  + 
1 )  e.  NN0 )
37 elfznn0 11085 . . . . . . . 8  |-  ( k  e.  ( 0 ... ( M  +  1 ) )  ->  k  e.  NN0 )
3837adantl 454 . . . . . . 7  |-  ( ( ( K  e.  NN0  /\  M  e.  NN0 )  /\  k  e.  (
0 ... ( M  + 
1 ) ) )  ->  k  e.  NN0 )
3938nn0cnd 10278 . . . . . 6  |-  ( ( ( K  e.  NN0  /\  M  e.  NN0 )  /\  k  e.  (
0 ... ( M  + 
1 ) ) )  ->  k  e.  CC )
40 bpolycl 26100 . . . . . 6  |-  ( ( ( K  +  1 )  e.  NN0  /\  k  e.  CC )  ->  ( ( K  + 
1 ) BernPoly  k )  e.  CC )
4136, 39, 40syl2anc 644 . . . . 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 12586 . . . 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 2476 . . 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 6098 . 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 2472 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 360    = wceq 1653    e. wcel 1726   ` cfv 5456  (class class class)co 6083   CCcc 8990   0cc0 8992   1c1 8993    + caddc 8995    x. cmul 8997    - cmin 9293    / cdiv 9679   NNcn 10002   NN0cn0 10223   ZZcz 10284   ZZ>=cuz 10490   ...cfz 11045   ^cexp 11384   sum_csu 12481   BernPoly cbp 26094
This theorem is referenced by:  fsumcube  26108
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-inf2 7598  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-se 4544  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-oadd 6730  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-sup 7448  df-oi 7481  df-card 7828  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-n0 10224  df-z 10285  df-uz 10491  df-rp 10615  df-fz 11046  df-fzo 11138  df-seq 11326  df-exp 11385  df-fac 11569  df-bc 11596  df-hash 11621  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-abs 12043  df-clim 12284  df-sum 12482  df-pred 25441  df-wrecs 25533  df-bpoly 26095
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