Theorem dvply1 23223

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 6317	. 2 |- ( ph -> ( CC _D F ) = ( CC _D ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( z ^ k ) ) ) ) )
3		eqid 2422	. . . . . 6 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
4	3	cnfldtop 21790	. . . . 5 |- ( TopOpen ` ℂfld ) e. Top
5	3	cnfldtopon 21789	. . . . . . 7 |- ( TopOpen ` ℂfld ) e. (TopOn ` CC )
6	5	toponunii 19933	. . . . . 6 |- CC = U. ( TopOpen ` ℂfld )
7	6	restid 15319	. . . . 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 2435	. . 3 |- ( TopOpen ` ℂfld ) = ( ( TopOpen ` ℂfld ) ↾t CC )
10		cnelprrecn 9632	. . . 4 |- CC e. { RR , CC }
11	10	a1i 11	. . 3 |- ( ph -> CC e. { RR , CC } )
12	6	topopn 19922	. . . 4 |- ( ( TopOpen ` ℂfld ) e. Top -> CC e. ( TopOpen ` ℂfld ) )
13	4, 12	mp1i 13	. . 3 |- ( ph -> CC e. ( TopOpen ` ℂfld ) )
14		fzfid 12185	. . 3 |- ( ph -> ( 0 ... N ) e. Fin )
15		dvply1.a	. . . . . . 7 |- ( ph -> A : NN0 --> CC )
16		elfznn0 11887	. . . . . . 7 |- ( k e. ( 0 ... N ) -> k e. NN0 )
17		ffvelrn 6031	. . . . . . 7 |- ( ( A : NN0 --> CC /\ k e. NN0 ) -> ( A ` k ) e. CC )
18	15, 16, 17	syl2an 479	. . . . . 6 |- ( ( ph /\ k e. ( 0 ... N ) ) -> ( A ` k ) e. CC )
19	18	adantr 466	. . . . 5 |- ( ( ( ph /\ k e. ( 0 ... N ) ) /\ z e. CC ) -> ( A ` k ) e. CC )
20		simpr 462	. . . . . 6 |- ( ( ( ph /\ k e. ( 0 ... N ) ) /\ z e. CC ) -> z e. CC )
21	16	ad2antlr 731	. . . . . 6 |- ( ( ( ph /\ k e. ( 0 ... N ) ) /\ z e. CC ) -> k e. NN0 )
22	20, 21	expcld 12415	. . . . 5 |- ( ( ( ph /\ k e. ( 0 ... N ) ) /\ z e. CC ) -> ( z ^ k ) e. CC )
23	19, 22	mulcld 9663	. . . 4 |- ( ( ( ph /\ k e. ( 0 ... N ) ) /\ z e. CC ) -> ( ( A ` k ) x. ( z ^ k ) ) e. CC )
24	23	3impa 1200	. . 3 |- ( ( ph /\ k e. ( 0 ... N ) /\ z e. CC ) -> ( ( A ` k ) x. ( z ^ k ) ) e. CC )
25	18	3adant3 1025	. . . 4 |- ( ( ph /\ k e. ( 0 ... N ) /\ z e. CC ) -> ( A ` k ) e. CC )
26		0cnd 9636	. . . . 5 |- ( ( ( ph /\ k e. ( 0 ... N ) /\ z e. CC ) /\ k = 0 ) -> 0 e. CC )
27		simpl2 1009	. . . . . . . 8 |- ( ( ( ph /\ k e. ( 0 ... N ) /\ z e. CC ) /\ -. k = 0 ) -> k e. ( 0 ... N ) )
28	27, 16	syl 17	. . . . . . 7 |- ( ( ( ph /\ k e. ( 0 ... N ) /\ z e. CC ) /\ -. k = 0 ) -> k e. NN0 )
29	28	nn0cnd 10927	. . . . . 6 |- ( ( ( ph /\ k e. ( 0 ... N ) /\ z e. CC ) /\ -. k = 0 ) -> k e. CC )
30		simpl3 1010	. . . . . . 7 |- ( ( ( ph /\ k e. ( 0 ... N ) /\ z e. CC ) /\ -. k = 0 ) -> z e. CC )
31		simpr 462	. . . . . . . . 9 |- ( ( ( ph /\ k e. ( 0 ... N ) /\ z e. CC ) /\ -. k = 0 ) -> -. k = 0 )
32		elnn0 10871	. . . . . . . . . 10 |- ( k e. NN0 <-> ( k e. NN \/ k = 0 ) )
33	28, 32	sylib 199	. . . . . . . . 9 |- ( ( ( ph /\ k e. ( 0 ... N ) /\ z e. CC ) /\ -. k = 0 ) -> ( k e. NN \/ k = 0 ) )
34		orel2 384	. . . . . . . . 9 |- ( -. k = 0 -> ( ( k e. NN \/ k = 0 ) -> k e. NN ) )
35	31, 33, 34	sylc 62	. . . . . . . 8 |- ( ( ( ph /\ k e. ( 0 ... N ) /\ z e. CC ) /\ -. k = 0 ) -> k e. NN )
36		nnm1nn0 10911	. . . . . . . 8 |- ( k e. NN -> ( k - 1 ) e. NN0 )
37	35, 36	syl 17	. . . . . . 7 |- ( ( ( ph /\ k e. ( 0 ... N ) /\ z e. CC ) /\ -. k = 0 ) -> ( k - 1 ) e. NN0 )
38	30, 37	expcld 12415	. . . . . 6 |- ( ( ( ph /\ k e. ( 0 ... N ) /\ z e. CC ) /\ -. k = 0 ) -> ( z ^ ( k - 1 ) ) e. CC )
39	29, 38	mulcld 9663	. . . . 5 |- ( ( ( ph /\ k e. ( 0 ... N ) /\ z e. CC ) /\ -. k = 0 ) -> ( k x. ( z ^ ( k - 1 ) ) ) e. CC )
40	26, 39	ifclda 3941	. . . 4 |- ( ( ph /\ k e. ( 0 ... N ) /\ z e. CC ) -> if ( k = 0 , 0 , ( k x. ( z ^ ( k - 1 ) ) ) ) e. CC )
41	25, 40	mulcld 9663	. . 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 9637	. . . . . 6 |- 0 e. _V
44		ovex 6329	. . . . . 6 |- ( k x. ( z ^ ( k - 1 ) ) ) e. _V
45	43, 44	ifex 3977	. . . . 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 467	. . . . 5 |- ( ( ph /\ k e. ( 0 ... N ) ) -> k e. NN0 )
48		dvexp2 22894	. . . . 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 17	. . . 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 22904	. . 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 22913	. 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 11828	. . . . . . . . . . 11 |- ( k e. ( 1 ... N ) -> k e. NN )
53	52	nnne0d 10654	. . . . . . . . . 10 |- ( k e. ( 1 ... N ) -> k =/= 0 )
54	53	neneqd 2625	. . . . . . . . 9 |- ( k e. ( 1 ... N ) -> -. k = 0 )
55	54	adantl 467	. . . . . . . 8 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> -. k = 0 )
56	55	iffalsed 3920	. . . . . . 7 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> if ( k = 0 , 0 , ( k x. ( z ^ ( k - 1 ) ) ) ) = ( k x. ( z ^ ( k - 1 ) ) ) )
57	56	oveq2d 6317	. . . . . 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 ) ) ) ) )
58	57	sumeq2dv 13756	. . . . 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 ) ) ) ) )
59		1eluzge0 11202	. . . . . . 7 |- 1 e. ( ZZ>= ` 0 )
60		fzss1 11837	. . . . . . 7 |- ( 1 e. ( ZZ>= ` 0 ) -> ( 1 ... N ) C_ ( 0 ... N ) )
61	59, 60	mp1i 13	. . . . . 6 |- ( ( ph /\ z e. CC ) -> ( 1 ... N ) C_ ( 0 ... N ) )
62	15	adantr 466	. . . . . . . 8 |- ( ( ph /\ z e. CC ) -> A : NN0 --> CC )
63	52	nnnn0d 10925	. . . . . . . 8 |- ( k e. ( 1 ... N ) -> k e. NN0 )
64	62, 63, 17	syl2an 479	. . . . . . 7 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> ( A ` k ) e. CC )
65	53	adantl 467	. . . . . . . . . 10 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> k =/= 0 )
66	65	neneqd 2625	. . . . . . . . 9 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> -. k = 0 )
67	66	iffalsed 3920	. . . . . . . 8 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> if ( k = 0 , 0 , ( k x. ( z ^ ( k - 1 ) ) ) ) = ( k x. ( z ^ ( k - 1 ) ) ) )
68	63	adantl 467	. . . . . . . . . 10 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> k e. NN0 )
69	68	nn0cnd 10927	. . . . . . . . 9 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> k e. CC )
70		simplr 760	. . . . . . . . . 10 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> z e. CC )
71	52, 36	syl 17	. . . . . . . . . . 11 |- ( k e. ( 1 ... N ) -> ( k - 1 ) e. NN0 )
72	71	adantl 467	. . . . . . . . . 10 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> ( k - 1 ) e. NN0 )
73	70, 72	expcld 12415	. . . . . . . . 9 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> ( z ^ ( k - 1 ) ) e. CC )
74	69, 73	mulcld 9663	. . . . . . . 8 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> ( k x. ( z ^ ( k - 1 ) ) ) e. CC )
75	67, 74	eqeltrd 2510	. . . . . . 7 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> if ( k = 0 , 0 , ( k x. ( z ^ ( k - 1 ) ) ) ) e. CC )
76	64, 75	mulcld 9663	. . . . . 6 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> ( ( A ` k ) x. if ( k = 0 , 0 , ( k x. ( z ^ ( k - 1 ) ) ) ) ) e. CC )
77		eldifn 3588	. . . . . . . . . . . 12 |- ( k e. ( ( 0 ... N ) \ ( 1 ... N ) ) -> -. k e. ( 1 ... N ) )
78		0p1e1 10721	. . . . . . . . . . . . . 14 |- ( 0 + 1 ) = 1
79	78	oveq1i 6311	. . . . . . . . . . . . 13 |- ( ( 0 + 1 ) ... N ) = ( 1 ... N )
80	79	eleq2i 2500	. . . . . . . . . . . 12 |- ( k e. ( ( 0 + 1 ) ... N ) <-> k e. ( 1 ... N ) )
81	77, 80	sylnibr 306	. . . . . . . . . . 11 |- ( k e. ( ( 0 ... N ) \ ( 1 ... N ) ) -> -. k e. ( ( 0 + 1 ) ... N ) )
82	81	adantl 467	. . . . . . . . . 10 |- ( ( ( ph /\ z e. CC ) /\ k e. ( ( 0 ... N ) \ ( 1 ... N ) ) ) -> -. k e. ( ( 0 + 1 ) ... N ) )
83		eldifi 3587	. . . . . . . . . . . 12 |- ( k e. ( ( 0 ... N ) \ ( 1 ... N ) ) -> k e. ( 0 ... N ) )
84	83	adantl 467	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. CC ) /\ k e. ( ( 0 ... N ) \ ( 1 ... N ) ) ) -> k e. ( 0 ... N ) )
85		dvply1.n	. . . . . . . . . . . . . 14 |- ( ph -> N e. NN0 )
86		nn0uz 11193	. . . . . . . . . . . . . 14 |- NN0 = ( ZZ>= ` 0 )
87	85, 86	syl6eleq 2520	. . . . . . . . . . . . 13 |- ( ph -> N e. ( ZZ>= ` 0 ) )
88	87	ad2antrr 730	. . . . . . . . . . . 12 |- ( ( ( ph /\ z e. CC ) /\ k e. ( ( 0 ... N ) \ ( 1 ... N ) ) ) -> N e. ( ZZ>= ` 0 ) )
89		elfzp12 11873	. . . . . . . . . . . 12 |- ( N e. ( ZZ>= ` 0 ) -> ( k e. ( 0 ... N ) <-> ( k = 0 \/ k e. ( ( 0 + 1 ) ... N ) ) ) )
90	88, 89	syl 17	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. CC ) /\ k e. ( ( 0 ... N ) \ ( 1 ... N ) ) ) -> ( k e. ( 0 ... N ) <-> ( k = 0 \/ k e. ( ( 0 + 1 ) ... N ) ) ) )
91	84, 90	mpbid 213	. . . . . . . . . 10 |- ( ( ( ph /\ z e. CC ) /\ k e. ( ( 0 ... N ) \ ( 1 ... N ) ) ) -> ( k = 0 \/ k e. ( ( 0 + 1 ) ... N ) ) )
92		orel2 384	. . . . . . . . . 10 |- ( -. k e. ( ( 0 + 1 ) ... N ) -> ( ( k = 0 \/ k e. ( ( 0 + 1 ) ... N ) ) -> k = 0 ) )
93	82, 91, 92	sylc 62	. . . . . . . . 9 |- ( ( ( ph /\ z e. CC ) /\ k e. ( ( 0 ... N ) \ ( 1 ... N ) ) ) -> k = 0 )
94	93	iftrued 3917	. . . . . . . 8 |- ( ( ( ph /\ z e. CC ) /\ k e. ( ( 0 ... N ) \ ( 1 ... N ) ) ) -> if ( k = 0 , 0 , ( k x. ( z ^ ( k - 1 ) ) ) ) = 0 )
95	94	oveq2d 6317	. . . . . . 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 ) )
96	62, 16, 17	syl2an 479	. . . . . . . . 9 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... N ) ) -> ( A ` k ) e. CC )
97	96	mul01d 9832	. . . . . . . 8 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... N ) ) -> ( ( A ` k ) x. 0 ) = 0 )
98	83, 97	sylan2 476	. . . . . . 7 |- ( ( ( ph /\ z e. CC ) /\ k e. ( ( 0 ... N ) \ ( 1 ... N ) ) ) -> ( ( A ` k ) x. 0 ) = 0 )
99	95, 98	eqtrd 2463	. . . . . 6 |- ( ( ( ph /\ z e. CC ) /\ k e. ( ( 0 ... N ) \ ( 1 ... N ) ) ) -> ( ( A ` k ) x. if ( k = 0 , 0 , ( k x. ( z ^ ( k - 1 ) ) ) ) ) = 0 )
100		fzfid 12185	. . . . . 6 |- ( ( ph /\ z e. CC ) -> ( 0 ... N ) e. Fin )
101	61, 76, 99, 100	fsumss 13778	. . . . 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 ) ) ) ) ) )
102		elfznn0 11887	. . . . . . . . . . . . . . 15 |- ( j e. ( 0 ... ( N - 1 ) ) -> j e. NN0 )
103	102	adantl 467	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ z e. CC ) /\ j e. ( 0 ... ( N - 1 ) ) ) -> j e. NN0 )
104	103	nn0cnd 10927	. . . . . . . . . . . . 13 |- ( ( ( ph /\ z e. CC ) /\ j e. ( 0 ... ( N - 1 ) ) ) -> j e. CC )
105		ax-1cn 9597	. . . . . . . . . . . . 13 |- 1 e. CC
106		pncan 9881	. . . . . . . . . . . . 13 |- ( ( j e. CC /\ 1 e. CC ) -> ( ( j + 1 ) - 1 ) = j )
107	104, 105, 106	sylancl 666	. . . . . . . . . . . 12 |- ( ( ( ph /\ z e. CC ) /\ j e. ( 0 ... ( N - 1 ) ) ) -> ( ( j + 1 ) - 1 ) = j )
108	107	oveq2d 6317	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. CC ) /\ j e. ( 0 ... ( N - 1 ) ) ) -> ( z ^ ( ( j + 1 ) - 1 ) ) = ( z ^ j ) )
109	108	oveq2d 6317	. . . . . . . . . 10 |- ( ( ( ph /\ z e. CC ) /\ j e. ( 0 ... ( N - 1 ) ) ) -> ( ( j + 1 ) x. ( z ^ ( ( j + 1 ) - 1 ) ) ) = ( ( j + 1 ) x. ( z ^ j ) ) )
110	109	oveq2d 6317	. . . . . . . . 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 ) ) ) )
111	15	ad2antrr 730	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. CC ) /\ j e. ( 0 ... ( N - 1 ) ) ) -> A : NN0 --> CC )
112		peano2nn0 10910	. . . . . . . . . . . . 13 |- ( j e. NN0 -> ( j + 1 ) e. NN0 )
113	102, 112	syl 17	. . . . . . . . . . . 12 |- ( j e. ( 0 ... ( N - 1 ) ) -> ( j + 1 ) e. NN0 )
114	113	adantl 467	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. CC ) /\ j e. ( 0 ... ( N - 1 ) ) ) -> ( j + 1 ) e. NN0 )
115	111, 114	ffvelrnd 6034	. . . . . . . . . 10 |- ( ( ( ph /\ z e. CC ) /\ j e. ( 0 ... ( N - 1 ) ) ) -> ( A ` ( j + 1 ) ) e. CC )
116	114	nn0cnd 10927	. . . . . . . . . 10 |- ( ( ( ph /\ z e. CC ) /\ j e. ( 0 ... ( N - 1 ) ) ) -> ( j + 1 ) e. CC )
117		simplr 760	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. CC ) /\ j e. ( 0 ... ( N - 1 ) ) ) -> z e. CC )
118	117, 103	expcld 12415	. . . . . . . . . 10 |- ( ( ( ph /\ z e. CC ) /\ j e. ( 0 ... ( N - 1 ) ) ) -> ( z ^ j ) e. CC )
119	115, 116, 118	mulassd 9666	. . . . . . . . 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 ) ) ) )
120	115, 116	mulcomd 9664	. . . . . . . . . 10 |- ( ( ( ph /\ z e. CC ) /\ j e. ( 0 ... ( N - 1 ) ) ) -> ( ( A ` ( j + 1 ) ) x. ( j + 1 ) ) = ( ( j + 1 ) x. ( A ` ( j + 1 ) ) ) )
121	120	oveq1d 6316	. . . . . . . . 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 ) ) )
122	110, 119, 121	3eqtr2d 2469	. . . . . . . 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 ) ) )
123	122	sumeq2dv 13756	. . . . . . 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 ) ) )
124		1m1e0 10678	. . . . . . . . 9 |- ( 1 - 1 ) = 0
125	124	oveq1i 6311	. . . . . . . 8 |- ( ( 1 - 1 ) ... ( N - 1 ) ) = ( 0 ... ( N - 1 ) )
126	125	sumeq1i 13751	. . . . . . 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 ) ) ) )
127		oveq1 6308	. . . . . . . . . 10 |- ( k = j -> ( k + 1 ) = ( j + 1 ) )
128	127	fveq2d 5881	. . . . . . . . . 10 |- ( k = j -> ( A ` ( k + 1 ) ) = ( A ` ( j + 1 ) ) )
129	127, 128	oveq12d 6319	. . . . . . . . 9 |- ( k = j -> ( ( k + 1 ) x. ( A ` ( k + 1 ) ) ) = ( ( j + 1 ) x. ( A ` ( j + 1 ) ) ) )
130		oveq2 6309	. . . . . . . . 9 |- ( k = j -> ( z ^ k ) = ( z ^ j ) )
131	129, 130	oveq12d 6319	. . . . . . . 8 |- ( k = j -> ( ( ( k + 1 ) x. ( A ` ( k + 1 ) ) ) x. ( z ^ k ) ) = ( ( ( j + 1 ) x. ( A ` ( j + 1 ) ) ) x. ( z ^ j ) ) )
132	131	cbvsumv 13749	. . . . . . 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 ) )
133	123, 126, 132	3eqtr4g 2488	. . . . . 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 ) ) )
134		1zzd 10968	. . . . . . 7 |- ( ( ph /\ z e. CC ) -> 1 e. ZZ )
135	85	adantr 466	. . . . . . . 8 |- ( ( ph /\ z e. CC ) -> N e. NN0 )
136	135	nn0zd 11038	. . . . . . 7 |- ( ( ph /\ z e. CC ) -> N e. ZZ )
137	64, 74	mulcld 9663	. . . . . . 7 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> ( ( A ` k ) x. ( k x. ( z ^ ( k - 1 ) ) ) ) e. CC )
138		fveq2 5877	. . . . . . . 8 |- ( k = ( j + 1 ) -> ( A ` k ) = ( A ` ( j + 1 ) ) )
139		id 23	. . . . . . . . 9 |- ( k = ( j + 1 ) -> k = ( j + 1 ) )
140		oveq1 6308	. . . . . . . . . 10 |- ( k = ( j + 1 ) -> ( k - 1 ) = ( ( j + 1 ) - 1 ) )
141	140	oveq2d 6317	. . . . . . . . 9 |- ( k = ( j + 1 ) -> ( z ^ ( k - 1 ) ) = ( z ^ ( ( j + 1 ) - 1 ) ) )
142	139, 141	oveq12d 6319	. . . . . . . 8 |- ( k = ( j + 1 ) -> ( k x. ( z ^ ( k - 1 ) ) ) = ( ( j + 1 ) x. ( z ^ ( ( j + 1 ) - 1 ) ) ) )
143	138, 142	oveq12d 6319	. . . . . . 7 |- ( k = ( j + 1 ) -> ( ( A ` k ) x. ( k x. ( z ^ ( k - 1 ) ) ) ) = ( ( A ` ( j + 1 ) ) x. ( ( j + 1 ) x. ( z ^ ( ( j + 1 ) - 1 ) ) ) ) )
144	134, 134, 136, 137, 143	fsumshftm 13829	. . . . . 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 ) ) ) ) )
145		elfznn0 11887	. . . . . . . . . 10 |- ( k e. ( 0 ... ( N - 1 ) ) -> k e. NN0 )
146	145	adantl 467	. . . . . . . . 9 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... ( N - 1 ) ) ) -> k e. NN0 )
147		ovex 6329	. . . . . . . . 9 |- ( ( k + 1 ) x. ( A ` ( k + 1 ) ) ) e. _V
148		dvply1.b	. . . . . . . . . 10 |- B = ( k e. NN0 |-> ( ( k + 1 ) x. ( A ` ( k + 1 ) ) ) )
149	148	fvmpt2 5969	. . . . . . . . 9 |- ( ( k e. NN0 /\ ( ( k + 1 ) x. ( A ` ( k + 1 ) ) ) e. _V ) -> ( B ` k ) = ( ( k + 1 ) x. ( A ` ( k + 1 ) ) ) )
150	146, 147, 149	sylancl 666	. . . . . . . 8 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... ( N - 1 ) ) ) -> ( B ` k ) = ( ( k + 1 ) x. ( A ` ( k + 1 ) ) ) )
151	150	oveq1d 6316	. . . . . . 7 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... ( N - 1 ) ) ) -> ( ( B ` k ) x. ( z ^ k ) ) = ( ( ( k + 1 ) x. ( A ` ( k + 1 ) ) ) x. ( z ^ k ) ) )
152	151	sumeq2dv 13756	. . . . . 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 ) ) )
153	133, 144, 152	3eqtr4d 2473	. . . . 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 ) ) )
154	58, 101, 153	3eqtr3d 2471	. . . 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 ) ) )
155	154	mpteq2dva 4507	. . 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 ) ) ) )
156		dvply1.g	. . 3 |- ( ph -> G = ( z e. CC |-> sum_ k e. ( 0 ... ( N - 1 ) ) ( ( B ` k ) x. ( z ^ k ) ) ) )
157	155, 156	eqtr4d 2466	. 2 |- ( ph -> ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( A ` k ) x. if ( k = 0 , 0 , ( k x. ( z ^ ( k - 1 ) ) ) ) ) ) = G )
158	2, 51, 157	3eqtrd 2467	1 |- ( ph -> ( CC _D F ) = G )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 187   \/ wo 369   /\ wa 370   /\ w3a 982   = wceq 1437   e. wcel 1868   =/= wne 2618   _Vcvv 3081   \ cdif 3433   C_ wss 3436   ifcif 3909   {cpr 3998   |-> cmpt 4479   -->wf 5593   ` cfv 5597  (class class class)co 6301   CCcc 9537   RRcr 9538   0cc0 9539   1c1 9540   + caddc 9542   x. cmul 9544   - cmin 9860   NNcn 10609   NN0cn0 10869   ZZ>=cuz 11159   ...cfz 11784   ^cexp 12271   sum_csu 13739   ↾_t crest 15306   TopOpenctopn 15307  ℂ_fldccnfld 18957   Topctop 19903   _D cdv 22804

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593  ax-inf2 8148  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617  ax-addf 9618  ax-mulf 9619

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-int 4253  df-iun 4298  df-iin 4299  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4760  df-id 4764  df-po 4770  df-so 4771  df-fr 4808  df-se 4809  df-we 4810  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-pred 5395  df-ord 5441  df-on 5442  df-lim 5443  df-suc 5444  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-f1 5602  df-fo 5603  df-f1o 5604  df-fv 5605  df-isom 5606  df-riota 6263  df-ov 6304  df-oprab 6305  df-mpt2 6306  df-of 6541  df-om 6703  df-1st 6803  df-2nd 6804  df-supp 6922  df-wrecs 7032  df-recs 7094  df-rdg 7132  df-1o 7186  df-2o 7187  df-oadd 7190  df-er 7367  df-map 7478  df-pm 7479  df-ixp 7527  df-en 7574  df-dom 7575  df-sdom 7576  df-fin 7577  df-fsupp 7886  df-fi 7927  df-sup 7958  df-inf 7959  df-oi 8027  df-card 8374  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-icc 11642  df-fz 11785  df-fzo 11916  df-seq 12213  df-exp 12272  df-hash 12515  df-cj 13150  df-re 13151  df-im 13152  df-sqrt 13286  df-abs 13287  df-clim 13539  df-sum 13740  df-struct 15110  df-ndx 15111  df-slot 15112  df-base 15113  df-sets 15114  df-ress 15115  df-plusg 15190  df-mulr 15191  df-starv 15192  df-sca 15193  df-vsca 15194  df-ip 15195  df-tset 15196  df-ple 15197  df-ds 15199  df-unif 15200  df-hom 15201  df-cco 15202  df-rest 15308  df-topn 15309  df-0g 15327  df-gsum 15328  df-topgen 15329  df-pt 15330  df-prds 15333  df-xrs 15387  df-qtop 15393  df-imas 15394  df-xps 15397  df-mre 15479  df-mrc 15480  df-acs 15482  df-mgm 16475  df-sgrp 16514  df-mnd 16524  df-submnd 16570  df-mulg 16663  df-cntz 16958  df-cmn 17419  df-psmet 18949  df-xmet 18950  df-met 18951  df-bl 18952  df-mopn 18953  df-fbas 18954  df-fg 18955  df-cnfld 18958  df-top 19907  df-bases 19908  df-topon 19909  df-topsp 19910  df-cld 20020  df-ntr 20021  df-cls 20022  df-nei 20100  df-lp 20138  df-perf 20139  df-cn 20229  df-cnp 20230  df-haus 20317  df-tx 20563  df-hmeo 20756  df-fil 20847  df-fm 20939  df-flim 20940  df-flf 20941  df-xms 21321  df-ms 21322  df-tms 21323  df-cncf 21896  df-limc 22807  df-dv 22808

This theorem is referenced by:  dvply2g  23224