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Theorem bpoly1 13998
Description: The value of the Bernoulli polynomials at one. (Contributed by Scott Fenton, 16-May-2014.)
Assertion
Ref Expression
bpoly1  |-  ( X  e.  CC  ->  (
1 BernPoly  X )  =  ( X  -  ( 1  /  2 ) ) )

Proof of Theorem bpoly1
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 1nn0 10854 . . 3  |-  1  e.  NN0
2 bpolyval 13996 . . 3  |-  ( ( 1  e.  NN0  /\  X  e.  CC )  ->  ( 1 BernPoly  X )  =  ( ( X ^ 1 )  -  sum_ k  e.  ( 0 ... ( 1  -  1 ) ) ( ( 1  _C  k
)  x.  ( ( k BernPoly  X )  /  (
( 1  -  k
)  +  1 ) ) ) ) )
31, 2mpan 670 . 2  |-  ( X  e.  CC  ->  (
1 BernPoly  X )  =  ( ( X ^ 1 )  -  sum_ k  e.  ( 0 ... (
1  -  1 ) ) ( ( 1  _C  k )  x.  ( ( k BernPoly  X
)  /  ( ( 1  -  k )  +  1 ) ) ) ) )
4 exp1 12218 . . 3  |-  ( X  e.  CC  ->  ( X ^ 1 )  =  X )
5 1m1e0 10647 . . . . . 6  |-  ( 1  -  1 )  =  0
65oveq2i 6291 . . . . 5  |-  ( 0 ... ( 1  -  1 ) )  =  ( 0 ... 0
)
76sumeq1i 13671 . . . 4  |-  sum_ k  e.  ( 0 ... (
1  -  1 ) ) ( ( 1  _C  k )  x.  ( ( k BernPoly  X
)  /  ( ( 1  -  k )  +  1 ) ) )  =  sum_ k  e.  ( 0 ... 0
) ( ( 1  _C  k )  x.  ( ( k BernPoly  X
)  /  ( ( 1  -  k )  +  1 ) ) )
8 0z 10918 . . . . . 6  |-  0  e.  ZZ
9 bpoly0 13997 . . . . . . . . . 10  |-  ( X  e.  CC  ->  (
0 BernPoly  X )  =  1 )
109oveq1d 6295 . . . . . . . . 9  |-  ( X  e.  CC  ->  (
( 0 BernPoly  X )  /  2 )  =  ( 1  /  2
) )
1110oveq2d 6296 . . . . . . . 8  |-  ( X  e.  CC  ->  (
1  x.  ( ( 0 BernPoly  X )  /  2
) )  =  ( 1  x.  ( 1  /  2 ) ) )
12 halfcn 10798 . . . . . . . . 9  |-  ( 1  /  2 )  e.  CC
1312mulid2i 9631 . . . . . . . 8  |-  ( 1  x.  ( 1  / 
2 ) )  =  ( 1  /  2
)
1411, 13syl6eq 2461 . . . . . . 7  |-  ( X  e.  CC  ->  (
1  x.  ( ( 0 BernPoly  X )  /  2
) )  =  ( 1  /  2 ) )
1514, 12syl6eqel 2500 . . . . . 6  |-  ( X  e.  CC  ->  (
1  x.  ( ( 0 BernPoly  X )  /  2
) )  e.  CC )
16 oveq2 6288 . . . . . . . . 9  |-  ( k  =  0  ->  (
1  _C  k )  =  ( 1  _C  0 ) )
17 bcn0 12434 . . . . . . . . . 10  |-  ( 1  e.  NN0  ->  ( 1  _C  0 )  =  1 )
181, 17ax-mp 5 . . . . . . . . 9  |-  ( 1  _C  0 )  =  1
1916, 18syl6eq 2461 . . . . . . . 8  |-  ( k  =  0  ->  (
1  _C  k )  =  1 )
20 oveq1 6287 . . . . . . . . 9  |-  ( k  =  0  ->  (
k BernPoly  X )  =  ( 0 BernPoly  X ) )
21 oveq2 6288 . . . . . . . . . . . 12  |-  ( k  =  0  ->  (
1  -  k )  =  ( 1  -  0 ) )
22 1m0e1 10689 . . . . . . . . . . . 12  |-  ( 1  -  0 )  =  1
2321, 22syl6eq 2461 . . . . . . . . . . 11  |-  ( k  =  0  ->  (
1  -  k )  =  1 )
2423oveq1d 6295 . . . . . . . . . 10  |-  ( k  =  0  ->  (
( 1  -  k
)  +  1 )  =  ( 1  +  1 ) )
25 df-2 10637 . . . . . . . . . 10  |-  2  =  ( 1  +  1 )
2624, 25syl6eqr 2463 . . . . . . . . 9  |-  ( k  =  0  ->  (
( 1  -  k
)  +  1 )  =  2 )
2720, 26oveq12d 6298 . . . . . . . 8  |-  ( k  =  0  ->  (
( k BernPoly  X )  /  ( ( 1  -  k )  +  1 ) )  =  ( ( 0 BernPoly  X
)  /  2 ) )
2819, 27oveq12d 6298 . . . . . . 7  |-  ( k  =  0  ->  (
( 1  _C  k
)  x.  ( ( k BernPoly  X )  /  (
( 1  -  k
)  +  1 ) ) )  =  ( 1  x.  ( ( 0 BernPoly  X )  /  2
) ) )
2928fsum1 13715 . . . . . 6  |-  ( ( 0  e.  ZZ  /\  ( 1  x.  (
( 0 BernPoly  X )  /  2 ) )  e.  CC )  ->  sum_ k  e.  ( 0 ... 0 ) ( ( 1  _C  k
)  x.  ( ( k BernPoly  X )  /  (
( 1  -  k
)  +  1 ) ) )  =  ( 1  x.  ( ( 0 BernPoly  X )  /  2
) ) )
308, 15, 29sylancr 663 . . . . 5  |-  ( X  e.  CC  ->  sum_ k  e.  ( 0 ... 0
) ( ( 1  _C  k )  x.  ( ( k BernPoly  X
)  /  ( ( 1  -  k )  +  1 ) ) )  =  ( 1  x.  ( ( 0 BernPoly  X )  /  2
) ) )
3130, 14eqtrd 2445 . . . 4  |-  ( X  e.  CC  ->  sum_ k  e.  ( 0 ... 0
) ( ( 1  _C  k )  x.  ( ( k BernPoly  X
)  /  ( ( 1  -  k )  +  1 ) ) )  =  ( 1  /  2 ) )
327, 31syl5eq 2457 . . 3  |-  ( X  e.  CC  ->  sum_ k  e.  ( 0 ... (
1  -  1 ) ) ( ( 1  _C  k )  x.  ( ( k BernPoly  X
)  /  ( ( 1  -  k )  +  1 ) ) )  =  ( 1  /  2 ) )
334, 32oveq12d 6298 . 2  |-  ( X  e.  CC  ->  (
( X ^ 1 )  -  sum_ k  e.  ( 0 ... (
1  -  1 ) ) ( ( 1  _C  k )  x.  ( ( k BernPoly  X
)  /  ( ( 1  -  k )  +  1 ) ) ) )  =  ( X  -  ( 1  /  2 ) ) )
343, 33eqtrd 2445 1  |-  ( X  e.  CC  ->  (
1 BernPoly  X )  =  ( X  -  ( 1  /  2 ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1407    e. wcel 1844  (class class class)co 6280   CCcc 9522   0cc0 9524   1c1 9525    + caddc 9527    x. cmul 9529    - cmin 9843    / cdiv 10249   2c2 10628   NN0cn0 10838   ZZcz 10907   ...cfz 11728   ^cexp 12212    _C cbc 12426   sum_csu 13659   BernPoly cbp 13993
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-inf2 8093  ax-cnex 9580  ax-resscn 9581  ax-1cn 9582  ax-icn 9583  ax-addcl 9584  ax-addrcl 9585  ax-mulcl 9586  ax-mulrcl 9587  ax-mulcom 9588  ax-addass 9589  ax-mulass 9590  ax-distr 9591  ax-i2m1 9592  ax-1ne0 9593  ax-1rid 9594  ax-rnegex 9595  ax-rrecex 9596  ax-cnre 9597  ax-pre-lttri 9598  ax-pre-lttrn 9599  ax-pre-ltadd 9600  ax-pre-mulgt0 9601  ax-pre-sup 9602
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-fal 1413  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-nel 2603  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-int 4230  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-se 4785  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-pred 5369  df-ord 5415  df-on 5416  df-lim 5417  df-suc 5418  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-isom 5580  df-riota 6242  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-om 6686  df-1st 6786  df-2nd 6787  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-1o 7169  df-oadd 7173  df-er 7350  df-en 7557  df-dom 7558  df-sdom 7559  df-fin 7560  df-sup 7937  df-oi 7971  df-card 8354  df-pnf 9662  df-mnf 9663  df-xr 9664  df-ltxr 9665  df-le 9666  df-sub 9845  df-neg 9846  df-div 10250  df-nn 10579  df-2 10637  df-3 10638  df-n0 10839  df-z 10908  df-uz 11130  df-rp 11268  df-fz 11729  df-fzo 11857  df-seq 12154  df-exp 12213  df-fac 12400  df-bc 12427  df-hash 12455  df-cj 13083  df-re 13084  df-im 13085  df-sqrt 13219  df-abs 13220  df-clim 13462  df-sum 13660  df-bpoly 13994
This theorem is referenced by:  bpoly2  14004  bpoly3  14005  bpoly4  14006
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