Theorem dvply1 20154

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 6056	. 2 |- ( ph -> ( CC _D F ) = ( CC _D ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( z ^ k ) ) ) ) )
3		eqid 2404	. . . . . 6 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
4	3	cnfldtop 18771	. . . . 5 |- ( TopOpen ` ℂfld ) e. Top
5	3	cnfldtopon 18770	. . . . . . 7 |- ( TopOpen ` ℂfld ) e. (TopOn ` CC )
6	5	toponunii 16952	. . . . . 6 |- CC = U. ( TopOpen ` ℂfld )
7	6	restid 13616	. . . . 5 |- ( ( TopOpen ` ℂfld ) e. Top -> ( ( TopOpen ` ℂfld ) ↾t CC ) = ( TopOpen ` ℂfld ) )
8	4, 7	ax-mp 8	. . . 4 |- ( ( TopOpen ` ℂfld ) ↾t CC ) = ( TopOpen ` ℂfld )
9	8	eqcomi 2408	. . 3 |- ( TopOpen ` ℂfld ) = ( ( TopOpen ` ℂfld ) ↾t CC )
10		cnex 9027	. . . . 5 |- CC e. _V
11	10	prid2 3873	. . . 4 |- CC e. { RR , CC }
12	11	a1i 11	. . 3 |- ( ph -> CC e. { RR , CC } )
13	6	topopn 16934	. . . 4 |- ( ( TopOpen ` ℂfld ) e. Top -> CC e. ( TopOpen ` ℂfld ) )
14	4, 13	mp1i 12	. . 3 |- ( ph -> CC e. ( TopOpen ` ℂfld ) )
15		fzfid 11267	. . 3 |- ( ph -> ( 0 ... N ) e. Fin )
16		dvply1.a	. . . . . . 7 |- ( ph -> A : NN0 --> CC )
17		elfznn0 11039	. . . . . . 7 |- ( k e. ( 0 ... N ) -> k e. NN0 )
18		ffvelrn 5827	. . . . . . 7 |- ( ( A : NN0 --> CC /\ k e. NN0 ) -> ( A ` k ) e. CC )
19	16, 17, 18	syl2an 464	. . . . . 6 |- ( ( ph /\ k e. ( 0 ... N ) ) -> ( A ` k ) e. CC )
20	19	adantr 452	. . . . 5 |- ( ( ( ph /\ k e. ( 0 ... N ) ) /\ z e. CC ) -> ( A ` k ) e. CC )
21		simpr 448	. . . . . 6 |- ( ( ( ph /\ k e. ( 0 ... N ) ) /\ z e. CC ) -> z e. CC )
22	17	ad2antlr 708	. . . . . 6 |- ( ( ( ph /\ k e. ( 0 ... N ) ) /\ z e. CC ) -> k e. NN0 )
23	21, 22	expcld 11478	. . . . 5 |- ( ( ( ph /\ k e. ( 0 ... N ) ) /\ z e. CC ) -> ( z ^ k ) e. CC )
24	20, 23	mulcld 9064	. . . 4 |- ( ( ( ph /\ k e. ( 0 ... N ) ) /\ z e. CC ) -> ( ( A ` k ) x. ( z ^ k ) ) e. CC )
25	24	3impa 1148	. . 3 |- ( ( ph /\ k e. ( 0 ... N ) /\ z e. CC ) -> ( ( A ` k ) x. ( z ^ k ) ) e. CC )
26	19	3adant3 977	. . . 4 |- ( ( ph /\ k e. ( 0 ... N ) /\ z e. CC ) -> ( A ` k ) e. CC )
27		0cn 9040	. . . . . 6 |- 0 e. CC
28	27	a1i 11	. . . . 5 |- ( ( ( ph /\ k e. ( 0 ... N ) /\ z e. CC ) /\ k = 0 ) -> 0 e. CC )
29		simpl2 961	. . . . . . . 8 |- ( ( ( ph /\ k e. ( 0 ... N ) /\ z e. CC ) /\ -. k = 0 ) -> k e. ( 0 ... N ) )
30	29, 17	syl 16	. . . . . . 7 |- ( ( ( ph /\ k e. ( 0 ... N ) /\ z e. CC ) /\ -. k = 0 ) -> k e. NN0 )
31	30	nn0cnd 10232	. . . . . 6 |- ( ( ( ph /\ k e. ( 0 ... N ) /\ z e. CC ) /\ -. k = 0 ) -> k e. CC )
32		simpl3 962	. . . . . . 7 |- ( ( ( ph /\ k e. ( 0 ... N ) /\ z e. CC ) /\ -. k = 0 ) -> z e. CC )
33		simpr 448	. . . . . . . . 9 |- ( ( ( ph /\ k e. ( 0 ... N ) /\ z e. CC ) /\ -. k = 0 ) -> -. k = 0 )
34		elnn0 10179	. . . . . . . . . 10 |- ( k e. NN0 <-> ( k e. NN \/ k = 0 ) )
35	30, 34	sylib 189	. . . . . . . . 9 |- ( ( ( ph /\ k e. ( 0 ... N ) /\ z e. CC ) /\ -. k = 0 ) -> ( k e. NN \/ k = 0 ) )
36		orel2 373	. . . . . . . . 9 |- ( -. k = 0 -> ( ( k e. NN \/ k = 0 ) -> k e. NN ) )
37	33, 35, 36	sylc 58	. . . . . . . 8 |- ( ( ( ph /\ k e. ( 0 ... N ) /\ z e. CC ) /\ -. k = 0 ) -> k e. NN )
38		nnm1nn0 10217	. . . . . . . 8 |- ( k e. NN -> ( k - 1 ) e. NN0 )
39	37, 38	syl 16	. . . . . . 7 |- ( ( ( ph /\ k e. ( 0 ... N ) /\ z e. CC ) /\ -. k = 0 ) -> ( k - 1 ) e. NN0 )
40	32, 39	expcld 11478	. . . . . 6 |- ( ( ( ph /\ k e. ( 0 ... N ) /\ z e. CC ) /\ -. k = 0 ) -> ( z ^ ( k - 1 ) ) e. CC )
41	31, 40	mulcld 9064	. . . . 5 |- ( ( ( ph /\ k e. ( 0 ... N ) /\ z e. CC ) /\ -. k = 0 ) -> ( k x. ( z ^ ( k - 1 ) ) ) e. CC )
42	28, 41	ifclda 3726	. . . 4 |- ( ( ph /\ k e. ( 0 ... N ) /\ z e. CC ) -> if ( k = 0 , 0 , ( k x. ( z ^ ( k - 1 ) ) ) ) e. CC )
43	26, 42	mulcld 9064	. . 3 |- ( ( ph /\ k e. ( 0 ... N ) /\ z e. CC ) -> ( ( A ` k ) x. if ( k = 0 , 0 , ( k x. ( z ^ ( k - 1 ) ) ) ) ) e. CC )
44	11	a1i 11	. . . 4 |- ( ( ph /\ k e. ( 0 ... N ) ) -> CC e. { RR , CC } )
45		c0ex 9041	. . . . . 6 |- 0 e. _V
46		ovex 6065	. . . . . 6 |- ( k x. ( z ^ ( k - 1 ) ) ) e. _V
47	45, 46	ifex 3757	. . . . 5 |- if ( k = 0 , 0 , ( k x. ( z ^ ( k - 1 ) ) ) ) e. _V
48	47	a1i 11	. . . 4 |- ( ( ( ph /\ k e. ( 0 ... N ) ) /\ z e. CC ) -> if ( k = 0 , 0 , ( k x. ( z ^ ( k - 1 ) ) ) ) e. _V )
49	17	adantl 453	. . . . 5 |- ( ( ph /\ k e. ( 0 ... N ) ) -> k e. NN0 )
50		dvexp2 19793	. . . . 5 |- ( k e. NN0 -> ( CC _D ( z e. CC |-> ( z ^ k ) ) ) = ( z e. CC |-> if ( k = 0 , 0 , ( k x. ( z ^ ( k - 1 ) ) ) ) ) )
51	49, 50	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 ) ) ) ) ) )
52	44, 23, 48, 51, 19	dvmptcmul 19803	. . 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 ) ) ) ) ) ) )
53	9, 3, 12, 14, 15, 25, 43, 52	dvmptfsum 19812	. 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 ) ) ) ) ) ) )
54		elfznn 11036	. . . . . . . . . . 11 |- ( k e. ( 1 ... N ) -> k e. NN )
55	54	nnne0d 10000	. . . . . . . . . 10 |- ( k e. ( 1 ... N ) -> k =/= 0 )
56	55	neneqd 2583	. . . . . . . . 9 |- ( k e. ( 1 ... N ) -> -. k = 0 )
57	56	adantl 453	. . . . . . . 8 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> -. k = 0 )
58		iffalse 3706	. . . . . . . 8 |- ( -. k = 0 -> if ( k = 0 , 0 , ( k x. ( z ^ ( k - 1 ) ) ) ) = ( k x. ( z ^ ( k - 1 ) ) ) )
59	57, 58	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 ) ) ) )
60	59	oveq2d 6056	. . . . . 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 ) ) ) ) )
61	60	sumeq2dv 12452	. . . . 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 ) ) ) ) )
62		1nn0 10193	. . . . . . . 8 |- 1 e. NN0
63		nn0uz 10476	. . . . . . . 8 |- NN0 = ( ZZ>= ` 0 )
64	62, 63	eleqtri 2476	. . . . . . 7 |- 1 e. ( ZZ>= ` 0 )
65		fzss1 11047	. . . . . . 7 |- ( 1 e. ( ZZ>= ` 0 ) -> ( 1 ... N ) C_ ( 0 ... N ) )
66	64, 65	mp1i 12	. . . . . 6 |- ( ( ph /\ z e. CC ) -> ( 1 ... N ) C_ ( 0 ... N ) )
67	16	adantr 452	. . . . . . . 8 |- ( ( ph /\ z e. CC ) -> A : NN0 --> CC )
68	54	nnnn0d 10230	. . . . . . . 8 |- ( k e. ( 1 ... N ) -> k e. NN0 )
69	67, 68, 18	syl2an 464	. . . . . . 7 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> ( A ` k ) e. CC )
70	55	adantl 453	. . . . . . . . . 10 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> k =/= 0 )
71	70	neneqd 2583	. . . . . . . . 9 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> -. k = 0 )
72	71, 58	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 ) ) ) )
73	68	adantl 453	. . . . . . . . . 10 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> k e. NN0 )
74	73	nn0cnd 10232	. . . . . . . . 9 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> k e. CC )
75		simplr 732	. . . . . . . . . 10 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> z e. CC )
76	54, 38	syl 16	. . . . . . . . . . 11 |- ( k e. ( 1 ... N ) -> ( k - 1 ) e. NN0 )
77	76	adantl 453	. . . . . . . . . 10 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> ( k - 1 ) e. NN0 )
78	75, 77	expcld 11478	. . . . . . . . 9 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> ( z ^ ( k - 1 ) ) e. CC )
79	74, 78	mulcld 9064	. . . . . . . 8 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> ( k x. ( z ^ ( k - 1 ) ) ) e. CC )
80	72, 79	eqeltrd 2478	. . . . . . 7 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> if ( k = 0 , 0 , ( k x. ( z ^ ( k - 1 ) ) ) ) e. CC )
81	69, 80	mulcld 9064	. . . . . 6 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> ( ( A ` k ) x. if ( k = 0 , 0 , ( k x. ( z ^ ( k - 1 ) ) ) ) ) e. CC )
82		eldifn 3430	. . . . . . . . . . . 12 |- ( k e. ( ( 0 ... N ) \ ( 1 ... N ) ) -> -. k e. ( 1 ... N ) )
83		0p1e1 10049	. . . . . . . . . . . . . 14 |- ( 0 + 1 ) = 1
84	83	oveq1i 6050	. . . . . . . . . . . . 13 |- ( ( 0 + 1 ) ... N ) = ( 1 ... N )
85	84	eleq2i 2468	. . . . . . . . . . . 12 |- ( k e. ( ( 0 + 1 ) ... N ) <-> k e. ( 1 ... N ) )
86	82, 85	sylnibr 297	. . . . . . . . . . 11 |- ( k e. ( ( 0 ... N ) \ ( 1 ... N ) ) -> -. k e. ( ( 0 + 1 ) ... N ) )
87	86	adantl 453	. . . . . . . . . 10 |- ( ( ( ph /\ z e. CC ) /\ k e. ( ( 0 ... N ) \ ( 1 ... N ) ) ) -> -. k e. ( ( 0 + 1 ) ... N ) )
88		eldifi 3429	. . . . . . . . . . . 12 |- ( k e. ( ( 0 ... N ) \ ( 1 ... N ) ) -> k e. ( 0 ... N ) )
89	88	adantl 453	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. CC ) /\ k e. ( ( 0 ... N ) \ ( 1 ... N ) ) ) -> k e. ( 0 ... N ) )
90		dvply1.n	. . . . . . . . . . . . . 14 |- ( ph -> N e. NN0 )
91	90, 63	syl6eleq 2494	. . . . . . . . . . . . 13 |- ( ph -> N e. ( ZZ>= ` 0 ) )
92	91	ad2antrr 707	. . . . . . . . . . . 12 |- ( ( ( ph /\ z e. CC ) /\ k e. ( ( 0 ... N ) \ ( 1 ... N ) ) ) -> N e. ( ZZ>= ` 0 ) )
93		elfzp12 11081	. . . . . . . . . . . 12 |- ( N e. ( ZZ>= ` 0 ) -> ( k e. ( 0 ... N ) <-> ( k = 0 \/ k e. ( ( 0 + 1 ) ... N ) ) ) )
94	92, 93	syl 16	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. CC ) /\ k e. ( ( 0 ... N ) \ ( 1 ... N ) ) ) -> ( k e. ( 0 ... N ) <-> ( k = 0 \/ k e. ( ( 0 + 1 ) ... N ) ) ) )
95	89, 94	mpbid 202	. . . . . . . . . 10 |- ( ( ( ph /\ z e. CC ) /\ k e. ( ( 0 ... N ) \ ( 1 ... N ) ) ) -> ( k = 0 \/ k e. ( ( 0 + 1 ) ... N ) ) )
96		orel2 373	. . . . . . . . . 10 |- ( -. k e. ( ( 0 + 1 ) ... N ) -> ( ( k = 0 \/ k e. ( ( 0 + 1 ) ... N ) ) -> k = 0 ) )
97	87, 95, 96	sylc 58	. . . . . . . . 9 |- ( ( ( ph /\ z e. CC ) /\ k e. ( ( 0 ... N ) \ ( 1 ... N ) ) ) -> k = 0 )
98		iftrue 3705	. . . . . . . . 9 |- ( k = 0 -> if ( k = 0 , 0 , ( k x. ( z ^ ( k - 1 ) ) ) ) = 0 )
99	97, 98	syl 16	. . . . . . . 8 |- ( ( ( ph /\ z e. CC ) /\ k e. ( ( 0 ... N ) \ ( 1 ... N ) ) ) -> if ( k = 0 , 0 , ( k x. ( z ^ ( k - 1 ) ) ) ) = 0 )
100	99	oveq2d 6056	. . . . . . 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 ) )
101	67, 17, 18	syl2an 464	. . . . . . . . 9 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... N ) ) -> ( A ` k ) e. CC )
102	101	mul01d 9221	. . . . . . . 8 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... N ) ) -> ( ( A ` k ) x. 0 ) = 0 )
103	88, 102	sylan2 461	. . . . . . 7 |- ( ( ( ph /\ z e. CC ) /\ k e. ( ( 0 ... N ) \ ( 1 ... N ) ) ) -> ( ( A ` k ) x. 0 ) = 0 )
104	100, 103	eqtrd 2436	. . . . . 6 |- ( ( ( ph /\ z e. CC ) /\ k e. ( ( 0 ... N ) \ ( 1 ... N ) ) ) -> ( ( A ` k ) x. if ( k = 0 , 0 , ( k x. ( z ^ ( k - 1 ) ) ) ) ) = 0 )
105		fzfid 11267	. . . . . 6 |- ( ( ph /\ z e. CC ) -> ( 0 ... N ) e. Fin )
106	66, 81, 104, 105	fsumss 12474	. . . . 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 ) ) ) ) ) )
107		elfznn0 11039	. . . . . . . . . . . . . . 15 |- ( j e. ( 0 ... ( N - 1 ) ) -> j e. NN0 )
108	107	adantl 453	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ z e. CC ) /\ j e. ( 0 ... ( N - 1 ) ) ) -> j e. NN0 )
109	108	nn0cnd 10232	. . . . . . . . . . . . 13 |- ( ( ( ph /\ z e. CC ) /\ j e. ( 0 ... ( N - 1 ) ) ) -> j e. CC )
110		ax-1cn 9004	. . . . . . . . . . . . 13 |- 1 e. CC
111		pncan 9267	. . . . . . . . . . . . 13 |- ( ( j e. CC /\ 1 e. CC ) -> ( ( j + 1 ) - 1 ) = j )
112	109, 110, 111	sylancl 644	. . . . . . . . . . . 12 |- ( ( ( ph /\ z e. CC ) /\ j e. ( 0 ... ( N - 1 ) ) ) -> ( ( j + 1 ) - 1 ) = j )
113	112	oveq2d 6056	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. CC ) /\ j e. ( 0 ... ( N - 1 ) ) ) -> ( z ^ ( ( j + 1 ) - 1 ) ) = ( z ^ j ) )
114	113	oveq2d 6056	. . . . . . . . . 10 |- ( ( ( ph /\ z e. CC ) /\ j e. ( 0 ... ( N - 1 ) ) ) -> ( ( j + 1 ) x. ( z ^ ( ( j + 1 ) - 1 ) ) ) = ( ( j + 1 ) x. ( z ^ j ) ) )
115	114	oveq2d 6056	. . . . . . . . 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 ) ) ) )
116	16	ad2antrr 707	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. CC ) /\ j e. ( 0 ... ( N - 1 ) ) ) -> A : NN0 --> CC )
117		peano2nn0 10216	. . . . . . . . . . . . 13 |- ( j e. NN0 -> ( j + 1 ) e. NN0 )
118	107, 117	syl 16	. . . . . . . . . . . 12 |- ( j e. ( 0 ... ( N - 1 ) ) -> ( j + 1 ) e. NN0 )
119	118	adantl 453	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. CC ) /\ j e. ( 0 ... ( N - 1 ) ) ) -> ( j + 1 ) e. NN0 )
120	116, 119	ffvelrnd 5830	. . . . . . . . . 10 |- ( ( ( ph /\ z e. CC ) /\ j e. ( 0 ... ( N - 1 ) ) ) -> ( A ` ( j + 1 ) ) e. CC )
121	119	nn0cnd 10232	. . . . . . . . . 10 |- ( ( ( ph /\ z e. CC ) /\ j e. ( 0 ... ( N - 1 ) ) ) -> ( j + 1 ) e. CC )
122		simplr 732	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. CC ) /\ j e. ( 0 ... ( N - 1 ) ) ) -> z e. CC )
123	122, 108	expcld 11478	. . . . . . . . . 10 |- ( ( ( ph /\ z e. CC ) /\ j e. ( 0 ... ( N - 1 ) ) ) -> ( z ^ j ) e. CC )
124	120, 121, 123	mulassd 9067	. . . . . . . . 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 ) ) ) )
125	120, 121	mulcomd 9065	. . . . . . . . . 10 |- ( ( ( ph /\ z e. CC ) /\ j e. ( 0 ... ( N - 1 ) ) ) -> ( ( A ` ( j + 1 ) ) x. ( j + 1 ) ) = ( ( j + 1 ) x. ( A ` ( j + 1 ) ) ) )
126	125	oveq1d 6055	. . . . . . . . 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 ) ) )
127	115, 124, 126	3eqtr2d 2442	. . . . . . . 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 ) ) )
128	127	sumeq2dv 12452	. . . . . . 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 ) ) )
129		1m1e0 10024	. . . . . . . . 9 |- ( 1 - 1 ) = 0
130	129	oveq1i 6050	. . . . . . . 8 |- ( ( 1 - 1 ) ... ( N - 1 ) ) = ( 0 ... ( N - 1 ) )
131	130	sumeq1i 12447	. . . . . . 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 ) ) ) )
132		oveq1 6047	. . . . . . . . . 10 |- ( k = j -> ( k + 1 ) = ( j + 1 ) )
133	132	fveq2d 5691	. . . . . . . . . 10 |- ( k = j -> ( A ` ( k + 1 ) ) = ( A ` ( j + 1 ) ) )
134	132, 133	oveq12d 6058	. . . . . . . . 9 |- ( k = j -> ( ( k + 1 ) x. ( A ` ( k + 1 ) ) ) = ( ( j + 1 ) x. ( A ` ( j + 1 ) ) ) )
135		oveq2 6048	. . . . . . . . 9 |- ( k = j -> ( z ^ k ) = ( z ^ j ) )
136	134, 135	oveq12d 6058	. . . . . . . 8 |- ( k = j -> ( ( ( k + 1 ) x. ( A ` ( k + 1 ) ) ) x. ( z ^ k ) ) = ( ( ( j + 1 ) x. ( A ` ( j + 1 ) ) ) x. ( z ^ j ) ) )
137	136	cbvsumv 12445	. . . . . . 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 ) )
138	128, 131, 137	3eqtr4g 2461	. . . . . 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 ) ) )
139		1z 10267	. . . . . . . 8 |- 1 e. ZZ
140	139	a1i 11	. . . . . . 7 |- ( ( ph /\ z e. CC ) -> 1 e. ZZ )
141	90	adantr 452	. . . . . . . 8 |- ( ( ph /\ z e. CC ) -> N e. NN0 )
142	141	nn0zd 10329	. . . . . . 7 |- ( ( ph /\ z e. CC ) -> N e. ZZ )
143	69, 79	mulcld 9064	. . . . . . 7 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> ( ( A ` k ) x. ( k x. ( z ^ ( k - 1 ) ) ) ) e. CC )
144		fveq2 5687	. . . . . . . 8 |- ( k = ( j + 1 ) -> ( A ` k ) = ( A ` ( j + 1 ) ) )
145		id 20	. . . . . . . . 9 |- ( k = ( j + 1 ) -> k = ( j + 1 ) )
146		oveq1 6047	. . . . . . . . . 10 |- ( k = ( j + 1 ) -> ( k - 1 ) = ( ( j + 1 ) - 1 ) )
147	146	oveq2d 6056	. . . . . . . . 9 |- ( k = ( j + 1 ) -> ( z ^ ( k - 1 ) ) = ( z ^ ( ( j + 1 ) - 1 ) ) )
148	145, 147	oveq12d 6058	. . . . . . . 8 |- ( k = ( j + 1 ) -> ( k x. ( z ^ ( k - 1 ) ) ) = ( ( j + 1 ) x. ( z ^ ( ( j + 1 ) - 1 ) ) ) )
149	144, 148	oveq12d 6058	. . . . . . 7 |- ( k = ( j + 1 ) -> ( ( A ` k ) x. ( k x. ( z ^ ( k - 1 ) ) ) ) = ( ( A ` ( j + 1 ) ) x. ( ( j + 1 ) x. ( z ^ ( ( j + 1 ) - 1 ) ) ) ) )
150	140, 140, 142, 143, 149	fsumshftm 12519	. . . . . 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 ) ) ) ) )
151		elfznn0 11039	. . . . . . . . . 10 |- ( k e. ( 0 ... ( N - 1 ) ) -> k e. NN0 )
152	151	adantl 453	. . . . . . . . 9 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... ( N - 1 ) ) ) -> k e. NN0 )
153		ovex 6065	. . . . . . . . 9 |- ( ( k + 1 ) x. ( A ` ( k + 1 ) ) ) e. _V
154		dvply1.b	. . . . . . . . . 10 |- B = ( k e. NN0 |-> ( ( k + 1 ) x. ( A ` ( k + 1 ) ) ) )
155	154	fvmpt2 5771	. . . . . . . . 9 |- ( ( k e. NN0 /\ ( ( k + 1 ) x. ( A ` ( k + 1 ) ) ) e. _V ) -> ( B ` k ) = ( ( k + 1 ) x. ( A ` ( k + 1 ) ) ) )
156	152, 153, 155	sylancl 644	. . . . . . . 8 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... ( N - 1 ) ) ) -> ( B ` k ) = ( ( k + 1 ) x. ( A ` ( k + 1 ) ) ) )
157	156	oveq1d 6055	. . . . . . 7 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... ( N - 1 ) ) ) -> ( ( B ` k ) x. ( z ^ k ) ) = ( ( ( k + 1 ) x. ( A ` ( k + 1 ) ) ) x. ( z ^ k ) ) )
158	157	sumeq2dv 12452	. . . . . 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 ) ) )
159	138, 150, 158	3eqtr4d 2446	. . . . 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 ) ) )
160	61, 106, 159	3eqtr3d 2444	. . . 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 ) ) )
161	160	mpteq2dva 4255	. . 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 ) ) ) )
162		dvply1.g	. . 3 |- ( ph -> G = ( z e. CC |-> sum_ k e. ( 0 ... ( N - 1 ) ) ( ( B ` k ) x. ( z ^ k ) ) ) )
163	161, 162	eqtr4d 2439	. 2 |- ( ph -> ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( A ` k ) x. if ( k = 0 , 0 , ( k x. ( z ^ ( k - 1 ) ) ) ) ) ) = G )
164	2, 53, 163	3eqtrd 2440	1 |- ( ph -> ( CC _D F ) = G )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 177   \/ wo 358   /\ wa 359   /\ w3a 936   = wceq 1649   e. wcel 1721   =/= wne 2567   _Vcvv 2916   \ cdif 3277   C_ wss 3280   ifcif 3699   {cpr 3775   e. cmpt 4226   -->wf 5409   ` cfv 5413  (class class class)co 6040   CCcc 8944   RRcr 8945   0cc0 8946   1c1 8947   + caddc 8949   x. cmul 8951   - cmin 9247   NNcn 9956   NN0cn0 10177   ZZcz 10238   ZZ>=cuz 10444   ...cfz 10999   ^cexp 11337   sum_csu 12434   ↾_t crest 13603   TopOpenctopn 13604  ℂ_fldccnfld 16658   Topctop 16913   _D cdv 19703

This theorem is referenced by:  dvply2g  20155

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-icc 10879  df-fz 11000  df-fzo 11091  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237  df-sum 12435  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-pt 13623  df-prds 13626  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-xps 13691  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-submnd 14694  df-mulg 14770  df-cntz 15071  df-cmn 15369  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-fbas 16654  df-fg 16655  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cld 17038  df-ntr 17039  df-cls 17040  df-nei 17117  df-lp 17155  df-perf 17156  df-cn 17245  df-cnp 17246  df-haus 17333  df-tx 17547  df-hmeo 17740  df-fil 17831  df-fm 17923  df-flim 17924  df-flf 17925  df-xms 18303  df-ms 18304  df-tms 18305  df-cncf 18861  df-limc 19706  df-dv 19707