Theorem bpolyval 14780

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.

Step	Hyp	Ref	Expression
1		fvex 6201	. . . . . 6 |- ( # ` dom c ) e. _V
2		oveq2 6658	. . . . . . 7 |- ( n = ( # ` dom c ) -> ( X ^ n ) = ( X ^ ( # ` dom c ) ) )
3		oveq1 6657	. . . . . . . . 9 |- ( n = ( # ` dom c ) -> ( n _C m ) = ( ( # ` dom c ) _C m ) )
4		oveq1 6657	. . . . . . . . . . 11 |- ( n = ( # ` dom c ) -> ( n - m ) = ( ( # ` dom c ) - m ) )
5	4	oveq1d 6665	. . . . . . . . . 10 |- ( n = ( # ` dom c ) -> ( ( n - m ) + 1 ) = ( ( ( # ` dom c ) - m ) + 1 ) )
6	5	oveq2d 6666	. . . . . . . . 9 |- ( n = ( # ` dom c ) -> ( ( c ` m ) / ( ( n - m ) + 1 ) ) = ( ( c ` m ) / ( ( ( # ` dom c ) - m ) + 1 ) ) )
7	3, 6	oveq12d 6668	. . . . . . . 8 |- ( n = ( # ` dom c ) -> ( ( n _C m ) x. ( ( c ` m ) / ( ( n - m ) + 1 ) ) ) = ( ( ( # ` dom c ) _C m ) x. ( ( c ` m ) / ( ( ( # ` dom c ) - m ) + 1 ) ) ) )
8	7	sumeq2sdv 14435	. . . . . . 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 ) ) ) )
9	2, 8	oveq12d 6668	. . . . . 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 ) ) ) ) )
10	1, 9	csbie 3559	. . . . 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 6658	. . . . . . . . . 10 |- ( m = k -> ( n _C m ) = ( n _C k ) )
12		fveq2 6191	. . . . . . . . . . 11 |- ( m = k -> ( c ` m ) = ( c ` k ) )
13		oveq2 6658	. . . . . . . . . . . 12 |- ( m = k -> ( n - m ) = ( n - k ) )
14	13	oveq1d 6665	. . . . . . . . . . 11 |- ( m = k -> ( ( n - m ) + 1 ) = ( ( n - k ) + 1 ) )
15	12, 14	oveq12d 6668	. . . . . . . . . 10 |- ( m = k -> ( ( c ` m ) / ( ( n - m ) + 1 ) ) = ( ( c ` k ) / ( ( n - k ) + 1 ) ) )
16	11, 15	oveq12d 6668	. . . . . . . . 9 |- ( m = k -> ( ( n _C m ) x. ( ( c ` m ) / ( ( n - m ) + 1 ) ) ) = ( ( n _C k ) x. ( ( c ` k ) / ( ( n - k ) + 1 ) ) ) )
17	16	cbvsumv 14426	. . . . . . . 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 5324	. . . . . . . . 9 |- ( c = g -> dom c = dom g )
19		fveq1 6190	. . . . . . . . . . . 12 |- ( c = g -> ( c ` k ) = ( g ` k ) )
20	19	oveq1d 6665	. . . . . . . . . . 11 |- ( c = g -> ( ( c ` k ) / ( ( n - k ) + 1 ) ) = ( ( g ` k ) / ( ( n - k ) + 1 ) ) )
21	20	oveq2d 6666	. . . . . . . . . 10 |- ( c = g -> ( ( n _C k ) x. ( ( c ` k ) / ( ( n - k ) + 1 ) ) ) = ( ( n _C k ) x. ( ( g ` k ) / ( ( n - k ) + 1 ) ) ) )
22	21	adantr 481	. . . . . . . . 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 ) ) ) )
23	18, 22	sumeq12dv 14437	. . . . . . . 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 ) ) ) )
24	17, 23	syl5eq 2668	. . . . . . 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 ) ) ) )
25	24	oveq2d 6666	. . . . . 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 ) ) ) ) )
26	25	csbeq2dv 3992	. . . . 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 ) ) ) ) )
27	10, 26	syl5eqr 2670	. . . 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 ) ) ) ) )
28	18	fveq2d 6195	. . . . 5 |- ( c = g -> ( # ` dom c ) = ( # ` dom g ) )
29	28	csbeq1d 3540	. . . 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 ) ) ) ) )
30	27, 29	eqtrd 2656	. . 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 ) ) ) ) )
31	30	cbvmptv 4750	. 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 2622	. 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 ) ) ) ) ) )
33	31, 32	bpolylem 14779	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 ) ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   [_csb 3533   |-> cmpt 4729   dom cdm 5114   ` cfv 5888  (class class class)co 6650  wrecscwrecs 7406   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   < clt 10074   - cmin 10266   / cdiv 10684   NN0cn0 11292   ...cfz 12326   ^cexp 12860   _C cbc 13089   #chash 13117   sum_csu 14416   BernPoly cbp 14777

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-seq 12802  df-hash 13118  df-sum 14417  df-bpoly 14778

This theorem is referenced by:  bpoly0  14781  bpoly1  14782  bpolycl  14783  bpolysum  14784  bpolydiflem  14785  bpoly2  14788  bpoly3  14789  bpoly4  14790