Theorem dvply1 21709

Description: Derivative of a polynomial, explicit sum version. (Contributed by Stefan O'Rear, 13-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

Hypotheses

Ref	Expression
dvply1.f	|- ( ph -> F = ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( z ^ k ) ) ) )
dvply1.g	|- ( ph -> G = ( z e. CC |-> sum_ k e. ( 0 ... ( N - 1 ) ) ( ( B ` k ) x. ( z ^ k ) ) ) )
dvply1.a	|- ( ph -> A : NN0 --> CC )
dvply1.b	|- B = ( k e. NN0 |-> ( ( k + 1 ) x. ( A ` ( k + 1 ) ) ) )
dvply1.n	|- ( ph -> N e. NN0 )

Assertion

Ref	Expression
dvply1	|- ( ph -> ( CC _D F ) = G )

Distinct variable groups:   ph, z, k   z, A, k   z, B   k, N, z

Allowed substitution hints:   B( k)   F( z, k)   G( z, k)

Proof of Theorem dvply1

Dummy variable j is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvply1.f	. . 3 |- ( ph -> F = ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( z ^ k ) ) ) )
2	1	oveq2d 6106	. 2 |- ( ph -> ( CC _D F ) = ( CC _D ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( z ^ k ) ) ) ) )
3		eqid 2441	. . . . . 6 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
4	3	cnfldtop 20322	. . . . 5 |- ( TopOpen ` ℂfld ) e. Top
5	3	cnfldtopon 20321	. . . . . . 7 |- ( TopOpen ` ℂfld ) e. (TopOn ` CC )
6	5	toponunii 18496	. . . . . 6 |- CC = U. ( TopOpen ` ℂfld )
7	6	restid 14368	. . . . 5 |- ( ( TopOpen ` ℂfld ) e. Top -> ( ( TopOpen ` ℂfld ) ↾t CC ) = ( TopOpen ` ℂfld ) )
8	4, 7	ax-mp 5	. . . 4 |- ( ( TopOpen ` ℂfld ) ↾t CC ) = ( TopOpen ` ℂfld )
9	8	eqcomi 2445	. . 3 |- ( TopOpen ` ℂfld ) = ( ( TopOpen ` ℂfld ) ↾t CC )
10		cnelprrecn 9371	. . . 4 |- CC e. { RR , CC }
11	10	a1i 11	. . 3 |- ( ph -> CC e. { RR , CC } )
12	6	topopn 18478	. . . 4 |- ( ( TopOpen ` ℂfld ) e. Top -> CC e. ( TopOpen ` ℂfld ) )
13	4, 12	mp1i 12	. . 3 |- ( ph -> CC e. ( TopOpen ` ℂfld ) )
14		fzfid 11791	. . 3 |- ( ph -> ( 0 ... N ) e. Fin )
15		dvply1.a	. . . . . . 7 |- ( ph -> A : NN0 --> CC )
16		elfznn0 11477	. . . . . . 7 |- ( k e. ( 0 ... N ) -> k e. NN0 )
17		ffvelrn 5838	. . . . . . 7 |- ( ( A : NN0 --> CC /\ k e. NN0 ) -> ( A ` k ) e. CC )
18	15, 16, 17	syl2an 474	. . . . . 6 |- ( ( ph /\ k e. ( 0 ... N ) ) -> ( A ` k ) e. CC )
19	18	adantr 462	. . . . 5 |- ( ( ( ph /\ k e. ( 0 ... N ) ) /\ z e. CC ) -> ( A ` k ) e. CC )
20		simpr 458	. . . . . 6 |- ( ( ( ph /\ k e. ( 0 ... N ) ) /\ z e. CC ) -> z e. CC )
21	16	ad2antlr 721	. . . . . 6 |- ( ( ( ph /\ k e. ( 0 ... N ) ) /\ z e. CC ) -> k e. NN0 )
22	20, 21	expcld 12004	. . . . 5 |- ( ( ( ph /\ k e. ( 0 ... N ) ) /\ z e. CC ) -> ( z ^ k ) e. CC )
23	19, 22	mulcld 9402	. . . 4 |- ( ( ( ph /\ k e. ( 0 ... N ) ) /\ z e. CC ) -> ( ( A ` k ) x. ( z ^ k ) ) e. CC )
24	23	3impa 1177	. . 3 |- ( ( ph /\ k e. ( 0 ... N ) /\ z e. CC ) -> ( ( A ` k ) x. ( z ^ k ) ) e. CC )
25	18	3adant3 1003	. . . 4 |- ( ( ph /\ k e. ( 0 ... N ) /\ z e. CC ) -> ( A ` k ) e. CC )
26		0cnd 9375	. . . . 5 |- ( ( ( ph /\ k e. ( 0 ... N ) /\ z e. CC ) /\ k = 0 ) -> 0 e. CC )
27		simpl2 987	. . . . . . . 8 |- ( ( ( ph /\ k e. ( 0 ... N ) /\ z e. CC ) /\ -. k = 0 ) -> k e. ( 0 ... N ) )
28	27, 16	syl 16	. . . . . . 7 |- ( ( ( ph /\ k e. ( 0 ... N ) /\ z e. CC ) /\ -. k = 0 ) -> k e. NN0 )
29	28	nn0cnd 10634	. . . . . 6 |- ( ( ( ph /\ k e. ( 0 ... N ) /\ z e. CC ) /\ -. k = 0 ) -> k e. CC )
30		simpl3 988	. . . . . . 7 |- ( ( ( ph /\ k e. ( 0 ... N ) /\ z e. CC ) /\ -. k = 0 ) -> z e. CC )
31		simpr 458	. . . . . . . . 9 |- ( ( ( ph /\ k e. ( 0 ... N ) /\ z e. CC ) /\ -. k = 0 ) -> -. k = 0 )
32		elnn0 10577	. . . . . . . . . 10 |- ( k e. NN0 <-> ( k e. NN \/ k = 0 ) )
33	28, 32	sylib 196	. . . . . . . . 9 |- ( ( ( ph /\ k e. ( 0 ... N ) /\ z e. CC ) /\ -. k = 0 ) -> ( k e. NN \/ k = 0 ) )
34		orel2 383	. . . . . . . . 9 |- ( -. k = 0 -> ( ( k e. NN \/ k = 0 ) -> k e. NN ) )
35	31, 33, 34	sylc 60	. . . . . . . 8 |- ( ( ( ph /\ k e. ( 0 ... N ) /\ z e. CC ) /\ -. k = 0 ) -> k e. NN )
36		nnm1nn0 10617	. . . . . . . 8 |- ( k e. NN -> ( k - 1 ) e. NN0 )
37	35, 36	syl 16	. . . . . . 7 |- ( ( ( ph /\ k e. ( 0 ... N ) /\ z e. CC ) /\ -. k = 0 ) -> ( k - 1 ) e. NN0 )
38	30, 37	expcld 12004	. . . . . 6 |- ( ( ( ph /\ k e. ( 0 ... N ) /\ z e. CC ) /\ -. k = 0 ) -> ( z ^ ( k - 1 ) ) e. CC )
39	29, 38	mulcld 9402	. . . . 5 |- ( ( ( ph /\ k e. ( 0 ... N ) /\ z e. CC ) /\ -. k = 0 ) -> ( k x. ( z ^ ( k - 1 ) ) ) e. CC )
40	26, 39	ifclda 3818	. . . 4 |- ( ( ph /\ k e. ( 0 ... N ) /\ z e. CC ) -> if ( k = 0 , 0 , ( k x. ( z ^ ( k - 1 ) ) ) ) e. CC )
41	25, 40	mulcld 9402	. . 3 |- ( ( ph /\ k e. ( 0 ... N ) /\ z e. CC ) -> ( ( A ` k ) x. if ( k = 0 , 0 , ( k x. ( z ^ ( k - 1 ) ) ) ) ) e. CC )
42	10	a1i 11	. . . 4 |- ( ( ph /\ k e. ( 0 ... N ) ) -> CC e. { RR , CC } )
43		c0ex 9376	. . . . . 6 |- 0 e. _V
44		ovex 6115	. . . . . 6 |- ( k x. ( z ^ ( k - 1 ) ) ) e. _V
45	43, 44	ifex 3855	. . . . 5 |- if ( k = 0 , 0 , ( k x. ( z ^ ( k - 1 ) ) ) ) e. _V
46	45	a1i 11	. . . 4 |- ( ( ( ph /\ k e. ( 0 ... N ) ) /\ z e. CC ) -> if ( k = 0 , 0 , ( k x. ( z ^ ( k - 1 ) ) ) ) e. _V )
47	16	adantl 463	. . . . 5 |- ( ( ph /\ k e. ( 0 ... N ) ) -> k e. NN0 )
48		dvexp2 21387	. . . . 5 |- ( k e. NN0 -> ( CC _D ( z e. CC |-> ( z ^ k ) ) ) = ( z e. CC |-> if ( k = 0 , 0 , ( k x. ( z ^ ( k - 1 ) ) ) ) ) )
49	47, 48	syl 16	. . . 4 |- ( ( ph /\ k e. ( 0 ... N ) ) -> ( CC _D ( z e. CC |-> ( z ^ k ) ) ) = ( z e. CC |-> if ( k = 0 , 0 , ( k x. ( z ^ ( k - 1 ) ) ) ) ) )
50	42, 22, 46, 49, 18	dvmptcmul 21397	. . 3 |- ( ( ph /\ k e. ( 0 ... N ) ) -> ( CC _D ( z e. CC |-> ( ( A ` k ) x. ( z ^ k ) ) ) ) = ( z e. CC |-> ( ( A ` k ) x. if ( k = 0 , 0 , ( k x. ( z ^ ( k - 1 ) ) ) ) ) ) )
51	9, 3, 11, 13, 14, 24, 41, 50	dvmptfsum 21406	. 2 |- ( ph -> ( CC _D ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( z ^ k ) ) ) ) = ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( A ` k ) x. if ( k = 0 , 0 , ( k x. ( z ^ ( k - 1 ) ) ) ) ) ) )
52		elfznn 11474	. . . . . . . . . . 11 |- ( k e. ( 1 ... N ) -> k e. NN )
53	52	nnne0d 10362	. . . . . . . . . 10 |- ( k e. ( 1 ... N ) -> k =/= 0 )
54	53	neneqd 2622	. . . . . . . . 9 |- ( k e. ( 1 ... N ) -> -. k = 0 )
55	54	adantl 463	. . . . . . . 8 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> -. k = 0 )
56		iffalse 3796	. . . . . . . 8 |- ( -. k = 0 -> if ( k = 0 , 0 , ( k x. ( z ^ ( k - 1 ) ) ) ) = ( k x. ( z ^ ( k - 1 ) ) ) )
57	55, 56	syl 16	. . . . . . 7 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> if ( k = 0 , 0 , ( k x. ( z ^ ( k - 1 ) ) ) ) = ( k x. ( z ^ ( k - 1 ) ) ) )
58	57	oveq2d 6106	. . . . . 6 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> ( ( A ` k ) x. if ( k = 0 , 0 , ( k x. ( z ^ ( k - 1 ) ) ) ) ) = ( ( A ` k ) x. ( k x. ( z ^ ( k - 1 ) ) ) ) )
59	58	sumeq2dv 13176	. . . . 5 |- ( ( ph /\ z e. CC ) -> sum_ k e. ( 1 ... N ) ( ( A ` k ) x. if ( k = 0 , 0 , ( k x. ( z ^ ( k - 1 ) ) ) ) ) = sum_ k e. ( 1 ... N ) ( ( A ` k ) x. ( k x. ( z ^ ( k - 1 ) ) ) ) )
60		1nn0 10591	. . . . . . . 8 |- 1 e. NN0
61		nn0uz 10891	. . . . . . . 8 |- NN0 = ( ZZ>= ` 0 )
62	60, 61	eleqtri 2513	. . . . . . 7 |- 1 e. ( ZZ>= ` 0 )
63		fzss1 11493	. . . . . . 7 |- ( 1 e. ( ZZ>= ` 0 ) -> ( 1 ... N ) C_ ( 0 ... N ) )
64	62, 63	mp1i 12	. . . . . 6 |- ( ( ph /\ z e. CC ) -> ( 1 ... N ) C_ ( 0 ... N ) )
65	15	adantr 462	. . . . . . . 8 |- ( ( ph /\ z e. CC ) -> A : NN0 --> CC )
66	52	nnnn0d 10632	. . . . . . . 8 |- ( k e. ( 1 ... N ) -> k e. NN0 )
67	65, 66, 17	syl2an 474	. . . . . . 7 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> ( A ` k ) e. CC )
68	53	adantl 463	. . . . . . . . . 10 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> k =/= 0 )
69	68	neneqd 2622	. . . . . . . . 9 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> -. k = 0 )
70	69, 56	syl 16	. . . . . . . 8 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> if ( k = 0 , 0 , ( k x. ( z ^ ( k - 1 ) ) ) ) = ( k x. ( z ^ ( k - 1 ) ) ) )
71	66	adantl 463	. . . . . . . . . 10 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> k e. NN0 )
72	71	nn0cnd 10634	. . . . . . . . 9 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> k e. CC )
73		simplr 749	. . . . . . . . . 10 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> z e. CC )
74	52, 36	syl 16	. . . . . . . . . . 11 |- ( k e. ( 1 ... N ) -> ( k - 1 ) e. NN0 )
75	74	adantl 463	. . . . . . . . . 10 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> ( k - 1 ) e. NN0 )
76	73, 75	expcld 12004	. . . . . . . . 9 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> ( z ^ ( k - 1 ) ) e. CC )
77	72, 76	mulcld 9402	. . . . . . . 8 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> ( k x. ( z ^ ( k - 1 ) ) ) e. CC )
78	70, 77	eqeltrd 2515	. . . . . . 7 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> if ( k = 0 , 0 , ( k x. ( z ^ ( k - 1 ) ) ) ) e. CC )
79	67, 78	mulcld 9402	. . . . . 6 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> ( ( A ` k ) x. if ( k = 0 , 0 , ( k x. ( z ^ ( k - 1 ) ) ) ) ) e. CC )
80		eldifn 3476	. . . . . . . . . . . 12 |- ( k e. ( ( 0 ... N ) \ ( 1 ... N ) ) -> -. k e. ( 1 ... N ) )
81		0p1e1 10429	. . . . . . . . . . . . . 14 |- ( 0 + 1 ) = 1
82	81	oveq1i 6100	. . . . . . . . . . . . 13 |- ( ( 0 + 1 ) ... N ) = ( 1 ... N )
83	82	eleq2i 2505	. . . . . . . . . . . 12 |- ( k e. ( ( 0 + 1 ) ... N ) <-> k e. ( 1 ... N ) )
84	80, 83	sylnibr 305	. . . . . . . . . . 11 |- ( k e. ( ( 0 ... N ) \ ( 1 ... N ) ) -> -. k e. ( ( 0 + 1 ) ... N ) )
85	84	adantl 463	. . . . . . . . . 10 |- ( ( ( ph /\ z e. CC ) /\ k e. ( ( 0 ... N ) \ ( 1 ... N ) ) ) -> -. k e. ( ( 0 + 1 ) ... N ) )
86		eldifi 3475	. . . . . . . . . . . 12 |- ( k e. ( ( 0 ... N ) \ ( 1 ... N ) ) -> k e. ( 0 ... N ) )
87	86	adantl 463	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. CC ) /\ k e. ( ( 0 ... N ) \ ( 1 ... N ) ) ) -> k e. ( 0 ... N ) )
88		dvply1.n	. . . . . . . . . . . . . 14 |- ( ph -> N e. NN0 )
89	88, 61	syl6eleq 2531	. . . . . . . . . . . . 13 |- ( ph -> N e. ( ZZ>= ` 0 ) )
90	89	ad2antrr 720	. . . . . . . . . . . 12 |- ( ( ( ph /\ z e. CC ) /\ k e. ( ( 0 ... N ) \ ( 1 ... N ) ) ) -> N e. ( ZZ>= ` 0 ) )
91		elfzp12 11535	. . . . . . . . . . . 12 |- ( N e. ( ZZ>= ` 0 ) -> ( k e. ( 0 ... N ) <-> ( k = 0 \/ k e. ( ( 0 + 1 ) ... N ) ) ) )
92	90, 91	syl 16	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. CC ) /\ k e. ( ( 0 ... N ) \ ( 1 ... N ) ) ) -> ( k e. ( 0 ... N ) <-> ( k = 0 \/ k e. ( ( 0 + 1 ) ... N ) ) ) )
93	87, 92	mpbid 210	. . . . . . . . . 10 |- ( ( ( ph /\ z e. CC ) /\ k e. ( ( 0 ... N ) \ ( 1 ... N ) ) ) -> ( k = 0 \/ k e. ( ( 0 + 1 ) ... N ) ) )
94		orel2 383	. . . . . . . . . 10 |- ( -. k e. ( ( 0 + 1 ) ... N ) -> ( ( k = 0 \/ k e. ( ( 0 + 1 ) ... N ) ) -> k = 0 ) )
95	85, 93, 94	sylc 60	. . . . . . . . 9 |- ( ( ( ph /\ z e. CC ) /\ k e. ( ( 0 ... N ) \ ( 1 ... N ) ) ) -> k = 0 )
96		iftrue 3794	. . . . . . . . 9 |- ( k = 0 -> if ( k = 0 , 0 , ( k x. ( z ^ ( k - 1 ) ) ) ) = 0 )
97	95, 96	syl 16	. . . . . . . 8 |- ( ( ( ph /\ z e. CC ) /\ k e. ( ( 0 ... N ) \ ( 1 ... N ) ) ) -> if ( k = 0 , 0 , ( k x. ( z ^ ( k - 1 ) ) ) ) = 0 )
98	97	oveq2d 6106	. . . . . . 7 |- ( ( ( ph /\ z e. CC ) /\ k e. ( ( 0 ... N ) \ ( 1 ... N ) ) ) -> ( ( A ` k ) x. if ( k = 0 , 0 , ( k x. ( z ^ ( k - 1 ) ) ) ) ) = ( ( A ` k ) x. 0 ) )
99	65, 16, 17	syl2an 474	. . . . . . . . 9 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... N ) ) -> ( A ` k ) e. CC )
100	99	mul01d 9564	. . . . . . . 8 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... N ) ) -> ( ( A ` k ) x. 0 ) = 0 )
101	86, 100	sylan2 471	. . . . . . 7 |- ( ( ( ph /\ z e. CC ) /\ k e. ( ( 0 ... N ) \ ( 1 ... N ) ) ) -> ( ( A ` k ) x. 0 ) = 0 )
102	98, 101	eqtrd 2473	. . . . . 6 |- ( ( ( ph /\ z e. CC ) /\ k e. ( ( 0 ... N ) \ ( 1 ... N ) ) ) -> ( ( A ` k ) x. if ( k = 0 , 0 , ( k x. ( z ^ ( k - 1 ) ) ) ) ) = 0 )
103		fzfid 11791	. . . . . 6 |- ( ( ph /\ z e. CC ) -> ( 0 ... N ) e. Fin )
104	64, 79, 102, 103	fsumss 13198	. . . . 5 |- ( ( ph /\ z e. CC ) -> sum_ k e. ( 1 ... N ) ( ( A ` k ) x. if ( k = 0 , 0 , ( k x. ( z ^ ( k - 1 ) ) ) ) ) = sum_ k e. ( 0 ... N ) ( ( A ` k ) x. if ( k = 0 , 0 , ( k x. ( z ^ ( k - 1 ) ) ) ) ) )
105		elfznn0 11477	. . . . . . . . . . . . . . 15 |- ( j e. ( 0 ... ( N - 1 ) ) -> j e. NN0 )
106	105	adantl 463	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ z e. CC ) /\ j e. ( 0 ... ( N - 1 ) ) ) -> j e. NN0 )
107	106	nn0cnd 10634	. . . . . . . . . . . . 13 |- ( ( ( ph /\ z e. CC ) /\ j e. ( 0 ... ( N - 1 ) ) ) -> j e. CC )
108		ax-1cn 9336	. . . . . . . . . . . . 13 |- 1 e. CC
109		pncan 9612	. . . . . . . . . . . . 13 |- ( ( j e. CC /\ 1 e. CC ) -> ( ( j + 1 ) - 1 ) = j )
110	107, 108, 109	sylancl 657	. . . . . . . . . . . 12 |- ( ( ( ph /\ z e. CC ) /\ j e. ( 0 ... ( N - 1 ) ) ) -> ( ( j + 1 ) - 1 ) = j )
111	110	oveq2d 6106	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. CC ) /\ j e. ( 0 ... ( N - 1 ) ) ) -> ( z ^ ( ( j + 1 ) - 1 ) ) = ( z ^ j ) )
112	111	oveq2d 6106	. . . . . . . . . 10 |- ( ( ( ph /\ z e. CC ) /\ j e. ( 0 ... ( N - 1 ) ) ) -> ( ( j + 1 ) x. ( z ^ ( ( j + 1 ) - 1 ) ) ) = ( ( j + 1 ) x. ( z ^ j ) ) )
113	112	oveq2d 6106	. . . . . . . . 9 |- ( ( ( ph /\ z e. CC ) /\ j e. ( 0 ... ( N - 1 ) ) ) -> ( ( A ` ( j + 1 ) ) x. ( ( j + 1 ) x. ( z ^ ( ( j + 1 ) - 1 ) ) ) ) = ( ( A ` ( j + 1 ) ) x. ( ( j + 1 ) x. ( z ^ j ) ) ) )
114	15	ad2antrr 720	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. CC ) /\ j e. ( 0 ... ( N - 1 ) ) ) -> A : NN0 --> CC )
115		peano2nn0 10616	. . . . . . . . . . . . 13 |- ( j e. NN0 -> ( j + 1 ) e. NN0 )
116	105, 115	syl 16	. . . . . . . . . . . 12 |- ( j e. ( 0 ... ( N - 1 ) ) -> ( j + 1 ) e. NN0 )
117	116	adantl 463	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. CC ) /\ j e. ( 0 ... ( N - 1 ) ) ) -> ( j + 1 ) e. NN0 )
118	114, 117	ffvelrnd 5841	. . . . . . . . . 10 |- ( ( ( ph /\ z e. CC ) /\ j e. ( 0 ... ( N - 1 ) ) ) -> ( A ` ( j + 1 ) ) e. CC )
119	117	nn0cnd 10634	. . . . . . . . . 10 |- ( ( ( ph /\ z e. CC ) /\ j e. ( 0 ... ( N - 1 ) ) ) -> ( j + 1 ) e. CC )
120		simplr 749	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. CC ) /\ j e. ( 0 ... ( N - 1 ) ) ) -> z e. CC )
121	120, 106	expcld 12004	. . . . . . . . . 10 |- ( ( ( ph /\ z e. CC ) /\ j e. ( 0 ... ( N - 1 ) ) ) -> ( z ^ j ) e. CC )
122	118, 119, 121	mulassd 9405	. . . . . . . . 9 |- ( ( ( ph /\ z e. CC ) /\ j e. ( 0 ... ( N - 1 ) ) ) -> ( ( ( A ` ( j + 1 ) ) x. ( j + 1 ) ) x. ( z ^ j ) ) = ( ( A ` ( j + 1 ) ) x. ( ( j + 1 ) x. ( z ^ j ) ) ) )
123	118, 119	mulcomd 9403	. . . . . . . . . 10 |- ( ( ( ph /\ z e. CC ) /\ j e. ( 0 ... ( N - 1 ) ) ) -> ( ( A ` ( j + 1 ) ) x. ( j + 1 ) ) = ( ( j + 1 ) x. ( A ` ( j + 1 ) ) ) )
124	123	oveq1d 6105	. . . . . . . . 9 |- ( ( ( ph /\ z e. CC ) /\ j e. ( 0 ... ( N - 1 ) ) ) -> ( ( ( A ` ( j + 1 ) ) x. ( j + 1 ) ) x. ( z ^ j ) ) = ( ( ( j + 1 ) x. ( A ` ( j + 1 ) ) ) x. ( z ^ j ) ) )
125	113, 122, 124	3eqtr2d 2479	. . . . . . . 8 |- ( ( ( ph /\ z e. CC ) /\ j e. ( 0 ... ( N - 1 ) ) ) -> ( ( A ` ( j + 1 ) ) x. ( ( j + 1 ) x. ( z ^ ( ( j + 1 ) - 1 ) ) ) ) = ( ( ( j + 1 ) x. ( A ` ( j + 1 ) ) ) x. ( z ^ j ) ) )
126	125	sumeq2dv 13176	. . . . . . 7 |- ( ( ph /\ z e. CC ) -> sum_ j e. ( 0 ... ( N - 1 ) ) ( ( A ` ( j + 1 ) ) x. ( ( j + 1 ) x. ( z ^ ( ( j + 1 ) - 1 ) ) ) ) = sum_ j e. ( 0 ... ( N - 1 ) ) ( ( ( j + 1 ) x. ( A ` ( j + 1 ) ) ) x. ( z ^ j ) ) )
127		1m1e0 10386	. . . . . . . . 9 |- ( 1 - 1 ) = 0
128	127	oveq1i 6100	. . . . . . . 8 |- ( ( 1 - 1 ) ... ( N - 1 ) ) = ( 0 ... ( N - 1 ) )
129	128	sumeq1i 13171	. . . . . . 7 |- sum_ j e. ( ( 1 - 1 ) ... ( N - 1 ) ) ( ( A ` ( j + 1 ) ) x. ( ( j + 1 ) x. ( z ^ ( ( j + 1 ) - 1 ) ) ) ) = sum_ j e. ( 0 ... ( N - 1 ) ) ( ( A ` ( j + 1 ) ) x. ( ( j + 1 ) x. ( z ^ ( ( j + 1 ) - 1 ) ) ) )
130		oveq1 6097	. . . . . . . . . 10 |- ( k = j -> ( k + 1 ) = ( j + 1 ) )
131	130	fveq2d 5692	. . . . . . . . . 10 |- ( k = j -> ( A ` ( k + 1 ) ) = ( A ` ( j + 1 ) ) )
132	130, 131	oveq12d 6108	. . . . . . . . 9 |- ( k = j -> ( ( k + 1 ) x. ( A ` ( k + 1 ) ) ) = ( ( j + 1 ) x. ( A ` ( j + 1 ) ) ) )
133		oveq2 6098	. . . . . . . . 9 |- ( k = j -> ( z ^ k ) = ( z ^ j ) )
134	132, 133	oveq12d 6108	. . . . . . . 8 |- ( k = j -> ( ( ( k + 1 ) x. ( A ` ( k + 1 ) ) ) x. ( z ^ k ) ) = ( ( ( j + 1 ) x. ( A ` ( j + 1 ) ) ) x. ( z ^ j ) ) )
135	134	cbvsumv 13169	. . . . . . 7 |- sum_ k e. ( 0 ... ( N - 1 ) ) ( ( ( k + 1 ) x. ( A ` ( k + 1 ) ) ) x. ( z ^ k ) ) = sum_ j e. ( 0 ... ( N - 1 ) ) ( ( ( j + 1 ) x. ( A ` ( j + 1 ) ) ) x. ( z ^ j ) )
136	126, 129, 135	3eqtr4g 2498	. . . . . 6 |- ( ( ph /\ z e. CC ) -> sum_ j e. ( ( 1 - 1 ) ... ( N - 1 ) ) ( ( A ` ( j + 1 ) ) x. ( ( j + 1 ) x. ( z ^ ( ( j + 1 ) - 1 ) ) ) ) = sum_ k e. ( 0 ... ( N - 1 ) ) ( ( ( k + 1 ) x. ( A ` ( k + 1 ) ) ) x. ( z ^ k ) ) )
137		1zzd 10673	. . . . . . 7 |- ( ( ph /\ z e. CC ) -> 1 e. ZZ )
138	88	adantr 462	. . . . . . . 8 |- ( ( ph /\ z e. CC ) -> N e. NN0 )
139	138	nn0zd 10741	. . . . . . 7 |- ( ( ph /\ z e. CC ) -> N e. ZZ )
140	67, 77	mulcld 9402	. . . . . . 7 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> ( ( A ` k ) x. ( k x. ( z ^ ( k - 1 ) ) ) ) e. CC )
141		fveq2 5688	. . . . . . . 8 |- ( k = ( j + 1 ) -> ( A ` k ) = ( A ` ( j + 1 ) ) )
142		id 22	. . . . . . . . 9 |- ( k = ( j + 1 ) -> k = ( j + 1 ) )
143		oveq1 6097	. . . . . . . . . 10 |- ( k = ( j + 1 ) -> ( k - 1 ) = ( ( j + 1 ) - 1 ) )
144	143	oveq2d 6106	. . . . . . . . 9 |- ( k = ( j + 1 ) -> ( z ^ ( k - 1 ) ) = ( z ^ ( ( j + 1 ) - 1 ) ) )
145	142, 144	oveq12d 6108	. . . . . . . 8 |- ( k = ( j + 1 ) -> ( k x. ( z ^ ( k - 1 ) ) ) = ( ( j + 1 ) x. ( z ^ ( ( j + 1 ) - 1 ) ) ) )
146	141, 145	oveq12d 6108	. . . . . . 7 |- ( k = ( j + 1 ) -> ( ( A ` k ) x. ( k x. ( z ^ ( k - 1 ) ) ) ) = ( ( A ` ( j + 1 ) ) x. ( ( j + 1 ) x. ( z ^ ( ( j + 1 ) - 1 ) ) ) ) )
147	137, 137, 139, 140, 146	fsumshftm 13244	. . . . . 6 |- ( ( ph /\ z e. CC ) -> sum_ k e. ( 1 ... N ) ( ( A ` k ) x. ( k x. ( z ^ ( k - 1 ) ) ) ) = sum_ j e. ( ( 1 - 1 ) ... ( N - 1 ) ) ( ( A ` ( j + 1 ) ) x. ( ( j + 1 ) x. ( z ^ ( ( j + 1 ) - 1 ) ) ) ) )
148		elfznn0 11477	. . . . . . . . . 10 |- ( k e. ( 0 ... ( N - 1 ) ) -> k e. NN0 )
149	148	adantl 463	. . . . . . . . 9 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... ( N - 1 ) ) ) -> k e. NN0 )
150		ovex 6115	. . . . . . . . 9 |- ( ( k + 1 ) x. ( A ` ( k + 1 ) ) ) e. _V
151		dvply1.b	. . . . . . . . . 10 |- B = ( k e. NN0 |-> ( ( k + 1 ) x. ( A ` ( k + 1 ) ) ) )
152	151	fvmpt2 5778	. . . . . . . . 9 |- ( ( k e. NN0 /\ ( ( k + 1 ) x. ( A ` ( k + 1 ) ) ) e. _V ) -> ( B ` k ) = ( ( k + 1 ) x. ( A ` ( k + 1 ) ) ) )
153	149, 150, 152	sylancl 657	. . . . . . . 8 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... ( N - 1 ) ) ) -> ( B ` k ) = ( ( k + 1 ) x. ( A ` ( k + 1 ) ) ) )
154	153	oveq1d 6105	. . . . . . 7 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... ( N - 1 ) ) ) -> ( ( B ` k ) x. ( z ^ k ) ) = ( ( ( k + 1 ) x. ( A ` ( k + 1 ) ) ) x. ( z ^ k ) ) )
155	154	sumeq2dv 13176	. . . . . 6 |- ( ( ph /\ z e. CC ) -> sum_ k e. ( 0 ... ( N - 1 ) ) ( ( B ` k ) x. ( z ^ k ) ) = sum_ k e. ( 0 ... ( N - 1 ) ) ( ( ( k + 1 ) x. ( A ` ( k + 1 ) ) ) x. ( z ^ k ) ) )
156	136, 147, 155	3eqtr4d 2483	. . . . 5 |- ( ( ph /\ z e. CC ) -> sum_ k e. ( 1 ... N ) ( ( A ` k ) x. ( k x. ( z ^ ( k - 1 ) ) ) ) = sum_ k e. ( 0 ... ( N - 1 ) ) ( ( B ` k ) x. ( z ^ k ) ) )
157	59, 104, 156	3eqtr3d 2481	. . . 4 |- ( ( ph /\ z e. CC ) -> sum_ k e. ( 0 ... N ) ( ( A ` k ) x. if ( k = 0 , 0 , ( k x. ( z ^ ( k - 1 ) ) ) ) ) = sum_ k e. ( 0 ... ( N - 1 ) ) ( ( B ` k ) x. ( z ^ k ) ) )
158	157	mpteq2dva 4375	. . 3 |- ( ph -> ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( A ` k ) x. if ( k = 0 , 0 , ( k x. ( z ^ ( k - 1 ) ) ) ) ) ) = ( z e. CC |-> sum_ k e. ( 0 ... ( N - 1 ) ) ( ( B ` k ) x. ( z ^ k ) ) ) )
159		dvply1.g	. . 3 |- ( ph -> G = ( z e. CC |-> sum_ k e. ( 0 ... ( N - 1 ) ) ( ( B ` k ) x. ( z ^ k ) ) ) )
160	158, 159	eqtr4d 2476	. 2 |- ( ph -> ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( A ` k ) x. if ( k = 0 , 0 , ( k x. ( z ^ ( k - 1 ) ) ) ) ) ) = G )
161	2, 51, 160	3eqtrd 2477	1 |- ( ph -> ( CC _D F ) = G )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   \/ wo 368   /\ wa 369   /\ w3a 960   = wceq 1364   e. wcel 1761   =/= wne 2604   _Vcvv 2970   \ cdif 3322   C_ wss 3325   ifcif 3788   {cpr 3876   e. cmpt 4347   -->wf 5411   ` cfv 5415  (class class class)co 6090   CCcc 9276   RRcr 9277   0cc0 9278   1c1 9279   + caddc 9281   x. cmul 9283   - cmin 9591   NNcn 10318   NN0cn0 10575   ZZ>=cuz 10857   ...cfz 11433   ^cexp 11861   sum_csu 13159   ↾_t crest 14355   TopOpenctopn 14356  ℂ_fldccnfld 17777   Topctop 18457   _D cdv 21297

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356  ax-addf 9357  ax-mulf 9358

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-supp 6690  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-er 7097  df-map 7212  df-pm 7213  df-ixp 7260  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-fsupp 7617  df-fi 7657  df-sup 7687  df-oi 7720  df-card 8105  df-cda 8333  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-q 10950  df-rp 10988  df-xneg 11085  df-xadd 11086  df-xmul 11087  df-icc 11303  df-fz 11434  df-fzo 11545  df-seq 11803  df-exp 11862  df-hash 12100  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-clim 12962  df-sum 13160  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-mulr 14248  df-starv 14249  df-sca 14250  df-vsca 14251  df-ip 14252  df-tset 14253  df-ple 14254  df-ds 14256  df-unif 14257  df-hom 14258  df-cco 14259  df-rest 14357  df-topn 14358  df-0g 14376  df-gsum 14377  df-topgen 14378  df-pt 14379  df-prds 14382  df-xrs 14436  df-qtop 14441  df-imas 14442  df-xps 14444  df-mre 14520  df-mrc 14521  df-acs 14523  df-mnd 15411  df-submnd 15461  df-mulg 15541  df-cntz 15828  df-cmn 16272  df-psmet 17768  df-xmet 17769  df-met 17770  df-bl 17771  df-mopn 17772  df-fbas 17773  df-fg 17774  df-cnfld 17778  df-top 18462  df-bases 18464  df-topon 18465  df-topsp 18466  df-cld 18582  df-ntr 18583  df-cls 18584  df-nei 18661  df-lp 18699  df-perf 18700  df-cn 18790  df-cnp 18791  df-haus 18878  df-tx 19094  df-hmeo 19287  df-fil 19378  df-fm 19470  df-flim 19471  df-flf 19472  df-xms 19854  df-ms 19855  df-tms 19856  df-cncf 20413  df-limc 21300  df-dv 21301

This theorem is referenced by:  dvply2g  21710