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Theorem fsumkthpow 23965
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
StepHypRef Expression
1 fzfid 10913 . . . 4  |-  ( ( K  e.  NN0  /\  M  e.  NN0 )  -> 
( 0 ... M
)  e.  Fin )
2 elfzelz 10676 . . . . . 6  |-  ( n  e.  ( 0 ... M )  ->  n  e.  ZZ )
32zcnd 9997 . . . . 5  |-  ( n  e.  ( 0 ... M )  ->  n  e.  CC )
4 simpl 445 . . . . 5  |-  ( ( K  e.  NN0  /\  M  e.  NN0 )  ->  K  e.  NN0 )
5 expcl 10999 . . . . 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 12083 . . 3  |-  ( ( K  e.  NN0  /\  M  e.  NN0 )  ->  sum_ n  e.  ( 0 ... M ) ( n ^ K )  e.  CC )
8 nn0p1nn 9882 . . . . 5  |-  ( K  e.  NN0  ->  ( K  +  1 )  e.  NN )
98adantr 453 . . . 4  |-  ( ( K  e.  NN0  /\  M  e.  NN0 )  -> 
( K  +  1 )  e.  NN )
109nncnd 9642 . . 3  |-  ( ( K  e.  NN0  /\  M  e.  NN0 )  -> 
( K  +  1 )  e.  CC )
119nnne0d 9670 . . 3  |-  ( ( K  e.  NN0  /\  M  e.  NN0 )  -> 
( K  +  1 )  =/=  0 )
127, 10, 11divcan3d 9421 . 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 12123 . . . 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 23964 . . . . . . 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 9854 . . . . . . . . . 10  |-  ( K  e.  NN0  ->  K  e.  CC )
1716ad2antrr 709 . . . . . . . . 9  |-  ( ( ( K  e.  NN0  /\  M  e.  NN0 )  /\  n  e.  (
0 ... M ) )  ->  K  e.  CC )
18 ax-1cn 8675 . . . . . . . . 9  |-  1  e.  CC
19 pncan 8937 . . . . . . . . 9  |-  ( ( K  e.  CC  /\  1  e.  CC )  ->  ( ( K  + 
1 )  -  1 )  =  K )
2017, 18, 19sylancl 646 . . . . . . . 8  |-  ( ( ( K  e.  NN0  /\  M  e.  NN0 )  /\  n  e.  (
0 ... M ) )  ->  ( ( K  +  1 )  - 
1 )  =  K )
2120oveq2d 5726 . . . . . . 7  |-  ( ( ( K  e.  NN0  /\  M  e.  NN0 )  /\  n  e.  (
0 ... M ) )  ->  ( n ^
( ( K  + 
1 )  -  1 ) )  =  ( n ^ K ) )
2221oveq2d 5726 . . . . . 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 2285 . . . . 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 12053 . . . 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 5718 . . . . 5  |-  ( k  =  n  ->  (
( K  +  1 ) BernPoly  k )  =  ( ( K  + 
1 ) BernPoly  n )
)
26 oveq2 5718 . . . . 5  |-  ( k  =  ( n  + 
1 )  ->  (
( K  +  1 ) BernPoly  k )  =  ( ( K  + 
1 ) BernPoly  ( n  +  1 ) ) )
27 oveq2 5718 . . . . 5  |-  ( k  =  0  ->  (
( K  +  1 ) BernPoly  k )  =  ( ( K  + 
1 ) BernPoly  0 ) )
28 oveq2 5718 . . . . 5  |-  ( k  =  ( M  + 
1 )  ->  (
( K  +  1 ) BernPoly  k )  =  ( ( K  + 
1 ) BernPoly  ( M  +  1 ) ) )
29 nn0z 9925 . . . . . 6  |-  ( M  e.  NN0  ->  M  e.  ZZ )
3029adantl 454 . . . . 5  |-  ( ( K  e.  NN0  /\  M  e.  NN0 )  ->  M  e.  ZZ )
31 peano2nn0 9883 . . . . . . 7  |-  ( M  e.  NN0  ->  ( M  +  1 )  e. 
NN0 )
3231adantl 454 . . . . . 6  |-  ( ( K  e.  NN0  /\  M  e.  NN0 )  -> 
( M  +  1 )  e.  NN0 )
33 nn0uz 10141 . . . . . 6  |-  NN0  =  ( ZZ>= `  0 )
3432, 33syl6eleq 2343 . . . . 5  |-  ( ( K  e.  NN0  /\  M  e.  NN0 )  -> 
( M  +  1 )  e.  ( ZZ>= ` 
0 ) )
35 peano2nn0 9883 . . . . . . 7  |-  ( K  e.  NN0  ->  ( K  +  1 )  e. 
NN0 )
3635ad2antrr 709 . . . . . 6  |-  ( ( ( K  e.  NN0  /\  M  e.  NN0 )  /\  k  e.  (
0 ... ( M  + 
1 ) ) )  ->  ( K  + 
1 )  e.  NN0 )
37 elfznn0 10700 . . . . . . . 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 9899 . . . . . 6  |-  ( ( ( K  e.  NN0  /\  M  e.  NN0 )  /\  k  e.  (
0 ... ( M  + 
1 ) ) )  ->  k  e.  CC )
40 bpolycl 23961 . . . . . 6  |-  ( ( ( K  +  1 )  e.  NN0  /\  k  e.  CC )  ->  ( ( K  + 
1 ) BernPoly  k )  e.  CC )
4136, 39, 40syl2anc 645 . . . . 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 12140 . . . 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 2291 . . 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 5725 . 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 2287 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 6    /\ wa 360    = wceq 1619    e. wcel 1621   ` cfv 4592  (class class class)co 5710   CCcc 8615   0cc0 8617   1c1 8618    + caddc 8620    x. cmul 8622    - cmin 8917    / cdiv 9303   NNcn 9626   NN0cn0 9844   ZZcz 9903   ZZ>=cuz 10109   ...cfz 10660   ^cexp 10982   sum_csu 12035   BernPoly cbp 23955
This theorem is referenced by:  fsumcube  23969
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-inf2 7226  ax-cnex 8673  ax-resscn 8674  ax-1cn 8675  ax-icn 8676  ax-addcl 8677  ax-addrcl 8678  ax-mulcl 8679  ax-mulrcl 8680  ax-mulcom 8681  ax-addass 8682  ax-mulass 8683  ax-distr 8684  ax-i2m1 8685  ax-1ne0 8686  ax-1rid 8687  ax-rnegex 8688  ax-rrecex 8689  ax-cnre 8690  ax-pre-lttri 8691  ax-pre-lttrn 8692  ax-pre-ltadd 8693  ax-pre-mulgt0 8694  ax-pre-sup 8695
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-int 3761  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-se 4246  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-isom 4609  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-riota 6190  df-recs 6274  df-rdg 6309  df-1o 6365  df-oadd 6369  df-er 6546  df-en 6750  df-dom 6751  df-sdom 6752  df-fin 6753  df-sup 7078  df-oi 7109  df-card 7456  df-pnf 8749  df-mnf 8750  df-xr 8751  df-ltxr 8752  df-le 8753  df-sub 8919  df-neg 8920  df-div 9304  df-n 9627  df-2 9684  df-3 9685  df-n0 9845  df-z 9904  df-uz 10110  df-rp 10234  df-fz 10661  df-fzo 10749  df-seq 10925  df-exp 10983  df-fac 11167  df-bc 11194  df-hash 11216  df-cj 11461  df-re 11462  df-im 11463  df-sqr 11597  df-abs 11598  df-clim 11839  df-sum 12036  df-pred 23336  df-bpoly 23956
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