Theorem dvply2g 15461

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 15430	. . . 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 15432	. . . . . . . . . 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 5692	. . . . . . . 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 6830	. . . . . . . . . . . . 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 14545	. . . . . . . . . . . . . 14 |- CC = ( Base ` ℂfld )
14	13	subrgss 14207	. . . . . . . . . . . . 13 |- ( S e. (SubRing ` ℂfld ) -> S C_ CC )
15		0cn 8154	. . . . . . . . . . . . . 14 |- 0 e. CC
16		snssi 3812	. . . . . . . . . . . . . 14 |- ( 0 e. CC -> { 0 } C_ CC )
17	15, 16	mp1i 10	. . . . . . . . . . . . 13 |- ( S e. (SubRing ` ℂfld ) -> { 0 } C_ CC )
18	14, 17	unssd 3380	. . . . . . . . . . . 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 5489	. . . . . . . . . 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 9482	. . . . . . . . . . . . 13 |- ( d e. NN0 -> d e. ZZ )
25	24	uzidd 9754	. . . . . . . . . . . 12 |- ( d e. NN0 -> d e. ( ZZ>= ` d ) )
26		peano2uz 9795	. . . . . . . . . . . 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 15452	. . . . . . . 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 4174	. . . . . . 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 2262	. . . . . 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 9440	. . . . . . . . . . 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 8178	. . . . . . . . . . 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 8474	. . . . . . . . . 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 2235	. . . . . . . . 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 6026	. . . . . . . 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 11898	. . . . . . 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 4175	. . . . . 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 6017	. . . . . . . 8 |- ( c = b -> ( c + 1 ) = ( b + 1 ) )
42		fvoveq1 6033	. . . . . . . 8 |- ( c = b -> ( p ` ( c + 1 ) ) = ( p ` ( b + 1 ) ) )
43	41, 42	oveq12d 6028	. . . . . . 7 |- ( c = b -> ( ( c + 1 ) x. ( p ` ( c + 1 ) ) ) = ( ( b + 1 ) x. ( p ` ( b + 1 ) ) ) )
44	43	cbvmptv 4180	. . . . . 6 |- ( c e. NN0 |-> ( ( c + 1 ) x. ( p ` ( c + 1 ) ) ) ) = ( b e. NN0 |-> ( ( b + 1 ) x. ( p ` ( b + 1 ) ) ) )
45		peano2nn0 9425	. . . . . . 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 15460	. . . . 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 10327	. . . . . . 7 |- ( b e. ( 0 ... d ) -> b e. NN0 )
50		peano2nn0 9425	. . . . . . . . . . . . 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 9440	. . . . . . . . . . 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 5776	. . . . . . . . . . 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 8183	. . . . . . . . . . 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 6017	. . . . . . . . . . . 12 |- ( u = ( c + 1 ) -> ( u x. v ) = ( ( c + 1 ) x. v ) )
57		oveq2 6018	. . . . . . . . . . . 12 |- ( v = ( p ` ( c + 1 ) ) -> ( ( c + 1 ) x. v ) = ( ( c + 1 ) x. ( p ` ( c + 1 ) ) ) )
58		eqid 2229	. . . . . . . . . . . 12 |- ( u e. CC , v e. CC |-> ( u x. v ) ) = ( u e. CC , v e. CC |-> ( u x. v ) )
59	56, 57, 58	ovmpog 6148	. . . . . . . . . . 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 1271	. . . . . . . . . 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 14570	. . . . . . . . . . . . 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 9583	. . . . . . . . . . . 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 3225	. . . . . . . . . . 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 14212	. . . . . . . . . . . . . . . . . 18 |- ( S e. (SubRing ` ℂfld ) -> S e. (SubGrp ` ℂfld ) )
68		cnfld0 14556	. . . . . . . . . . . . . . . . . . 19 |- 0 = ( 0g ` ℂfld )
69	68	subg0cl 13740	. . . . . . . . . . . . . . . . . 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 3813	. . . . . . . . . . . . . . 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 3377	. . . . . . . . . . . . . . 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 5465	. . . . . . . . . . . . 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 5776	. . . . . . . . . . 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 14548	. . . . . . . . . . . 12 |- ( u e. CC , v e. CC |-> ( u x. v ) ) = ( .r ` ℂfld )
79	78	subrgmcl 14218	. . . . . . . . . . 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 1271	. . . . . . . . . 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 2307	. . . . . . . . 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 5795	. . . . . . . 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 5775	. . . . . . 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 15436	. . . . 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 2306	. . . 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 2656	. 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 1395   e. wcel 2200   E.wrex 2509   u. cun 3195   C_ wss 3197   {csn 3666   |-> cmpt 4145   "cima 4723   -->wf 5317   ` cfv 5321  (class class class)co 6010   e. cmpo 6012   ^m cmap 6808   CCcc 8013   0cc0 8015   1c1 8016   + caddc 8018   x. cmul 8020   - cmin 8333   NN0cn0 9385   ZZcz 9462   ZZ>=cuz 9738   ...cfz 10221   ^cexp 10777   sum_csu 11885  SubGrpcsubg 13725  SubRingcsubrg 14202  ℂ_fldccnfld 14541   _D cdv 15350  Polycply 15423

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4259  ax-pr 4294  ax-un 4525  ax-setind 4630  ax-iinf 4681  ax-cnex 8106  ax-resscn 8107  ax-1cn 8108  ax-1re 8109  ax-icn 8110  ax-addcl 8111  ax-addrcl 8112  ax-mulcl 8113  ax-mulrcl 8114  ax-addcom 8115  ax-mulcom 8116  ax-addass 8117  ax-mulass 8118  ax-distr 8119  ax-i2m1 8120  ax-0lt1 8121  ax-1rid 8122  ax-0id 8123  ax-rnegex 8124  ax-precex 8125  ax-cnre 8126  ax-pre-ltirr 8127  ax-pre-ltwlin 8128  ax-pre-lttrn 8129  ax-pre-apti 8130  ax-pre-ltadd 8131  ax-pre-mulgt0 8132  ax-pre-mulext 8133  ax-arch 8134  ax-caucvg 8135  ax-addf 8137  ax-mulf 8138

This theorem depends on definitions:  df-bi 117  df-stab 836  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-tp 3674  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4385  df-po 4388  df-iso 4389  df-iord 4458  df-on 4460  df-ilim 4461  df-suc 4463  df-iom 4684  df-xp 4726  df-rel 4727  df-cnv 4728  df-co 4729  df-dm 4730  df-rn 4731  df-res 4732  df-ima 4733  df-iota 5281  df-fun 5323  df-fn 5324  df-f 5325  df-f1 5326  df-fo 5327  df-f1o 5328  df-fv 5329  df-isom 5330  df-riota 5963  df-ov 6013  df-oprab 6014  df-mpo 6015  df-of 6227  df-1st 6295  df-2nd 6296  df-recs 6462  df-irdg 6527  df-frec 6548  df-1o 6573  df-oadd 6577  df-er 6693  df-map 6810  df-pm 6811  df-en 6901  df-dom 6902  df-fin 6903  df-sup 7167  df-inf 7168  df-pnf 8199  df-mnf 8200  df-xr 8201  df-ltxr 8202  df-le 8203  df-sub 8335  df-neg 8336  df-reap 8738  df-ap 8745  df-div 8836  df-inn 9127  df-2 9185  df-3 9186  df-4 9187  df-5 9188  df-6 9189  df-7 9190  df-8 9191  df-9 9192  df-n0 9386  df-z 9463  df-dec 9595  df-uz 9739  df-q 9832  df-rp 9867  df-xneg 9985  df-xadd 9986  df-fz 10222  df-fzo 10356  df-seqfrec 10687  df-exp 10778  df-ihash 11015  df-cj 11374  df-re 11375  df-im 11376  df-rsqrt 11530  df-abs 11531  df-clim 11811  df-sumdc 11886  df-struct 13055  df-ndx 13056  df-slot 13057  df-base 13059  df-sets 13060  df-iress 13061  df-plusg 13144  df-mulr 13145  df-starv 13146  df-tset 13150  df-ple 13151  df-ds 13153  df-unif 13154  df-rest 13295  df-topn 13296  df-0g 13312  df-topgen 13314  df-mgm 13410  df-sgrp 13456  df-mnd 13471  df-grp 13557  df-minusg 13558  df-mulg 13678  df-subg 13728  df-cmn 13844  df-mgp 13905  df-ur 13944  df-ring 13982  df-cring 13983  df-subrg 14204  df-psmet 14528  df-xmet 14529  df-met 14530  df-bl 14531  df-mopn 14532  df-fg 14534  df-metu 14535  df-cnfld 14542  df-top 14693  df-topon 14706  df-topsp 14726  df-bases 14738  df-ntr 14791  df-cn 14883  df-cnp 14884  df-tx 14948  df-xms 15034  df-ms 15035  df-cncf 15266  df-limced 15351  df-dvap 15352  df-ply 15425

This theorem is referenced by:  dvply2  15462