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Theorem bpoly1 25172
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 10028 . . 3  |-  1  e.  NN0
2 bpolyval 25170 . . 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 651 . 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 11156 . . 3  |-  ( X  e.  CC  ->  ( X ^ 1 )  =  X )
5 1m1e0 9859 . . . . . 6  |-  ( 1  -  1 )  =  0
65oveq2i 5911 . . . . 5  |-  ( 0 ... ( 1  -  1 ) )  =  ( 0 ... 0
)
76sumeq1i 12218 . . . 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 10082 . . . . . 6  |-  0  e.  ZZ
9 bpoly0 25171 . . . . . . . . . 10  |-  ( X  e.  CC  ->  (
0 BernPoly  X )  =  1 )
109oveq1d 5915 . . . . . . . . 9  |-  ( X  e.  CC  ->  (
( 0 BernPoly  X )  /  2 )  =  ( 1  /  2
) )
1110oveq2d 5916 . . . . . . . 8  |-  ( X  e.  CC  ->  (
1  x.  ( ( 0 BernPoly  X )  /  2
) )  =  ( 1  x.  ( 1  /  2 ) ) )
12 ax-1cn 8840 . . . . . . . . . 10  |-  1  e.  CC
13 halfcl 9984 . . . . . . . . . 10  |-  ( 1  e.  CC  ->  (
1  /  2 )  e.  CC )
1412, 13ax-mp 8 . . . . . . . . 9  |-  ( 1  /  2 )  e.  CC
1514mulid2i 8885 . . . . . . . 8  |-  ( 1  x.  ( 1  / 
2 ) )  =  ( 1  /  2
)
1611, 15syl6eq 2364 . . . . . . 7  |-  ( X  e.  CC  ->  (
1  x.  ( ( 0 BernPoly  X )  /  2
) )  =  ( 1  /  2 ) )
1716, 14syl6eqel 2404 . . . . . 6  |-  ( X  e.  CC  ->  (
1  x.  ( ( 0 BernPoly  X )  /  2
) )  e.  CC )
18 oveq2 5908 . . . . . . . . 9  |-  ( k  =  0  ->  (
1  _C  k )  =  ( 1  _C  0 ) )
19 bcn0 11370 . . . . . . . . . 10  |-  ( 1  e.  NN0  ->  ( 1  _C  0 )  =  1 )
201, 19ax-mp 8 . . . . . . . . 9  |-  ( 1  _C  0 )  =  1
2118, 20syl6eq 2364 . . . . . . . 8  |-  ( k  =  0  ->  (
1  _C  k )  =  1 )
22 oveq1 5907 . . . . . . . . 9  |-  ( k  =  0  ->  (
k BernPoly  X )  =  ( 0 BernPoly  X ) )
23 oveq2 5908 . . . . . . . . . . . 12  |-  ( k  =  0  ->  (
1  -  k )  =  ( 1  -  0 ) )
2412subid1i 9163 . . . . . . . . . . . 12  |-  ( 1  -  0 )  =  1
2523, 24syl6eq 2364 . . . . . . . . . . 11  |-  ( k  =  0  ->  (
1  -  k )  =  1 )
2625oveq1d 5915 . . . . . . . . . 10  |-  ( k  =  0  ->  (
( 1  -  k
)  +  1 )  =  ( 1  +  1 ) )
27 df-2 9849 . . . . . . . . . 10  |-  2  =  ( 1  +  1 )
2826, 27syl6eqr 2366 . . . . . . . . 9  |-  ( k  =  0  ->  (
( 1  -  k
)  +  1 )  =  2 )
2922, 28oveq12d 5918 . . . . . . . 8  |-  ( k  =  0  ->  (
( k BernPoly  X )  /  ( ( 1  -  k )  +  1 ) )  =  ( ( 0 BernPoly  X
)  /  2 ) )
3021, 29oveq12d 5918 . . . . . . 7  |-  ( k  =  0  ->  (
( 1  _C  k
)  x.  ( ( k BernPoly  X )  /  (
( 1  -  k
)  +  1 ) ) )  =  ( 1  x.  ( ( 0 BernPoly  X )  /  2
) ) )
3130fsum1 12261 . . . . . 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 644 . . . . 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 2348 . . . 4  |-  ( X  e.  CC  ->  sum_ k  e.  ( 0 ... 0
) ( ( 1  _C  k )  x.  ( ( k BernPoly  X
)  /  ( ( 1  -  k )  +  1 ) ) )  =  ( 1  /  2 ) )
347, 33syl5eq 2360 . . 3  |-  ( X  e.  CC  ->  sum_ k  e.  ( 0 ... (
1  -  1 ) ) ( ( 1  _C  k )  x.  ( ( k BernPoly  X
)  /  ( ( 1  -  k )  +  1 ) ) )  =  ( 1  /  2 ) )
354, 34oveq12d 5918 . 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 2348 1  |-  ( X  e.  CC  ->  (
1 BernPoly  X )  =  ( X  -  ( 1  /  2 ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1633    e. wcel 1701  (class class class)co 5900   CCcc 8780   0cc0 8782   1c1 8783    + caddc 8785    x. cmul 8787    - cmin 9082    / cdiv 9468   2c2 9840   NN0cn0 10012   ZZcz 10071   ...cfz 10829   ^cexp 11151    _C cbc 11362   sum_csu 12205   BernPoly cbp 25167
This theorem is referenced by:  bpoly2  25178  bpoly3  25179  bpoly4  25180
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-inf2 7387  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859  ax-pre-sup 8860
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-se 4390  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-isom 5301  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-riota 6346  df-recs 6430  df-rdg 6465  df-1o 6521  df-oadd 6525  df-er 6702  df-en 6907  df-dom 6908  df-sdom 6909  df-fin 6910  df-sup 7239  df-oi 7270  df-card 7617  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-div 9469  df-nn 9792  df-2 9849  df-3 9850  df-n0 10013  df-z 10072  df-uz 10278  df-rp 10402  df-fz 10830  df-fzo 10918  df-seq 11094  df-exp 11152  df-fac 11336  df-bc 11363  df-hash 11385  df-cj 11631  df-re 11632  df-im 11633  df-sqr 11767  df-abs 11768  df-clim 12009  df-sum 12206  df-pred 24553  df-bpoly 25168
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