Theorem plyco 20113

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 20070	. . . . 5 |- ( G e. (Poly ` S ) -> G : CC --> CC )
3	1, 2	syl 16	. . . 4 |- ( ph -> G : CC --> CC )
4	3	ffvelrnda 5829	. . 3 |- ( ( ph /\ z e. CC ) -> ( G ` z ) e. CC )
5	3	feqmptd 5738	. . 3 |- ( ph -> G = ( z e. CC |-> ( G ` z ) ) )
6		plyco.1	. . . 4 |- ( ph -> F e. (Poly ` S ) )
7		eqid 2404	. . . . 5 |- (coeff ` F ) = (coeff ` F )
8		eqid 2404	. . . . 5 |- (deg ` F ) = (deg ` F )
9	7, 8	coeid 20110	. . . 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 16	. . 3 |- ( ph -> F = ( x e. CC |-> sum_ k e. ( 0 ... (deg ` F ) ) ( ( (coeff ` F ) ` k ) x. ( x ^ k ) ) ) )
11		oveq1 6047	. . . . 5 |- ( x = ( G ` z ) -> ( x ^ k ) = ( ( G ` z ) ^ k ) )
12	11	oveq2d 6056	. . . 4 |- ( x = ( G ` z ) -> ( ( (coeff ` F ) ` k ) x. ( x ^ k ) ) = ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) )
13	12	sumeq2sdv 12453	. . 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 5860	. 2 |- ( ph -> ( F o. G ) = ( z e. CC |-> sum_ k e. ( 0 ... (deg ` F ) ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) ) )
15		dgrcl 20105	. . . 4 |- ( F e. (Poly ` S ) -> (deg ` F ) e. NN0 )
16	6, 15	syl 16	. . 3 |- ( ph -> (deg ` F ) e. NN0 )
17		oveq2 6048	. . . . . . . 8 |- ( x = 0 -> ( 0 ... x ) = ( 0 ... 0 ) )
18	17	sumeq1d 12450	. . . . . . 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 4256	. . . . . 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 2470	. . . . 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 308	. . . 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 6048	. . . . . . . 8 |- ( x = d -> ( 0 ... x ) = ( 0 ... d ) )
23	22	sumeq1d 12450	. . . . . . 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 4256	. . . . . 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 2470	. . . . 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 308	. . . 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 6048	. . . . . . . 8 |- ( x = ( d + 1 ) -> ( 0 ... x ) = ( 0 ... ( d + 1 ) ) )
28	27	sumeq1d 12450	. . . . . . 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 4256	. . . . . 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 2470	. . . . 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 308	. . . 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 6048	. . . . . . . 8 |- ( x = (deg ` F ) -> ( 0 ... x ) = ( 0 ... (deg ` F ) ) )
33	32	sumeq1d 12450	. . . . . . 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 4256	. . . . . 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 2470	. . . . 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 308	. . . 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 10249	. . . . . . . . 9 |- 0 e. ZZ
38	4	exp0d 11472	. . . . . . . . . . . 12 |- ( ( ph /\ z e. CC ) -> ( ( G ` z ) ^ 0 ) = 1 )
39	38	oveq2d 6056	. . . . . . . . . . 11 |- ( ( ph /\ z e. CC ) -> ( ( (coeff ` F ) ` 0 ) x. ( ( G ` z ) ^ 0 ) ) = ( ( (coeff ` F ) ` 0 ) x. 1 ) )
40		plybss 20066	. . . . . . . . . . . . . . . 16 |- ( F e. (Poly ` S ) -> S C_ CC )
41	6, 40	syl 16	. . . . . . . . . . . . . . 15 |- ( ph -> S C_ CC )
42		0cn 9040	. . . . . . . . . . . . . . . . 17 |- 0 e. CC
43	42	a1i 11	. . . . . . . . . . . . . . . 16 |- ( ph -> 0 e. CC )
44	43	snssd 3903	. . . . . . . . . . . . . . 15 |- ( ph -> { 0 } C_ CC )
45	41, 44	unssd 3483	. . . . . . . . . . . . . 14 |- ( ph -> ( S u. { 0 } ) C_ CC )
46	7	coef 20102	. . . . . . . . . . . . . . . 16 |- ( F e. (Poly ` S ) -> (coeff ` F ) : NN0 --> ( S u. { 0 } ) )
47	6, 46	syl 16	. . . . . . . . . . . . . . 15 |- ( ph -> (coeff ` F ) : NN0 --> ( S u. { 0 } ) )
48		0nn0 10192	. . . . . . . . . . . . . . 15 |- 0 e. NN0
49		ffvelrn 5827	. . . . . . . . . . . . . . 15 |- ( ( (coeff ` F ) : NN0 --> ( S u. { 0 } ) /\ 0 e. NN0 ) -> ( (coeff ` F ) ` 0 ) e. ( S u. { 0 } ) )
50	47, 48, 49	sylancl 644	. . . . . . . . . . . . . 14 |- ( ph -> ( (coeff ` F ) ` 0 ) e. ( S u. { 0 } ) )
51	45, 50	sseldd 3309	. . . . . . . . . . . . 13 |- ( ph -> ( (coeff ` F ) ` 0 ) e. CC )
52	51	adantr 452	. . . . . . . . . . . 12 |- ( ( ph /\ z e. CC ) -> ( (coeff ` F ) ` 0 ) e. CC )
53	52	mulid1d 9061	. . . . . . . . . . 11 |- ( ( ph /\ z e. CC ) -> ( ( (coeff ` F ) ` 0 ) x. 1 ) = ( (coeff ` F ) ` 0 ) )
54	39, 53	eqtrd 2436	. . . . . . . . . 10 |- ( ( ph /\ z e. CC ) -> ( ( (coeff ` F ) ` 0 ) x. ( ( G ` z ) ^ 0 ) ) = ( (coeff ` F ) ` 0 ) )
55	54, 52	eqeltrd 2478	. . . . . . . . 9 |- ( ( ph /\ z e. CC ) -> ( ( (coeff ` F ) ` 0 ) x. ( ( G ` z ) ^ 0 ) ) e. CC )
56		fveq2 5687	. . . . . . . . . . 11 |- ( k = 0 -> ( (coeff ` F ) ` k ) = ( (coeff ` F ) ` 0 ) )
57		oveq2 6048	. . . . . . . . . . 11 |- ( k = 0 -> ( ( G ` z ) ^ k ) = ( ( G ` z ) ^ 0 ) )
58	56, 57	oveq12d 6058	. . . . . . . . . 10 |- ( k = 0 -> ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) = ( ( (coeff ` F ) ` 0 ) x. ( ( G ` z ) ^ 0 ) ) )
59	58	fsum1 12490	. . . . . . . . 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 ) ) )
60	37, 55, 59	sylancr 645	. . . . . . . 8 |- ( ( ph /\ z e. CC ) -> sum_ k e. ( 0 ... 0 ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) = ( ( (coeff ` F ) ` 0 ) x. ( ( G ` z ) ^ 0 ) ) )
61	60, 54	eqtrd 2436	. . . . . . 7 |- ( ( ph /\ z e. CC ) -> sum_ k e. ( 0 ... 0 ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) = ( (coeff ` F ) ` 0 ) )
62	61	mpteq2dva 4255	. . . . . 6 |- ( ph -> ( z e. CC |-> sum_ k e. ( 0 ... 0 ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) ) = ( z e. CC |-> ( (coeff ` F ) ` 0 ) ) )
63		fconstmpt 4880	. . . . . 6 |- ( CC X. { ( (coeff ` F ) ` 0 ) } ) = ( z e. CC |-> ( (coeff ` F ) ` 0 ) )
64	62, 63	syl6eqr 2454	. . . . 5 |- ( ph -> ( z e. CC |-> sum_ k e. ( 0 ... 0 ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) ) = ( CC X. { ( (coeff ` F ) ` 0 ) } ) )
65		plyconst 20078	. . . . . . 7 |- ( ( ( S u. { 0 } ) C_ CC /\ ( (coeff ` F ) ` 0 ) e. ( S u. { 0 } ) ) -> ( CC X. { ( (coeff ` F ) ` 0 ) } ) e. (Poly ` ( S u. { 0 } ) ) )
66	45, 50, 65	syl2anc 643	. . . . . 6 |- ( ph -> ( CC X. { ( (coeff ` F ) ` 0 ) } ) e. (Poly ` ( S u. { 0 } ) ) )
67		plyun0 20069	. . . . . 6 |- (Poly ` ( S u. { 0 } ) ) = (Poly ` S )
68	66, 67	syl6eleq 2494	. . . . 5 |- ( ph -> ( CC X. { ( (coeff ` F ) ` 0 ) } ) e. (Poly ` S ) )
69	64, 68	eqeltrd 2478	. . . 4 |- ( ph -> ( z e. CC |-> sum_ k e. ( 0 ... 0 ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) ) e. (Poly ` S ) )
70		simprr 734	. . . . . . . . 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 ) )
71	45	adantr 452	. . . . . . . . . . . . 13 |- ( ( ph /\ d e. NN0 ) -> ( S u. { 0 } ) C_ CC )
72		peano2nn0 10216	. . . . . . . . . . . . . 14 |- ( d e. NN0 -> ( d + 1 ) e. NN0 )
73		ffvelrn 5827	. . . . . . . . . . . . . 14 |- ( ( (coeff ` F ) : NN0 --> ( S u. { 0 } ) /\ ( d + 1 ) e. NN0 ) -> ( (coeff ` F ) ` ( d + 1 ) ) e. ( S u. { 0 } ) )
74	47, 72, 73	syl2an 464	. . . . . . . . . . . . 13 |- ( ( ph /\ d e. NN0 ) -> ( (coeff ` F ) ` ( d + 1 ) ) e. ( S u. { 0 } ) )
75		plyconst 20078	. . . . . . . . . . . . 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 } ) ) )
76	71, 74, 75	syl2anc 643	. . . . . . . . . . . 12 |- ( ( ph /\ d e. NN0 ) -> ( CC X. { ( (coeff ` F ) ` ( d + 1 ) ) } ) e. (Poly ` ( S u. { 0 } ) ) )
77	76, 67	syl6eleq 2494	. . . . . . . . . . 11 |- ( ( ph /\ d e. NN0 ) -> ( CC X. { ( (coeff ` F ) ` ( d + 1 ) ) } ) e. (Poly ` S ) )
78		nn0p1nn 10215	. . . . . . . . . . . . 13 |- ( d e. NN0 -> ( d + 1 ) e. NN )
79		oveq2 6048	. . . . . . . . . . . . . . . . 17 |- ( x = 1 -> ( ( G ` z ) ^ x ) = ( ( G ` z ) ^ 1 ) )
80	79	mpteq2dv 4256	. . . . . . . . . . . . . . . 16 |- ( x = 1 -> ( z e. CC |-> ( ( G ` z ) ^ x ) ) = ( z e. CC |-> ( ( G ` z ) ^ 1 ) ) )
81	80	eleq1d 2470	. . . . . . . . . . . . . . 15 |- ( x = 1 -> ( ( z e. CC |-> ( ( G ` z ) ^ x ) ) e. (Poly ` S ) <-> ( z e. CC |-> ( ( G ` z ) ^ 1 ) ) e. (Poly ` S ) ) )
82	81	imbi2d 308	. . . . . . . . . . . . . 14 |- ( x = 1 -> ( ( ph -> ( z e. CC |-> ( ( G ` z ) ^ x ) ) e. (Poly ` S ) ) <-> ( ph -> ( z e. CC |-> ( ( G ` z ) ^ 1 ) ) e. (Poly ` S ) ) ) )
83		oveq2 6048	. . . . . . . . . . . . . . . . 17 |- ( x = d -> ( ( G ` z ) ^ x ) = ( ( G ` z ) ^ d ) )
84	83	mpteq2dv 4256	. . . . . . . . . . . . . . . 16 |- ( x = d -> ( z e. CC |-> ( ( G ` z ) ^ x ) ) = ( z e. CC |-> ( ( G ` z ) ^ d ) ) )
85	84	eleq1d 2470	. . . . . . . . . . . . . . 15 |- ( x = d -> ( ( z e. CC |-> ( ( G ` z ) ^ x ) ) e. (Poly ` S ) <-> ( z e. CC |-> ( ( G ` z ) ^ d ) ) e. (Poly ` S ) ) )
86	85	imbi2d 308	. . . . . . . . . . . . . 14 |- ( x = d -> ( ( ph -> ( z e. CC |-> ( ( G ` z ) ^ x ) ) e. (Poly ` S ) ) <-> ( ph -> ( z e. CC |-> ( ( G ` z ) ^ d ) ) e. (Poly ` S ) ) ) )
87		oveq2 6048	. . . . . . . . . . . . . . . . 17 |- ( x = ( d + 1 ) -> ( ( G ` z ) ^ x ) = ( ( G ` z ) ^ ( d + 1 ) ) )
88	87	mpteq2dv 4256	. . . . . . . . . . . . . . . 16 |- ( x = ( d + 1 ) -> ( z e. CC |-> ( ( G ` z ) ^ x ) ) = ( z e. CC |-> ( ( G ` z ) ^ ( d + 1 ) ) ) )
89	88	eleq1d 2470	. . . . . . . . . . . . . . 15 |- ( x = ( d + 1 ) -> ( ( z e. CC |-> ( ( G ` z ) ^ x ) ) e. (Poly ` S ) <-> ( z e. CC |-> ( ( G ` z ) ^ ( d + 1 ) ) ) e. (Poly ` S ) ) )
90	89	imbi2d 308	. . . . . . . . . . . . . 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 ) ) ) )
91	4	exp1d 11473	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ z e. CC ) -> ( ( G ` z ) ^ 1 ) = ( G ` z ) )
92	91	mpteq2dva 4255	. . . . . . . . . . . . . . . 16 |- ( ph -> ( z e. CC |-> ( ( G ` z ) ^ 1 ) ) = ( z e. CC |-> ( G ` z ) ) )
93	92, 5	eqtr4d 2439	. . . . . . . . . . . . . . 15 |- ( ph -> ( z e. CC |-> ( ( G ` z ) ^ 1 ) ) = G )
94	93, 1	eqeltrd 2478	. . . . . . . . . . . . . 14 |- ( ph -> ( z e. CC |-> ( ( G ` z ) ^ 1 ) ) e. (Poly ` S ) )
95		simprr 734	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ ( d e. NN /\ ( z e. CC |-> ( ( G ` z ) ^ d ) ) e. (Poly ` S ) ) ) -> ( z e. CC |-> ( ( G ` z ) ^ d ) ) e. (Poly ` S ) )
96	1	adantr 452	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ ( d e. NN /\ ( z e. CC |-> ( ( G ` z ) ^ d ) ) e. (Poly ` S ) ) ) -> G e. (Poly ` S ) )
97		plyco.3	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x + y ) e. S )
98	97	adantlr 696	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ph /\ ( d e. NN /\ ( z e. CC |-> ( ( G ` z ) ^ d ) ) e. (Poly ` S ) ) ) /\ ( x e. S /\ y e. S ) ) -> ( x + y ) e. S )
99		plyco.4	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x x. y ) e. S )
100	99	adantlr 696	. . . . . . . . . . . . . . . . . . 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 )
101	95, 96, 98, 100	plymul 20090	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ ( d e. NN /\ ( z e. CC |-> ( ( G ` z ) ^ d ) ) e. (Poly ` S ) ) ) -> ( ( z e. CC |-> ( ( G ` z ) ^ d ) ) o F x. G ) e. (Poly ` S ) )
102	101	expr 599	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ d e. NN ) -> ( ( z e. CC |-> ( ( G ` z ) ^ d ) ) e. (Poly ` S ) -> ( ( z e. CC |-> ( ( G ` z ) ^ d ) ) o F x. G ) e. (Poly ` S ) ) )
103		cnex 9027	. . . . . . . . . . . . . . . . . . . . 21 |- CC e. _V
104	103	a1i 11	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ d e. NN ) -> CC e. _V )
105		ovex 6065	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( G ` z ) ^ d ) e. _V
106	105	a1i 11	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ph /\ d e. NN ) /\ z e. CC ) -> ( ( G ` z ) ^ d ) e. _V )
107	4	adantlr 696	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ph /\ d e. NN ) /\ z e. CC ) -> ( G ` z ) e. CC )
108		eqidd 2405	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ d e. NN ) -> ( z e. CC |-> ( ( G ` z ) ^ d ) ) = ( z e. CC |-> ( ( G ` z ) ^ d ) ) )
109	5	adantr 452	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ d e. NN ) -> G = ( z e. CC |-> ( G ` z ) ) )
110	104, 106, 107, 108, 109	offval2 6281	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ d e. NN ) -> ( ( z e. CC |-> ( ( G ` z ) ^ d ) ) o F x. G ) = ( z e. CC |-> ( ( ( G ` z ) ^ d ) x. ( G ` z ) ) ) )
111		nnnn0 10184	. . . . . . . . . . . . . . . . . . . . . 22 |- ( d e. NN -> d e. NN0 )
112	111	ad2antlr 708	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ph /\ d e. NN ) /\ z e. CC ) -> d e. NN0 )
113	107, 112	expp1d 11479	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ph /\ d e. NN ) /\ z e. CC ) -> ( ( G ` z ) ^ ( d + 1 ) ) = ( ( ( G ` z ) ^ d ) x. ( G ` z ) ) )
114	113	mpteq2dva 4255	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ d e. NN ) -> ( z e. CC |-> ( ( G ` z ) ^ ( d + 1 ) ) ) = ( z e. CC |-> ( ( ( G ` z ) ^ d ) x. ( G ` z ) ) ) )
115	110, 114	eqtr4d 2439	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ d e. NN ) -> ( ( z e. CC |-> ( ( G ` z ) ^ d ) ) o F x. G ) = ( z e. CC |-> ( ( G ` z ) ^ ( d + 1 ) ) ) )
116	115	eleq1d 2470	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ d e. NN ) -> ( ( ( z e. CC |-> ( ( G ` z ) ^ d ) ) o F x. G ) e. (Poly ` S ) <-> ( z e. CC |-> ( ( G ` z ) ^ ( d + 1 ) ) ) e. (Poly ` S ) ) )
117	102, 116	sylibd 206	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ d e. NN ) -> ( ( z e. CC |-> ( ( G ` z ) ^ d ) ) e. (Poly ` S ) -> ( z e. CC |-> ( ( G ` z ) ^ ( d + 1 ) ) ) e. (Poly ` S ) ) )
118	117	expcom 425	. . . . . . . . . . . . . . 15 |- ( d e. NN -> ( ph -> ( ( z e. CC |-> ( ( G ` z ) ^ d ) ) e. (Poly ` S ) -> ( z e. CC |-> ( ( G ` z ) ^ ( d + 1 ) ) ) e. (Poly ` S ) ) ) )
119	118	a2d 24	. . . . . . . . . . . . . 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 ) ) ) )
120	82, 86, 90, 90, 94, 119	nnind 9974	. . . . . . . . . . . . 13 |- ( ( d + 1 ) e. NN -> ( ph -> ( z e. CC |-> ( ( G ` z ) ^ ( d + 1 ) ) ) e. (Poly ` S ) ) )
121	78, 120	syl 16	. . . . . . . . . . . 12 |- ( d e. NN0 -> ( ph -> ( z e. CC |-> ( ( G ` z ) ^ ( d + 1 ) ) ) e. (Poly ` S ) ) )
122	121	impcom 420	. . . . . . . . . . 11 |- ( ( ph /\ d e. NN0 ) -> ( z e. CC |-> ( ( G ` z ) ^ ( d + 1 ) ) ) e. (Poly ` S ) )
123	97	adantlr 696	. . . . . . . . . . 11 |- ( ( ( ph /\ d e. NN0 ) /\ ( x e. S /\ y e. S ) ) -> ( x + y ) e. S )
124	99	adantlr 696	. . . . . . . . . . 11 |- ( ( ( ph /\ d e. NN0 ) /\ ( x e. S /\ y e. S ) ) -> ( x x. y ) e. S )
125	77, 122, 123, 124	plymul 20090	. . . . . . . . . 10 |- ( ( ph /\ d e. NN0 ) -> ( ( CC X. { ( (coeff ` F ) ` ( d + 1 ) ) } ) o F x. ( z e. CC |-> ( ( G ` z ) ^ ( d + 1 ) ) ) ) e. (Poly ` S ) )
126	125	adantrr 698	. . . . . . . . 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 ) ) } ) o F x. ( z e. CC |-> ( ( G ` z ) ^ ( d + 1 ) ) ) ) e. (Poly ` S ) )
127	97	adantlr 696	. . . . . . . . 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 )
128	70, 126, 127	plyadd 20089	. . . . . . . 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 ) ) ) o F + ( ( CC X. { ( (coeff ` F ) ` ( d + 1 ) ) } ) o F x. ( z e. CC |-> ( ( G ` z ) ^ ( d + 1 ) ) ) ) ) e. (Poly ` S ) )
129	128	expr 599	. . . . . . 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 ) ) ) o F + ( ( CC X. { ( (coeff ` F ) ` ( d + 1 ) ) } ) o F x. ( z e. CC |-> ( ( G ` z ) ^ ( d + 1 ) ) ) ) ) e. (Poly ` S ) ) )
130	103	a1i 11	. . . . . . . . . 10 |- ( ( ph /\ d e. NN0 ) -> CC e. _V )
131		sumex 12436	. . . . . . . . . . 11 |- sum_ k e. ( 0 ... d ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) e. _V
132	131	a1i 11	. . . . . . . . . 10 |- ( ( ( ph /\ d e. NN0 ) /\ z e. CC ) -> sum_ k e. ( 0 ... d ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) e. _V )
133		ovex 6065	. . . . . . . . . . 11 |- ( ( (coeff ` F ) ` ( d + 1 ) ) x. ( ( G ` z ) ^ ( d + 1 ) ) ) e. _V
134	133	a1i 11	. . . . . . . . . 10 |- ( ( ( ph /\ d e. NN0 ) /\ z e. CC ) -> ( ( (coeff ` F ) ` ( d + 1 ) ) x. ( ( G ` z ) ^ ( d + 1 ) ) ) e. _V )
135		eqidd 2405	. . . . . . . . . 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 ) ) ) )
136		fvex 5701	. . . . . . . . . . . 12 |- ( (coeff ` F ) ` ( d + 1 ) ) e. _V
137	136	a1i 11	. . . . . . . . . . 11 |- ( ( ( ph /\ d e. NN0 ) /\ z e. CC ) -> ( (coeff ` F ) ` ( d + 1 ) ) e. _V )
138		ovex 6065	. . . . . . . . . . . 12 |- ( ( G ` z ) ^ ( d + 1 ) ) e. _V
139	138	a1i 11	. . . . . . . . . . 11 |- ( ( ( ph /\ d e. NN0 ) /\ z e. CC ) -> ( ( G ` z ) ^ ( d + 1 ) ) e. _V )
140		fconstmpt 4880	. . . . . . . . . . . 12 |- ( CC X. { ( (coeff ` F ) ` ( d + 1 ) ) } ) = ( z e. CC |-> ( (coeff ` F ) ` ( d + 1 ) ) )
141	140	a1i 11	. . . . . . . . . . 11 |- ( ( ph /\ d e. NN0 ) -> ( CC X. { ( (coeff ` F ) ` ( d + 1 ) ) } ) = ( z e. CC |-> ( (coeff ` F ) ` ( d + 1 ) ) ) )
142		eqidd 2405	. . . . . . . . . . 11 |- ( ( ph /\ d e. NN0 ) -> ( z e. CC |-> ( ( G ` z ) ^ ( d + 1 ) ) ) = ( z e. CC |-> ( ( G ` z ) ^ ( d + 1 ) ) ) )
143	130, 137, 139, 141, 142	offval2 6281	. . . . . . . . . 10 |- ( ( ph /\ d e. NN0 ) -> ( ( CC X. { ( (coeff ` F ) ` ( d + 1 ) ) } ) o F x. ( z e. CC |-> ( ( G ` z ) ^ ( d + 1 ) ) ) ) = ( z e. CC |-> ( ( (coeff ` F ) ` ( d + 1 ) ) x. ( ( G ` z ) ^ ( d + 1 ) ) ) ) )
144	130, 132, 134, 135, 143	offval2 6281	. . . . . . . . 9 |- ( ( ph /\ d e. NN0 ) -> ( ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) ) o F + ( ( CC X. { ( (coeff ` F ) ` ( d + 1 ) ) } ) o F 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 ) ) ) ) ) )
145		simplr 732	. . . . . . . . . . . 12 |- ( ( ( ph /\ d e. NN0 ) /\ z e. CC ) -> d e. NN0 )
146		nn0uz 10476	. . . . . . . . . . . 12 |- NN0 = ( ZZ>= ` 0 )
147	145, 146	syl6eleq 2494	. . . . . . . . . . 11 |- ( ( ( ph /\ d e. NN0 ) /\ z e. CC ) -> d e. ( ZZ>= ` 0 ) )
148	7	coef3 20104	. . . . . . . . . . . . . . 15 |- ( F e. (Poly ` S ) -> (coeff ` F ) : NN0 --> CC )
149	6, 148	syl 16	. . . . . . . . . . . . . 14 |- ( ph -> (coeff ` F ) : NN0 --> CC )
150	149	ad2antrr 707	. . . . . . . . . . . . 13 |- ( ( ( ph /\ d e. NN0 ) /\ z e. CC ) -> (coeff ` F ) : NN0 --> CC )
151		elfznn0 11039	. . . . . . . . . . . . 13 |- ( k e. ( 0 ... ( d + 1 ) ) -> k e. NN0 )
152		ffvelrn 5827	. . . . . . . . . . . . 13 |- ( ( (coeff ` F ) : NN0 --> CC /\ k e. NN0 ) -> ( (coeff ` F ) ` k ) e. CC )
153	150, 151, 152	syl2an 464	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ d e. NN0 ) /\ z e. CC ) /\ k e. ( 0 ... ( d + 1 ) ) ) -> ( (coeff ` F ) ` k ) e. CC )
154	4	adantlr 696	. . . . . . . . . . . . 13 |- ( ( ( ph /\ d e. NN0 ) /\ z e. CC ) -> ( G ` z ) e. CC )
155		expcl 11354	. . . . . . . . . . . . 13 |- ( ( ( G ` z ) e. CC /\ k e. NN0 ) -> ( ( G ` z ) ^ k ) e. CC )
156	154, 151, 155	syl2an 464	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ d e. NN0 ) /\ z e. CC ) /\ k e. ( 0 ... ( d + 1 ) ) ) -> ( ( G ` z ) ^ k ) e. CC )
157	153, 156	mulcld 9064	. . . . . . . . . . 11 |- ( ( ( ( ph /\ d e. NN0 ) /\ z e. CC ) /\ k e. ( 0 ... ( d + 1 ) ) ) -> ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) e. CC )
158		fveq2 5687	. . . . . . . . . . . 12 |- ( k = ( d + 1 ) -> ( (coeff ` F ) ` k ) = ( (coeff ` F ) ` ( d + 1 ) ) )
159		oveq2 6048	. . . . . . . . . . . 12 |- ( k = ( d + 1 ) -> ( ( G ` z ) ^ k ) = ( ( G ` z ) ^ ( d + 1 ) ) )
160	158, 159	oveq12d 6058	. . . . . . . . . . 11 |- ( k = ( d + 1 ) -> ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) = ( ( (coeff ` F ) ` ( d + 1 ) ) x. ( ( G ` z ) ^ ( d + 1 ) ) ) )
161	147, 157, 160	fsump1 12495	. . . . . . . . . 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 ) ) ) ) )
162	161	mpteq2dva 4255	. . . . . . . . 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 ) ) ) ) ) )
163	144, 162	eqtr4d 2439	. . . . . . . 8 |- ( ( ph /\ d e. NN0 ) -> ( ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) ) o F + ( ( CC X. { ( (coeff ` F ) ` ( d + 1 ) ) } ) o F x. ( z e. CC |-> ( ( G ` z ) ^ ( d + 1 ) ) ) ) ) = ( z e. CC |-> sum_ k e. ( 0 ... ( d + 1 ) ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) ) )
164	163	eleq1d 2470	. . . . . . 7 |- ( ( ph /\ d e. NN0 ) -> ( ( ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) ) o F + ( ( CC X. { ( (coeff ` F ) ` ( d + 1 ) ) } ) o F 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 ) ) )
165	129, 164	sylibd 206	. . . . . 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 ) ) )
166	165	expcom 425	. . . . 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 ) ) ) )
167	166	a2d 24	. . . 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 ) ) ) )
168	21, 26, 31, 36, 69, 167	nn0ind 10322	. . 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 ) ) )
169	16, 168	mpcom 34	. 2 |- ( ph -> ( z e. CC |-> sum_ k e. ( 0 ... (deg ` F ) ) ( ( (coeff ` F ) ` k ) x. ( ( G ` z ) ^ k ) ) ) e. (Poly ` S ) )
170	14, 169	eqeltrd 2478	1 |- ( ph -> ( F o. G ) e. (Poly ` S ) )

Syntax hints:   -> wi 4   /\ wa 359   = wceq 1649   e. wcel 1721   _Vcvv 2916   u. cun 3278   C_ wss 3280   {csn 3774   e. cmpt 4226   X. cxp 4835   o. ccom 4841   -->wf 5409   ` cfv 5413  (class class class)co 6040   o Fcof 6262   CCcc 8944   0cc0 8946   1c1 8947   + caddc 8949   x. cmul 8951   NNcn 9956   NN0cn0 10177   ZZcz 10238   ZZ>=cuz 10444   ...cfz 10999   ^cexp 11337   sum_csu 12434  Polycply 20056  coeffccoe 20058  degcdgr 20059

This theorem is referenced by:  dgrcolem1  20144  dgrcolem2  20145  taylply2  20237  ftalem7  20814

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-oi 7435  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-fz 11000  df-fzo 11091  df-fl 11157  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237  df-rlim 12238  df-sum 12435  df-0p 19515  df-ply 20060  df-coe 20062  df-dgr 20063