Theorem plyco 23274

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 23231	. . . . 5 |- ( G e. (Poly ` S ) -> G : CC --> CC )
3	1, 2	syl 17	. . . 4 |- ( ph -> G : CC --> CC )
4	3	ffvelrnda 6037	. . 3 |- ( ( ph /\ z e. CC ) -> ( G ` z ) e. CC )
5	3	feqmptd 5932	. . 3 |- ( ph -> G = ( z e. CC |-> ( G ` z ) ) )
6		plyco.1	. . . 4 |- ( ph -> F e. (Poly ` S ) )
7		eqid 2471	. . . . 5 |- (coeff ` F ) = (coeff ` F )
8		eqid 2471	. . . . 5 |- (deg ` F ) = (deg ` F )
9	7, 8	coeid 23271	. . . 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 6315	. . . . 5 |- ( x = ( G ` z ) -> ( x ^ k ) = ( ( G ` z ) ^ k ) )
12	11	oveq2d 6324	. . . 4 |- ( x = ( G ` z ) -> ( ( (coeff ` F ) ` k ) x. ( x ^ k ) ) = ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) )
13	12	sumeq2sdv 13847	. . 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 6072	. 2 |- ( ph -> ( F o. G ) = ( z e. CC |-> sum_ k e. ( 0 ... (deg ` F ) ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) ) )
15		dgrcl 23266	. . . 4 |- ( F e. (Poly ` S ) -> (deg ` F ) e. NN0 )
16	6, 15	syl 17	. . 3 |- ( ph -> (deg ` F ) e. NN0 )
17		oveq2 6316	. . . . . . . 8 |- ( x = 0 -> ( 0 ... x ) = ( 0 ... 0 ) )
18	17	sumeq1d 13844	. . . . . . 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 4483	. . . . . 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 2533	. . . . 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 323	. . . 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 6316	. . . . . . . 8 |- ( x = d -> ( 0 ... x ) = ( 0 ... d ) )
23	22	sumeq1d 13844	. . . . . . 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 4483	. . . . . 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 2533	. . . . 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 323	. . . 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 6316	. . . . . . . 8 |- ( x = ( d + 1 ) -> ( 0 ... x ) = ( 0 ... ( d + 1 ) ) )
28	27	sumeq1d 13844	. . . . . . 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 4483	. . . . . 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 2533	. . . . 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 323	. . . 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 6316	. . . . . . . 8 |- ( x = (deg ` F ) -> ( 0 ... x ) = ( 0 ... (deg ` F ) ) )
33	32	sumeq1d 13844	. . . . . . 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 4483	. . . . . 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 2533	. . . . 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 323	. . . 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 10972	. . . . . . . . 9 |- 0 e. ZZ
38	4	exp0d 12448	. . . . . . . . . . . 12 |- ( ( ph /\ z e. CC ) -> ( ( G ` z ) ^ 0 ) = 1 )
39	38	oveq2d 6324	. . . . . . . . . . 11 |- ( ( ph /\ z e. CC ) -> ( ( (coeff ` F ) ` 0 ) x. ( ( G ` z ) ^ 0 ) ) = ( ( (coeff ` F ) ` 0 ) x. 1 ) )
40		plybss 23227	. . . . . . . . . . . . . . . 16 |- ( F e. (Poly ` S ) -> S C_ CC )
41	6, 40	syl 17	. . . . . . . . . . . . . . 15 |- ( ph -> S C_ CC )
42		0cnd 9654	. . . . . . . . . . . . . . . 16 |- ( ph -> 0 e. CC )
43	42	snssd 4108	. . . . . . . . . . . . . . 15 |- ( ph -> { 0 } C_ CC )
44	41, 43	unssd 3601	. . . . . . . . . . . . . 14 |- ( ph -> ( S u. { 0 } ) C_ CC )
45	7	coef 23263	. . . . . . . . . . . . . . . 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 10908	. . . . . . . . . . . . . . 15 |- 0 e. NN0
48		ffvelrn 6035	. . . . . . . . . . . . . . 15 |- ( ( (coeff ` F ) : NN0 --> ( S u. { 0 } ) /\ 0 e. NN0 ) -> ( (coeff ` F ) ` 0 ) e. ( S u. { 0 } ) )
49	46, 47, 48	sylancl 675	. . . . . . . . . . . . . 14 |- ( ph -> ( (coeff ` F ) ` 0 ) e. ( S u. { 0 } ) )
50	44, 49	sseldd 3419	. . . . . . . . . . . . 13 |- ( ph -> ( (coeff ` F ) ` 0 ) e. CC )
51	50	adantr 472	. . . . . . . . . . . 12 |- ( ( ph /\ z e. CC ) -> ( (coeff ` F ) ` 0 ) e. CC )
52	51	mulid1d 9678	. . . . . . . . . . 11 |- ( ( ph /\ z e. CC ) -> ( ( (coeff ` F ) ` 0 ) x. 1 ) = ( (coeff ` F ) ` 0 ) )
53	39, 52	eqtrd 2505	. . . . . . . . . 10 |- ( ( ph /\ z e. CC ) -> ( ( (coeff ` F ) ` 0 ) x. ( ( G ` z ) ^ 0 ) ) = ( (coeff ` F ) ` 0 ) )
54	53, 51	eqeltrd 2549	. . . . . . . . 9 |- ( ( ph /\ z e. CC ) -> ( ( (coeff ` F ) ` 0 ) x. ( ( G ` z ) ^ 0 ) ) e. CC )
55		fveq2 5879	. . . . . . . . . . 11 |- ( k = 0 -> ( (coeff ` F ) ` k ) = ( (coeff ` F ) ` 0 ) )
56		oveq2 6316	. . . . . . . . . . 11 |- ( k = 0 -> ( ( G ` z ) ^ k ) = ( ( G ` z ) ^ 0 ) )
57	55, 56	oveq12d 6326	. . . . . . . . . 10 |- ( k = 0 -> ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) = ( ( (coeff ` F ) ` 0 ) x. ( ( G ` z ) ^ 0 ) ) )
58	57	fsum1 13885	. . . . . . . . 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 676	. . . . . . . 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 2505	. . . . . . 7 |- ( ( ph /\ z e. CC ) -> sum_ k e. ( 0 ... 0 ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) = ( (coeff ` F ) ` 0 ) )
61	60	mpteq2dva 4482	. . . . . 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 4883	. . . . . 6 |- ( CC X. { ( (coeff ` F ) ` 0 ) } ) = ( z e. CC |-> ( (coeff ` F ) ` 0 ) )
63	61, 62	syl6eqr 2523	. . . . 5 |- ( ph -> ( z e. CC |-> sum_ k e. ( 0 ... 0 ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) ) = ( CC X. { ( (coeff ` F ) ` 0 ) } ) )
64		plyconst 23239	. . . . . . 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 673	. . . . . 6 |- ( ph -> ( CC X. { ( (coeff ` F ) ` 0 ) } ) e. (Poly ` ( S u. { 0 } ) ) )
66		plyun0 23230	. . . . . 6 |- (Poly ` ( S u. { 0 } ) ) = (Poly ` S )
67	65, 66	syl6eleq 2559	. . . . 5 |- ( ph -> ( CC X. { ( (coeff ` F ) ` 0 ) } ) e. (Poly ` S ) )
68	63, 67	eqeltrd 2549	. . . 4 |- ( ph -> ( z e. CC |-> sum_ k e. ( 0 ... 0 ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) ) e. (Poly ` S ) )
69		simprr 774	. . . . . . . . 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 472	. . . . . . . . . . . . 13 |- ( ( ph /\ d e. NN0 ) -> ( S u. { 0 } ) C_ CC )
71		peano2nn0 10934	. . . . . . . . . . . . . 14 |- ( d e. NN0 -> ( d + 1 ) e. NN0 )
72		ffvelrn 6035	. . . . . . . . . . . . . 14 |- ( ( (coeff ` F ) : NN0 --> ( S u. { 0 } ) /\ ( d + 1 ) e. NN0 ) -> ( (coeff ` F ) ` ( d + 1 ) ) e. ( S u. { 0 } ) )
73	46, 71, 72	syl2an 485	. . . . . . . . . . . . 13 |- ( ( ph /\ d e. NN0 ) -> ( (coeff ` F ) ` ( d + 1 ) ) e. ( S u. { 0 } ) )
74		plyconst 23239	. . . . . . . . . . . . 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 673	. . . . . . . . . . . 12 |- ( ( ph /\ d e. NN0 ) -> ( CC X. { ( (coeff ` F ) ` ( d + 1 ) ) } ) e. (Poly ` ( S u. { 0 } ) ) )
76	75, 66	syl6eleq 2559	. . . . . . . . . . 11 |- ( ( ph /\ d e. NN0 ) -> ( CC X. { ( (coeff ` F ) ` ( d + 1 ) ) } ) e. (Poly ` S ) )
77		nn0p1nn 10933	. . . . . . . . . . . . 13 |- ( d e. NN0 -> ( d + 1 ) e. NN )
78		oveq2 6316	. . . . . . . . . . . . . . . . 17 |- ( x = 1 -> ( ( G ` z ) ^ x ) = ( ( G ` z ) ^ 1 ) )
79	78	mpteq2dv 4483	. . . . . . . . . . . . . . . 16 |- ( x = 1 -> ( z e. CC |-> ( ( G ` z ) ^ x ) ) = ( z e. CC |-> ( ( G ` z ) ^ 1 ) ) )
80	79	eleq1d 2533	. . . . . . . . . . . . . . 15 |- ( x = 1 -> ( ( z e. CC |-> ( ( G ` z ) ^ x ) ) e. (Poly ` S ) <-> ( z e. CC |-> ( ( G ` z ) ^ 1 ) ) e. (Poly ` S ) ) )
81	80	imbi2d 323	. . . . . . . . . . . . . 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 6316	. . . . . . . . . . . . . . . . 17 |- ( x = d -> ( ( G ` z ) ^ x ) = ( ( G ` z ) ^ d ) )
83	82	mpteq2dv 4483	. . . . . . . . . . . . . . . 16 |- ( x = d -> ( z e. CC |-> ( ( G ` z ) ^ x ) ) = ( z e. CC |-> ( ( G ` z ) ^ d ) ) )
84	83	eleq1d 2533	. . . . . . . . . . . . . . 15 |- ( x = d -> ( ( z e. CC |-> ( ( G ` z ) ^ x ) ) e. (Poly ` S ) <-> ( z e. CC |-> ( ( G ` z ) ^ d ) ) e. (Poly ` S ) ) )
85	84	imbi2d 323	. . . . . . . . . . . . . 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 6316	. . . . . . . . . . . . . . . . 17 |- ( x = ( d + 1 ) -> ( ( G ` z ) ^ x ) = ( ( G ` z ) ^ ( d + 1 ) ) )
87	86	mpteq2dv 4483	. . . . . . . . . . . . . . . 16 |- ( x = ( d + 1 ) -> ( z e. CC |-> ( ( G ` z ) ^ x ) ) = ( z e. CC |-> ( ( G ` z ) ^ ( d + 1 ) ) ) )
88	87	eleq1d 2533	. . . . . . . . . . . . . . 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 323	. . . . . . . . . . . . . 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 12449	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ z e. CC ) -> ( ( G ` z ) ^ 1 ) = ( G ` z ) )
91	90	mpteq2dva 4482	. . . . . . . . . . . . . . . 16 |- ( ph -> ( z e. CC |-> ( ( G ` z ) ^ 1 ) ) = ( z e. CC |-> ( G ` z ) ) )
92	91, 5	eqtr4d 2508	. . . . . . . . . . . . . . 15 |- ( ph -> ( z e. CC |-> ( ( G ` z ) ^ 1 ) ) = G )
93	92, 1	eqeltrd 2549	. . . . . . . . . . . . . 14 |- ( ph -> ( z e. CC |-> ( ( G ` z ) ^ 1 ) ) e. (Poly ` S ) )
94		simprr 774	. . . . . . . . . . . . . . . . . . 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 472	. . . . . . . . . . . . . . . . . . 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 729	. . . . . . . . . . . . . . . . . . 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 729	. . . . . . . . . . . . . . . . . . 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 23251	. . . . . . . . . . . . . . . . . 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 626	. . . . . . . . . . . . . . . . 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 9638	. . . . . . . . . . . . . . . . . . . . 21 |- CC e. _V
103	102	a1i 11	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ d e. NN ) -> CC e. _V )
104		ovex 6336	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( G ` z ) ^ d ) e. _V
105	104	a1i 11	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ph /\ d e. NN ) /\ z e. CC ) -> ( ( G ` z ) ^ d ) e. _V )
106	4	adantlr 729	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ph /\ d e. NN ) /\ z e. CC ) -> ( G ` z ) e. CC )
107		eqidd 2472	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ d e. NN ) -> ( z e. CC |-> ( ( G ` z ) ^ d ) ) = ( z e. CC |-> ( ( G ` z ) ^ d ) ) )
108	5	adantr 472	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ d e. NN ) -> G = ( z e. CC |-> ( G ` z ) ) )
109	103, 105, 106, 107, 108	offval2 6567	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ d e. NN ) -> ( ( z e. CC |-> ( ( G ` z ) ^ d ) ) oF x. G ) = ( z e. CC |-> ( ( ( G ` z ) ^ d ) x. ( G ` z ) ) ) )
110		nnnn0 10900	. . . . . . . . . . . . . . . . . . . . . 22 |- ( d e. NN -> d e. NN0 )
111	110	ad2antlr 741	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ph /\ d e. NN ) /\ z e. CC ) -> d e. NN0 )
112	106, 111	expp1d 12455	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ph /\ d e. NN ) /\ z e. CC ) -> ( ( G ` z ) ^ ( d + 1 ) ) = ( ( ( G ` z ) ^ d ) x. ( G ` z ) ) )
113	112	mpteq2dva 4482	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ d e. NN ) -> ( z e. CC |-> ( ( G ` z ) ^ ( d + 1 ) ) ) = ( z e. CC |-> ( ( ( G ` z ) ^ d ) x. ( G ` z ) ) ) )
114	109, 113	eqtr4d 2508	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ d e. NN ) -> ( ( z e. CC |-> ( ( G ` z ) ^ d ) ) oF x. G ) = ( z e. CC |-> ( ( G ` z ) ^ ( d + 1 ) ) ) )
115	114	eleq1d 2533	. . . . . . . . . . . . . . . . 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 ) ) )
116	101, 115	sylibd 222	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ d e. NN ) -> ( ( z e. CC |-> ( ( G ` z ) ^ d ) ) e. (Poly ` S ) -> ( z e. CC |-> ( ( G ` z ) ^ ( d + 1 ) ) ) e. (Poly ` S ) ) )
117	116	expcom 442	. . . . . . . . . . . . . . 15 |- ( d e. NN -> ( ph -> ( ( z e. CC |-> ( ( G ` z ) ^ d ) ) e. (Poly ` S ) -> ( z e. CC |-> ( ( G ` z ) ^ ( d + 1 ) ) ) e. (Poly ` S ) ) ) )
118	117	a2d 28	. . . . . . . . . . . . . 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 ) ) ) )
119	81, 85, 89, 89, 93, 118	nnind 10649	. . . . . . . . . . . . 13 |- ( ( d + 1 ) e. NN -> ( ph -> ( z e. CC |-> ( ( G ` z ) ^ ( d + 1 ) ) ) e. (Poly ` S ) ) )
120	77, 119	syl 17	. . . . . . . . . . . 12 |- ( d e. NN0 -> ( ph -> ( z e. CC |-> ( ( G ` z ) ^ ( d + 1 ) ) ) e. (Poly ` S ) ) )
121	120	impcom 437	. . . . . . . . . . 11 |- ( ( ph /\ d e. NN0 ) -> ( z e. CC |-> ( ( G ` z ) ^ ( d + 1 ) ) ) e. (Poly ` S ) )
122	96	adantlr 729	. . . . . . . . . . 11 |- ( ( ( ph /\ d e. NN0 ) /\ ( x e. S /\ y e. S ) ) -> ( x + y ) e. S )
123	98	adantlr 729	. . . . . . . . . . 11 |- ( ( ( ph /\ d e. NN0 ) /\ ( x e. S /\ y e. S ) ) -> ( x x. y ) e. S )
124	76, 121, 122, 123	plymul 23251	. . . . . . . . . 10 |- ( ( ph /\ d e. NN0 ) -> ( ( CC X. { ( (coeff ` F ) ` ( d + 1 ) ) } ) oF x. ( z e. CC |-> ( ( G ` z ) ^ ( d + 1 ) ) ) ) e. (Poly ` S ) )
125	124	adantrr 731	. . . . . . . . 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 ) )
126	96	adantlr 729	. . . . . . . . 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 )
127	69, 125, 126	plyadd 23250	. . . . . . . 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 ) )
128	127	expr 626	. . . . . . 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 ) ) )
129	102	a1i 11	. . . . . . . . . 10 |- ( ( ph /\ d e. NN0 ) -> CC e. _V )
130		sumex 13831	. . . . . . . . . . 11 |- sum_ k e. ( 0 ... d ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) e. _V
131	130	a1i 11	. . . . . . . . . 10 |- ( ( ( ph /\ d e. NN0 ) /\ z e. CC ) -> sum_ k e. ( 0 ... d ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) e. _V )
132		ovex 6336	. . . . . . . . . . 11 |- ( ( (coeff ` F ) ` ( d + 1 ) ) x. ( ( G ` z ) ^ ( d + 1 ) ) ) e. _V
133	132	a1i 11	. . . . . . . . . 10 |- ( ( ( ph /\ d e. NN0 ) /\ z e. CC ) -> ( ( (coeff ` F ) ` ( d + 1 ) ) x. ( ( G ` z ) ^ ( d + 1 ) ) ) e. _V )
134		eqidd 2472	. . . . . . . . . 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 ) ) ) )
135		fvex 5889	. . . . . . . . . . . 12 |- ( (coeff ` F ) ` ( d + 1 ) ) e. _V
136	135	a1i 11	. . . . . . . . . . 11 |- ( ( ( ph /\ d e. NN0 ) /\ z e. CC ) -> ( (coeff ` F ) ` ( d + 1 ) ) e. _V )
137		ovex 6336	. . . . . . . . . . . 12 |- ( ( G ` z ) ^ ( d + 1 ) ) e. _V
138	137	a1i 11	. . . . . . . . . . 11 |- ( ( ( ph /\ d e. NN0 ) /\ z e. CC ) -> ( ( G ` z ) ^ ( d + 1 ) ) e. _V )
139		fconstmpt 4883	. . . . . . . . . . . 12 |- ( CC X. { ( (coeff ` F ) ` ( d + 1 ) ) } ) = ( z e. CC |-> ( (coeff ` F ) ` ( d + 1 ) ) )
140	139	a1i 11	. . . . . . . . . . 11 |- ( ( ph /\ d e. NN0 ) -> ( CC X. { ( (coeff ` F ) ` ( d + 1 ) ) } ) = ( z e. CC |-> ( (coeff ` F ) ` ( d + 1 ) ) ) )
141		eqidd 2472	. . . . . . . . . . 11 |- ( ( ph /\ d e. NN0 ) -> ( z e. CC |-> ( ( G ` z ) ^ ( d + 1 ) ) ) = ( z e. CC |-> ( ( G ` z ) ^ ( d + 1 ) ) ) )
142	129, 136, 138, 140, 141	offval2 6567	. . . . . . . . . 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 ) ) ) ) )
143	129, 131, 133, 134, 142	offval2 6567	. . . . . . . . 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 ) ) ) ) ) )
144		simplr 770	. . . . . . . . . . . 12 |- ( ( ( ph /\ d e. NN0 ) /\ z e. CC ) -> d e. NN0 )
145		nn0uz 11217	. . . . . . . . . . . 12 |- NN0 = ( ZZ>= ` 0 )
146	144, 145	syl6eleq 2559	. . . . . . . . . . 11 |- ( ( ( ph /\ d e. NN0 ) /\ z e. CC ) -> d e. ( ZZ>= ` 0 ) )
147	7	coef3 23265	. . . . . . . . . . . . . . 15 |- ( F e. (Poly ` S ) -> (coeff ` F ) : NN0 --> CC )
148	6, 147	syl 17	. . . . . . . . . . . . . 14 |- ( ph -> (coeff ` F ) : NN0 --> CC )
149	148	ad2antrr 740	. . . . . . . . . . . . 13 |- ( ( ( ph /\ d e. NN0 ) /\ z e. CC ) -> (coeff ` F ) : NN0 --> CC )
150		elfznn0 11913	. . . . . . . . . . . . 13 |- ( k e. ( 0 ... ( d + 1 ) ) -> k e. NN0 )
151		ffvelrn 6035	. . . . . . . . . . . . 13 |- ( ( (coeff ` F ) : NN0 --> CC /\ k e. NN0 ) -> ( (coeff ` F ) ` k ) e. CC )
152	149, 150, 151	syl2an 485	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ d e. NN0 ) /\ z e. CC ) /\ k e. ( 0 ... ( d + 1 ) ) ) -> ( (coeff ` F ) ` k ) e. CC )
153	4	adantlr 729	. . . . . . . . . . . . 13 |- ( ( ( ph /\ d e. NN0 ) /\ z e. CC ) -> ( G ` z ) e. CC )
154		expcl 12328	. . . . . . . . . . . . 13 |- ( ( ( G ` z ) e. CC /\ k e. NN0 ) -> ( ( G ` z ) ^ k ) e. CC )
155	153, 150, 154	syl2an 485	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ d e. NN0 ) /\ z e. CC ) /\ k e. ( 0 ... ( d + 1 ) ) ) -> ( ( G ` z ) ^ k ) e. CC )
156	152, 155	mulcld 9681	. . . . . . . . . . 11 |- ( ( ( ( ph /\ d e. NN0 ) /\ z e. CC ) /\ k e. ( 0 ... ( d + 1 ) ) ) -> ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) e. CC )
157		fveq2 5879	. . . . . . . . . . . 12 |- ( k = ( d + 1 ) -> ( (coeff ` F ) ` k ) = ( (coeff ` F ) ` ( d + 1 ) ) )
158		oveq2 6316	. . . . . . . . . . . 12 |- ( k = ( d + 1 ) -> ( ( G ` z ) ^ k ) = ( ( G ` z ) ^ ( d + 1 ) ) )
159	157, 158	oveq12d 6326	. . . . . . . . . . 11 |- ( k = ( d + 1 ) -> ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) = ( ( (coeff ` F ) ` ( d + 1 ) ) x. ( ( G ` z ) ^ ( d + 1 ) ) ) )
160	146, 156, 159	fsump1 13894	. . . . . . . . . 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 ) ) ) ) )
161	160	mpteq2dva 4482	. . . . . . . . 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 ) ) ) ) ) )
162	143, 161	eqtr4d 2508	. . . . . . . 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 ) ) ) )
163	162	eleq1d 2533	. . . . . . 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 ) ) )
164	128, 163	sylibd 222	. . . . . 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 ) ) )
165	164	expcom 442	. . . . 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 ) ) ) )
166	165	a2d 28	. . . 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 ) ) ) )
167	21, 26, 31, 36, 68, 166	nn0ind 11053	. . 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 ) ) )
168	16, 167	mpcom 36	. 2 |- ( ph -> ( z e. CC |-> sum_ k e. ( 0 ... (deg ` F ) ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) ) e. (Poly ` S ) )
169	14, 168	eqeltrd 2549	1 |- ( ph -> ( F o. G ) e. (Poly ` S ) )

Syntax hints:   -> wi 4   /\ wa 376   = wceq 1452   e. wcel 1904   _Vcvv 3031   u. cun 3388   C_ wss 3390   {csn 3959   |-> cmpt 4454   X. cxp 4837   o. ccom 4843   -->wf 5585   ` cfv 5589  (class class class)co 6308   oFcof 6548   CCcc 9555   0cc0 9557   1c1 9558   + caddc 9560   x. cmul 9562   NNcn 10631   NN0cn0 10893   ZZcz 10961   ZZ>=cuz 11182   ...cfz 11810   ^cexp 12310   sum_csu 13829  Polycply 23217  coeffccoe 23219  degcdgr 23220

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-addf 9636

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-rp 11326  df-fz 11811  df-fzo 11943  df-fl 12061  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-clim 13629  df-rlim 13630  df-sum 13830  df-0p 22707  df-ply 23221  df-coe 23223  df-dgr 23224

This theorem is referenced by:  dgrcolem1  23306  dgrcolem2  23307  taylply2  23402  ftalem7  24084