Theorem dvply2g 15743

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 15712	. . . 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 15714	. . . . . . . . . 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 5735	. . . . . . . 8 |- ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) -> F = ( a e. CC |-> ( F ` a ) ) )
7	6	ad2antrr 488	. . . . . . 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 537	. . . . . . . . . 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 6917	. . . . . . . . . . . . 13 |- ( p e. ( ( S u. { 0 } ) ^m NN0 ) -> p : NN0 --> ( S u. { 0 } ) )
11	10	ad2antll 491	. . . . . . . . . . . 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 14820	. . . . . . . . . . . . . 14 |- CC = ( Base ` ℂfld )
14	13	subrgss 14453	. . . . . . . . . . . . 13 |- ( S e. (SubRing ` ℂfld ) -> S C_ CC )
15		0cn 8282	. . . . . . . . . . . . . 14 |- 0 e. CC
16		snssi 3843	. . . . . . . . . . . . . 14 |- ( 0 e. CC -> { 0 } C_ CC )
17	15, 16	mp1i 10	. . . . . . . . . . . . 13 |- ( S e. (SubRing ` ℂfld ) -> { 0 } C_ CC )
18	14, 17	unssd 3399	. . . . . . . . . . . 12 |- ( S e. (SubRing ` ℂfld ) -> ( S u. { 0 } ) C_ CC )
19	18	ad3antrrr 492	. . . . . . . . . . 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 5527	. . . . . . . . . 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 537	. . . . . . . . 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 538	. . . . . . . . 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 9614	. . . . . . . . . . . . 13 |- ( d e. NN0 -> d e. ZZ )
25	24	uzidd 9887	. . . . . . . . . . . 12 |- ( d e. NN0 -> d e. ( ZZ>= ` d ) )
26		peano2uz 9933	. . . . . . . . . . . 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 15734	. . . . . . . 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 4205	. . . . . . 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 2267	. . . . . 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 9572	. . . . . . . . . . 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 8306	. . . . . . . . . . 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 8601	. . . . . . . . . 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 2240	. . . . . . . . 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 6074	. . . . . . . 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 12076	. . . . . . 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 4206	. . . . . 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 6065	. . . . . . . 8 |- ( c = b -> ( c + 1 ) = ( b + 1 ) )
42		fvoveq1 6081	. . . . . . . 8 |- ( c = b -> ( p ` ( c + 1 ) ) = ( p ` ( b + 1 ) ) )
43	41, 42	oveq12d 6076	. . . . . . 7 |- ( c = b -> ( ( c + 1 ) x. ( p ` ( c + 1 ) ) ) = ( ( b + 1 ) x. ( p ` ( b + 1 ) ) ) )
44	43	cbvmptv 4211	. . . . . 6 |- ( c e. NN0 |-> ( ( c + 1 ) x. ( p ` ( c + 1 ) ) ) ) = ( b e. NN0 |-> ( ( b + 1 ) x. ( p ` ( b + 1 ) ) ) )
45		peano2nn0 9553	. . . . . . 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 15742	. . . . 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 492	. . . . . 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 10470	. . . . . . 7 |- ( b e. ( 0 ... d ) -> b e. NN0 )
50		peano2nn0 9553	. . . . . . . . . . . . 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 9572	. . . . . . . . . . 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 5818	. . . . . . . . . . 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 8310	. . . . . . . . . . 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 6065	. . . . . . . . . . . 12 |- ( u = ( c + 1 ) -> ( u x. v ) = ( ( c + 1 ) x. v ) )
57		oveq2 6066	. . . . . . . . . . . 12 |- ( v = ( p ` ( c + 1 ) ) -> ( ( c + 1 ) x. v ) = ( ( c + 1 ) x. ( p ` ( c + 1 ) ) ) )
58		eqid 2234	. . . . . . . . . . . 12 |- ( u e. CC , v e. CC |-> ( u x. v ) ) = ( u e. CC , v e. CC |-> ( u x. v ) )
59	56, 57, 58	ovmpog 6196	. . . . . . . . . . 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 1274	. . . . . . . . . 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 543	. . . . . . . . . . 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 14845	. . . . . . . . . . . . 13 |- ( S e. (SubRing ` ℂfld ) -> ZZ C_ S )
63	62	ad4antr 494	. . . . . . . . . . . 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 9716	. . . . . . . . . . . 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 3243	. . . . . . . . . . 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 14458	. . . . . . . . . . . . . . . . . 18 |- ( S e. (SubRing ` ℂfld ) -> S e. (SubGrp ` ℂfld ) )
68		cnfld0 14831	. . . . . . . . . . . . . . . . . . 19 |- 0 = ( 0g ` ℂfld )
69	68	subg0cl 13983	. . . . . . . . . . . . . . . . . 18 |- ( S e. (SubGrp ` ℂfld ) -> 0 e. S )
70	67, 69	syl 14	. . . . . . . . . . . . . . . . 17 |- ( S e. (SubRing ` ℂfld ) -> 0 e. S )
71	70	ad4antr 494	. . . . . . . . . . . . . . . 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 3844	. . . . . . . . . . . . . . 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 3396	. . . . . . . . . . . . . . 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 5502	. . . . . . . . . . . . 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 5818	. . . . . . . . . . 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 14823	. . . . . . . . . . . 12 |- ( u e. CC , v e. CC |-> ( u x. v ) ) = ( .r ` ℂfld )
79	78	subrgmcl 14464	. . . . . . . . . . 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 1274	. . . . . . . . . 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 2312	. . . . . . . . 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 5837	. . . . . . . 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 5817	. . . . . . 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 15718	. . . . 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 2311	. . . 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 2670	. 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 ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2205   E.wrex 2523   u. cun 3212   C_ wss 3214   {csn 3694   |-> cmpt 4176   "cima 4757   -->wf 5353   ` cfv 5357  (class class class)co 6058   e. cmpo 6060   ^m cmap 6895   CCcc 8141   0cc0 8143   1c1 8144   + caddc 8146   x. cmul 8148   - cmin 8460   NN0cn0 9513   ZZcz 9594   ZZ>=cuz 9871   ...cfz 10361   ^cexp 10924   sum_csu 12063  SubGrpcsubg 13968  SubRingcsubrg 14448  ℂ_fldccnfld 14816   _D cdv 15632  Polycply 15705

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261  ax-arch 8262  ax-caucvg 8263  ax-addf 8265  ax-mulf 8266

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-tp 3702  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-isom 5366  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-of 6275  df-1st 6347  df-2nd 6348  df-recs 6549  df-irdg 6614  df-frec 6635  df-1o 6660  df-oadd 6664  df-er 6780  df-map 6897  df-pm 6898  df-en 6989  df-dom 6990  df-fin 6991  df-sup 7288  df-inf 7289  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-5 9316  df-6 9317  df-7 9318  df-8 9319  df-9 9320  df-n0 9514  df-z 9595  df-dec 9728  df-uz 9872  df-q 9970  df-rp 10005  df-xneg 10124  df-xadd 10125  df-fz 10362  df-fzo 10499  df-seqfrec 10834  df-exp 10925  df-ihash 11164  df-cj 11552  df-re 11553  df-im 11554  df-rsqrt 11708  df-abs 11709  df-clim 11989  df-sumdc 12064  df-struct 13298  df-ndx 13299  df-slot 13300  df-base 13302  df-sets 13303  df-iress 13304  df-plusg 13387  df-mulr 13388  df-starv 13389  df-tset 13393  df-ple 13394  df-ds 13396  df-unif 13397  df-rest 13538  df-topn 13539  df-0g 13555  df-topgen 13557  df-mgm 13653  df-sgrp 13699  df-mnd 13714  df-grp 13800  df-minusg 13801  df-mulg 13921  df-subg 13971  df-cmn 14087  df-mgp 14149  df-ur 14188  df-ring 14226  df-cring 14227  df-subrg 14450  df-psmet 14803  df-xmet 14804  df-met 14805  df-bl 14806  df-mopn 14807  df-fg 14809  df-metu 14810  df-cnfld 14817  df-top 14975  df-topon 14988  df-topsp 15008  df-bases 15020  df-ntr 15073  df-cn 15165  df-cnp 15166  df-tx 15230  df-xms 15316  df-ms 15317  df-cncf 15548  df-limced 15633  df-dvap 15634  df-ply 15707

This theorem is referenced by:  dvply2  15744