Theorem dvply2g 15086

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 15055	. . . 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 15057	. . . . . . . . . 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 5617	. . . . . . . 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 535	. . . . . . . . . 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 6738	. . . . . . . . . . . . 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 14192	. . . . . . . . . . . . . 14 |- CC = ( Base ` ℂfld )
14	13	subrgss 13854	. . . . . . . . . . . . 13 |- ( S e. (SubRing ` ℂfld ) -> S C_ CC )
15		0cn 8035	. . . . . . . . . . . . . 14 |- 0 e. CC
16		snssi 3767	. . . . . . . . . . . . . 14 |- ( 0 e. CC -> { 0 } C_ CC )
17	15, 16	mp1i 10	. . . . . . . . . . . . 13 |- ( S e. (SubRing ` ℂfld ) -> { 0 } C_ CC )
18	14, 17	unssd 3340	. . . . . . . . . . . 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 5423	. . . . . . . . . 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 535	. . . . . . . . 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 536	. . . . . . . . 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 9363	. . . . . . . . . . . . 13 |- ( d e. NN0 -> d e. ZZ )
25	24	uzidd 9633	. . . . . . . . . . . 12 |- ( d e. NN0 -> d e. ( ZZ>= ` d ) )
26		peano2uz 9674	. . . . . . . . . . . 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 15077	. . . . . . . 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 4124	. . . . . . 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 2229	. . . . . 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 9321	. . . . . . . . . . 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 8059	. . . . . . . . . . 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 8355	. . . . . . . . . 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 2202	. . . . . . . . 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 5941	. . . . . . . 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 11548	. . . . . . 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 4125	. . . . . 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 5932	. . . . . . . 8 |- ( c = b -> ( c + 1 ) = ( b + 1 ) )
42		fvoveq1 5948	. . . . . . . 8 |- ( c = b -> ( p ` ( c + 1 ) ) = ( p ` ( b + 1 ) ) )
43	41, 42	oveq12d 5943	. . . . . . 7 |- ( c = b -> ( ( c + 1 ) x. ( p ` ( c + 1 ) ) ) = ( ( b + 1 ) x. ( p ` ( b + 1 ) ) ) )
44	43	cbvmptv 4130	. . . . . 6 |- ( c e. NN0 |-> ( ( c + 1 ) x. ( p ` ( c + 1 ) ) ) ) = ( b e. NN0 |-> ( ( b + 1 ) x. ( p ` ( b + 1 ) ) ) )
45		peano2nn0 9306	. . . . . . 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 15085	. . . . 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 10206	. . . . . . 7 |- ( b e. ( 0 ... d ) -> b e. NN0 )
50		peano2nn0 9306	. . . . . . . . . . . . 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 9321	. . . . . . . . . . 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 5701	. . . . . . . . . . 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 8064	. . . . . . . . . . 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 5932	. . . . . . . . . . . 12 |- ( u = ( c + 1 ) -> ( u x. v ) = ( ( c + 1 ) x. v ) )
57		oveq2 5933	. . . . . . . . . . . 12 |- ( v = ( p ` ( c + 1 ) ) -> ( ( c + 1 ) x. v ) = ( ( c + 1 ) x. ( p ` ( c + 1 ) ) ) )
58		eqid 2196	. . . . . . . . . . . 12 |- ( u e. CC , v e. CC |-> ( u x. v ) ) = ( u e. CC , v e. CC |-> ( u x. v ) )
59	56, 57, 58	ovmpog 6061	. . . . . . . . . . 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 1249	. . . . . . . . . 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 541	. . . . . . . . . . 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 14217	. . . . . . . . . . . . 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 9463	. . . . . . . . . . . 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 3185	. . . . . . . . . . 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 13859	. . . . . . . . . . . . . . . . . 18 |- ( S e. (SubRing ` ℂfld ) -> S e. (SubGrp ` ℂfld ) )
68		cnfld0 14203	. . . . . . . . . . . . . . . . . . 19 |- 0 = ( 0g ` ℂfld )
69	68	subg0cl 13388	. . . . . . . . . . . . . . . . . 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 3768	. . . . . . . . . . . . . . 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 3337	. . . . . . . . . . . . . . 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 5399	. . . . . . . . . . . . 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 5701	. . . . . . . . . . 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 14195	. . . . . . . . . . . 12 |- ( u e. CC , v e. CC |-> ( u x. v ) ) = ( .r ` ℂfld )
79	78	subrgmcl 13865	. . . . . . . . . . 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 1249	. . . . . . . . . 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 2274	. . . . . . . . 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 5720	. . . . . . . 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 5700	. . . . . . 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 15061	. . . . 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 2273	. . . 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 2622	. 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 1364   e. wcel 2167   E.wrex 2476   u. cun 3155   C_ wss 3157   {csn 3623   |-> cmpt 4095   "cima 4667   -->wf 5255   ` cfv 5259  (class class class)co 5925   e. cmpo 5927   ^m cmap 6716   CCcc 7894   0cc0 7896   1c1 7897   + caddc 7899   x. cmul 7901   - cmin 8214   NN0cn0 9266   ZZcz 9343   ZZ>=cuz 9618   ...cfz 10100   ^cexp 10647   sum_csu 11535  SubGrpcsubg 13373  SubRingcsubrg 13849  ℂ_fldccnfld 14188   _D cdv 14975  Polycply 15048

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-mulrcl 7995  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-precex 8006  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012  ax-pre-mulgt0 8013  ax-pre-mulext 8014  ax-arch 8015  ax-caucvg 8016  ax-addf 8018  ax-mulf 8019

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-tp 3631  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-isom 5268  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-of 6139  df-1st 6207  df-2nd 6208  df-recs 6372  df-irdg 6437  df-frec 6458  df-1o 6483  df-oadd 6487  df-er 6601  df-map 6718  df-pm 6719  df-en 6809  df-dom 6810  df-fin 6811  df-sup 7059  df-inf 7060  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-reap 8619  df-ap 8626  df-div 8717  df-inn 9008  df-2 9066  df-3 9067  df-4 9068  df-5 9069  df-6 9070  df-7 9071  df-8 9072  df-9 9073  df-n0 9267  df-z 9344  df-dec 9475  df-uz 9619  df-q 9711  df-rp 9746  df-xneg 9864  df-xadd 9865  df-fz 10101  df-fzo 10235  df-seqfrec 10557  df-exp 10648  df-ihash 10885  df-cj 11024  df-re 11025  df-im 11026  df-rsqrt 11180  df-abs 11181  df-clim 11461  df-sumdc 11536  df-struct 12705  df-ndx 12706  df-slot 12707  df-base 12709  df-sets 12710  df-iress 12711  df-plusg 12793  df-mulr 12794  df-starv 12795  df-tset 12799  df-ple 12800  df-ds 12802  df-unif 12803  df-rest 12943  df-topn 12944  df-0g 12960  df-topgen 12962  df-mgm 13058  df-sgrp 13104  df-mnd 13119  df-grp 13205  df-minusg 13206  df-mulg 13326  df-subg 13376  df-cmn 13492  df-mgp 13553  df-ur 13592  df-ring 13630  df-cring 13631  df-subrg 13851  df-psmet 14175  df-xmet 14176  df-met 14177  df-bl 14178  df-mopn 14179  df-fg 14181  df-metu 14182  df-cnfld 14189  df-top 14318  df-topon 14331  df-topsp 14351  df-bases 14363  df-ntr 14416  df-cn 14508  df-cnp 14509  df-tx 14573  df-xms 14659  df-ms 14660  df-cncf 14891  df-limced 14976  df-dvap 14977  df-ply 15050

This theorem is referenced by:  dvply2  15087