Theorem plyco 23997

Description: The composition of two polynomials is a polynomial. (Contributed by Mario Carneiro, 23-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

Hypotheses

Ref	Expression
plyco.1	|- ( ph -> F e. (Poly ` S ) )
plyco.2	|- ( ph -> G e. (Poly ` S ) )
plyco.3	|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x + y ) e. S )
plyco.4	|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x x. y ) e. S )

Assertion

Ref	Expression
plyco	|- ( ph -> ( F o. G ) e. (Poly ` S ) )

Distinct variable groups:   x, y, F   x, G, y   ph, x, y   x, S, y

Proof of Theorem plyco

Dummy variables k d z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		plyco.2	. . . . 5 |- ( ph -> G e. (Poly ` S ) )
2		plyf 23954	. . . . 5 |- ( G e. (Poly ` S ) -> G : CC --> CC )
3	1, 2	syl 17	. . . 4 |- ( ph -> G : CC --> CC )
4	3	ffvelrnda 6359	. . 3 |- ( ( ph /\ z e. CC ) -> ( G ` z ) e. CC )
5	3	feqmptd 6249	. . 3 |- ( ph -> G = ( z e. CC |-> ( G ` z ) ) )
6		plyco.1	. . . 4 |- ( ph -> F e. (Poly ` S ) )
7		eqid 2622	. . . . 5 |- (coeff ` F ) = (coeff ` F )
8		eqid 2622	. . . . 5 |- (deg ` F ) = (deg ` F )
9	7, 8	coeid 23994	. . . 4 |- ( F e. (Poly ` S ) -> F = ( x e. CC |-> sum_ k e. ( 0 ... (deg ` F ) ) ( ( (coeff ` F ) ` k ) x. ( x ^ k ) ) ) )
10	6, 9	syl 17	. . 3 |- ( ph -> F = ( x e. CC |-> sum_ k e. ( 0 ... (deg ` F ) ) ( ( (coeff ` F ) ` k ) x. ( x ^ k ) ) ) )
11		oveq1 6657	. . . . 5 |- ( x = ( G ` z ) -> ( x ^ k ) = ( ( G ` z ) ^ k ) )
12	11	oveq2d 6666	. . . 4 |- ( x = ( G ` z ) -> ( ( (coeff ` F ) ` k ) x. ( x ^ k ) ) = ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) )
13	12	sumeq2sdv 14435	. . 3 |- ( x = ( G ` z ) -> sum_ k e. ( 0 ... (deg ` F ) ) ( ( (coeff ` F ) ` k ) x. ( x ^ k ) ) = sum_ k e. ( 0 ... (deg ` F ) ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) )
14	4, 5, 10, 13	fmptco 6396	. 2 |- ( ph -> ( F o. G ) = ( z e. CC |-> sum_ k e. ( 0 ... (deg ` F ) ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) ) )
15		dgrcl 23989	. . . 4 |- ( F e. (Poly ` S ) -> (deg ` F ) e. NN0 )
16	6, 15	syl 17	. . 3 |- ( ph -> (deg ` F ) e. NN0 )
17		oveq2 6658	. . . . . . . 8 |- ( x = 0 -> ( 0 ... x ) = ( 0 ... 0 ) )
18	17	sumeq1d 14431	. . . . . . 7 |- ( x = 0 -> sum_ k e. ( 0 ... x ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) = sum_ k e. ( 0 ... 0 ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) )
19	18	mpteq2dv 4745	. . . . . 6 |- ( x = 0 -> ( z e. CC |-> sum_ k e. ( 0 ... x ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) ) = ( z e. CC |-> sum_ k e. ( 0 ... 0 ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) ) )
20	19	eleq1d 2686	. . . . 5 |- ( x = 0 -> ( ( z e. CC |-> sum_ k e. ( 0 ... x ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) ) e. (Poly ` S ) <-> ( z e. CC |-> sum_ k e. ( 0 ... 0 ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) ) e. (Poly ` S ) ) )
21	20	imbi2d 330	. . . 4 |- ( x = 0 -> ( ( ph -> ( z e. CC |-> sum_ k e. ( 0 ... x ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) ) e. (Poly ` S ) ) <-> ( ph -> ( z e. CC |-> sum_ k e. ( 0 ... 0 ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) ) e. (Poly ` S ) ) ) )
22		oveq2 6658	. . . . . . . 8 |- ( x = d -> ( 0 ... x ) = ( 0 ... d ) )
23	22	sumeq1d 14431	. . . . . . 7 |- ( x = d -> sum_ k e. ( 0 ... x ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) = sum_ k e. ( 0 ... d ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) )
24	23	mpteq2dv 4745	. . . . . 6 |- ( x = d -> ( z e. CC |-> sum_ k e. ( 0 ... x ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) ) = ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) ) )
25	24	eleq1d 2686	. . . . 5 |- ( x = d -> ( ( z e. CC |-> sum_ k e. ( 0 ... x ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) ) e. (Poly ` S ) <-> ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) ) e. (Poly ` S ) ) )
26	25	imbi2d 330	. . . 4 |- ( x = d -> ( ( ph -> ( z e. CC |-> sum_ k e. ( 0 ... x ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) ) e. (Poly ` S ) ) <-> ( ph -> ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) ) e. (Poly ` S ) ) ) )
27		oveq2 6658	. . . . . . . 8 |- ( x = ( d + 1 ) -> ( 0 ... x ) = ( 0 ... ( d + 1 ) ) )
28	27	sumeq1d 14431	. . . . . . 7 |- ( x = ( d + 1 ) -> sum_ k e. ( 0 ... x ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) = sum_ k e. ( 0 ... ( d + 1 ) ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) )
29	28	mpteq2dv 4745	. . . . . 6 |- ( x = ( d + 1 ) -> ( z e. CC |-> sum_ k e. ( 0 ... x ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) ) = ( z e. CC |-> sum_ k e. ( 0 ... ( d + 1 ) ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) ) )
30	29	eleq1d 2686	. . . . 5 |- ( x = ( d + 1 ) -> ( ( z e. CC |-> sum_ k e. ( 0 ... x ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) ) e. (Poly ` S ) <-> ( z e. CC |-> sum_ k e. ( 0 ... ( d + 1 ) ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) ) e. (Poly ` S ) ) )
31	30	imbi2d 330	. . . 4 |- ( x = ( d + 1 ) -> ( ( ph -> ( z e. CC |-> sum_ k e. ( 0 ... x ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) ) e. (Poly ` S ) ) <-> ( ph -> ( z e. CC |-> sum_ k e. ( 0 ... ( d + 1 ) ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) ) e. (Poly ` S ) ) ) )
32		oveq2 6658	. . . . . . . 8 |- ( x = (deg ` F ) -> ( 0 ... x ) = ( 0 ... (deg ` F ) ) )
33	32	sumeq1d 14431	. . . . . . 7 |- ( x = (deg ` F ) -> sum_ k e. ( 0 ... x ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) = sum_ k e. ( 0 ... (deg ` F ) ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) )
34	33	mpteq2dv 4745	. . . . . 6 |- ( x = (deg ` F ) -> ( z e. CC |-> sum_ k e. ( 0 ... x ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) ) = ( z e. CC |-> sum_ k e. ( 0 ... (deg ` F ) ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) ) )
35	34	eleq1d 2686	. . . . 5 |- ( x = (deg ` F ) -> ( ( z e. CC |-> sum_ k e. ( 0 ... x ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) ) e. (Poly ` S ) <-> ( z e. CC |-> sum_ k e. ( 0 ... (deg ` F ) ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) ) e. (Poly ` S ) ) )
36	35	imbi2d 330	. . . 4 |- ( x = (deg ` F ) -> ( ( ph -> ( z e. CC |-> sum_ k e. ( 0 ... x ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) ) e. (Poly ` S ) ) <-> ( ph -> ( z e. CC |-> sum_ k e. ( 0 ... (deg ` F ) ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) ) e. (Poly ` S ) ) ) )
37		0z 11388	. . . . . . . . 9 |- 0 e. ZZ
38	4	exp0d 13002	. . . . . . . . . . . 12 |- ( ( ph /\ z e. CC ) -> ( ( G ` z ) ^ 0 ) = 1 )
39	38	oveq2d 6666	. . . . . . . . . . 11 |- ( ( ph /\ z e. CC ) -> ( ( (coeff ` F ) ` 0 ) x. ( ( G ` z ) ^ 0 ) ) = ( ( (coeff ` F ) ` 0 ) x. 1 ) )
40		plybss 23950	. . . . . . . . . . . . . . . 16 |- ( F e. (Poly ` S ) -> S C_ CC )
41	6, 40	syl 17	. . . . . . . . . . . . . . 15 |- ( ph -> S C_ CC )
42		0cnd 10033	. . . . . . . . . . . . . . . 16 |- ( ph -> 0 e. CC )
43	42	snssd 4340	. . . . . . . . . . . . . . 15 |- ( ph -> { 0 } C_ CC )
44	41, 43	unssd 3789	. . . . . . . . . . . . . 14 |- ( ph -> ( S u. { 0 } ) C_ CC )
45	7	coef 23986	. . . . . . . . . . . . . . . 16 |- ( F e. (Poly ` S ) -> (coeff ` F ) : NN0 --> ( S u. { 0 } ) )
46	6, 45	syl 17	. . . . . . . . . . . . . . 15 |- ( ph -> (coeff ` F ) : NN0 --> ( S u. { 0 } ) )
47		0nn0 11307	. . . . . . . . . . . . . . 15 |- 0 e. NN0
48		ffvelrn 6357	. . . . . . . . . . . . . . 15 |- ( ( (coeff ` F ) : NN0 --> ( S u. { 0 } ) /\ 0 e. NN0 ) -> ( (coeff ` F ) ` 0 ) e. ( S u. { 0 } ) )
49	46, 47, 48	sylancl 694	. . . . . . . . . . . . . 14 |- ( ph -> ( (coeff ` F ) ` 0 ) e. ( S u. { 0 } ) )
50	44, 49	sseldd 3604	. . . . . . . . . . . . 13 |- ( ph -> ( (coeff ` F ) ` 0 ) e. CC )
51	50	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ z e. CC ) -> ( (coeff ` F ) ` 0 ) e. CC )
52	51	mulid1d 10057	. . . . . . . . . . 11 |- ( ( ph /\ z e. CC ) -> ( ( (coeff ` F ) ` 0 ) x. 1 ) = ( (coeff ` F ) ` 0 ) )
53	39, 52	eqtrd 2656	. . . . . . . . . 10 |- ( ( ph /\ z e. CC ) -> ( ( (coeff ` F ) ` 0 ) x. ( ( G ` z ) ^ 0 ) ) = ( (coeff ` F ) ` 0 ) )
54	53, 51	eqeltrd 2701	. . . . . . . . 9 |- ( ( ph /\ z e. CC ) -> ( ( (coeff ` F ) ` 0 ) x. ( ( G ` z ) ^ 0 ) ) e. CC )
55		fveq2 6191	. . . . . . . . . . 11 |- ( k = 0 -> ( (coeff ` F ) ` k ) = ( (coeff ` F ) ` 0 ) )
56		oveq2 6658	. . . . . . . . . . 11 |- ( k = 0 -> ( ( G ` z ) ^ k ) = ( ( G ` z ) ^ 0 ) )
57	55, 56	oveq12d 6668	. . . . . . . . . 10 |- ( k = 0 -> ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) = ( ( (coeff ` F ) ` 0 ) x. ( ( G ` z ) ^ 0 ) ) )
58	57	fsum1 14476	. . . . . . . . 9 |- ( ( 0 e. ZZ /\ ( ( (coeff ` F ) ` 0 ) x. ( ( G ` z ) ^ 0 ) ) e. CC ) -> sum_ k e. ( 0 ... 0 ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) = ( ( (coeff ` F ) ` 0 ) x. ( ( G ` z ) ^ 0 ) ) )
59	37, 54, 58	sylancr 695	. . . . . . . 8 |- ( ( ph /\ z e. CC ) -> sum_ k e. ( 0 ... 0 ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) = ( ( (coeff ` F ) ` 0 ) x. ( ( G ` z ) ^ 0 ) ) )
60	59, 53	eqtrd 2656	. . . . . . 7 |- ( ( ph /\ z e. CC ) -> sum_ k e. ( 0 ... 0 ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) = ( (coeff ` F ) ` 0 ) )
61	60	mpteq2dva 4744	. . . . . 6 |- ( ph -> ( z e. CC |-> sum_ k e. ( 0 ... 0 ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) ) = ( z e. CC |-> ( (coeff ` F ) ` 0 ) ) )
62		fconstmpt 5163	. . . . . 6 |- ( CC X. { ( (coeff ` F ) ` 0 ) } ) = ( z e. CC |-> ( (coeff ` F ) ` 0 ) )
63	61, 62	syl6eqr 2674	. . . . 5 |- ( ph -> ( z e. CC |-> sum_ k e. ( 0 ... 0 ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) ) = ( CC X. { ( (coeff ` F ) ` 0 ) } ) )
64		plyconst 23962	. . . . . . 7 |- ( ( ( S u. { 0 } ) C_ CC /\ ( (coeff ` F ) ` 0 ) e. ( S u. { 0 } ) ) -> ( CC X. { ( (coeff ` F ) ` 0 ) } ) e. (Poly ` ( S u. { 0 } ) ) )
65	44, 49, 64	syl2anc 693	. . . . . 6 |- ( ph -> ( CC X. { ( (coeff ` F ) ` 0 ) } ) e. (Poly ` ( S u. { 0 } ) ) )
66		plyun0 23953	. . . . . 6 |- (Poly ` ( S u. { 0 } ) ) = (Poly ` S )
67	65, 66	syl6eleq 2711	. . . . 5 |- ( ph -> ( CC X. { ( (coeff ` F ) ` 0 ) } ) e. (Poly ` S ) )
68	63, 67	eqeltrd 2701	. . . 4 |- ( ph -> ( z e. CC |-> sum_ k e. ( 0 ... 0 ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) ) e. (Poly ` S ) )
69		simprr 796	. . . . . . . . 9 |- ( ( ph /\ ( d e. NN0 /\ ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) ) e. (Poly ` S ) ) ) -> ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) ) e. (Poly ` S ) )
70	44	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ d e. NN0 ) -> ( S u. { 0 } ) C_ CC )
71		peano2nn0 11333	. . . . . . . . . . . . . 14 |- ( d e. NN0 -> ( d + 1 ) e. NN0 )
72		ffvelrn 6357	. . . . . . . . . . . . . 14 |- ( ( (coeff ` F ) : NN0 --> ( S u. { 0 } ) /\ ( d + 1 ) e. NN0 ) -> ( (coeff ` F ) ` ( d + 1 ) ) e. ( S u. { 0 } ) )
73	46, 71, 72	syl2an 494	. . . . . . . . . . . . 13 |- ( ( ph /\ d e. NN0 ) -> ( (coeff ` F ) ` ( d + 1 ) ) e. ( S u. { 0 } ) )
74		plyconst 23962	. . . . . . . . . . . . 13 |- ( ( ( S u. { 0 } ) C_ CC /\ ( (coeff ` F ) ` ( d + 1 ) ) e. ( S u. { 0 } ) ) -> ( CC X. { ( (coeff ` F ) ` ( d + 1 ) ) } ) e. (Poly ` ( S u. { 0 } ) ) )
75	70, 73, 74	syl2anc 693	. . . . . . . . . . . 12 |- ( ( ph /\ d e. NN0 ) -> ( CC X. { ( (coeff ` F ) ` ( d + 1 ) ) } ) e. (Poly ` ( S u. { 0 } ) ) )
76	75, 66	syl6eleq 2711	. . . . . . . . . . 11 |- ( ( ph /\ d e. NN0 ) -> ( CC X. { ( (coeff ` F ) ` ( d + 1 ) ) } ) e. (Poly ` S ) )
77		nn0p1nn 11332	. . . . . . . . . . . . 13 |- ( d e. NN0 -> ( d + 1 ) e. NN )
78		oveq2 6658	. . . . . . . . . . . . . . . . 17 |- ( x = 1 -> ( ( G ` z ) ^ x ) = ( ( G ` z ) ^ 1 ) )
79	78	mpteq2dv 4745	. . . . . . . . . . . . . . . 16 |- ( x = 1 -> ( z e. CC |-> ( ( G ` z ) ^ x ) ) = ( z e. CC |-> ( ( G ` z ) ^ 1 ) ) )
80	79	eleq1d 2686	. . . . . . . . . . . . . . 15 |- ( x = 1 -> ( ( z e. CC |-> ( ( G ` z ) ^ x ) ) e. (Poly ` S ) <-> ( z e. CC |-> ( ( G ` z ) ^ 1 ) ) e. (Poly ` S ) ) )
81	80	imbi2d 330	. . . . . . . . . . . . . 14 |- ( x = 1 -> ( ( ph -> ( z e. CC |-> ( ( G ` z ) ^ x ) ) e. (Poly ` S ) ) <-> ( ph -> ( z e. CC |-> ( ( G ` z ) ^ 1 ) ) e. (Poly ` S ) ) ) )
82		oveq2 6658	. . . . . . . . . . . . . . . . 17 |- ( x = d -> ( ( G ` z ) ^ x ) = ( ( G ` z ) ^ d ) )
83	82	mpteq2dv 4745	. . . . . . . . . . . . . . . 16 |- ( x = d -> ( z e. CC |-> ( ( G ` z ) ^ x ) ) = ( z e. CC |-> ( ( G ` z ) ^ d ) ) )
84	83	eleq1d 2686	. . . . . . . . . . . . . . 15 |- ( x = d -> ( ( z e. CC |-> ( ( G ` z ) ^ x ) ) e. (Poly ` S ) <-> ( z e. CC |-> ( ( G ` z ) ^ d ) ) e. (Poly ` S ) ) )
85	84	imbi2d 330	. . . . . . . . . . . . . 14 |- ( x = d -> ( ( ph -> ( z e. CC |-> ( ( G ` z ) ^ x ) ) e. (Poly ` S ) ) <-> ( ph -> ( z e. CC |-> ( ( G ` z ) ^ d ) ) e. (Poly ` S ) ) ) )
86		oveq2 6658	. . . . . . . . . . . . . . . . 17 |- ( x = ( d + 1 ) -> ( ( G ` z ) ^ x ) = ( ( G ` z ) ^ ( d + 1 ) ) )
87	86	mpteq2dv 4745	. . . . . . . . . . . . . . . 16 |- ( x = ( d + 1 ) -> ( z e. CC |-> ( ( G ` z ) ^ x ) ) = ( z e. CC |-> ( ( G ` z ) ^ ( d + 1 ) ) ) )
88	87	eleq1d 2686	. . . . . . . . . . . . . . 15 |- ( x = ( d + 1 ) -> ( ( z e. CC |-> ( ( G ` z ) ^ x ) ) e. (Poly ` S ) <-> ( z e. CC |-> ( ( G ` z ) ^ ( d + 1 ) ) ) e. (Poly ` S ) ) )
89	88	imbi2d 330	. . . . . . . . . . . . . 14 |- ( x = ( d + 1 ) -> ( ( ph -> ( z e. CC |-> ( ( G ` z ) ^ x ) ) e. (Poly ` S ) ) <-> ( ph -> ( z e. CC |-> ( ( G ` z ) ^ ( d + 1 ) ) ) e. (Poly ` S ) ) ) )
90	4	exp1d 13003	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ z e. CC ) -> ( ( G ` z ) ^ 1 ) = ( G ` z ) )
91	90	mpteq2dva 4744	. . . . . . . . . . . . . . . 16 |- ( ph -> ( z e. CC |-> ( ( G ` z ) ^ 1 ) ) = ( z e. CC |-> ( G ` z ) ) )
92	91, 5	eqtr4d 2659	. . . . . . . . . . . . . . 15 |- ( ph -> ( z e. CC |-> ( ( G ` z ) ^ 1 ) ) = G )
93	92, 1	eqeltrd 2701	. . . . . . . . . . . . . 14 |- ( ph -> ( z e. CC |-> ( ( G ` z ) ^ 1 ) ) e. (Poly ` S ) )
94		simprr 796	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ ( d e. NN /\ ( z e. CC |-> ( ( G ` z ) ^ d ) ) e. (Poly ` S ) ) ) -> ( z e. CC |-> ( ( G ` z ) ^ d ) ) e. (Poly ` S ) )
95	1	adantr 481	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ ( d e. NN /\ ( z e. CC |-> ( ( G ` z ) ^ d ) ) e. (Poly ` S ) ) ) -> G e. (Poly ` S ) )
96		plyco.3	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x + y ) e. S )
97	96	adantlr 751	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ph /\ ( d e. NN /\ ( z e. CC |-> ( ( G ` z ) ^ d ) ) e. (Poly ` S ) ) ) /\ ( x e. S /\ y e. S ) ) -> ( x + y ) e. S )
98		plyco.4	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x x. y ) e. S )
99	98	adantlr 751	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ph /\ ( d e. NN /\ ( z e. CC |-> ( ( G ` z ) ^ d ) ) e. (Poly ` S ) ) ) /\ ( x e. S /\ y e. S ) ) -> ( x x. y ) e. S )
100	94, 95, 97, 99	plymul 23974	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ ( d e. NN /\ ( z e. CC |-> ( ( G ` z ) ^ d ) ) e. (Poly ` S ) ) ) -> ( ( z e. CC |-> ( ( G ` z ) ^ d ) ) oF x. G ) e. (Poly ` S ) )
101	100	expr 643	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ d e. NN ) -> ( ( z e. CC |-> ( ( G ` z ) ^ d ) ) e. (Poly ` S ) -> ( ( z e. CC |-> ( ( G ` z ) ^ d ) ) oF x. G ) e. (Poly ` S ) ) )
102		cnex 10017	. . . . . . . . . . . . . . . . . . . . 21 |- CC e. _V
103	102	a1i 11	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ d e. NN ) -> CC e. _V )
104		ovexd 6680	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ph /\ d e. NN ) /\ z e. CC ) -> ( ( G ` z ) ^ d ) e. _V )
105	4	adantlr 751	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ph /\ d e. NN ) /\ z e. CC ) -> ( G ` z ) e. CC )
106		eqidd 2623	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ d e. NN ) -> ( z e. CC |-> ( ( G ` z ) ^ d ) ) = ( z e. CC |-> ( ( G ` z ) ^ d ) ) )
107	5	adantr 481	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ d e. NN ) -> G = ( z e. CC |-> ( G ` z ) ) )
108	103, 104, 105, 106, 107	offval2 6914	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ d e. NN ) -> ( ( z e. CC |-> ( ( G ` z ) ^ d ) ) oF x. G ) = ( z e. CC |-> ( ( ( G ` z ) ^ d ) x. ( G ` z ) ) ) )
109		nnnn0 11299	. . . . . . . . . . . . . . . . . . . . . 22 |- ( d e. NN -> d e. NN0 )
110	109	ad2antlr 763	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ph /\ d e. NN ) /\ z e. CC ) -> d e. NN0 )
111	105, 110	expp1d 13009	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ph /\ d e. NN ) /\ z e. CC ) -> ( ( G ` z ) ^ ( d + 1 ) ) = ( ( ( G ` z ) ^ d ) x. ( G ` z ) ) )
112	111	mpteq2dva 4744	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ d e. NN ) -> ( z e. CC |-> ( ( G ` z ) ^ ( d + 1 ) ) ) = ( z e. CC |-> ( ( ( G ` z ) ^ d ) x. ( G ` z ) ) ) )
113	108, 112	eqtr4d 2659	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ d e. NN ) -> ( ( z e. CC |-> ( ( G ` z ) ^ d ) ) oF x. G ) = ( z e. CC |-> ( ( G ` z ) ^ ( d + 1 ) ) ) )
114	113	eleq1d 2686	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ d e. NN ) -> ( ( ( z e. CC |-> ( ( G ` z ) ^ d ) ) oF x. G ) e. (Poly ` S ) <-> ( z e. CC |-> ( ( G ` z ) ^ ( d + 1 ) ) ) e. (Poly ` S ) ) )
115	101, 114	sylibd 229	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ d e. NN ) -> ( ( z e. CC |-> ( ( G ` z ) ^ d ) ) e. (Poly ` S ) -> ( z e. CC |-> ( ( G ` z ) ^ ( d + 1 ) ) ) e. (Poly ` S ) ) )
116	115	expcom 451	. . . . . . . . . . . . . . 15 |- ( d e. NN -> ( ph -> ( ( z e. CC |-> ( ( G ` z ) ^ d ) ) e. (Poly ` S ) -> ( z e. CC |-> ( ( G ` z ) ^ ( d + 1 ) ) ) e. (Poly ` S ) ) ) )
117	116	a2d 29	. . . . . . . . . . . . . 14 |- ( d e. NN -> ( ( ph -> ( z e. CC |-> ( ( G ` z ) ^ d ) ) e. (Poly ` S ) ) -> ( ph -> ( z e. CC |-> ( ( G ` z ) ^ ( d + 1 ) ) ) e. (Poly ` S ) ) ) )
118	81, 85, 89, 89, 93, 117	nnind 11038	. . . . . . . . . . . . 13 |- ( ( d + 1 ) e. NN -> ( ph -> ( z e. CC |-> ( ( G ` z ) ^ ( d + 1 ) ) ) e. (Poly ` S ) ) )
119	77, 118	syl 17	. . . . . . . . . . . 12 |- ( d e. NN0 -> ( ph -> ( z e. CC |-> ( ( G ` z ) ^ ( d + 1 ) ) ) e. (Poly ` S ) ) )
120	119	impcom 446	. . . . . . . . . . 11 |- ( ( ph /\ d e. NN0 ) -> ( z e. CC |-> ( ( G ` z ) ^ ( d + 1 ) ) ) e. (Poly ` S ) )
121	96	adantlr 751	. . . . . . . . . . 11 |- ( ( ( ph /\ d e. NN0 ) /\ ( x e. S /\ y e. S ) ) -> ( x + y ) e. S )
122	98	adantlr 751	. . . . . . . . . . 11 |- ( ( ( ph /\ d e. NN0 ) /\ ( x e. S /\ y e. S ) ) -> ( x x. y ) e. S )
123	76, 120, 121, 122	plymul 23974	. . . . . . . . . 10 |- ( ( ph /\ d e. NN0 ) -> ( ( CC X. { ( (coeff ` F ) ` ( d + 1 ) ) } ) oF x. ( z e. CC |-> ( ( G ` z ) ^ ( d + 1 ) ) ) ) e. (Poly ` S ) )
124	123	adantrr 753	. . . . . . . . 9 |- ( ( ph /\ ( d e. NN0 /\ ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) ) e. (Poly ` S ) ) ) -> ( ( CC X. { ( (coeff ` F ) ` ( d + 1 ) ) } ) oF x. ( z e. CC |-> ( ( G ` z ) ^ ( d + 1 ) ) ) ) e. (Poly ` S ) )
125	96	adantlr 751	. . . . . . . . 9 |- ( ( ( ph /\ ( d e. NN0 /\ ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) ) e. (Poly ` S ) ) ) /\ ( x e. S /\ y e. S ) ) -> ( x + y ) e. S )
126	69, 124, 125	plyadd 23973	. . . . . . . 8 |- ( ( ph /\ ( d e. NN0 /\ ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) ) e. (Poly ` S ) ) ) -> ( ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) ) oF + ( ( CC X. { ( (coeff ` F ) ` ( d + 1 ) ) } ) oF x. ( z e. CC |-> ( ( G ` z ) ^ ( d + 1 ) ) ) ) ) e. (Poly ` S ) )
127	126	expr 643	. . . . . . 7 |- ( ( ph /\ d e. NN0 ) -> ( ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) ) e. (Poly ` S ) -> ( ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) ) oF + ( ( CC X. { ( (coeff ` F ) ` ( d + 1 ) ) } ) oF x. ( z e. CC |-> ( ( G ` z ) ^ ( d + 1 ) ) ) ) ) e. (Poly ` S ) ) )
128	102	a1i 11	. . . . . . . . . 10 |- ( ( ph /\ d e. NN0 ) -> CC e. _V )
129		sumex 14418	. . . . . . . . . . 11 |- sum_ k e. ( 0 ... d ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) e. _V
130	129	a1i 11	. . . . . . . . . 10 |- ( ( ( ph /\ d e. NN0 ) /\ z e. CC ) -> sum_ k e. ( 0 ... d ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) e. _V )
131		ovexd 6680	. . . . . . . . . 10 |- ( ( ( ph /\ d e. NN0 ) /\ z e. CC ) -> ( ( (coeff ` F ) ` ( d + 1 ) ) x. ( ( G ` z ) ^ ( d + 1 ) ) ) e. _V )
132		eqidd 2623	. . . . . . . . . 10 |- ( ( ph /\ d e. NN0 ) -> ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) ) = ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) ) )
133		fvexd 6203	. . . . . . . . . . 11 |- ( ( ( ph /\ d e. NN0 ) /\ z e. CC ) -> ( (coeff ` F ) ` ( d + 1 ) ) e. _V )
134		ovexd 6680	. . . . . . . . . . 11 |- ( ( ( ph /\ d e. NN0 ) /\ z e. CC ) -> ( ( G ` z ) ^ ( d + 1 ) ) e. _V )
135		fconstmpt 5163	. . . . . . . . . . . 12 |- ( CC X. { ( (coeff ` F ) ` ( d + 1 ) ) } ) = ( z e. CC |-> ( (coeff ` F ) ` ( d + 1 ) ) )
136	135	a1i 11	. . . . . . . . . . 11 |- ( ( ph /\ d e. NN0 ) -> ( CC X. { ( (coeff ` F ) ` ( d + 1 ) ) } ) = ( z e. CC |-> ( (coeff ` F ) ` ( d + 1 ) ) ) )
137		eqidd 2623	. . . . . . . . . . 11 |- ( ( ph /\ d e. NN0 ) -> ( z e. CC |-> ( ( G ` z ) ^ ( d + 1 ) ) ) = ( z e. CC |-> ( ( G ` z ) ^ ( d + 1 ) ) ) )
138	128, 133, 134, 136, 137	offval2 6914	. . . . . . . . . 10 |- ( ( ph /\ d e. NN0 ) -> ( ( CC X. { ( (coeff ` F ) ` ( d + 1 ) ) } ) oF x. ( z e. CC |-> ( ( G ` z ) ^ ( d + 1 ) ) ) ) = ( z e. CC |-> ( ( (coeff ` F ) ` ( d + 1 ) ) x. ( ( G ` z ) ^ ( d + 1 ) ) ) ) )
139	128, 130, 131, 132, 138	offval2 6914	. . . . . . . . 9 |- ( ( ph /\ d e. NN0 ) -> ( ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) ) oF + ( ( CC X. { ( (coeff ` F ) ` ( d + 1 ) ) } ) oF x. ( z e. CC |-> ( ( G ` z ) ^ ( d + 1 ) ) ) ) ) = ( z e. CC |-> ( sum_ k e. ( 0 ... d ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) + ( ( (coeff ` F ) ` ( d + 1 ) ) x. ( ( G ` z ) ^ ( d + 1 ) ) ) ) ) )
140		simplr 792	. . . . . . . . . . . 12 |- ( ( ( ph /\ d e. NN0 ) /\ z e. CC ) -> d e. NN0 )
141		nn0uz 11722	. . . . . . . . . . . 12 |- NN0 = ( ZZ>= ` 0 )
142	140, 141	syl6eleq 2711	. . . . . . . . . . 11 |- ( ( ( ph /\ d e. NN0 ) /\ z e. CC ) -> d e. ( ZZ>= ` 0 ) )
143	7	coef3 23988	. . . . . . . . . . . . . . 15 |- ( F e. (Poly ` S ) -> (coeff ` F ) : NN0 --> CC )
144	6, 143	syl 17	. . . . . . . . . . . . . 14 |- ( ph -> (coeff ` F ) : NN0 --> CC )
145	144	ad2antrr 762	. . . . . . . . . . . . 13 |- ( ( ( ph /\ d e. NN0 ) /\ z e. CC ) -> (coeff ` F ) : NN0 --> CC )
146		elfznn0 12433	. . . . . . . . . . . . 13 |- ( k e. ( 0 ... ( d + 1 ) ) -> k e. NN0 )
147		ffvelrn 6357	. . . . . . . . . . . . 13 |- ( ( (coeff ` F ) : NN0 --> CC /\ k e. NN0 ) -> ( (coeff ` F ) ` k ) e. CC )
148	145, 146, 147	syl2an 494	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ d e. NN0 ) /\ z e. CC ) /\ k e. ( 0 ... ( d + 1 ) ) ) -> ( (coeff ` F ) ` k ) e. CC )
149	4	adantlr 751	. . . . . . . . . . . . 13 |- ( ( ( ph /\ d e. NN0 ) /\ z e. CC ) -> ( G ` z ) e. CC )
150		expcl 12878	. . . . . . . . . . . . 13 |- ( ( ( G ` z ) e. CC /\ k e. NN0 ) -> ( ( G ` z ) ^ k ) e. CC )
151	149, 146, 150	syl2an 494	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ d e. NN0 ) /\ z e. CC ) /\ k e. ( 0 ... ( d + 1 ) ) ) -> ( ( G ` z ) ^ k ) e. CC )
152	148, 151	mulcld 10060	. . . . . . . . . . 11 |- ( ( ( ( ph /\ d e. NN0 ) /\ z e. CC ) /\ k e. ( 0 ... ( d + 1 ) ) ) -> ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) e. CC )
153		fveq2 6191	. . . . . . . . . . . 12 |- ( k = ( d + 1 ) -> ( (coeff ` F ) ` k ) = ( (coeff ` F ) ` ( d + 1 ) ) )
154		oveq2 6658	. . . . . . . . . . . 12 |- ( k = ( d + 1 ) -> ( ( G ` z ) ^ k ) = ( ( G ` z ) ^ ( d + 1 ) ) )
155	153, 154	oveq12d 6668	. . . . . . . . . . 11 |- ( k = ( d + 1 ) -> ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) = ( ( (coeff ` F ) ` ( d + 1 ) ) x. ( ( G ` z ) ^ ( d + 1 ) ) ) )
156	142, 152, 155	fsump1 14487	. . . . . . . . . 10 |- ( ( ( ph /\ d e. NN0 ) /\ z e. CC ) -> sum_ k e. ( 0 ... ( d + 1 ) ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) = ( sum_ k e. ( 0 ... d ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) + ( ( (coeff ` F ) ` ( d + 1 ) ) x. ( ( G ` z ) ^ ( d + 1 ) ) ) ) )
157	156	mpteq2dva 4744	. . . . . . . . 9 |- ( ( ph /\ d e. NN0 ) -> ( z e. CC |-> sum_ k e. ( 0 ... ( d + 1 ) ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) ) = ( z e. CC |-> ( sum_ k e. ( 0 ... d ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) + ( ( (coeff ` F ) ` ( d + 1 ) ) x. ( ( G ` z ) ^ ( d + 1 ) ) ) ) ) )
158	139, 157	eqtr4d 2659	. . . . . . . 8 |- ( ( ph /\ d e. NN0 ) -> ( ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) ) oF + ( ( CC X. { ( (coeff ` F ) ` ( d + 1 ) ) } ) oF x. ( z e. CC |-> ( ( G ` z ) ^ ( d + 1 ) ) ) ) ) = ( z e. CC |-> sum_ k e. ( 0 ... ( d + 1 ) ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) ) )
159	158	eleq1d 2686	. . . . . . 7 |- ( ( ph /\ d e. NN0 ) -> ( ( ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) ) oF + ( ( CC X. { ( (coeff ` F ) ` ( d + 1 ) ) } ) oF x. ( z e. CC |-> ( ( G ` z ) ^ ( d + 1 ) ) ) ) ) e. (Poly ` S ) <-> ( z e. CC |-> sum_ k e. ( 0 ... ( d + 1 ) ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) ) e. (Poly ` S ) ) )
160	127, 159	sylibd 229	. . . . . 6 |- ( ( ph /\ d e. NN0 ) -> ( ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) ) e. (Poly ` S ) -> ( z e. CC |-> sum_ k e. ( 0 ... ( d + 1 ) ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) ) e. (Poly ` S ) ) )
161	160	expcom 451	. . . . 5 |- ( d e. NN0 -> ( ph -> ( ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) ) e. (Poly ` S ) -> ( z e. CC |-> sum_ k e. ( 0 ... ( d + 1 ) ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) ) e. (Poly ` S ) ) ) )
162	161	a2d 29	. . . 4 |- ( d e. NN0 -> ( ( ph -> ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) ) e. (Poly ` S ) ) -> ( ph -> ( z e. CC |-> sum_ k e. ( 0 ... ( d + 1 ) ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) ) e. (Poly ` S ) ) ) )
163	21, 26, 31, 36, 68, 162	nn0ind 11472	. . 3 |- ( (deg ` F ) e. NN0 -> ( ph -> ( z e. CC |-> sum_ k e. ( 0 ... (deg ` F ) ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) ) e. (Poly ` S ) ) )
164	16, 163	mpcom 38	. 2 |- ( ph -> ( z e. CC |-> sum_ k e. ( 0 ... (deg ` F ) ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) ) e. (Poly ` S ) )
165	14, 164	eqeltrd 2701	1 |- ( ph -> ( F o. G ) e. (Poly ` S ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   u. cun 3572   C_ wss 3574   {csn 4177   |-> cmpt 4729   X. cxp 5112   o. ccom 5118   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   NNcn 11020   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   ...cfz 12326   ^cexp 12860   sum_csu 14416  Polycply 23940  coeffccoe 23942  degcdgr 23943

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-rlim 14220  df-sum 14417  df-0p 23437  df-ply 23944  df-coe 23946  df-dgr 23947

This theorem is referenced by:  dgrcolem1  24029  dgrcolem2  24030  taylply2  24122  ftalem7  24805