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Theorem bpoly1 26097
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 10237 . . 3  |-  1  e.  NN0
2 bpolyval 26095 . . 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 652 . 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 11387 . . 3  |-  ( X  e.  CC  ->  ( X ^ 1 )  =  X )
5 1m1e0 10068 . . . . . 6  |-  ( 1  -  1 )  =  0
65oveq2i 6092 . . . . 5  |-  ( 0 ... ( 1  -  1 ) )  =  ( 0 ... 0
)
76sumeq1i 12492 . . . 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 10293 . . . . . 6  |-  0  e.  ZZ
9 bpoly0 26096 . . . . . . . . . 10  |-  ( X  e.  CC  ->  (
0 BernPoly  X )  =  1 )
109oveq1d 6096 . . . . . . . . 9  |-  ( X  e.  CC  ->  (
( 0 BernPoly  X )  /  2 )  =  ( 1  /  2
) )
1110oveq2d 6097 . . . . . . . 8  |-  ( X  e.  CC  ->  (
1  x.  ( ( 0 BernPoly  X )  /  2
) )  =  ( 1  x.  ( 1  /  2 ) ) )
12 ax-1cn 9048 . . . . . . . . . 10  |-  1  e.  CC
13 halfcl 10193 . . . . . . . . . 10  |-  ( 1  e.  CC  ->  (
1  /  2 )  e.  CC )
1412, 13ax-mp 8 . . . . . . . . 9  |-  ( 1  /  2 )  e.  CC
1514mulid2i 9093 . . . . . . . 8  |-  ( 1  x.  ( 1  / 
2 ) )  =  ( 1  /  2
)
1611, 15syl6eq 2484 . . . . . . 7  |-  ( X  e.  CC  ->  (
1  x.  ( ( 0 BernPoly  X )  /  2
) )  =  ( 1  /  2 ) )
1716, 14syl6eqel 2524 . . . . . 6  |-  ( X  e.  CC  ->  (
1  x.  ( ( 0 BernPoly  X )  /  2
) )  e.  CC )
18 oveq2 6089 . . . . . . . . 9  |-  ( k  =  0  ->  (
1  _C  k )  =  ( 1  _C  0 ) )
19 bcn0 11601 . . . . . . . . . 10  |-  ( 1  e.  NN0  ->  ( 1  _C  0 )  =  1 )
201, 19ax-mp 8 . . . . . . . . 9  |-  ( 1  _C  0 )  =  1
2118, 20syl6eq 2484 . . . . . . . 8  |-  ( k  =  0  ->  (
1  _C  k )  =  1 )
22 oveq1 6088 . . . . . . . . 9  |-  ( k  =  0  ->  (
k BernPoly  X )  =  ( 0 BernPoly  X ) )
23 oveq2 6089 . . . . . . . . . . . 12  |-  ( k  =  0  ->  (
1  -  k )  =  ( 1  -  0 ) )
2412subid1i 9372 . . . . . . . . . . . 12  |-  ( 1  -  0 )  =  1
2523, 24syl6eq 2484 . . . . . . . . . . 11  |-  ( k  =  0  ->  (
1  -  k )  =  1 )
2625oveq1d 6096 . . . . . . . . . 10  |-  ( k  =  0  ->  (
( 1  -  k
)  +  1 )  =  ( 1  +  1 ) )
27 df-2 10058 . . . . . . . . . 10  |-  2  =  ( 1  +  1 )
2826, 27syl6eqr 2486 . . . . . . . . 9  |-  ( k  =  0  ->  (
( 1  -  k
)  +  1 )  =  2 )
2922, 28oveq12d 6099 . . . . . . . 8  |-  ( k  =  0  ->  (
( k BernPoly  X )  /  ( ( 1  -  k )  +  1 ) )  =  ( ( 0 BernPoly  X
)  /  2 ) )
3021, 29oveq12d 6099 . . . . . . 7  |-  ( k  =  0  ->  (
( 1  _C  k
)  x.  ( ( k BernPoly  X )  /  (
( 1  -  k
)  +  1 ) ) )  =  ( 1  x.  ( ( 0 BernPoly  X )  /  2
) ) )
3130fsum1 12535 . . . . . 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
) ) )
328, 17, 31sylancr 645 . . . . 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
) ) )
3332, 16eqtrd 2468 . . . 4  |-  ( X  e.  CC  ->  sum_ k  e.  ( 0 ... 0
) ( ( 1  _C  k )  x.  ( ( k BernPoly  X
)  /  ( ( 1  -  k )  +  1 ) ) )  =  ( 1  /  2 ) )
347, 33syl5eq 2480 . . 3  |-  ( X  e.  CC  ->  sum_ k  e.  ( 0 ... (
1  -  1 ) ) ( ( 1  _C  k )  x.  ( ( k BernPoly  X
)  /  ( ( 1  -  k )  +  1 ) ) )  =  ( 1  /  2 ) )
354, 34oveq12d 6099 . 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 ) ) )
363, 35eqtrd 2468 1  |-  ( X  e.  CC  ->  (
1 BernPoly  X )  =  ( X  -  ( 1  /  2 ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725  (class class class)co 6081   CCcc 8988   0cc0 8990   1c1 8991    + caddc 8993    x. cmul 8995    - cmin 9291    / cdiv 9677   2c2 10049   NN0cn0 10221   ZZcz 10282   ...cfz 11043   ^cexp 11382    _C cbc 11593   sum_csu 12479   BernPoly cbp 26092
This theorem is referenced by:  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  ax-pre-sup 9068
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-oadd 6728  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-oi 7479  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-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-rp 10613  df-fz 11044  df-fzo 11136  df-seq 11324  df-exp 11383  df-fac 11567  df-bc 11594  df-hash 11619  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-clim 12282  df-sum 12480  df-pred 25439  df-wrecs 25531  df-bpoly 26093
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