Theorem dvply2g 16004

Description: The derivative of a polynomial with coefficients in a subring is a polynomial with coefficients in the same ring. (Contributed by Mario Carneiro, 1-Jan-2017.) (Revised by GG, 30-Apr-2025.)

Assertion

Ref	Expression
dvply2g	|- ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) -> ( CC _D F ) e. (Poly ` S ) )

Proof of Theorem dvply2g

Dummy variables a b c d p u v k z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elply2 15973	. . . 4 |- ( F e. (Poly ` S ) <-> ( S C_ CC /\ E. d e. NN0 E. p e. ( ( S u. { 0 } ) ^m NN0 ) ( ( p " ( ZZ>= ` ( d + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( p ` k ) x. ( z ^ k ) ) ) ) ) )
2	1	simprbi 275	. . 3 |- ( F e. (Poly ` S ) -> E. d e. NN0 E. p e. ( ( S u. { 0 } ) ^m NN0 ) ( ( p " ( ZZ>= ` ( d + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( p ` k ) x. ( z ^ k ) ) ) ) )
3	2	adantl 277	. 2 |- ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) -> E. d e. NN0 E. p e. ( ( S u. { 0 } ) ^m NN0 ) ( ( p " ( ZZ>= ` ( d + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( p ` k ) x. ( z ^ k ) ) ) ) )
4		plyf 15975	. . . . . . . . . 10 |- ( F e. (Poly ` S ) -> F : CC --> CC )
5	4	adantl 277	. . . . . . . . 9 |- ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) -> F : CC --> CC )
6	5	feqmptd 5759	. . . . . . . 8 |- ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) -> F = ( a e. CC |-> ( F ` a ) ) )
7	6	ad2antrr 492	. . . . . . 7 |- ( ( ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) /\ ( d e. NN0 /\ p e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ ( ( p " ( ZZ>= ` ( d + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( p ` k ) x. ( z ^ k ) ) ) ) ) -> F = ( a e. CC |-> ( F ` a ) ) )
8		simplrl 541	. . . . . . . . . 10 |- ( ( ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) /\ ( d e. NN0 /\ p e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ ( ( p " ( ZZ>= ` ( d + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( p ` k ) x. ( z ^ k ) ) ) ) ) -> d e. NN0 )
9	8	adantr 276	. . . . . . . . 9 |- ( ( ( ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) /\ ( d e. NN0 /\ p e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ ( ( p " ( ZZ>= ` ( d + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( p ` k ) x. ( z ^ k ) ) ) ) ) /\ a e. CC ) -> d e. NN0 )
10		elmapi 6948	. . . . . . . . . . . . 13 |- ( p e. ( ( S u. { 0 } ) ^m NN0 ) -> p : NN0 --> ( S u. { 0 } ) )
11	10	ad2antll 495	. . . . . . . . . . . 12 |- ( ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) /\ ( d e. NN0 /\ p e. ( ( S u. { 0 } ) ^m NN0 ) ) ) -> p : NN0 --> ( S u. { 0 } ) )
12	11	adantr 276	. . . . . . . . . . 11 |- ( ( ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) /\ ( d e. NN0 /\ p e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ ( ( p " ( ZZ>= ` ( d + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( p ` k ) x. ( z ^ k ) ) ) ) ) -> p : NN0 --> ( S u. { 0 } ) )
13		cnfldbas 15007	. . . . . . . . . . . . . 14 |- CC = ( Base ` ℂfld )
14	13	subrgss 14640	. . . . . . . . . . . . 13 |- ( S e. (SubRing ` ℂfld ) -> S C_ CC )
15		0cn 8325	. . . . . . . . . . . . . 14 |- 0 e. CC
16		snssi 3859	. . . . . . . . . . . . . 14 |- ( 0 e. CC -> { 0 } C_ CC )
17	15, 16	mp1i 10	. . . . . . . . . . . . 13 |- ( S e. (SubRing ` ℂfld ) -> { 0 } C_ CC )
18	14, 17	unssd 3405	. . . . . . . . . . . 12 |- ( S e. (SubRing ` ℂfld ) -> ( S u. { 0 } ) C_ CC )
19	18	ad3antrrr 496	. . . . . . . . . . 11 |- ( ( ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) /\ ( d e. NN0 /\ p e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ ( ( p " ( ZZ>= ` ( d + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( p ` k ) x. ( z ^ k ) ) ) ) ) -> ( S u. { 0 } ) C_ CC )
20	12, 19	fssd 5550	. . . . . . . . . 10 |- ( ( ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) /\ ( d e. NN0 /\ p e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ ( ( p " ( ZZ>= ` ( d + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( p ` k ) x. ( z ^ k ) ) ) ) ) -> p : NN0 --> CC )
21	20	adantr 276	. . . . . . . . 9 |- ( ( ( ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) /\ ( d e. NN0 /\ p e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ ( ( p " ( ZZ>= ` ( d + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( p ` k ) x. ( z ^ k ) ) ) ) ) /\ a e. CC ) -> p : NN0 --> CC )
22		simplrl 541	. . . . . . . . 9 |- ( ( ( ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) /\ ( d e. NN0 /\ p e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ ( ( p " ( ZZ>= ` ( d + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( p ` k ) x. ( z ^ k ) ) ) ) ) /\ a e. CC ) -> ( p " ( ZZ>= ` ( d + 1 ) ) ) = { 0 } )
23		simplrr 542	. . . . . . . . 9 |- ( ( ( ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) /\ ( d e. NN0 /\ p e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ ( ( p " ( ZZ>= ` ( d + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( p ` k ) x. ( z ^ k ) ) ) ) ) /\ a e. CC ) -> F = ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( p ` k ) x. ( z ^ k ) ) ) )
24		nn0z 9676	. . . . . . . . . . . . 13 |- ( d e. NN0 -> d e. ZZ )
25	24	uzidd 9954	. . . . . . . . . . . 12 |- ( d e. NN0 -> d e. ( ZZ>= ` d ) )
26		peano2uz 10000	. . . . . . . . . . . 12 |- ( d e. ( ZZ>= ` d ) -> ( d + 1 ) e. ( ZZ>= ` d ) )
27	25, 26	syl 14	. . . . . . . . . . 11 |- ( d e. NN0 -> ( d + 1 ) e. ( ZZ>= ` d ) )
28	8, 27	syl 14	. . . . . . . . . 10 |- ( ( ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) /\ ( d e. NN0 /\ p e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ ( ( p " ( ZZ>= ` ( d + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( p ` k ) x. ( z ^ k ) ) ) ) ) -> ( d + 1 ) e. ( ZZ>= ` d ) )
29	28	adantr 276	. . . . . . . . 9 |- ( ( ( ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) /\ ( d e. NN0 /\ p e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ ( ( p " ( ZZ>= ` ( d + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( p ` k ) x. ( z ^ k ) ) ) ) ) /\ a e. CC ) -> ( d + 1 ) e. ( ZZ>= ` d ) )
30		simpr 110	. . . . . . . . 9 |- ( ( ( ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) /\ ( d e. NN0 /\ p e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ ( ( p " ( ZZ>= ` ( d + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( p ` k ) x. ( z ^ k ) ) ) ) ) /\ a e. CC ) -> a e. CC )
31	9, 21, 22, 23, 29, 30	plycoeid3 15995	. . . . . . . 8 |- ( ( ( ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) /\ ( d e. NN0 /\ p e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ ( ( p " ( ZZ>= ` ( d + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( p ` k ) x. ( z ^ k ) ) ) ) ) /\ a e. CC ) -> ( F ` a ) = sum_ b e. ( 0 ... ( d + 1 ) ) ( ( p ` b ) x. ( a ^ b ) ) )
32	31	mpteq2dva 4223	. . . . . . 7 |- ( ( ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) /\ ( d e. NN0 /\ p e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ ( ( p " ( ZZ>= ` ( d + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( p ` k ) x. ( z ^ k ) ) ) ) ) -> ( a e. CC |-> ( F ` a ) ) = ( a e. CC |-> sum_ b e. ( 0 ... ( d + 1 ) ) ( ( p ` b ) x. ( a ^ b ) ) ) )
33	7, 32	eqtrd 2271	. . . . . 6 |- ( ( ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) /\ ( d e. NN0 /\ p e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ ( ( p " ( ZZ>= ` ( d + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( p ` k ) x. ( z ^ k ) ) ) ) ) -> F = ( a e. CC |-> sum_ b e. ( 0 ... ( d + 1 ) ) ( ( p ` b ) x. ( a ^ b ) ) ) )
34	8	nn0cnd 9634	. . . . . . . . . . 11 |- ( ( ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) /\ ( d e. NN0 /\ p e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ ( ( p " ( ZZ>= ` ( d + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( p ` k ) x. ( z ^ k ) ) ) ) ) -> d e. CC )
35		1cnd 8349	. . . . . . . . . . 11 |- ( ( ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) /\ ( d e. NN0 /\ p e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ ( ( p " ( ZZ>= ` ( d + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( p ` k ) x. ( z ^ k ) ) ) ) ) -> 1 e. CC )
36	34, 35	pncand 8646	. . . . . . . . . 10 |- ( ( ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) /\ ( d e. NN0 /\ p e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ ( ( p " ( ZZ>= ` ( d + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( p ` k ) x. ( z ^ k ) ) ) ) ) -> ( ( d + 1 ) - 1 ) = d )
37	36	eqcomd 2244	. . . . . . . . 9 |- ( ( ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) /\ ( d e. NN0 /\ p e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ ( ( p " ( ZZ>= ` ( d + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( p ` k ) x. ( z ^ k ) ) ) ) ) -> d = ( ( d + 1 ) - 1 ) )
38	37	oveq2d 6104	. . . . . . . 8 |- ( ( ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) /\ ( d e. NN0 /\ p e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ ( ( p " ( ZZ>= ` ( d + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( p ` k ) x. ( z ^ k ) ) ) ) ) -> ( 0 ... d ) = ( 0 ... ( ( d + 1 ) - 1 ) ) )
39	38	sumeq1d 12158	. . . . . . 7 |- ( ( ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) /\ ( d e. NN0 /\ p e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ ( ( p " ( ZZ>= ` ( d + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( p ` k ) x. ( z ^ k ) ) ) ) ) -> sum_ b e. ( 0 ... d ) ( ( ( c e. NN0 |-> ( ( c + 1 ) x. ( p ` ( c + 1 ) ) ) ) ` b ) x. ( a ^ b ) ) = sum_ b e. ( 0 ... ( ( d + 1 ) - 1 ) ) ( ( ( c e. NN0 |-> ( ( c + 1 ) x. ( p ` ( c + 1 ) ) ) ) ` b ) x. ( a ^ b ) ) )
40	39	mpteq2dv 4224	. . . . . 6 |- ( ( ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) /\ ( d e. NN0 /\ p e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ ( ( p " ( ZZ>= ` ( d + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( p ` k ) x. ( z ^ k ) ) ) ) ) -> ( a e. CC |-> sum_ b e. ( 0 ... d ) ( ( ( c e. NN0 |-> ( ( c + 1 ) x. ( p ` ( c + 1 ) ) ) ) ` b ) x. ( a ^ b ) ) ) = ( a e. CC |-> sum_ b e. ( 0 ... ( ( d + 1 ) - 1 ) ) ( ( ( c e. NN0 |-> ( ( c + 1 ) x. ( p ` ( c + 1 ) ) ) ) ` b ) x. ( a ^ b ) ) ) )
41		oveq1 6095	. . . . . . . 8 |- ( c = b -> ( c + 1 ) = ( b + 1 ) )
42		fvoveq1 6111	. . . . . . . 8 |- ( c = b -> ( p ` ( c + 1 ) ) = ( p ` ( b + 1 ) ) )
43	41, 42	oveq12d 6106	. . . . . . 7 |- ( c = b -> ( ( c + 1 ) x. ( p ` ( c + 1 ) ) ) = ( ( b + 1 ) x. ( p ` ( b + 1 ) ) ) )
44	43	cbvmptv 4229	. . . . . 6 |- ( c e. NN0 |-> ( ( c + 1 ) x. ( p ` ( c + 1 ) ) ) ) = ( b e. NN0 |-> ( ( b + 1 ) x. ( p ` ( b + 1 ) ) ) )
45		peano2nn0 9615	. . . . . . 7 |- ( d e. NN0 -> ( d + 1 ) e. NN0 )
46	8, 45	syl 14	. . . . . 6 |- ( ( ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) /\ ( d e. NN0 /\ p e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ ( ( p " ( ZZ>= ` ( d + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( p ` k ) x. ( z ^ k ) ) ) ) ) -> ( d + 1 ) e. NN0 )
47	33, 40, 20, 44, 46	dvply1 16003	. . . . 5 |- ( ( ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) /\ ( d e. NN0 /\ p e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ ( ( p " ( ZZ>= ` ( d + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( p ` k ) x. ( z ^ k ) ) ) ) ) -> ( CC _D F ) = ( a e. CC |-> sum_ b e. ( 0 ... d ) ( ( ( c e. NN0 |-> ( ( c + 1 ) x. ( p ` ( c + 1 ) ) ) ) ` b ) x. ( a ^ b ) ) ) )
48	14	ad3antrrr 496	. . . . . 6 |- ( ( ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) /\ ( d e. NN0 /\ p e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ ( ( p " ( ZZ>= ` ( d + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( p ` k ) x. ( z ^ k ) ) ) ) ) -> S C_ CC )
49		elfznn0 10539	. . . . . . 7 |- ( b e. ( 0 ... d ) -> b e. NN0 )
50		peano2nn0 9615	. . . . . . . . . . . . 13 |- ( c e. NN0 -> ( c + 1 ) e. NN0 )
51	50	adantl 277	. . . . . . . . . . . 12 |- ( ( ( ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) /\ ( d e. NN0 /\ p e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ ( ( p " ( ZZ>= ` ( d + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( p ` k ) x. ( z ^ k ) ) ) ) ) /\ c e. NN0 ) -> ( c + 1 ) e. NN0 )
52	51	nn0cnd 9634	. . . . . . . . . . 11 |- ( ( ( ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) /\ ( d e. NN0 /\ p e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ ( ( p " ( ZZ>= ` ( d + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( p ` k ) x. ( z ^ k ) ) ) ) ) /\ c e. NN0 ) -> ( c + 1 ) e. CC )
53	20	adantr 276	. . . . . . . . . . . 12 |- ( ( ( ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) /\ ( d e. NN0 /\ p e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ ( ( p " ( ZZ>= ` ( d + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( p ` k ) x. ( z ^ k ) ) ) ) ) /\ c e. NN0 ) -> p : NN0 --> CC )
54	53, 51	ffvelcdmd 5847	. . . . . . . . . . 11 |- ( ( ( ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) /\ ( d e. NN0 /\ p e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ ( ( p " ( ZZ>= ` ( d + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( p ` k ) x. ( z ^ k ) ) ) ) ) /\ c e. NN0 ) -> ( p ` ( c + 1 ) ) e. CC )
55	52, 54	mulcld 8353	. . . . . . . . . . 11 |- ( ( ( ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) /\ ( d e. NN0 /\ p e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ ( ( p " ( ZZ>= ` ( d + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( p ` k ) x. ( z ^ k ) ) ) ) ) /\ c e. NN0 ) -> ( ( c + 1 ) x. ( p ` ( c + 1 ) ) ) e. CC )
56		oveq1 6095	. . . . . . . . . . . 12 |- ( u = ( c + 1 ) -> ( u x. v ) = ( ( c + 1 ) x. v ) )
57		oveq2 6096	. . . . . . . . . . . 12 |- ( v = ( p ` ( c + 1 ) ) -> ( ( c + 1 ) x. v ) = ( ( c + 1 ) x. ( p ` ( c + 1 ) ) ) )
58		eqid 2238	. . . . . . . . . . . 12 |- ( u e. CC , v e. CC |-> ( u x. v ) ) = ( u e. CC , v e. CC |-> ( u x. v ) )
59	56, 57, 58	ovmpog 6226	. . . . . . . . . . 11 |- ( ( ( c + 1 ) e. CC /\ ( p ` ( c + 1 ) ) e. CC /\ ( ( c + 1 ) x. ( p ` ( c + 1 ) ) ) e. CC ) -> ( ( c + 1 ) ( u e. CC , v e. CC |-> ( u x. v ) ) ( p ` ( c + 1 ) ) ) = ( ( c + 1 ) x. ( p ` ( c + 1 ) ) ) )
60	52, 54, 55, 59	syl3anc 1278	. . . . . . . . . 10 |- ( ( ( ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) /\ ( d e. NN0 /\ p e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ ( ( p " ( ZZ>= ` ( d + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( p ` k ) x. ( z ^ k ) ) ) ) ) /\ c e. NN0 ) -> ( ( c + 1 ) ( u e. CC , v e. CC |-> ( u x. v ) ) ( p ` ( c + 1 ) ) ) = ( ( c + 1 ) x. ( p ` ( c + 1 ) ) ) )
61		simp-4l 547	. . . . . . . . . . 11 |- ( ( ( ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) /\ ( d e. NN0 /\ p e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ ( ( p " ( ZZ>= ` ( d + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( p ` k ) x. ( z ^ k ) ) ) ) ) /\ c e. NN0 ) -> S e. (SubRing ` ℂfld ) )
62		zsssubrg 15032	. . . . . . . . . . . . 13 |- ( S e. (SubRing ` ℂfld ) -> ZZ C_ S )
63	62	ad4antr 498	. . . . . . . . . . . 12 |- ( ( ( ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) /\ ( d e. NN0 /\ p e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ ( ( p " ( ZZ>= ` ( d + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( p ` k ) x. ( z ^ k ) ) ) ) ) /\ c e. NN0 ) -> ZZ C_ S )
64	51	nn0zd 9778	. . . . . . . . . . . 12 |- ( ( ( ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) /\ ( d e. NN0 /\ p e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ ( ( p " ( ZZ>= ` ( d + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( p ` k ) x. ( z ^ k ) ) ) ) ) /\ c e. NN0 ) -> ( c + 1 ) e. ZZ )
65	63, 64	sseldd 3249	. . . . . . . . . . 11 |- ( ( ( ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) /\ ( d e. NN0 /\ p e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ ( ( p " ( ZZ>= ` ( d + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( p ` k ) x. ( z ^ k ) ) ) ) ) /\ c e. NN0 ) -> ( c + 1 ) e. S )
66	12	adantr 276	. . . . . . . . . . . . 13 |- ( ( ( ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) /\ ( d e. NN0 /\ p e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ ( ( p " ( ZZ>= ` ( d + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( p ` k ) x. ( z ^ k ) ) ) ) ) /\ c e. NN0 ) -> p : NN0 --> ( S u. { 0 } ) )
67		subrgsubg 14645	. . . . . . . . . . . . . . . . . 18 |- ( S e. (SubRing ` ℂfld ) -> S e. (SubGrp ` ℂfld ) )
68		cnfld0 15018	. . . . . . . . . . . . . . . . . . 19 |- 0 = ( 0g ` ℂfld )
69	68	subg0cl 14045	. . . . . . . . . . . . . . . . . 18 |- ( S e. (SubGrp ` ℂfld ) -> 0 e. S )
70	67, 69	syl 14	. . . . . . . . . . . . . . . . 17 |- ( S e. (SubRing ` ℂfld ) -> 0 e. S )
71	70	ad4antr 498	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) /\ ( d e. NN0 /\ p e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ ( ( p " ( ZZ>= ` ( d + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( p ` k ) x. ( z ^ k ) ) ) ) ) /\ c e. NN0 ) -> 0 e. S )
72	71	snssd 3860	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) /\ ( d e. NN0 /\ p e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ ( ( p " ( ZZ>= ` ( d + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( p ` k ) x. ( z ^ k ) ) ) ) ) /\ c e. NN0 ) -> { 0 } C_ S )
73		ssequn2 3402	. . . . . . . . . . . . . . 15 |- ( { 0 } C_ S <-> ( S u. { 0 } ) = S )
74	72, 73	sylib 122	. . . . . . . . . . . . . 14 |- ( ( ( ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) /\ ( d e. NN0 /\ p e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ ( ( p " ( ZZ>= ` ( d + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( p ` k ) x. ( z ^ k ) ) ) ) ) /\ c e. NN0 ) -> ( S u. { 0 } ) = S )
75	74	feq3d 5525	. . . . . . . . . . . . 13 |- ( ( ( ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) /\ ( d e. NN0 /\ p e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ ( ( p " ( ZZ>= ` ( d + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( p ` k ) x. ( z ^ k ) ) ) ) ) /\ c e. NN0 ) -> ( p : NN0 --> ( S u. { 0 } ) <-> p : NN0 --> S ) )
76	66, 75	mpbid 147	. . . . . . . . . . . 12 |- ( ( ( ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) /\ ( d e. NN0 /\ p e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ ( ( p " ( ZZ>= ` ( d + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( p ` k ) x. ( z ^ k ) ) ) ) ) /\ c e. NN0 ) -> p : NN0 --> S )
77	76, 51	ffvelcdmd 5847	. . . . . . . . . . 11 |- ( ( ( ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) /\ ( d e. NN0 /\ p e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ ( ( p " ( ZZ>= ` ( d + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( p ` k ) x. ( z ^ k ) ) ) ) ) /\ c e. NN0 ) -> ( p ` ( c + 1 ) ) e. S )
78		mpocnfldmul 15010	. . . . . . . . . . . 12 |- ( u e. CC , v e. CC |-> ( u x. v ) ) = ( .r ` ℂfld )
79	78	subrgmcl 14651	. . . . . . . . . . 11 |- ( ( S e. (SubRing ` ℂfld ) /\ ( c + 1 ) e. S /\ ( p ` ( c + 1 ) ) e. S ) -> ( ( c + 1 ) ( u e. CC , v e. CC |-> ( u x. v ) ) ( p ` ( c + 1 ) ) ) e. S )
80	61, 65, 77, 79	syl3anc 1278	. . . . . . . . . 10 |- ( ( ( ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) /\ ( d e. NN0 /\ p e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ ( ( p " ( ZZ>= ` ( d + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( p ` k ) x. ( z ^ k ) ) ) ) ) /\ c e. NN0 ) -> ( ( c + 1 ) ( u e. CC , v e. CC |-> ( u x. v ) ) ( p ` ( c + 1 ) ) ) e. S )
81	60, 80	eqeltrrd 2316	. . . . . . . . 9 |- ( ( ( ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) /\ ( d e. NN0 /\ p e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ ( ( p " ( ZZ>= ` ( d + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( p ` k ) x. ( z ^ k ) ) ) ) ) /\ c e. NN0 ) -> ( ( c + 1 ) x. ( p ` ( c + 1 ) ) ) e. S )
82	81	fmpttd 5866	. . . . . . . 8 |- ( ( ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) /\ ( d e. NN0 /\ p e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ ( ( p " ( ZZ>= ` ( d + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( p ` k ) x. ( z ^ k ) ) ) ) ) -> ( c e. NN0 |-> ( ( c + 1 ) x. ( p ` ( c + 1 ) ) ) ) : NN0 --> S )
83	82	ffvelcdmda 5846	. . . . . . 7 |- ( ( ( ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) /\ ( d e. NN0 /\ p e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ ( ( p " ( ZZ>= ` ( d + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( p ` k ) x. ( z ^ k ) ) ) ) ) /\ b e. NN0 ) -> ( ( c e. NN0 |-> ( ( c + 1 ) x. ( p ` ( c + 1 ) ) ) ) ` b ) e. S )
84	49, 83	sylan2 286	. . . . . 6 |- ( ( ( ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) /\ ( d e. NN0 /\ p e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ ( ( p " ( ZZ>= ` ( d + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( p ` k ) x. ( z ^ k ) ) ) ) ) /\ b e. ( 0 ... d ) ) -> ( ( c e. NN0 |-> ( ( c + 1 ) x. ( p ` ( c + 1 ) ) ) ) ` b ) e. S )
85	48, 8, 84	elplyd 15979	. . . . 5 |- ( ( ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) /\ ( d e. NN0 /\ p e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ ( ( p " ( ZZ>= ` ( d + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( p ` k ) x. ( z ^ k ) ) ) ) ) -> ( a e. CC |-> sum_ b e. ( 0 ... d ) ( ( ( c e. NN0 |-> ( ( c + 1 ) x. ( p ` ( c + 1 ) ) ) ) ` b ) x. ( a ^ b ) ) ) e. (Poly ` S ) )
86	47, 85	eqeltrd 2315	. . . 4 |- ( ( ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) /\ ( d e. NN0 /\ p e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ ( ( p " ( ZZ>= ` ( d + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( p ` k ) x. ( z ^ k ) ) ) ) ) -> ( CC _D F ) e. (Poly ` S ) )
87	86	ex 115	. . 3 |- ( ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) /\ ( d e. NN0 /\ p e. ( ( S u. { 0 } ) ^m NN0 ) ) ) -> ( ( ( p " ( ZZ>= ` ( d + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( p ` k ) x. ( z ^ k ) ) ) ) -> ( CC _D F ) e. (Poly ` S ) ) )
88	87	rexlimdvva 2676	. 2 |- ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) -> ( E. d e. NN0 E. p e. ( ( S u. { 0 } ) ^m NN0 ) ( ( p " ( ZZ>= ` ( d + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( p ` k ) x. ( z ^ k ) ) ) ) -> ( CC _D F ) e. (Poly ` S ) ) )
89	3, 88	mpd 13	1 |- ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) -> ( CC _D F ) e. (Poly ` S ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   E.wrex 2529   u. cun 3218   C_ wss 3220   {csn 3709   |-> cmpt 4193   "cima 4779   -->wf 5376   ` cfv 5380  (class class class)co 6088   e. cmpo 6090   ^m cmap 6926   CCcc 8184   0cc0 8186   1c1 8187   + caddc 8189   x. cmul 8191   - cmin 8505   NN0cn0 9575   ZZcz 9656   ZZ>=cuz 9938   ...cfz 10429   ^cexp 10997   sum_csu 12145  SubGrpcsubg 14030  SubRingcsubrg 14635  ℂ_fldccnfld 15003   _D cdv 15893  Polycply 15966

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8277  ax-resscn 8278  ax-1cn 8279  ax-1re 8280  ax-icn 8281  ax-addcl 8282  ax-addrcl 8283  ax-mulcl 8284  ax-mulrcl 8285  ax-addcom 8286  ax-mulcom 8287  ax-addass 8288  ax-mulass 8289  ax-distr 8290  ax-i2m1 8291  ax-0lt1 8292  ax-1rid 8293  ax-0id 8294  ax-rnegex 8295  ax-precex 8296  ax-cnre 8297  ax-pre-ltirr 8298  ax-pre-ltwlin 8299  ax-pre-lttrn 8300  ax-pre-apti 8301  ax-pre-ltadd 8302  ax-pre-mulgt0 8303  ax-pre-mulext 8304  ax-arch 8305  ax-caucvg 8306  ax-addf 8308  ax-mulf 8309

This proof depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-tp 3717  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-isom 5389  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-of 6305  df-1st 6378  df-2nd 6379  df-recs 6580  df-irdg 6645  df-frec 6666  df-1o 6691  df-oadd 6695  df-er 6811  df-map 6928  df-pm 6929  df-en 7028  df-dom 7029  df-fin 7030  df-sup 7331  df-inf 7332  df-pnf 8369  df-mnf 8370  df-xr 8371  df-ltxr 8372  df-le 8373  df-sub 8507  df-neg 8508  df-reap 8912  df-ap 8919  df-div 9012  df-inn 9315  df-2 9373  df-3 9374  df-4 9375  df-5 9376  df-6 9377  df-7 9378  df-8 9379  df-9 9380  df-n0 9576  df-z 9657  df-dec 9790  df-uz 9939  df-q 10037  df-rp 10073  df-xneg 10192  df-xadd 10193  df-fz 10430  df-fzo 10568  df-seqfrec 10907  df-exp 10998  df-ihash 11238  df-cj 11630  df-re 11631  df-im 11632  df-rsqrt 11787  df-abs 11788  df-clim 12071  df-sumdc 12146  df-struct 13413  df-ndx 13414  df-slot 13415  df-base 13417  df-sets 13418  df-iress 13419  df-plusg 13504  df-mulr 13505  df-starv 13506  df-tset 13510  df-ple 13511  df-ds 13513  df-unif 13514  df-rest 13655  df-topn 13656  df-0g 13672  df-topgen 13674  df-mgm 13736  df-sgrp 13777  df-mnd 13790  df-grp 13868  df-minusg 13869  df-mulg 13983  df-subg 14033  df-cmn 14180  df-mgp 14326  df-ur 14371  df-ring 14410  df-cring 14411  df-subrg 14637  df-psmet 14990  df-xmet 14991  df-met 14992  df-bl 14993  df-mopn 14994  df-fg 14996  df-metu 14997  df-cnfld 15004  df-top 15236  df-topon 15249  df-topsp 15269  df-bases 15281  df-ntr 15334  df-cn 15426  df-cnp 15427  df-tx 15491  df-xms 15577  df-ms 15578  df-cncf 15809  df-limced 15894  df-dvap 15895  df-ply 15968

This theorem is used by:  dvply2  16005