Theorem dvply1 23316

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 6324	. 2 |- ( ph -> ( CC _D F ) = ( CC _D ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( z ^ k ) ) ) ) )
3		eqid 2471	. . . . . 6 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
4	3	cnfldtop 21882	. . . . 5 |- ( TopOpen ` ℂfld ) e. Top
5	3	cnfldtopon 21881	. . . . . . 7 |- ( TopOpen ` ℂfld ) e. (TopOn ` CC )
6	5	toponunii 20024	. . . . . 6 |- CC = U. ( TopOpen ` ℂfld )
7	6	restid 15410	. . . . 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 2480	. . 3 |- ( TopOpen ` ℂfld ) = ( ( TopOpen ` ℂfld ) ↾t CC )
10		cnelprrecn 9650	. . . 4 |- CC e. { RR , CC }
11	10	a1i 11	. . 3 |- ( ph -> CC e. { RR , CC } )
12	6	topopn 20013	. . . 4 |- ( ( TopOpen ` ℂfld ) e. Top -> CC e. ( TopOpen ` ℂfld ) )
13	4, 12	mp1i 13	. . 3 |- ( ph -> CC e. ( TopOpen ` ℂfld ) )
14		fzfid 12224	. . 3 |- ( ph -> ( 0 ... N ) e. Fin )
15		dvply1.a	. . . . . . 7 |- ( ph -> A : NN0 --> CC )
16		elfznn0 11913	. . . . . . 7 |- ( k e. ( 0 ... N ) -> k e. NN0 )
17		ffvelrn 6035	. . . . . . 7 |- ( ( A : NN0 --> CC /\ k e. NN0 ) -> ( A ` k ) e. CC )
18	15, 16, 17	syl2an 485	. . . . . 6 |- ( ( ph /\ k e. ( 0 ... N ) ) -> ( A ` k ) e. CC )
19	18	adantr 472	. . . . 5 |- ( ( ( ph /\ k e. ( 0 ... N ) ) /\ z e. CC ) -> ( A ` k ) e. CC )
20		simpr 468	. . . . . 6 |- ( ( ( ph /\ k e. ( 0 ... N ) ) /\ z e. CC ) -> z e. CC )
21	16	ad2antlr 741	. . . . . 6 |- ( ( ( ph /\ k e. ( 0 ... N ) ) /\ z e. CC ) -> k e. NN0 )
22	20, 21	expcld 12454	. . . . 5 |- ( ( ( ph /\ k e. ( 0 ... N ) ) /\ z e. CC ) -> ( z ^ k ) e. CC )
23	19, 22	mulcld 9681	. . . 4 |- ( ( ( ph /\ k e. ( 0 ... N ) ) /\ z e. CC ) -> ( ( A ` k ) x. ( z ^ k ) ) e. CC )
24	23	3impa 1226	. . 3 |- ( ( ph /\ k e. ( 0 ... N ) /\ z e. CC ) -> ( ( A ` k ) x. ( z ^ k ) ) e. CC )
25	18	3adant3 1050	. . . 4 |- ( ( ph /\ k e. ( 0 ... N ) /\ z e. CC ) -> ( A ` k ) e. CC )
26		0cnd 9654	. . . . 5 |- ( ( ( ph /\ k e. ( 0 ... N ) /\ z e. CC ) /\ k = 0 ) -> 0 e. CC )
27		simpl2 1034	. . . . . . . 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 10951	. . . . . 6 |- ( ( ( ph /\ k e. ( 0 ... N ) /\ z e. CC ) /\ -. k = 0 ) -> k e. CC )
30		simpl3 1035	. . . . . . 7 |- ( ( ( ph /\ k e. ( 0 ... N ) /\ z e. CC ) /\ -. k = 0 ) -> z e. CC )
31		simpr 468	. . . . . . . . 9 |- ( ( ( ph /\ k e. ( 0 ... N ) /\ z e. CC ) /\ -. k = 0 ) -> -. k = 0 )
32		elnn0 10895	. . . . . . . . . 10 |- ( k e. NN0 <-> ( k e. NN \/ k = 0 ) )
33	28, 32	sylib 201	. . . . . . . . 9 |- ( ( ( ph /\ k e. ( 0 ... N ) /\ z e. CC ) /\ -. k = 0 ) -> ( k e. NN \/ k = 0 ) )
34		orel2 390	. . . . . . . . 9 |- ( -. k = 0 -> ( ( k e. NN \/ k = 0 ) -> k e. NN ) )
35	31, 33, 34	sylc 61	. . . . . . . 8 |- ( ( ( ph /\ k e. ( 0 ... N ) /\ z e. CC ) /\ -. k = 0 ) -> k e. NN )
36		nnm1nn0 10935	. . . . . . . 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 12454	. . . . . 6 |- ( ( ( ph /\ k e. ( 0 ... N ) /\ z e. CC ) /\ -. k = 0 ) -> ( z ^ ( k - 1 ) ) e. CC )
39	29, 38	mulcld 9681	. . . . 5 |- ( ( ( ph /\ k e. ( 0 ... N ) /\ z e. CC ) /\ -. k = 0 ) -> ( k x. ( z ^ ( k - 1 ) ) ) e. CC )
40	26, 39	ifclda 3904	. . . 4 |- ( ( ph /\ k e. ( 0 ... N ) /\ z e. CC ) -> if ( k = 0 , 0 , ( k x. ( z ^ ( k - 1 ) ) ) ) e. CC )
41	25, 40	mulcld 9681	. . 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 9655	. . . . . 6 |- 0 e. _V
44		ovex 6336	. . . . . 6 |- ( k x. ( z ^ ( k - 1 ) ) ) e. _V
45	43, 44	ifex 3940	. . . . 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 473	. . . . 5 |- ( ( ph /\ k e. ( 0 ... N ) ) -> k e. NN0 )
48		dvexp2 22987	. . . . 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 22997	. . 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 23006	. 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 11854	. . . . . . . . . . 11 |- ( k e. ( 1 ... N ) -> k e. NN )
53	52	nnne0d 10676	. . . . . . . . . 10 |- ( k e. ( 1 ... N ) -> k =/= 0 )
54	53	neneqd 2648	. . . . . . . . 9 |- ( k e. ( 1 ... N ) -> -. k = 0 )
55	54	adantl 473	. . . . . . . 8 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> -. k = 0 )
56	55	iffalsed 3883	. . . . . . 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 6324	. . . . . 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 13846	. . . . 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 11226	. . . . . . 7 |- 1 e. ( ZZ>= ` 0 )
60		fzss1 11863	. . . . . . 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 472	. . . . . . . 8 |- ( ( ph /\ z e. CC ) -> A : NN0 --> CC )
63	52	nnnn0d 10949	. . . . . . . 8 |- ( k e. ( 1 ... N ) -> k e. NN0 )
64	62, 63, 17	syl2an 485	. . . . . . 7 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> ( A ` k ) e. CC )
65	53	adantl 473	. . . . . . . . . 10 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> k =/= 0 )
66	65	neneqd 2648	. . . . . . . . 9 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> -. k = 0 )
67	66	iffalsed 3883	. . . . . . . 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 473	. . . . . . . . . 10 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> k e. NN0 )
69	68	nn0cnd 10951	. . . . . . . . 9 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> k e. CC )
70		simplr 770	. . . . . . . . . 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 473	. . . . . . . . . 10 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> ( k - 1 ) e. NN0 )
73	70, 72	expcld 12454	. . . . . . . . 9 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> ( z ^ ( k - 1 ) ) e. CC )
74	69, 73	mulcld 9681	. . . . . . . 8 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> ( k x. ( z ^ ( k - 1 ) ) ) e. CC )
75	67, 74	eqeltrd 2549	. . . . . . 7 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> if ( k = 0 , 0 , ( k x. ( z ^ ( k - 1 ) ) ) ) e. CC )
76	64, 75	mulcld 9681	. . . . . 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 3545	. . . . . . . . . . . 12 |- ( k e. ( ( 0 ... N ) \ ( 1 ... N ) ) -> -. k e. ( 1 ... N ) )
78		0p1e1 10743	. . . . . . . . . . . . . 14 |- ( 0 + 1 ) = 1
79	78	oveq1i 6318	. . . . . . . . . . . . 13 |- ( ( 0 + 1 ) ... N ) = ( 1 ... N )
80	79	eleq2i 2541	. . . . . . . . . . . 12 |- ( k e. ( ( 0 + 1 ) ... N ) <-> k e. ( 1 ... N ) )
81	77, 80	sylnibr 312	. . . . . . . . . . 11 |- ( k e. ( ( 0 ... N ) \ ( 1 ... N ) ) -> -. k e. ( ( 0 + 1 ) ... N ) )
82	81	adantl 473	. . . . . . . . . 10 |- ( ( ( ph /\ z e. CC ) /\ k e. ( ( 0 ... N ) \ ( 1 ... N ) ) ) -> -. k e. ( ( 0 + 1 ) ... N ) )
83		eldifi 3544	. . . . . . . . . . . 12 |- ( k e. ( ( 0 ... N ) \ ( 1 ... N ) ) -> k e. ( 0 ... N ) )
84	83	adantl 473	. . . . . . . . . . 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 11217	. . . . . . . . . . . . . 14 |- NN0 = ( ZZ>= ` 0 )
87	85, 86	syl6eleq 2559	. . . . . . . . . . . . 13 |- ( ph -> N e. ( ZZ>= ` 0 ) )
88	87	ad2antrr 740	. . . . . . . . . . . 12 |- ( ( ( ph /\ z e. CC ) /\ k e. ( ( 0 ... N ) \ ( 1 ... N ) ) ) -> N e. ( ZZ>= ` 0 ) )
89		elfzp12 11899	. . . . . . . . . . . 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 215	. . . . . . . . . 10 |- ( ( ( ph /\ z e. CC ) /\ k e. ( ( 0 ... N ) \ ( 1 ... N ) ) ) -> ( k = 0 \/ k e. ( ( 0 + 1 ) ... N ) ) )
92		orel2 390	. . . . . . . . . 10 |- ( -. k e. ( ( 0 + 1 ) ... N ) -> ( ( k = 0 \/ k e. ( ( 0 + 1 ) ... N ) ) -> k = 0 ) )
93	82, 91, 92	sylc 61	. . . . . . . . 9 |- ( ( ( ph /\ z e. CC ) /\ k e. ( ( 0 ... N ) \ ( 1 ... N ) ) ) -> k = 0 )
94	93	iftrued 3880	. . . . . . . 8 |- ( ( ( ph /\ z e. CC ) /\ k e. ( ( 0 ... N ) \ ( 1 ... N ) ) ) -> if ( k = 0 , 0 , ( k x. ( z ^ ( k - 1 ) ) ) ) = 0 )
95	94	oveq2d 6324	. . . . . . 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 485	. . . . . . . . 9 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... N ) ) -> ( A ` k ) e. CC )
97	96	mul01d 9850	. . . . . . . 8 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... N ) ) -> ( ( A ` k ) x. 0 ) = 0 )
98	83, 97	sylan2 482	. . . . . . 7 |- ( ( ( ph /\ z e. CC ) /\ k e. ( ( 0 ... N ) \ ( 1 ... N ) ) ) -> ( ( A ` k ) x. 0 ) = 0 )
99	95, 98	eqtrd 2505	. . . . . 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 12224	. . . . . 6 |- ( ( ph /\ z e. CC ) -> ( 0 ... N ) e. Fin )
101	61, 76, 99, 100	fsumss 13868	. . . . 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 11913	. . . . . . . . . . . . . . 15 |- ( j e. ( 0 ... ( N - 1 ) ) -> j e. NN0 )
103	102	adantl 473	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ z e. CC ) /\ j e. ( 0 ... ( N - 1 ) ) ) -> j e. NN0 )
104	103	nn0cnd 10951	. . . . . . . . . . . . 13 |- ( ( ( ph /\ z e. CC ) /\ j e. ( 0 ... ( N - 1 ) ) ) -> j e. CC )
105		ax-1cn 9615	. . . . . . . . . . . . 13 |- 1 e. CC
106		pncan 9901	. . . . . . . . . . . . 13 |- ( ( j e. CC /\ 1 e. CC ) -> ( ( j + 1 ) - 1 ) = j )
107	104, 105, 106	sylancl 675	. . . . . . . . . . . 12 |- ( ( ( ph /\ z e. CC ) /\ j e. ( 0 ... ( N - 1 ) ) ) -> ( ( j + 1 ) - 1 ) = j )
108	107	oveq2d 6324	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. CC ) /\ j e. ( 0 ... ( N - 1 ) ) ) -> ( z ^ ( ( j + 1 ) - 1 ) ) = ( z ^ j ) )
109	108	oveq2d 6324	. . . . . . . . . 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 6324	. . . . . . . . 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 740	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. CC ) /\ j e. ( 0 ... ( N - 1 ) ) ) -> A : NN0 --> CC )
112		peano2nn0 10934	. . . . . . . . . . . . 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 473	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. CC ) /\ j e. ( 0 ... ( N - 1 ) ) ) -> ( j + 1 ) e. NN0 )
115	111, 114	ffvelrnd 6038	. . . . . . . . . 10 |- ( ( ( ph /\ z e. CC ) /\ j e. ( 0 ... ( N - 1 ) ) ) -> ( A ` ( j + 1 ) ) e. CC )
116	114	nn0cnd 10951	. . . . . . . . . 10 |- ( ( ( ph /\ z e. CC ) /\ j e. ( 0 ... ( N - 1 ) ) ) -> ( j + 1 ) e. CC )
117		simplr 770	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. CC ) /\ j e. ( 0 ... ( N - 1 ) ) ) -> z e. CC )
118	117, 103	expcld 12454	. . . . . . . . . 10 |- ( ( ( ph /\ z e. CC ) /\ j e. ( 0 ... ( N - 1 ) ) ) -> ( z ^ j ) e. CC )
119	115, 116, 118	mulassd 9684	. . . . . . . . 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 9682	. . . . . . . . . 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 6323	. . . . . . . . 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 2511	. . . . . . . 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 13846	. . . . . . 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 10700	. . . . . . . . 9 |- ( 1 - 1 ) = 0
125	124	oveq1i 6318	. . . . . . . 8 |- ( ( 1 - 1 ) ... ( N - 1 ) ) = ( 0 ... ( N - 1 ) )
126	125	sumeq1i 13841	. . . . . . 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 6315	. . . . . . . . . 10 |- ( k = j -> ( k + 1 ) = ( j + 1 ) )
128	127	fveq2d 5883	. . . . . . . . . 10 |- ( k = j -> ( A ` ( k + 1 ) ) = ( A ` ( j + 1 ) ) )
129	127, 128	oveq12d 6326	. . . . . . . . 9 |- ( k = j -> ( ( k + 1 ) x. ( A ` ( k + 1 ) ) ) = ( ( j + 1 ) x. ( A ` ( j + 1 ) ) ) )
130		oveq2 6316	. . . . . . . . 9 |- ( k = j -> ( z ^ k ) = ( z ^ j ) )
131	129, 130	oveq12d 6326	. . . . . . . 8 |- ( k = j -> ( ( ( k + 1 ) x. ( A ` ( k + 1 ) ) ) x. ( z ^ k ) ) = ( ( ( j + 1 ) x. ( A ` ( j + 1 ) ) ) x. ( z ^ j ) ) )
132	131	cbvsumv 13839	. . . . . . 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 2530	. . . . . 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 10992	. . . . . . 7 |- ( ( ph /\ z e. CC ) -> 1 e. ZZ )
135	85	adantr 472	. . . . . . . 8 |- ( ( ph /\ z e. CC ) -> N e. NN0 )
136	135	nn0zd 11061	. . . . . . 7 |- ( ( ph /\ z e. CC ) -> N e. ZZ )
137	64, 74	mulcld 9681	. . . . . . 7 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> ( ( A ` k ) x. ( k x. ( z ^ ( k - 1 ) ) ) ) e. CC )
138		fveq2 5879	. . . . . . . 8 |- ( k = ( j + 1 ) -> ( A ` k ) = ( A ` ( j + 1 ) ) )
139		id 22	. . . . . . . . 9 |- ( k = ( j + 1 ) -> k = ( j + 1 ) )
140		oveq1 6315	. . . . . . . . . 10 |- ( k = ( j + 1 ) -> ( k - 1 ) = ( ( j + 1 ) - 1 ) )
141	140	oveq2d 6324	. . . . . . . . 9 |- ( k = ( j + 1 ) -> ( z ^ ( k - 1 ) ) = ( z ^ ( ( j + 1 ) - 1 ) ) )
142	139, 141	oveq12d 6326	. . . . . . . 8 |- ( k = ( j + 1 ) -> ( k x. ( z ^ ( k - 1 ) ) ) = ( ( j + 1 ) x. ( z ^ ( ( j + 1 ) - 1 ) ) ) )
143	138, 142	oveq12d 6326	. . . . . . 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 13919	. . . . . 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 11913	. . . . . . . . . 10 |- ( k e. ( 0 ... ( N - 1 ) ) -> k e. NN0 )
146	145	adantl 473	. . . . . . . . 9 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... ( N - 1 ) ) ) -> k e. NN0 )
147		ovex 6336	. . . . . . . . 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 5972	. . . . . . . . 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 675	. . . . . . . 8 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... ( N - 1 ) ) ) -> ( B ` k ) = ( ( k + 1 ) x. ( A ` ( k + 1 ) ) ) )
151	150	oveq1d 6323	. . . . . . 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 13846	. . . . . 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 2515	. . . . 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 2513	. . . 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 4482	. . 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 2508	. 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 2509	1 |- ( ph -> ( CC _D F ) = G )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 189   \/ wo 375   /\ wa 376   /\ w3a 1007   = wceq 1452   e. wcel 1904   =/= wne 2641   _Vcvv 3031   \ cdif 3387   C_ wss 3390   ifcif 3872   {cpr 3961   |-> cmpt 4454   -->wf 5585   ` cfv 5589  (class class class)co 6308   CCcc 9555   RRcr 9556   0cc0 9557   1c1 9558   + caddc 9560   x. cmul 9562   - cmin 9880   NNcn 10631   NN0cn0 10893   ZZ>=cuz 11182   ...cfz 11810   ^cexp 12310   sum_csu 13829   ↾_t crest 15397   TopOpenctopn 15398  ℂ_fldccnfld 19047   Topctop 19994   _D cdv 22897

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-addf 9636  ax-mulf 9637

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-fi 7943  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-icc 11667  df-fz 11811  df-fzo 11943  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-clim 13629  df-sum 13830  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-starv 15283  df-sca 15284  df-vsca 15285  df-ip 15286  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-hom 15292  df-cco 15293  df-rest 15399  df-topn 15400  df-0g 15418  df-gsum 15419  df-topgen 15420  df-pt 15421  df-prds 15424  df-xrs 15478  df-qtop 15484  df-imas 15485  df-xps 15488  df-mre 15570  df-mrc 15571  df-acs 15573  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-mulg 16754  df-cntz 17049  df-cmn 17510  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-fbas 19044  df-fg 19045  df-cnfld 19048  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-cld 20111  df-ntr 20112  df-cls 20113  df-nei 20191  df-lp 20229  df-perf 20230  df-cn 20320  df-cnp 20321  df-haus 20408  df-tx 20654  df-hmeo 20847  df-fil 20939  df-fm 21031  df-flim 21032  df-flf 21033  df-xms 21413  df-ms 21414  df-tms 21415  df-cncf 21988  df-limc 22900  df-dv 22901

This theorem is referenced by:  dvply2g  23317