Theorem dvply1 22806

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 6312	. 2 |- ( ph -> ( CC _D F ) = ( CC _D ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( z ^ k ) ) ) ) )
3		eqid 2457	. . . . . 6 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
4	3	cnfldtop 21417	. . . . 5 |- ( TopOpen ` ℂfld ) e. Top
5	3	cnfldtopon 21416	. . . . . . 7 |- ( TopOpen ` ℂfld ) e. (TopOn ` CC )
6	5	toponunii 19560	. . . . . 6 |- CC = U. ( TopOpen ` ℂfld )
7	6	restid 14851	. . . . 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 2470	. . 3 |- ( TopOpen ` ℂfld ) = ( ( TopOpen ` ℂfld ) ↾t CC )
10		cnelprrecn 9602	. . . 4 |- CC e. { RR , CC }
11	10	a1i 11	. . 3 |- ( ph -> CC e. { RR , CC } )
12	6	topopn 19542	. . . 4 |- ( ( TopOpen ` ℂfld ) e. Top -> CC e. ( TopOpen ` ℂfld ) )
13	4, 12	mp1i 12	. . 3 |- ( ph -> CC e. ( TopOpen ` ℂfld ) )
14		fzfid 12086	. . 3 |- ( ph -> ( 0 ... N ) e. Fin )
15		dvply1.a	. . . . . . 7 |- ( ph -> A : NN0 --> CC )
16		elfznn0 11797	. . . . . . 7 |- ( k e. ( 0 ... N ) -> k e. NN0 )
17		ffvelrn 6030	. . . . . . 7 |- ( ( A : NN0 --> CC /\ k e. NN0 ) -> ( A ` k ) e. CC )
18	15, 16, 17	syl2an 477	. . . . . 6 |- ( ( ph /\ k e. ( 0 ... N ) ) -> ( A ` k ) e. CC )
19	18	adantr 465	. . . . 5 |- ( ( ( ph /\ k e. ( 0 ... N ) ) /\ z e. CC ) -> ( A ` k ) e. CC )
20		simpr 461	. . . . . 6 |- ( ( ( ph /\ k e. ( 0 ... N ) ) /\ z e. CC ) -> z e. CC )
21	16	ad2antlr 726	. . . . . 6 |- ( ( ( ph /\ k e. ( 0 ... N ) ) /\ z e. CC ) -> k e. NN0 )
22	20, 21	expcld 12313	. . . . 5 |- ( ( ( ph /\ k e. ( 0 ... N ) ) /\ z e. CC ) -> ( z ^ k ) e. CC )
23	19, 22	mulcld 9633	. . . 4 |- ( ( ( ph /\ k e. ( 0 ... N ) ) /\ z e. CC ) -> ( ( A ` k ) x. ( z ^ k ) ) e. CC )
24	23	3impa 1191	. . 3 |- ( ( ph /\ k e. ( 0 ... N ) /\ z e. CC ) -> ( ( A ` k ) x. ( z ^ k ) ) e. CC )
25	18	3adant3 1016	. . . 4 |- ( ( ph /\ k e. ( 0 ... N ) /\ z e. CC ) -> ( A ` k ) e. CC )
26		0cnd 9606	. . . . 5 |- ( ( ( ph /\ k e. ( 0 ... N ) /\ z e. CC ) /\ k = 0 ) -> 0 e. CC )
27		simpl2 1000	. . . . . . . 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 10875	. . . . . 6 |- ( ( ( ph /\ k e. ( 0 ... N ) /\ z e. CC ) /\ -. k = 0 ) -> k e. CC )
30		simpl3 1001	. . . . . . 7 |- ( ( ( ph /\ k e. ( 0 ... N ) /\ z e. CC ) /\ -. k = 0 ) -> z e. CC )
31		simpr 461	. . . . . . . . 9 |- ( ( ( ph /\ k e. ( 0 ... N ) /\ z e. CC ) /\ -. k = 0 ) -> -. k = 0 )
32		elnn0 10818	. . . . . . . . . 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 10858	. . . . . . . 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 12313	. . . . . 6 |- ( ( ( ph /\ k e. ( 0 ... N ) /\ z e. CC ) /\ -. k = 0 ) -> ( z ^ ( k - 1 ) ) e. CC )
39	29, 38	mulcld 9633	. . . . 5 |- ( ( ( ph /\ k e. ( 0 ... N ) /\ z e. CC ) /\ -. k = 0 ) -> ( k x. ( z ^ ( k - 1 ) ) ) e. CC )
40	26, 39	ifclda 3976	. . . 4 |- ( ( ph /\ k e. ( 0 ... N ) /\ z e. CC ) -> if ( k = 0 , 0 , ( k x. ( z ^ ( k - 1 ) ) ) ) e. CC )
41	25, 40	mulcld 9633	. . 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 9607	. . . . . 6 |- 0 e. _V
44		ovex 6324	. . . . . 6 |- ( k x. ( z ^ ( k - 1 ) ) ) e. _V
45	43, 44	ifex 4013	. . . . 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 466	. . . . 5 |- ( ( ph /\ k e. ( 0 ... N ) ) -> k e. NN0 )
48		dvexp2 22483	. . . . 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 22493	. . 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 22502	. 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 11739	. . . . . . . . . . 11 |- ( k e. ( 1 ... N ) -> k e. NN )
53	52	nnne0d 10601	. . . . . . . . . 10 |- ( k e. ( 1 ... N ) -> k =/= 0 )
54	53	neneqd 2659	. . . . . . . . 9 |- ( k e. ( 1 ... N ) -> -. k = 0 )
55	54	adantl 466	. . . . . . . 8 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> -. k = 0 )
56	55	iffalsed 3955	. . . . . . 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 6312	. . . . . 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 13537	. . . . 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 11149	. . . . . . 7 |- 1 e. ( ZZ>= ` 0 )
60		fzss1 11748	. . . . . . 7 |- ( 1 e. ( ZZ>= ` 0 ) -> ( 1 ... N ) C_ ( 0 ... N ) )
61	59, 60	mp1i 12	. . . . . 6 |- ( ( ph /\ z e. CC ) -> ( 1 ... N ) C_ ( 0 ... N ) )
62	15	adantr 465	. . . . . . . 8 |- ( ( ph /\ z e. CC ) -> A : NN0 --> CC )
63	52	nnnn0d 10873	. . . . . . . 8 |- ( k e. ( 1 ... N ) -> k e. NN0 )
64	62, 63, 17	syl2an 477	. . . . . . 7 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> ( A ` k ) e. CC )
65	53	adantl 466	. . . . . . . . . 10 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> k =/= 0 )
66	65	neneqd 2659	. . . . . . . . 9 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> -. k = 0 )
67	66	iffalsed 3955	. . . . . . . 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 466	. . . . . . . . . 10 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> k e. NN0 )
69	68	nn0cnd 10875	. . . . . . . . 9 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> k e. CC )
70		simplr 755	. . . . . . . . . 10 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> z e. CC )
71	52, 36	syl 16	. . . . . . . . . . 11 |- ( k e. ( 1 ... N ) -> ( k - 1 ) e. NN0 )
72	71	adantl 466	. . . . . . . . . 10 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> ( k - 1 ) e. NN0 )
73	70, 72	expcld 12313	. . . . . . . . 9 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> ( z ^ ( k - 1 ) ) e. CC )
74	69, 73	mulcld 9633	. . . . . . . 8 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> ( k x. ( z ^ ( k - 1 ) ) ) e. CC )
75	67, 74	eqeltrd 2545	. . . . . . 7 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> if ( k = 0 , 0 , ( k x. ( z ^ ( k - 1 ) ) ) ) e. CC )
76	64, 75	mulcld 9633	. . . . . 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 3623	. . . . . . . . . . . 12 |- ( k e. ( ( 0 ... N ) \ ( 1 ... N ) ) -> -. k e. ( 1 ... N ) )
78		0p1e1 10668	. . . . . . . . . . . . . 14 |- ( 0 + 1 ) = 1
79	78	oveq1i 6306	. . . . . . . . . . . . 13 |- ( ( 0 + 1 ) ... N ) = ( 1 ... N )
80	79	eleq2i 2535	. . . . . . . . . . . 12 |- ( k e. ( ( 0 + 1 ) ... N ) <-> k e. ( 1 ... N ) )
81	77, 80	sylnibr 305	. . . . . . . . . . 11 |- ( k e. ( ( 0 ... N ) \ ( 1 ... N ) ) -> -. k e. ( ( 0 + 1 ) ... N ) )
82	81	adantl 466	. . . . . . . . . 10 |- ( ( ( ph /\ z e. CC ) /\ k e. ( ( 0 ... N ) \ ( 1 ... N ) ) ) -> -. k e. ( ( 0 + 1 ) ... N ) )
83		eldifi 3622	. . . . . . . . . . . 12 |- ( k e. ( ( 0 ... N ) \ ( 1 ... N ) ) -> k e. ( 0 ... N ) )
84	83	adantl 466	. . . . . . . . . . 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 11140	. . . . . . . . . . . . . 14 |- NN0 = ( ZZ>= ` 0 )
87	85, 86	syl6eleq 2555	. . . . . . . . . . . . 13 |- ( ph -> N e. ( ZZ>= ` 0 ) )
88	87	ad2antrr 725	. . . . . . . . . . . 12 |- ( ( ( ph /\ z e. CC ) /\ k e. ( ( 0 ... N ) \ ( 1 ... N ) ) ) -> N e. ( ZZ>= ` 0 ) )
89		elfzp12 11783	. . . . . . . . . . . 12 |- ( N e. ( ZZ>= ` 0 ) -> ( k e. ( 0 ... N ) <-> ( k = 0 \/ k e. ( ( 0 + 1 ) ... N ) ) ) )
90	88, 89	syl 16	. . . . . . . . . . 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 210	. . . . . . . . . 10 |- ( ( ( ph /\ z e. CC ) /\ k e. ( ( 0 ... N ) \ ( 1 ... N ) ) ) -> ( k = 0 \/ k e. ( ( 0 + 1 ) ... N ) ) )
92		orel2 383	. . . . . . . . . 10 |- ( -. k e. ( ( 0 + 1 ) ... N ) -> ( ( k = 0 \/ k e. ( ( 0 + 1 ) ... N ) ) -> k = 0 ) )
93	82, 91, 92	sylc 60	. . . . . . . . 9 |- ( ( ( ph /\ z e. CC ) /\ k e. ( ( 0 ... N ) \ ( 1 ... N ) ) ) -> k = 0 )
94	93	iftrued 3952	. . . . . . . 8 |- ( ( ( ph /\ z e. CC ) /\ k e. ( ( 0 ... N ) \ ( 1 ... N ) ) ) -> if ( k = 0 , 0 , ( k x. ( z ^ ( k - 1 ) ) ) ) = 0 )
95	94	oveq2d 6312	. . . . . . 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 477	. . . . . . . . 9 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... N ) ) -> ( A ` k ) e. CC )
97	96	mul01d 9796	. . . . . . . 8 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... N ) ) -> ( ( A ` k ) x. 0 ) = 0 )
98	83, 97	sylan2 474	. . . . . . 7 |- ( ( ( ph /\ z e. CC ) /\ k e. ( ( 0 ... N ) \ ( 1 ... N ) ) ) -> ( ( A ` k ) x. 0 ) = 0 )
99	95, 98	eqtrd 2498	. . . . . 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 12086	. . . . . 6 |- ( ( ph /\ z e. CC ) -> ( 0 ... N ) e. Fin )
101	61, 76, 99, 100	fsumss 13559	. . . . 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 11797	. . . . . . . . . . . . . . 15 |- ( j e. ( 0 ... ( N - 1 ) ) -> j e. NN0 )
103	102	adantl 466	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ z e. CC ) /\ j e. ( 0 ... ( N - 1 ) ) ) -> j e. NN0 )
104	103	nn0cnd 10875	. . . . . . . . . . . . 13 |- ( ( ( ph /\ z e. CC ) /\ j e. ( 0 ... ( N - 1 ) ) ) -> j e. CC )
105		ax-1cn 9567	. . . . . . . . . . . . 13 |- 1 e. CC
106		pncan 9845	. . . . . . . . . . . . 13 |- ( ( j e. CC /\ 1 e. CC ) -> ( ( j + 1 ) - 1 ) = j )
107	104, 105, 106	sylancl 662	. . . . . . . . . . . 12 |- ( ( ( ph /\ z e. CC ) /\ j e. ( 0 ... ( N - 1 ) ) ) -> ( ( j + 1 ) - 1 ) = j )
108	107	oveq2d 6312	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. CC ) /\ j e. ( 0 ... ( N - 1 ) ) ) -> ( z ^ ( ( j + 1 ) - 1 ) ) = ( z ^ j ) )
109	108	oveq2d 6312	. . . . . . . . . 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 6312	. . . . . . . . 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 725	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. CC ) /\ j e. ( 0 ... ( N - 1 ) ) ) -> A : NN0 --> CC )
112		peano2nn0 10857	. . . . . . . . . . . . 13 |- ( j e. NN0 -> ( j + 1 ) e. NN0 )
113	102, 112	syl 16	. . . . . . . . . . . 12 |- ( j e. ( 0 ... ( N - 1 ) ) -> ( j + 1 ) e. NN0 )
114	113	adantl 466	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. CC ) /\ j e. ( 0 ... ( N - 1 ) ) ) -> ( j + 1 ) e. NN0 )
115	111, 114	ffvelrnd 6033	. . . . . . . . . 10 |- ( ( ( ph /\ z e. CC ) /\ j e. ( 0 ... ( N - 1 ) ) ) -> ( A ` ( j + 1 ) ) e. CC )
116	114	nn0cnd 10875	. . . . . . . . . 10 |- ( ( ( ph /\ z e. CC ) /\ j e. ( 0 ... ( N - 1 ) ) ) -> ( j + 1 ) e. CC )
117		simplr 755	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. CC ) /\ j e. ( 0 ... ( N - 1 ) ) ) -> z e. CC )
118	117, 103	expcld 12313	. . . . . . . . . 10 |- ( ( ( ph /\ z e. CC ) /\ j e. ( 0 ... ( N - 1 ) ) ) -> ( z ^ j ) e. CC )
119	115, 116, 118	mulassd 9636	. . . . . . . . 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 9634	. . . . . . . . . 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 6311	. . . . . . . . 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 2504	. . . . . . . 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 13537	. . . . . . 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 10625	. . . . . . . . 9 |- ( 1 - 1 ) = 0
125	124	oveq1i 6306	. . . . . . . 8 |- ( ( 1 - 1 ) ... ( N - 1 ) ) = ( 0 ... ( N - 1 ) )
126	125	sumeq1i 13532	. . . . . . 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 6303	. . . . . . . . . 10 |- ( k = j -> ( k + 1 ) = ( j + 1 ) )
128	127	fveq2d 5876	. . . . . . . . . 10 |- ( k = j -> ( A ` ( k + 1 ) ) = ( A ` ( j + 1 ) ) )
129	127, 128	oveq12d 6314	. . . . . . . . 9 |- ( k = j -> ( ( k + 1 ) x. ( A ` ( k + 1 ) ) ) = ( ( j + 1 ) x. ( A ` ( j + 1 ) ) ) )
130		oveq2 6304	. . . . . . . . 9 |- ( k = j -> ( z ^ k ) = ( z ^ j ) )
131	129, 130	oveq12d 6314	. . . . . . . 8 |- ( k = j -> ( ( ( k + 1 ) x. ( A ` ( k + 1 ) ) ) x. ( z ^ k ) ) = ( ( ( j + 1 ) x. ( A ` ( j + 1 ) ) ) x. ( z ^ j ) ) )
132	131	cbvsumv 13530	. . . . . . 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 2523	. . . . . 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 10916	. . . . . . 7 |- ( ( ph /\ z e. CC ) -> 1 e. ZZ )
135	85	adantr 465	. . . . . . . 8 |- ( ( ph /\ z e. CC ) -> N e. NN0 )
136	135	nn0zd 10988	. . . . . . 7 |- ( ( ph /\ z e. CC ) -> N e. ZZ )
137	64, 74	mulcld 9633	. . . . . . 7 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> ( ( A ` k ) x. ( k x. ( z ^ ( k - 1 ) ) ) ) e. CC )
138		fveq2 5872	. . . . . . . 8 |- ( k = ( j + 1 ) -> ( A ` k ) = ( A ` ( j + 1 ) ) )
139		id 22	. . . . . . . . 9 |- ( k = ( j + 1 ) -> k = ( j + 1 ) )
140		oveq1 6303	. . . . . . . . . 10 |- ( k = ( j + 1 ) -> ( k - 1 ) = ( ( j + 1 ) - 1 ) )
141	140	oveq2d 6312	. . . . . . . . 9 |- ( k = ( j + 1 ) -> ( z ^ ( k - 1 ) ) = ( z ^ ( ( j + 1 ) - 1 ) ) )
142	139, 141	oveq12d 6314	. . . . . . . 8 |- ( k = ( j + 1 ) -> ( k x. ( z ^ ( k - 1 ) ) ) = ( ( j + 1 ) x. ( z ^ ( ( j + 1 ) - 1 ) ) ) )
143	138, 142	oveq12d 6314	. . . . . . 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 13608	. . . . . 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 11797	. . . . . . . . . 10 |- ( k e. ( 0 ... ( N - 1 ) ) -> k e. NN0 )
146	145	adantl 466	. . . . . . . . 9 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... ( N - 1 ) ) ) -> k e. NN0 )
147		ovex 6324	. . . . . . . . 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 5964	. . . . . . . . 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 662	. . . . . . . 8 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... ( N - 1 ) ) ) -> ( B ` k ) = ( ( k + 1 ) x. ( A ` ( k + 1 ) ) ) )
151	150	oveq1d 6311	. . . . . . 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 13537	. . . . . 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 2508	. . . . 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 2506	. . . 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 4543	. . 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 2501	. 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 2502	1 |- ( ph -> ( CC _D F ) = G )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   \/ wo 368   /\ wa 369   /\ w3a 973   = wceq 1395   e. wcel 1819   =/= wne 2652   _Vcvv 3109   \ cdif 3468   C_ wss 3471   ifcif 3944   {cpr 4034   |-> cmpt 4515   -->wf 5590   ` cfv 5594  (class class class)co 6296   CCcc 9507   RRcr 9508   0cc0 9509   1c1 9510   + caddc 9512   x. cmul 9514   - cmin 9824   NNcn 10556   NN0cn0 10816   ZZ>=cuz 11106   ...cfz 11697   ^cexp 12169   sum_csu 13520   ↾_t crest 14838   TopOpenctopn 14839  ℂ_fldccnfld 18547   Topctop 19521   _D cdv 22393

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587  ax-addf 9588  ax-mulf 9589

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-er 7329  df-map 7440  df-pm 7441  df-ixp 7489  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fsupp 7848  df-fi 7889  df-sup 7919  df-oi 7953  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-q 11208  df-rp 11246  df-xneg 11343  df-xadd 11344  df-xmul 11345  df-icc 11561  df-fz 11698  df-fzo 11822  df-seq 12111  df-exp 12170  df-hash 12409  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081  df-clim 13323  df-sum 13521  df-struct 14646  df-ndx 14647  df-slot 14648  df-base 14649  df-sets 14650  df-ress 14651  df-plusg 14725  df-mulr 14726  df-starv 14727  df-sca 14728  df-vsca 14729  df-ip 14730  df-tset 14731  df-ple 14732  df-ds 14734  df-unif 14735  df-hom 14736  df-cco 14737  df-rest 14840  df-topn 14841  df-0g 14859  df-gsum 14860  df-topgen 14861  df-pt 14862  df-prds 14865  df-xrs 14919  df-qtop 14924  df-imas 14925  df-xps 14927  df-mre 15003  df-mrc 15004  df-acs 15006  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-submnd 16094  df-mulg 16187  df-cntz 16482  df-cmn 16927  df-psmet 18538  df-xmet 18539  df-met 18540  df-bl 18541  df-mopn 18542  df-fbas 18543  df-fg 18544  df-cnfld 18548  df-top 19526  df-bases 19528  df-topon 19529  df-topsp 19530  df-cld 19647  df-ntr 19648  df-cls 19649  df-nei 19726  df-lp 19764  df-perf 19765  df-cn 19855  df-cnp 19856  df-haus 19943  df-tx 20189  df-hmeo 20382  df-fil 20473  df-fm 20565  df-flim 20566  df-flf 20567  df-xms 20949  df-ms 20950  df-tms 20951  df-cncf 21508  df-limc 22396  df-dv 22397

This theorem is referenced by:  dvply2g  22807