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Theorem bpolyval 28039
Description: The value of the Bernoulli polynomials. (Contributed by Scott Fenton, 16-May-2014.)
Assertion
Ref Expression
bpolyval  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( N BernPoly  X )  =  ( ( X ^ N )  -  sum_ k  e.  ( 0 ... ( N  - 
1 ) ) ( ( N  _C  k
)  x.  ( ( k BernPoly  X )  /  (
( N  -  k
)  +  1 ) ) ) ) )
Distinct variable groups:    k, N    k, X

Proof of Theorem bpolyval
Dummy variables  g  m  n  c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvex 5689 . . . . . 6  |-  ( # `  dom  c )  e. 
_V
2 nfcv 2569 . . . . . 6  |-  F/_ n
( ( X ^
( # `  dom  c
) )  -  sum_ m  e.  dom  c ( ( ( # `  dom  c )  _C  m
)  x.  ( ( c `  m )  /  ( ( (
# `  dom  c )  -  m )  +  1 ) ) ) )
3 oveq2 6088 . . . . . . 7  |-  ( n  =  ( # `  dom  c )  ->  ( X ^ n )  =  ( X ^ ( # `
 dom  c )
) )
4 oveq1 6087 . . . . . . . . 9  |-  ( n  =  ( # `  dom  c )  ->  (
n  _C  m )  =  ( ( # `  dom  c )  _C  m ) )
5 oveq1 6087 . . . . . . . . . . 11  |-  ( n  =  ( # `  dom  c )  ->  (
n  -  m )  =  ( ( # `  dom  c )  -  m ) )
65oveq1d 6095 . . . . . . . . . 10  |-  ( n  =  ( # `  dom  c )  ->  (
( n  -  m
)  +  1 )  =  ( ( (
# `  dom  c )  -  m )  +  1 ) )
76oveq2d 6096 . . . . . . . . 9  |-  ( n  =  ( # `  dom  c )  ->  (
( c `  m
)  /  ( ( n  -  m )  +  1 ) )  =  ( ( c `
 m )  / 
( ( ( # `  dom  c )  -  m )  +  1 ) ) )
84, 7oveq12d 6098 . . . . . . . 8  |-  ( n  =  ( # `  dom  c )  ->  (
( n  _C  m
)  x.  ( ( c `  m )  /  ( ( n  -  m )  +  1 ) ) )  =  ( ( (
# `  dom  c )  _C  m )  x.  ( ( c `  m )  /  (
( ( # `  dom  c )  -  m
)  +  1 ) ) ) )
98sumeq2sdv 13165 . . . . . . 7  |-  ( n  =  ( # `  dom  c )  ->  sum_ m  e.  dom  c ( ( n  _C  m )  x.  ( ( c `
 m )  / 
( ( n  -  m )  +  1 ) ) )  = 
sum_ m  e.  dom  c ( ( (
# `  dom  c )  _C  m )  x.  ( ( c `  m )  /  (
( ( # `  dom  c )  -  m
)  +  1 ) ) ) )
103, 9oveq12d 6098 . . . . . 6  |-  ( n  =  ( # `  dom  c )  ->  (
( X ^ n
)  -  sum_ m  e.  dom  c ( ( n  _C  m )  x.  ( ( c `
 m )  / 
( ( n  -  m )  +  1 ) ) ) )  =  ( ( X ^ ( # `  dom  c ) )  -  sum_ m  e.  dom  c
( ( ( # `  dom  c )  _C  m )  x.  (
( c `  m
)  /  ( ( ( # `  dom  c )  -  m
)  +  1 ) ) ) ) )
111, 2, 10csbief 3301 . . . . 5  |-  [_ ( # `
 dom  c )  /  n ]_ ( ( X ^ n )  -  sum_ m  e.  dom  c ( ( n  _C  m )  x.  ( ( c `  m )  /  (
( n  -  m
)  +  1 ) ) ) )  =  ( ( X ^
( # `  dom  c
) )  -  sum_ m  e.  dom  c ( ( ( # `  dom  c )  _C  m
)  x.  ( ( c `  m )  /  ( ( (
# `  dom  c )  -  m )  +  1 ) ) ) )
12 oveq2 6088 . . . . . . . . . 10  |-  ( m  =  k  ->  (
n  _C  m )  =  ( n  _C  k ) )
13 fveq2 5679 . . . . . . . . . . 11  |-  ( m  =  k  ->  (
c `  m )  =  ( c `  k ) )
14 oveq2 6088 . . . . . . . . . . . 12  |-  ( m  =  k  ->  (
n  -  m )  =  ( n  -  k ) )
1514oveq1d 6095 . . . . . . . . . . 11  |-  ( m  =  k  ->  (
( n  -  m
)  +  1 )  =  ( ( n  -  k )  +  1 ) )
1613, 15oveq12d 6098 . . . . . . . . . 10  |-  ( m  =  k  ->  (
( c `  m
)  /  ( ( n  -  m )  +  1 ) )  =  ( ( c `
 k )  / 
( ( n  -  k )  +  1 ) ) )
1712, 16oveq12d 6098 . . . . . . . . 9  |-  ( m  =  k  ->  (
( n  _C  m
)  x.  ( ( c `  m )  /  ( ( n  -  m )  +  1 ) ) )  =  ( ( n  _C  k )  x.  ( ( c `  k )  /  (
( n  -  k
)  +  1 ) ) ) )
1817cbvsumv 13157 . . . . . . . 8  |-  sum_ m  e.  dom  c ( ( n  _C  m )  x.  ( ( c `
 m )  / 
( ( n  -  m )  +  1 ) ) )  = 
sum_ k  e.  dom  c ( ( n  _C  k )  x.  ( ( c `  k )  /  (
( n  -  k
)  +  1 ) ) )
19 dmeq 5027 . . . . . . . . 9  |-  ( c  =  g  ->  dom  c  =  dom  g )
20 fveq1 5678 . . . . . . . . . . . 12  |-  ( c  =  g  ->  (
c `  k )  =  ( g `  k ) )
2120oveq1d 6095 . . . . . . . . . . 11  |-  ( c  =  g  ->  (
( c `  k
)  /  ( ( n  -  k )  +  1 ) )  =  ( ( g `
 k )  / 
( ( n  -  k )  +  1 ) ) )
2221oveq2d 6096 . . . . . . . . . 10  |-  ( c  =  g  ->  (
( n  _C  k
)  x.  ( ( c `  k )  /  ( ( n  -  k )  +  1 ) ) )  =  ( ( n  _C  k )  x.  ( ( g `  k )  /  (
( n  -  k
)  +  1 ) ) ) )
2322adantr 462 . . . . . . . . 9  |-  ( ( c  =  g  /\  k  e.  dom  c )  ->  ( ( n  _C  k )  x.  ( ( c `  k )  /  (
( n  -  k
)  +  1 ) ) )  =  ( ( n  _C  k
)  x.  ( ( g `  k )  /  ( ( n  -  k )  +  1 ) ) ) )
2419, 23sumeq12dv 13167 . . . . . . . 8  |-  ( c  =  g  ->  sum_ k  e.  dom  c ( ( n  _C  k )  x.  ( ( c `
 k )  / 
( ( n  -  k )  +  1 ) ) )  = 
sum_ k  e.  dom  g ( ( n  _C  k )  x.  ( ( g `  k )  /  (
( n  -  k
)  +  1 ) ) ) )
2518, 24syl5eq 2477 . . . . . . 7  |-  ( c  =  g  ->  sum_ m  e.  dom  c ( ( n  _C  m )  x.  ( ( c `
 m )  / 
( ( n  -  m )  +  1 ) ) )  = 
sum_ k  e.  dom  g ( ( n  _C  k )  x.  ( ( g `  k )  /  (
( n  -  k
)  +  1 ) ) ) )
2625oveq2d 6096 . . . . . 6  |-  ( c  =  g  ->  (
( X ^ n
)  -  sum_ m  e.  dom  c ( ( n  _C  m )  x.  ( ( c `
 m )  / 
( ( n  -  m )  +  1 ) ) ) )  =  ( ( X ^ n )  -  sum_ k  e.  dom  g
( ( n  _C  k )  x.  (
( g `  k
)  /  ( ( n  -  k )  +  1 ) ) ) ) )
2726csbeq2dv 3675 . . . . 5  |-  ( c  =  g  ->  [_ ( # `
 dom  c )  /  n ]_ ( ( X ^ n )  -  sum_ m  e.  dom  c ( ( n  _C  m )  x.  ( ( c `  m )  /  (
( n  -  m
)  +  1 ) ) ) )  = 
[_ ( # `  dom  c )  /  n ]_ ( ( X ^
n )  -  sum_ k  e.  dom  g ( ( n  _C  k
)  x.  ( ( g `  k )  /  ( ( n  -  k )  +  1 ) ) ) ) )
2811, 27syl5eqr 2479 . . . 4  |-  ( c  =  g  ->  (
( X ^ ( # `
 dom  c )
)  -  sum_ m  e.  dom  c ( ( ( # `  dom  c )  _C  m
)  x.  ( ( c `  m )  /  ( ( (
# `  dom  c )  -  m )  +  1 ) ) ) )  =  [_ ( # `
 dom  c )  /  n ]_ ( ( X ^ n )  -  sum_ k  e.  dom  g ( ( n  _C  k )  x.  ( ( g `  k )  /  (
( n  -  k
)  +  1 ) ) ) ) )
2919fveq2d 5683 . . . . 5  |-  ( c  =  g  ->  ( # `
 dom  c )  =  ( # `  dom  g ) )
3029csbeq1d 3283 . . . 4  |-  ( c  =  g  ->  [_ ( # `
 dom  c )  /  n ]_ ( ( X ^ n )  -  sum_ k  e.  dom  g ( ( n  _C  k )  x.  ( ( g `  k )  /  (
( n  -  k
)  +  1 ) ) ) )  = 
[_ ( # `  dom  g )  /  n ]_ ( ( X ^
n )  -  sum_ k  e.  dom  g ( ( n  _C  k
)  x.  ( ( g `  k )  /  ( ( n  -  k )  +  1 ) ) ) ) )
3128, 30eqtrd 2465 . . 3  |-  ( c  =  g  ->  (
( X ^ ( # `
 dom  c )
)  -  sum_ m  e.  dom  c ( ( ( # `  dom  c )  _C  m
)  x.  ( ( c `  m )  /  ( ( (
# `  dom  c )  -  m )  +  1 ) ) ) )  =  [_ ( # `
 dom  g )  /  n ]_ ( ( X ^ n )  -  sum_ k  e.  dom  g ( ( n  _C  k )  x.  ( ( g `  k )  /  (
( n  -  k
)  +  1 ) ) ) ) )
3231cbvmptv 4371 . 2  |-  ( c  e.  _V  |->  ( ( X ^ ( # `  dom  c ) )  -  sum_ m  e.  dom  c ( ( (
# `  dom  c )  _C  m )  x.  ( ( c `  m )  /  (
( ( # `  dom  c )  -  m
)  +  1 ) ) ) ) )  =  ( g  e. 
_V  |->  [_ ( # `  dom  g )  /  n ]_ ( ( X ^
n )  -  sum_ k  e.  dom  g ( ( n  _C  k
)  x.  ( ( g `  k )  /  ( ( n  -  k )  +  1 ) ) ) ) )
33 eqid 2433 . 2  |- wrecs (  <  ,  NN0 ,  ( c  e.  _V  |->  ( ( X ^ ( # `  dom  c ) )  -  sum_ m  e.  dom  c ( ( (
# `  dom  c )  _C  m )  x.  ( ( c `  m )  /  (
( ( # `  dom  c )  -  m
)  +  1 ) ) ) ) ) )  = wrecs (  <  ,  NN0 ,  ( c  e.  _V  |->  ( ( X ^ ( # `  dom  c ) )  -  sum_ m  e.  dom  c ( ( (
# `  dom  c )  _C  m )  x.  ( ( c `  m )  /  (
( ( # `  dom  c )  -  m
)  +  1 ) ) ) ) ) )
3432, 33bpolylem 28038 1  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( N BernPoly  X )  =  ( ( X ^ N )  -  sum_ k  e.  ( 0 ... ( N  - 
1 ) ) ( ( N  _C  k
)  x.  ( ( k BernPoly  X )  /  (
( N  -  k
)  +  1 ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1362    e. wcel 1755   _Vcvv 2962   [_csb 3276    e. cmpt 4338   dom cdm 4827   ` cfv 5406  (class class class)co 6080   CCcc 9268   0cc0 9270   1c1 9271    + caddc 9273    x. cmul 9275    < clt 9406    - cmin 9583    / cdiv 9981   NN0cn0 10567   ...cfz 11424   ^cexp 11849    _C cbc 12062   #chash 12087   sum_csu 13147  wrecscwrecs 27563   BernPoly cbp 28036
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-inf2 7835  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-fal 1368  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-se 4667  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-isom 5415  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-om 6466  df-1st 6566  df-2nd 6567  df-recs 6818  df-rdg 6852  df-1o 6908  df-er 7089  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-card 8097  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-nn 10311  df-n0 10568  df-z 10635  df-uz 10850  df-fz 11425  df-seq 11791  df-hash 12088  df-sum 13148  df-pred 27472  df-wrecs 27564  df-bpoly 28037
This theorem is referenced by:  bpoly0  28040  bpoly1  28041  bpolycl  28042  bpolysum  28043  bpolydiflem  28044  bpoly2  28047  bpoly3  28048  bpoly4  28049
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