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Theorem bpoly1 29390
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 10807 . . 3  |-  1  e.  NN0
2 bpolyval 29388 . . 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 12136 . . 3  |-  ( X  e.  CC  ->  ( X ^ 1 )  =  X )
5 1m1e0 10600 . . . . . 6  |-  ( 1  -  1 )  =  0
65oveq2i 6293 . . . . 5  |-  ( 0 ... ( 1  -  1 ) )  =  ( 0 ... 0
)
76sumeq1i 13479 . . . 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 10871 . . . . . 6  |-  0  e.  ZZ
9 bpoly0 29389 . . . . . . . . . 10  |-  ( X  e.  CC  ->  (
0 BernPoly  X )  =  1 )
109oveq1d 6297 . . . . . . . . 9  |-  ( X  e.  CC  ->  (
( 0 BernPoly  X )  /  2 )  =  ( 1  /  2
) )
1110oveq2d 6298 . . . . . . . 8  |-  ( X  e.  CC  ->  (
1  x.  ( ( 0 BernPoly  X )  /  2
) )  =  ( 1  x.  ( 1  /  2 ) ) )
12 halfcn 10751 . . . . . . . . 9  |-  ( 1  /  2 )  e.  CC
1312mulid2i 9595 . . . . . . . 8  |-  ( 1  x.  ( 1  / 
2 ) )  =  ( 1  /  2
)
1411, 13syl6eq 2524 . . . . . . 7  |-  ( X  e.  CC  ->  (
1  x.  ( ( 0 BernPoly  X )  /  2
) )  =  ( 1  /  2 ) )
1514, 12syl6eqel 2563 . . . . . 6  |-  ( X  e.  CC  ->  (
1  x.  ( ( 0 BernPoly  X )  /  2
) )  e.  CC )
16 oveq2 6290 . . . . . . . . 9  |-  ( k  =  0  ->  (
1  _C  k )  =  ( 1  _C  0 ) )
17 bcn0 12352 . . . . . . . . . 10  |-  ( 1  e.  NN0  ->  ( 1  _C  0 )  =  1 )
181, 17ax-mp 5 . . . . . . . . 9  |-  ( 1  _C  0 )  =  1
1916, 18syl6eq 2524 . . . . . . . 8  |-  ( k  =  0  ->  (
1  _C  k )  =  1 )
20 oveq1 6289 . . . . . . . . 9  |-  ( k  =  0  ->  (
k BernPoly  X )  =  ( 0 BernPoly  X ) )
21 oveq2 6290 . . . . . . . . . . . 12  |-  ( k  =  0  ->  (
1  -  k )  =  ( 1  -  0 ) )
22 1m0e1 10642 . . . . . . . . . . . 12  |-  ( 1  -  0 )  =  1
2321, 22syl6eq 2524 . . . . . . . . . . 11  |-  ( k  =  0  ->  (
1  -  k )  =  1 )
2423oveq1d 6297 . . . . . . . . . 10  |-  ( k  =  0  ->  (
( 1  -  k
)  +  1 )  =  ( 1  +  1 ) )
25 df-2 10590 . . . . . . . . . 10  |-  2  =  ( 1  +  1 )
2624, 25syl6eqr 2526 . . . . . . . . 9  |-  ( k  =  0  ->  (
( 1  -  k
)  +  1 )  =  2 )
2720, 26oveq12d 6300 . . . . . . . 8  |-  ( k  =  0  ->  (
( k BernPoly  X )  /  ( ( 1  -  k )  +  1 ) )  =  ( ( 0 BernPoly  X
)  /  2 ) )
2819, 27oveq12d 6300 . . . . . . 7  |-  ( k  =  0  ->  (
( 1  _C  k
)  x.  ( ( k BernPoly  X )  /  (
( 1  -  k
)  +  1 ) ) )  =  ( 1  x.  ( ( 0 BernPoly  X )  /  2
) ) )
2928fsum1 13523 . . . . . 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 2508 . . . 4  |-  ( X  e.  CC  ->  sum_ k  e.  ( 0 ... 0
) ( ( 1  _C  k )  x.  ( ( k BernPoly  X
)  /  ( ( 1  -  k )  +  1 ) ) )  =  ( 1  /  2 ) )
327, 31syl5eq 2520 . . 3  |-  ( X  e.  CC  ->  sum_ k  e.  ( 0 ... (
1  -  1 ) ) ( ( 1  _C  k )  x.  ( ( k BernPoly  X
)  /  ( ( 1  -  k )  +  1 ) ) )  =  ( 1  /  2 ) )
334, 32oveq12d 6300 . 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 2508 1  |-  ( X  e.  CC  ->  (
1 BernPoly  X )  =  ( X  -  ( 1  /  2 ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767  (class class class)co 6282   CCcc 9486   0cc0 9488   1c1 9489    + caddc 9491    x. cmul 9493    - cmin 9801    / cdiv 10202   2c2 10581   NN0cn0 10791   ZZcz 10860   ...cfz 11668   ^cexp 12130    _C cbc 12344   sum_csu 13467   BernPoly cbp 29385
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-sup 7897  df-oi 7931  df-card 8316  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11079  df-rp 11217  df-fz 11669  df-fzo 11789  df-seq 12072  df-exp 12131  df-fac 12318  df-bc 12345  df-hash 12370  df-cj 12891  df-re 12892  df-im 12893  df-sqrt 13027  df-abs 13028  df-clim 13270  df-sum 13468  df-pred 28821  df-wrecs 28913  df-bpoly 29386
This theorem is referenced by:  bpoly2  29396  bpoly3  29397  bpoly4  29398
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