Theorem bpolylem 14779

Description: Lemma for bpolyval 14780. (Contributed by Scott Fenton, 22-May-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

Hypotheses

Ref	Expression
bpoly.1	|- G = ( g e. _V |-> [_ ( # ` dom g ) / n ]_ ( ( X ^ n ) - sum_ k e. dom g ( ( n _C k ) x. ( ( g ` k ) / ( ( n - k ) + 1 ) ) ) ) )
bpoly.2	|- F = wrecs ( < , NN0 , G )

Assertion

Ref	Expression
bpolylem	|- ( ( 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:   g, k, n, F   g, N, k, n   g, X, k, n

Allowed substitution hints:   G( g, k, n)

Proof of Theorem bpolylem

Dummy variables m x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq1 6657	. . . . . . . . . . 11 |- ( x = X -> ( x ^ n ) = ( X ^ n ) )
2	1	oveq1d 6665	. . . . . . . . . 10 |- ( x = X -> ( ( x ^ n ) - sum_ k e. dom g ( ( n _C k ) x. ( ( g ` k ) / ( ( n - k ) + 1 ) ) ) ) = ( ( X ^ n ) - sum_ k e. dom g ( ( n _C k ) x. ( ( g ` k ) / ( ( n - k ) + 1 ) ) ) ) )
3	2	csbeq2dv 3992	. . . . . . . . 9 |- ( x = X -> [_ ( # ` dom g ) / 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 ) ) ) ) )
4	3	mpteq2dv 4745	. . . . . . . 8 |- ( x = X -> ( g e. _V |-> [_ ( # ` dom g ) / n ]_ ( ( x ^ n ) - sum_ k e. dom g ( ( n _C k ) x. ( ( g ` k ) / ( ( n - k ) + 1 ) ) ) ) ) = ( g e. _V |-> [_ ( # ` dom g ) / n ]_ ( ( X ^ n ) - sum_ k e. dom g ( ( n _C k ) x. ( ( g ` k ) / ( ( n - k ) + 1 ) ) ) ) ) )
5		bpoly.1	. . . . . . . 8 |- G = ( g e. _V |-> [_ ( # ` dom g ) / n ]_ ( ( X ^ n ) - sum_ k e. dom g ( ( n _C k ) x. ( ( g ` k ) / ( ( n - k ) + 1 ) ) ) ) )
6	4, 5	syl6eqr 2674	. . . . . . 7 |- ( x = X -> ( g e. _V |-> [_ ( # ` dom g ) / n ]_ ( ( x ^ n ) - sum_ k e. dom g ( ( n _C k ) x. ( ( g ` k ) / ( ( n - k ) + 1 ) ) ) ) ) = G )
7		wrecseq3 7412	. . . . . . 7 |- ( ( g e. _V |-> [_ ( # ` dom g ) / n ]_ ( ( x ^ n ) - sum_ k e. dom g ( ( n _C k ) x. ( ( g ` k ) / ( ( n - k ) + 1 ) ) ) ) ) = G -> wrecs ( < , NN0 , ( g e. _V |-> [_ ( # ` dom g ) / n ]_ ( ( x ^ n ) - sum_ k e. dom g ( ( n _C k ) x. ( ( g ` k ) / ( ( n - k ) + 1 ) ) ) ) ) ) = wrecs ( < , NN0 , G ) )
8	6, 7	syl 17	. . . . . 6 |- ( x = X -> wrecs ( < , NN0 , ( g e. _V |-> [_ ( # ` dom g ) / n ]_ ( ( x ^ n ) - sum_ k e. dom g ( ( n _C k ) x. ( ( g ` k ) / ( ( n - k ) + 1 ) ) ) ) ) ) = wrecs ( < , NN0 , G ) )
9		bpoly.2	. . . . . 6 |- F = wrecs ( < , NN0 , G )
10	8, 9	syl6eqr 2674	. . . . 5 |- ( x = X -> wrecs ( < , NN0 , ( g e. _V |-> [_ ( # ` dom g ) / n ]_ ( ( x ^ n ) - sum_ k e. dom g ( ( n _C k ) x. ( ( g ` k ) / ( ( n - k ) + 1 ) ) ) ) ) ) = F )
11	10	fveq1d 6193	. . . 4 |- ( x = X -> (wrecs ( < , NN0 , ( g e. _V |-> [_ ( # ` dom g ) / n ]_ ( ( x ^ n ) - sum_ k e. dom g ( ( n _C k ) x. ( ( g ` k ) / ( ( n - k ) + 1 ) ) ) ) ) ) ` m ) = ( F ` m ) )
12		fveq2 6191	. . . 4 |- ( m = N -> ( F ` m ) = ( F ` N ) )
13	11, 12	sylan9eqr 2678	. . 3 |- ( ( m = N /\ x = X ) -> (wrecs ( < , NN0 , ( g e. _V |-> [_ ( # ` dom g ) / n ]_ ( ( x ^ n ) - sum_ k e. dom g ( ( n _C k ) x. ( ( g ` k ) / ( ( n - k ) + 1 ) ) ) ) ) ) ` m ) = ( F ` N ) )
14		df-bpoly 14778	. . 3 |- BernPoly = ( m e. NN0 , x e. CC |-> (wrecs ( < , NN0 , ( g e. _V |-> [_ ( # ` dom g ) / n ]_ ( ( x ^ n ) - sum_ k e. dom g ( ( n _C k ) x. ( ( g ` k ) / ( ( n - k ) + 1 ) ) ) ) ) ) ` m ) )
15		fvex 6201	. . 3 |- ( F ` N ) e. _V
16	13, 14, 15	ovmpt2a 6791	. 2 |- ( ( N e. NN0 /\ X e. CC ) -> ( N BernPoly X ) = ( F ` N ) )
17		ltweuz 12760	. . . . 5 |- < We ( ZZ>= ` 0 )
18		nn0uz 11722	. . . . . 6 |- NN0 = ( ZZ>= ` 0 )
19		weeq2 5103	. . . . . 6 |- ( NN0 = ( ZZ>= ` 0 ) -> ( < We NN0 <-> < We ( ZZ>= ` 0 ) ) )
20	18, 19	ax-mp 5	. . . . 5 |- ( < We NN0 <-> < We ( ZZ>= ` 0 ) )
21	17, 20	mpbir 221	. . . 4 |- < We NN0
22		nn0ex 11298	. . . . 5 |- NN0 e. _V
23		exse 5078	. . . . 5 |- ( NN0 e. _V -> < Se NN0 )
24	22, 23	ax-mp 5	. . . 4 |- < Se NN0
25	21, 24, 9	wfr2 7434	. . 3 |- ( N e. NN0 -> ( F ` N ) = ( G ` ( F |` Pred ( < , NN0 , N ) ) ) )
26	25	adantr 481	. 2 |- ( ( N e. NN0 /\ X e. CC ) -> ( F ` N ) = ( G ` ( F |` Pred ( < , NN0 , N ) ) ) )
27		prednn0 12463	. . . . . 6 |- ( N e. NN0 -> Pred ( < , NN0 , N ) = ( 0 ... ( N - 1 ) ) )
28	27	adantr 481	. . . . 5 |- ( ( N e. NN0 /\ X e. CC ) -> Pred ( < , NN0 , N ) = ( 0 ... ( N - 1 ) ) )
29	28	reseq2d 5396	. . . 4 |- ( ( N e. NN0 /\ X e. CC ) -> ( F |` Pred ( < , NN0 , N ) ) = ( F |` ( 0 ... ( N - 1 ) ) ) )
30	29	fveq2d 6195	. . 3 |- ( ( N e. NN0 /\ X e. CC ) -> ( G ` ( F |` Pred ( < , NN0 , N ) ) ) = ( G ` ( F |` ( 0 ... ( N - 1 ) ) ) ) )
31	21, 24, 9	wfrfun 7425	. . . . . 6 |- Fun F
32		ovex 6678	. . . . . 6 |- ( 0 ... ( N - 1 ) ) e. _V
33		resfunexg 6479	. . . . . 6 |- ( ( Fun F /\ ( 0 ... ( N - 1 ) ) e. _V ) -> ( F |` ( 0 ... ( N - 1 ) ) ) e. _V )
34	31, 32, 33	mp2an 708	. . . . 5 |- ( F |` ( 0 ... ( N - 1 ) ) ) e. _V
35		dmeq 5324	. . . . . . . . . . 11 |- ( g = ( F |` ( 0 ... ( N - 1 ) ) ) -> dom g = dom ( F |` ( 0 ... ( N - 1 ) ) ) )
36	21, 24, 9	wfr1 7433	. . . . . . . . . . . . 13 |- F Fn NN0
37		fz0ssnn0 12435	. . . . . . . . . . . . 13 |- ( 0 ... ( N - 1 ) ) C_ NN0
38		fnssres 6004	. . . . . . . . . . . . 13 |- ( ( F Fn NN0 /\ ( 0 ... ( N - 1 ) ) C_ NN0 ) -> ( F |` ( 0 ... ( N - 1 ) ) ) Fn ( 0 ... ( N - 1 ) ) )
39	36, 37, 38	mp2an 708	. . . . . . . . . . . 12 |- ( F |` ( 0 ... ( N - 1 ) ) ) Fn ( 0 ... ( N - 1 ) )
40		fndm 5990	. . . . . . . . . . . 12 |- ( ( F |` ( 0 ... ( N - 1 ) ) ) Fn ( 0 ... ( N - 1 ) ) -> dom ( F |` ( 0 ... ( N - 1 ) ) ) = ( 0 ... ( N - 1 ) ) )
41	39, 40	ax-mp 5	. . . . . . . . . . 11 |- dom ( F |` ( 0 ... ( N - 1 ) ) ) = ( 0 ... ( N - 1 ) )
42	35, 41	syl6eq 2672	. . . . . . . . . 10 |- ( g = ( F |` ( 0 ... ( N - 1 ) ) ) -> dom g = ( 0 ... ( N - 1 ) ) )
43		fveq1 6190	. . . . . . . . . . . . 13 |- ( g = ( F |` ( 0 ... ( N - 1 ) ) ) -> ( g ` k ) = ( ( F |` ( 0 ... ( N - 1 ) ) ) ` k ) )
44		fvres 6207	. . . . . . . . . . . . 13 |- ( k e. ( 0 ... ( N - 1 ) ) -> ( ( F |` ( 0 ... ( N - 1 ) ) ) ` k ) = ( F ` k ) )
45	43, 44	sylan9eq 2676	. . . . . . . . . . . 12 |- ( ( g = ( F |` ( 0 ... ( N - 1 ) ) ) /\ k e. ( 0 ... ( N - 1 ) ) ) -> ( g ` k ) = ( F ` k ) )
46	45	oveq1d 6665	. . . . . . . . . . 11 |- ( ( g = ( F |` ( 0 ... ( N - 1 ) ) ) /\ k e. ( 0 ... ( N - 1 ) ) ) -> ( ( g ` k ) / ( ( n - k ) + 1 ) ) = ( ( F ` k ) / ( ( n - k ) + 1 ) ) )
47	46	oveq2d 6666	. . . . . . . . . 10 |- ( ( g = ( F |` ( 0 ... ( N - 1 ) ) ) /\ k e. ( 0 ... ( N - 1 ) ) ) -> ( ( n _C k ) x. ( ( g ` k ) / ( ( n - k ) + 1 ) ) ) = ( ( n _C k ) x. ( ( F ` k ) / ( ( n - k ) + 1 ) ) ) )
48	42, 47	sumeq12rdv 14438	. . . . . . . . 9 |- ( g = ( F |` ( 0 ... ( N - 1 ) ) ) -> sum_ k e. dom g ( ( n _C k ) x. ( ( g ` k ) / ( ( n - k ) + 1 ) ) ) = sum_ k e. ( 0 ... ( N - 1 ) ) ( ( n _C k ) x. ( ( F ` k ) / ( ( n - k ) + 1 ) ) ) )
49	48	oveq2d 6666	. . . . . . . 8 |- ( g = ( F |` ( 0 ... ( N - 1 ) ) ) -> ( ( X ^ n ) - sum_ k e. dom g ( ( n _C k ) x. ( ( g ` k ) / ( ( n - k ) + 1 ) ) ) ) = ( ( X ^ n ) - sum_ k e. ( 0 ... ( N - 1 ) ) ( ( n _C k ) x. ( ( F ` k ) / ( ( n - k ) + 1 ) ) ) ) )
50	49	csbeq2dv 3992	. . . . . . 7 |- ( g = ( F |` ( 0 ... ( N - 1 ) ) ) -> [_ ( # ` dom g ) / 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. ( 0 ... ( N - 1 ) ) ( ( n _C k ) x. ( ( F ` k ) / ( ( n - k ) + 1 ) ) ) ) )
51	42	fveq2d 6195	. . . . . . . 8 |- ( g = ( F |` ( 0 ... ( N - 1 ) ) ) -> ( # ` dom g ) = ( # ` ( 0 ... ( N - 1 ) ) ) )
52	51	csbeq1d 3540	. . . . . . 7 |- ( g = ( F |` ( 0 ... ( N - 1 ) ) ) -> [_ ( # ` dom g ) / n ]_ ( ( X ^ n ) - sum_ k e. ( 0 ... ( N - 1 ) ) ( ( n _C k ) x. ( ( F ` k ) / ( ( n - k ) + 1 ) ) ) ) = [_ ( # ` ( 0 ... ( N - 1 ) ) ) / n ]_ ( ( X ^ n ) - sum_ k e. ( 0 ... ( N - 1 ) ) ( ( n _C k ) x. ( ( F ` k ) / ( ( n - k ) + 1 ) ) ) ) )
53	50, 52	eqtrd 2656	. . . . . 6 |- ( g = ( F |` ( 0 ... ( N - 1 ) ) ) -> [_ ( # ` dom g ) / n ]_ ( ( X ^ n ) - sum_ k e. dom g ( ( n _C k ) x. ( ( g ` k ) / ( ( n - k ) + 1 ) ) ) ) = [_ ( # ` ( 0 ... ( N - 1 ) ) ) / n ]_ ( ( X ^ n ) - sum_ k e. ( 0 ... ( N - 1 ) ) ( ( n _C k ) x. ( ( F ` k ) / ( ( n - k ) + 1 ) ) ) ) )
54		ovex 6678	. . . . . . 7 |- ( ( X ^ n ) - sum_ k e. ( 0 ... ( N - 1 ) ) ( ( n _C k ) x. ( ( F ` k ) / ( ( n - k ) + 1 ) ) ) ) e. _V
55	54	csbex 4793	. . . . . 6 |- [_ ( # ` ( 0 ... ( N - 1 ) ) ) / n ]_ ( ( X ^ n ) - sum_ k e. ( 0 ... ( N - 1 ) ) ( ( n _C k ) x. ( ( F ` k ) / ( ( n - k ) + 1 ) ) ) ) e. _V
56	53, 5, 55	fvmpt 6282	. . . . 5 |- ( ( F |` ( 0 ... ( N - 1 ) ) ) e. _V -> ( G ` ( F |` ( 0 ... ( N - 1 ) ) ) ) = [_ ( # ` ( 0 ... ( N - 1 ) ) ) / n ]_ ( ( X ^ n ) - sum_ k e. ( 0 ... ( N - 1 ) ) ( ( n _C k ) x. ( ( F ` k ) / ( ( n - k ) + 1 ) ) ) ) )
57	34, 56	ax-mp 5	. . . 4 |- ( G ` ( F |` ( 0 ... ( N - 1 ) ) ) ) = [_ ( # ` ( 0 ... ( N - 1 ) ) ) / n ]_ ( ( X ^ n ) - sum_ k e. ( 0 ... ( N - 1 ) ) ( ( n _C k ) x. ( ( F ` k ) / ( ( n - k ) + 1 ) ) ) )
58		nfcvd 2765	. . . . . . 7 |- ( N e. NN0 -> F/_ n ( ( X ^ N ) - sum_ k e. ( 0 ... ( N - 1 ) ) ( ( N _C k ) x. ( ( F ` k ) / ( ( N - k ) + 1 ) ) ) ) )
59		oveq2 6658	. . . . . . . 8 |- ( n = N -> ( X ^ n ) = ( X ^ N ) )
60		oveq1 6657	. . . . . . . . . 10 |- ( n = N -> ( n _C k ) = ( N _C k ) )
61		oveq1 6657	. . . . . . . . . . . 12 |- ( n = N -> ( n - k ) = ( N - k ) )
62	61	oveq1d 6665	. . . . . . . . . . 11 |- ( n = N -> ( ( n - k ) + 1 ) = ( ( N - k ) + 1 ) )
63	62	oveq2d 6666	. . . . . . . . . 10 |- ( n = N -> ( ( F ` k ) / ( ( n - k ) + 1 ) ) = ( ( F ` k ) / ( ( N - k ) + 1 ) ) )
64	60, 63	oveq12d 6668	. . . . . . . . 9 |- ( n = N -> ( ( n _C k ) x. ( ( F ` k ) / ( ( n - k ) + 1 ) ) ) = ( ( N _C k ) x. ( ( F ` k ) / ( ( N - k ) + 1 ) ) ) )
65	64	sumeq2sdv 14435	. . . . . . . 8 |- ( n = N -> sum_ k e. ( 0 ... ( N - 1 ) ) ( ( n _C k ) x. ( ( F ` k ) / ( ( n - k ) + 1 ) ) ) = sum_ k e. ( 0 ... ( N - 1 ) ) ( ( N _C k ) x. ( ( F ` k ) / ( ( N - k ) + 1 ) ) ) )
66	59, 65	oveq12d 6668	. . . . . . 7 |- ( n = N -> ( ( X ^ n ) - sum_ k e. ( 0 ... ( N - 1 ) ) ( ( n _C k ) x. ( ( F ` k ) / ( ( n - k ) + 1 ) ) ) ) = ( ( X ^ N ) - sum_ k e. ( 0 ... ( N - 1 ) ) ( ( N _C k ) x. ( ( F ` k ) / ( ( N - k ) + 1 ) ) ) ) )
67	58, 66	csbiegf 3557	. . . . . 6 |- ( N e. NN0 -> [_ N / n ]_ ( ( X ^ n ) - sum_ k e. ( 0 ... ( N - 1 ) ) ( ( n _C k ) x. ( ( F ` k ) / ( ( n - k ) + 1 ) ) ) ) = ( ( X ^ N ) - sum_ k e. ( 0 ... ( N - 1 ) ) ( ( N _C k ) x. ( ( F ` k ) / ( ( N - k ) + 1 ) ) ) ) )
68	67	adantr 481	. . . . 5 |- ( ( N e. NN0 /\ X e. CC ) -> [_ N / n ]_ ( ( X ^ n ) - sum_ k e. ( 0 ... ( N - 1 ) ) ( ( n _C k ) x. ( ( F ` k ) / ( ( n - k ) + 1 ) ) ) ) = ( ( X ^ N ) - sum_ k e. ( 0 ... ( N - 1 ) ) ( ( N _C k ) x. ( ( F ` k ) / ( ( N - k ) + 1 ) ) ) ) )
69		nn0z 11400	. . . . . . . . . 10 |- ( N e. NN0 -> N e. ZZ )
70		fz01en 12369	. . . . . . . . . 10 |- ( N e. ZZ -> ( 0 ... ( N - 1 ) ) ~~ ( 1 ... N ) )
71	69, 70	syl 17	. . . . . . . . 9 |- ( N e. NN0 -> ( 0 ... ( N - 1 ) ) ~~ ( 1 ... N ) )
72		fzfi 12771	. . . . . . . . . 10 |- ( 0 ... ( N - 1 ) ) e. Fin
73		fzfi 12771	. . . . . . . . . 10 |- ( 1 ... N ) e. Fin
74		hashen 13135	. . . . . . . . . 10 |- ( ( ( 0 ... ( N - 1 ) ) e. Fin /\ ( 1 ... N ) e. Fin ) -> ( ( # ` ( 0 ... ( N - 1 ) ) ) = ( # ` ( 1 ... N ) ) <-> ( 0 ... ( N - 1 ) ) ~~ ( 1 ... N ) ) )
75	72, 73, 74	mp2an 708	. . . . . . . . 9 |- ( ( # ` ( 0 ... ( N - 1 ) ) ) = ( # ` ( 1 ... N ) ) <-> ( 0 ... ( N - 1 ) ) ~~ ( 1 ... N ) )
76	71, 75	sylibr 224	. . . . . . . 8 |- ( N e. NN0 -> ( # ` ( 0 ... ( N - 1 ) ) ) = ( # ` ( 1 ... N ) ) )
77		hashfz1 13134	. . . . . . . 8 |- ( N e. NN0 -> ( # ` ( 1 ... N ) ) = N )
78	76, 77	eqtrd 2656	. . . . . . 7 |- ( N e. NN0 -> ( # ` ( 0 ... ( N - 1 ) ) ) = N )
79	78	adantr 481	. . . . . 6 |- ( ( N e. NN0 /\ X e. CC ) -> ( # ` ( 0 ... ( N - 1 ) ) ) = N )
80	79	csbeq1d 3540	. . . . 5 |- ( ( N e. NN0 /\ X e. CC ) -> [_ ( # ` ( 0 ... ( N - 1 ) ) ) / n ]_ ( ( X ^ n ) - sum_ k e. ( 0 ... ( N - 1 ) ) ( ( n _C k ) x. ( ( F ` k ) / ( ( n - k ) + 1 ) ) ) ) = [_ N / n ]_ ( ( X ^ n ) - sum_ k e. ( 0 ... ( N - 1 ) ) ( ( n _C k ) x. ( ( F ` k ) / ( ( n - k ) + 1 ) ) ) ) )
81		elfznn0 12433	. . . . . . . . . 10 |- ( k e. ( 0 ... ( N - 1 ) ) -> k e. NN0 )
82		simpr 477	. . . . . . . . . 10 |- ( ( N e. NN0 /\ X e. CC ) -> X e. CC )
83		fveq2 6191	. . . . . . . . . . . 12 |- ( m = k -> ( F ` m ) = ( F ` k ) )
84	11, 83	sylan9eqr 2678	. . . . . . . . . . 11 |- ( ( m = k /\ x = X ) -> (wrecs ( < , NN0 , ( g e. _V |-> [_ ( # ` dom g ) / n ]_ ( ( x ^ n ) - sum_ k e. dom g ( ( n _C k ) x. ( ( g ` k ) / ( ( n - k ) + 1 ) ) ) ) ) ) ` m ) = ( F ` k ) )
85		fvex 6201	. . . . . . . . . . 11 |- ( F ` k ) e. _V
86	84, 14, 85	ovmpt2a 6791	. . . . . . . . . 10 |- ( ( k e. NN0 /\ X e. CC ) -> ( k BernPoly X ) = ( F ` k ) )
87	81, 82, 86	syl2anr 495	. . . . . . . . 9 |- ( ( ( N e. NN0 /\ X e. CC ) /\ k e. ( 0 ... ( N - 1 ) ) ) -> ( k BernPoly X ) = ( F ` k ) )
88	87	oveq1d 6665	. . . . . . . 8 |- ( ( ( N e. NN0 /\ X e. CC ) /\ k e. ( 0 ... ( N - 1 ) ) ) -> ( ( k BernPoly X ) / ( ( N - k ) + 1 ) ) = ( ( F ` k ) / ( ( N - k ) + 1 ) ) )
89	88	oveq2d 6666	. . . . . . 7 |- ( ( ( N e. NN0 /\ X e. CC ) /\ k e. ( 0 ... ( N - 1 ) ) ) -> ( ( N _C k ) x. ( ( k BernPoly X ) / ( ( N - k ) + 1 ) ) ) = ( ( N _C k ) x. ( ( F ` k ) / ( ( N - k ) + 1 ) ) ) )
90	89	sumeq2dv 14433	. . . . . 6 |- ( ( N e. NN0 /\ X e. CC ) -> sum_ k e. ( 0 ... ( N - 1 ) ) ( ( N _C k ) x. ( ( k BernPoly X ) / ( ( N - k ) + 1 ) ) ) = sum_ k e. ( 0 ... ( N - 1 ) ) ( ( N _C k ) x. ( ( F ` k ) / ( ( N - k ) + 1 ) ) ) )
91	90	oveq2d 6666	. . . . 5 |- ( ( N e. NN0 /\ X e. CC ) -> ( ( X ^ N ) - sum_ k e. ( 0 ... ( N - 1 ) ) ( ( N _C k ) x. ( ( k BernPoly X ) / ( ( N - k ) + 1 ) ) ) ) = ( ( X ^ N ) - sum_ k e. ( 0 ... ( N - 1 ) ) ( ( N _C k ) x. ( ( F ` k ) / ( ( N - k ) + 1 ) ) ) ) )
92	68, 80, 91	3eqtr4d 2666	. . . 4 |- ( ( N e. NN0 /\ X e. CC ) -> [_ ( # ` ( 0 ... ( N - 1 ) ) ) / n ]_ ( ( X ^ n ) - sum_ k e. ( 0 ... ( N - 1 ) ) ( ( n _C k ) x. ( ( F ` k ) / ( ( n - k ) + 1 ) ) ) ) = ( ( X ^ N ) - sum_ k e. ( 0 ... ( N - 1 ) ) ( ( N _C k ) x. ( ( k BernPoly X ) / ( ( N - k ) + 1 ) ) ) ) )
93	57, 92	syl5eq 2668	. . 3 |- ( ( N e. NN0 /\ X e. CC ) -> ( G ` ( F |` ( 0 ... ( N - 1 ) ) ) ) = ( ( X ^ N ) - sum_ k e. ( 0 ... ( N - 1 ) ) ( ( N _C k ) x. ( ( k BernPoly X ) / ( ( N - k ) + 1 ) ) ) ) )
94	30, 93	eqtrd 2656	. 2 |- ( ( N e. NN0 /\ X e. CC ) -> ( G ` ( F |` Pred ( < , NN0 , N ) ) ) = ( ( X ^ N ) - sum_ k e. ( 0 ... ( N - 1 ) ) ( ( N _C k ) x. ( ( k BernPoly X ) / ( ( N - k ) + 1 ) ) ) ) )
95	16, 26, 94	3eqtrd 2660	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   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   [_csb 3533   C_ wss 3574   class class class wbr 4653   |-> cmpt 4729   Se wse 5071   We wwe 5072   dom cdm 5114   |` cres 5116   Predcpred 5679   Fun wfun 5882   Fn wfn 5883   ` cfv 5888  (class class class)co 6650  wrecscwrecs 7406   ~~ cen 7952   Fincfn 7955   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   < clt 10074   - cmin 10266   / cdiv 10684   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   ...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:  bpolyval  14780