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Theorem bpolyval 13992
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 5858 . . . . . 6  |-  ( # `  dom  c )  e. 
_V
2 oveq2 6285 . . . . . . 7  |-  ( n  =  ( # `  dom  c )  ->  ( X ^ n )  =  ( X ^ ( # `
 dom  c )
) )
3 oveq1 6284 . . . . . . . . 9  |-  ( n  =  ( # `  dom  c )  ->  (
n  _C  m )  =  ( ( # `  dom  c )  _C  m ) )
4 oveq1 6284 . . . . . . . . . . 11  |-  ( n  =  ( # `  dom  c )  ->  (
n  -  m )  =  ( ( # `  dom  c )  -  m ) )
54oveq1d 6292 . . . . . . . . . 10  |-  ( n  =  ( # `  dom  c )  ->  (
( n  -  m
)  +  1 )  =  ( ( (
# `  dom  c )  -  m )  +  1 ) )
65oveq2d 6293 . . . . . . . . 9  |-  ( n  =  ( # `  dom  c )  ->  (
( c `  m
)  /  ( ( n  -  m )  +  1 ) )  =  ( ( c `
 m )  / 
( ( ( # `  dom  c )  -  m )  +  1 ) ) )
73, 6oveq12d 6295 . . . . . . . 8  |-  ( n  =  ( # `  dom  c )  ->  (
( n  _C  m
)  x.  ( ( c `  m )  /  ( ( n  -  m )  +  1 ) ) )  =  ( ( (
# `  dom  c )  _C  m )  x.  ( ( c `  m )  /  (
( ( # `  dom  c )  -  m
)  +  1 ) ) ) )
87sumeq2sdv 13673 . . . . . . 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 ) ) ) )
92, 8oveq12d 6295 . . . . . 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 ) ) ) ) )
101, 9csbie 3398 . . . . 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 ) ) ) )
11 oveq2 6285 . . . . . . . . . 10  |-  ( m  =  k  ->  (
n  _C  m )  =  ( n  _C  k ) )
12 fveq2 5848 . . . . . . . . . . 11  |-  ( m  =  k  ->  (
c `  m )  =  ( c `  k ) )
13 oveq2 6285 . . . . . . . . . . . 12  |-  ( m  =  k  ->  (
n  -  m )  =  ( n  -  k ) )
1413oveq1d 6292 . . . . . . . . . . 11  |-  ( m  =  k  ->  (
( n  -  m
)  +  1 )  =  ( ( n  -  k )  +  1 ) )
1512, 14oveq12d 6295 . . . . . . . . . 10  |-  ( m  =  k  ->  (
( c `  m
)  /  ( ( n  -  m )  +  1 ) )  =  ( ( c `
 k )  / 
( ( n  -  k )  +  1 ) ) )
1611, 15oveq12d 6295 . . . . . . . . 9  |-  ( m  =  k  ->  (
( n  _C  m
)  x.  ( ( c `  m )  /  ( ( n  -  m )  +  1 ) ) )  =  ( ( n  _C  k )  x.  ( ( c `  k )  /  (
( n  -  k
)  +  1 ) ) ) )
1716cbvsumv 13665 . . . . . . . 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 ) ) )
18 dmeq 5023 . . . . . . . . 9  |-  ( c  =  g  ->  dom  c  =  dom  g )
19 fveq1 5847 . . . . . . . . . . . 12  |-  ( c  =  g  ->  (
c `  k )  =  ( g `  k ) )
2019oveq1d 6292 . . . . . . . . . . 11  |-  ( c  =  g  ->  (
( c `  k
)  /  ( ( n  -  k )  +  1 ) )  =  ( ( g `
 k )  / 
( ( n  -  k )  +  1 ) ) )
2120oveq2d 6293 . . . . . . . . . 10  |-  ( c  =  g  ->  (
( n  _C  k
)  x.  ( ( c `  k )  /  ( ( n  -  k )  +  1 ) ) )  =  ( ( n  _C  k )  x.  ( ( g `  k )  /  (
( n  -  k
)  +  1 ) ) ) )
2221adantr 463 . . . . . . . . 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 ) ) ) )
2318, 22sumeq12dv 13675 . . . . . . . 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 ) ) ) )
2417, 23syl5eq 2455 . . . . . . 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 ) ) ) )
2524oveq2d 6293 . . . . . 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 ) ) ) ) )
2625csbeq2dv 3785 . . . . 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 ) ) ) ) )
2710, 26syl5eqr 2457 . . . 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 ) ) ) ) )
2818fveq2d 5852 . . . . 5  |-  ( c  =  g  ->  ( # `
 dom  c )  =  ( # `  dom  g ) )
2928csbeq1d 3379 . . . 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 ) ) ) ) )
3027, 29eqtrd 2443 . . 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 ) ) ) ) )
3130cbvmptv 4486 . 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 ) ) ) ) )
32 eqid 2402 . 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 ) ) ) ) ) )
3331, 32bpolylem 13991 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 367    = wceq 1405    e. wcel 1842   _Vcvv 3058   [_csb 3372    |-> cmpt 4452   dom cdm 4822   ` cfv 5568  (class class class)co 6277  wrecscwrecs 7011   CCcc 9519   0cc0 9521   1c1 9522    + caddc 9524    x. cmul 9526    < clt 9657    - cmin 9840    / cdiv 10246   NN0cn0 10835   ...cfz 11724   ^cexp 12208    _C cbc 12422   #chash 12450   sum_csu 13655   BernPoly cbp 13989
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-inf2 8090  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-se 4782  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-isom 5577  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6683  df-1st 6783  df-2nd 6784  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-1o 7166  df-er 7347  df-en 7554  df-dom 7555  df-sdom 7556  df-fin 7557  df-card 8351  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-nn 10576  df-n0 10836  df-z 10905  df-uz 11127  df-fz 11725  df-seq 12150  df-hash 12451  df-sum 13656  df-bpoly 13990
This theorem is referenced by:  bpoly0  13993  bpoly1  13994  bpolycl  13995  bpolysum  13996  bpolydiflem  13997  bpoly2  14000  bpoly3  14001  bpoly4  14002
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