Theorem dvply2g 15002

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 14971	. . . 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 14973	. . . . . . . . . 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 5614	. . . . . . . 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 6729	. . . . . . . . . . . . 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 14116	. . . . . . . . . . . . . 14 |- CC = ( Base ` ℂfld )
14	13	subrgss 13778	. . . . . . . . . . . . 13 |- ( S e. (SubRing ` ℂfld ) -> S C_ CC )
15		0cn 8018	. . . . . . . . . . . . . 14 |- 0 e. CC
16		snssi 3766	. . . . . . . . . . . . . 14 |- ( 0 e. CC -> { 0 } C_ CC )
17	15, 16	mp1i 10	. . . . . . . . . . . . 13 |- ( S e. (SubRing ` ℂfld ) -> { 0 } C_ CC )
18	14, 17	unssd 3339	. . . . . . . . . . . 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 5420	. . . . . . . . . 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 9346	. . . . . . . . . . . . 13 |- ( d e. NN0 -> d e. ZZ )
25	24	uzidd 9616	. . . . . . . . . . . 12 |- ( d e. NN0 -> d e. ( ZZ>= ` d ) )
26		peano2uz 9657	. . . . . . . . . . . 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 14993	. . . . . . . 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 4123	. . . . . . 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 9304	. . . . . . . . . . 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 8042	. . . . . . . . . . 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 8338	. . . . . . . . . 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 5938	. . . . . . . 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 11531	. . . . . . 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 4124	. . . . . 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 5929	. . . . . . . 8 |- ( c = b -> ( c + 1 ) = ( b + 1 ) )
42		fvoveq1 5945	. . . . . . . 8 |- ( c = b -> ( p ` ( c + 1 ) ) = ( p ` ( b + 1 ) ) )
43	41, 42	oveq12d 5940	. . . . . . 7 |- ( c = b -> ( ( c + 1 ) x. ( p ` ( c + 1 ) ) ) = ( ( b + 1 ) x. ( p ` ( b + 1 ) ) ) )
44	43	cbvmptv 4129	. . . . . 6 |- ( c e. NN0 |-> ( ( c + 1 ) x. ( p ` ( c + 1 ) ) ) ) = ( b e. NN0 |-> ( ( b + 1 ) x. ( p ` ( b + 1 ) ) ) )
45		peano2nn0 9289	. . . . . . 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 15001	. . . . 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 10189	. . . . . . 7 |- ( b e. ( 0 ... d ) -> b e. NN0 )
50		peano2nn0 9289	. . . . . . . . . . . . 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 9304	. . . . . . . . . . 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 5698	. . . . . . . . . . 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 8047	. . . . . . . . . . 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 5929	. . . . . . . . . . . 12 |- ( u = ( c + 1 ) -> ( u x. v ) = ( ( c + 1 ) x. v ) )
57		oveq2 5930	. . . . . . . . . . . 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 6057	. . . . . . . . . . 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 14141	. . . . . . . . . . . . 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 9446	. . . . . . . . . . . 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 3184	. . . . . . . . . . 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 13783	. . . . . . . . . . . . . . . . . 18 |- ( S e. (SubRing ` ℂfld ) -> S e. (SubGrp ` ℂfld ) )
68		cnfld0 14127	. . . . . . . . . . . . . . . . . . 19 |- 0 = ( 0g ` ℂfld )
69	68	subg0cl 13312	. . . . . . . . . . . . . . . . . 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 3767	. . . . . . . . . . . . . . 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 3336	. . . . . . . . . . . . . . 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 5396	. . . . . . . . . . . . 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 5698	. . . . . . . . . . 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 14119	. . . . . . . . . . . 12 |- ( u e. CC , v e. CC |-> ( u x. v ) ) = ( .r ` ℂfld )
79	78	subrgmcl 13789	. . . . . . . . . . 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 5717	. . . . . . . 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 5697	. . . . . . 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 14977	. . . . 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 3622   |-> cmpt 4094   "cima 4666   -->wf 5254   ` cfv 5258  (class class class)co 5922   e. cmpo 5924   ^m cmap 6707   CCcc 7877   0cc0 7879   1c1 7880   + caddc 7882   x. cmul 7884   - cmin 8197   NN0cn0 9249   ZZcz 9326   ZZ>=cuz 9601   ...cfz 10083   ^cexp 10630   sum_csu 11518  SubGrpcsubg 13297  SubRingcsubrg 13773  ℂ_fldccnfld 14112   _D cdv 14891  Polycply 14964

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 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995  ax-pre-mulgt0 7996  ax-pre-mulext 7997  ax-arch 7998  ax-caucvg 7999  ax-addf 8001  ax-mulf 8002

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 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-tp 3630  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-isom 5267  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-of 6135  df-1st 6198  df-2nd 6199  df-recs 6363  df-irdg 6428  df-frec 6449  df-1o 6474  df-oadd 6478  df-er 6592  df-map 6709  df-pm 6710  df-en 6800  df-dom 6801  df-fin 6802  df-sup 7050  df-inf 7051  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-reap 8602  df-ap 8609  df-div 8700  df-inn 8991  df-2 9049  df-3 9050  df-4 9051  df-5 9052  df-6 9053  df-7 9054  df-8 9055  df-9 9056  df-n0 9250  df-z 9327  df-dec 9458  df-uz 9602  df-q 9694  df-rp 9729  df-xneg 9847  df-xadd 9848  df-fz 10084  df-fzo 10218  df-seqfrec 10540  df-exp 10631  df-ihash 10868  df-cj 11007  df-re 11008  df-im 11009  df-rsqrt 11163  df-abs 11164  df-clim 11444  df-sumdc 11519  df-struct 12680  df-ndx 12681  df-slot 12682  df-base 12684  df-sets 12685  df-iress 12686  df-plusg 12768  df-mulr 12769  df-starv 12770  df-tset 12774  df-ple 12775  df-ds 12777  df-unif 12778  df-rest 12912  df-topn 12913  df-0g 12929  df-topgen 12931  df-mgm 12999  df-sgrp 13045  df-mnd 13058  df-grp 13135  df-minusg 13136  df-mulg 13250  df-subg 13300  df-cmn 13416  df-mgp 13477  df-ur 13516  df-ring 13554  df-cring 13555  df-subrg 13775  df-psmet 14099  df-xmet 14100  df-met 14101  df-bl 14102  df-mopn 14103  df-fg 14105  df-metu 14106  df-cnfld 14113  df-top 14234  df-topon 14247  df-topsp 14267  df-bases 14279  df-ntr 14332  df-cn 14424  df-cnp 14425  df-tx 14489  df-xms 14575  df-ms 14576  df-cncf 14807  df-limced 14892  df-dvap 14893  df-ply 14966

This theorem is referenced by:  dvply2  15003