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Theorem bpolyval 26095
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 5742 . . . . . 6  |-  ( # `  dom  c )  e. 
_V
2 nfcv 2572 . . . . . 6  |-  F/_ n
( ( X ^
( # `  dom  c
) )  -  sum_ m  e.  dom  c ( ( ( # `  dom  c )  _C  m
)  x.  ( ( c `  m )  /  ( ( (
# `  dom  c )  -  m )  +  1 ) ) ) )
3 oveq2 6089 . . . . . . 7  |-  ( n  =  ( # `  dom  c )  ->  ( X ^ n )  =  ( X ^ ( # `
 dom  c )
) )
4 oveq1 6088 . . . . . . . . 9  |-  ( n  =  ( # `  dom  c )  ->  (
n  _C  m )  =  ( ( # `  dom  c )  _C  m ) )
5 oveq1 6088 . . . . . . . . . . 11  |-  ( n  =  ( # `  dom  c )  ->  (
n  -  m )  =  ( ( # `  dom  c )  -  m ) )
65oveq1d 6096 . . . . . . . . . 10  |-  ( n  =  ( # `  dom  c )  ->  (
( n  -  m
)  +  1 )  =  ( ( (
# `  dom  c )  -  m )  +  1 ) )
76oveq2d 6097 . . . . . . . . 9  |-  ( n  =  ( # `  dom  c )  ->  (
( c `  m
)  /  ( ( n  -  m )  +  1 ) )  =  ( ( c `
 m )  / 
( ( ( # `  dom  c )  -  m )  +  1 ) ) )
84, 7oveq12d 6099 . . . . . . . 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 12498 . . . . . . 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 6099 . . . . . 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 3292 . . . . 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 6089 . . . . . . . . . 10  |-  ( m  =  k  ->  (
n  _C  m )  =  ( n  _C  k ) )
13 fveq2 5728 . . . . . . . . . . 11  |-  ( m  =  k  ->  (
c `  m )  =  ( c `  k ) )
14 oveq2 6089 . . . . . . . . . . . 12  |-  ( m  =  k  ->  (
n  -  m )  =  ( n  -  k ) )
1514oveq1d 6096 . . . . . . . . . . 11  |-  ( m  =  k  ->  (
( n  -  m
)  +  1 )  =  ( ( n  -  k )  +  1 ) )
1613, 15oveq12d 6099 . . . . . . . . . 10  |-  ( m  =  k  ->  (
( c `  m
)  /  ( ( n  -  m )  +  1 ) )  =  ( ( c `
 k )  / 
( ( n  -  k )  +  1 ) ) )
1712, 16oveq12d 6099 . . . . . . . . 9  |-  ( m  =  k  ->  (
( n  _C  m
)  x.  ( ( c `  m )  /  ( ( n  -  m )  +  1 ) ) )  =  ( ( n  _C  k )  x.  ( ( c `  k )  /  (
( n  -  k
)  +  1 ) ) ) )
1817cbvsumv 12490 . . . . . . . 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 5070 . . . . . . . . 9  |-  ( c  =  g  ->  dom  c  =  dom  g )
20 fveq1 5727 . . . . . . . . . . . 12  |-  ( c  =  g  ->  (
c `  k )  =  ( g `  k ) )
2120oveq1d 6096 . . . . . . . . . . 11  |-  ( c  =  g  ->  (
( c `  k
)  /  ( ( n  -  k )  +  1 ) )  =  ( ( g `
 k )  / 
( ( n  -  k )  +  1 ) ) )
2221oveq2d 6097 . . . . . . . . . 10  |-  ( c  =  g  ->  (
( n  _C  k
)  x.  ( ( c `  k )  /  ( ( n  -  k )  +  1 ) ) )  =  ( ( n  _C  k )  x.  ( ( g `  k )  /  (
( n  -  k
)  +  1 ) ) ) )
2322adantr 452 . . . . . . . . 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 12500 . . . . . . . 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 2480 . . . . . . 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 6097 . . . . . 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 3276 . . . . 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 2482 . . . 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 5732 . . . . 5  |-  ( c  =  g  ->  ( # `
 dom  c )  =  ( # `  dom  g ) )
3029csbeq1d 3257 . . . 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 2468 . . 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 4300 . 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 2436 . 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 26094 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 set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2956   [_csb 3251    e. cmpt 4266   dom cdm 4878   ` cfv 5454  (class class class)co 6081   CCcc 8988   0cc0 8990   1c1 8991    + caddc 8993    x. cmul 8995    < clt 9120    - cmin 9291    / cdiv 9677   NN0cn0 10221   ...cfz 11043   ^cexp 11382    _C cbc 11593   #chash 11618   sum_csu 12479  wrecscwrecs 25530   BernPoly cbp 26092
This theorem is referenced by:  bpoly0  26096  bpoly1  26097  bpolycl  26098  bpolysum  26099  bpolydiflem  26100  bpoly2  26103  bpoly3  26104  bpoly4  26105
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-card 7826  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-n0 10222  df-z 10283  df-uz 10489  df-fz 11044  df-seq 11324  df-hash 11619  df-sum 12480  df-pred 25439  df-wrecs 25531  df-bpoly 26093
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