Theorem plydivlem4 24051

Description: Lemma for plydivex 24052. Induction step. (Contributed by Mario Carneiro, 26-Jul-2014.)

Hypotheses

Ref	Expression
plydiv.pl	|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x + y ) e. S )
plydiv.tm	|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x x. y ) e. S )
plydiv.rc	|- ( ( ph /\ ( x e. S /\ x =/= 0 ) ) -> ( 1 / x ) e. S )
plydiv.m1	|- ( ph -> -u 1 e. S )
plydiv.f	|- ( ph -> F e. (Poly ` S ) )
plydiv.g	|- ( ph -> G e. (Poly ` S ) )
plydiv.z	|- ( ph -> G =/= 0p )
plydiv.r	|- R = ( F oF - ( G oF x. q ) )
plydiv.d	|- ( ph -> D e. NN0 )
plydiv.e	|- ( ph -> ( M - N ) = D )
plydiv.fz	|- ( ph -> F =/= 0p )
plydiv.u	|- U = ( f oF - ( G oF x. p ) )
plydiv.h	|- H = ( z e. CC |-> ( ( ( A ` M ) / ( B ` N ) ) x. ( z ^ D ) ) )
plydiv.al	|- ( ph -> A. f e. (Poly ` S ) ( ( f = 0p \/ ( (deg ` f ) - N ) < D ) -> E. p e. (Poly ` S ) ( U = 0p \/ (deg ` U ) < N ) ) )
plydiv.a	|- A = (coeff ` F )
plydiv.b	|- B = (coeff ` G )
plydiv.m	|- M = (deg ` F )
plydiv.n	|- N = (deg ` G )

Assertion

Ref	Expression
plydivlem4	|- ( ph -> E. q e. (Poly ` S ) ( R = 0p \/ (deg ` R ) < N ) )

Distinct variable groups:   x, y, z, A   f, p, q, x, y, z, F   f, H, p, q, x, y, z   ph, x, y, z   x, B, y, z   D, f, z   x, M, y, z   f, N, p, q, x, y, z   f, G, p, q, x, y, z   R, f, p, x, y   S, f, p, q, x, y, z   ph, p

Allowed substitution hints:   ph( f, q)   A( f, q, p)   B( f, q, p)   D( x, y, q, p)   R( z, q)   U( x, y, z, f, q, p)   M( f, q, p)

Proof of Theorem plydivlem4

Step	Hyp	Ref	Expression
1		plydiv.f	. . . . . . . 8 |- ( ph -> F e. (Poly ` S ) )
2		plybss 23950	. . . . . . . 8 |- ( F e. (Poly ` S ) -> S C_ CC )
3	1, 2	syl 17	. . . . . . 7 |- ( ph -> S C_ CC )
4		plydiv.pl	. . . . . . . . . . . . 13 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x + y ) e. S )
5		plydiv.tm	. . . . . . . . . . . . 13 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x x. y ) e. S )
6		plydiv.rc	. . . . . . . . . . . . 13 |- ( ( ph /\ ( x e. S /\ x =/= 0 ) ) -> ( 1 / x ) e. S )
7		plydiv.m1	. . . . . . . . . . . . 13 |- ( ph -> -u 1 e. S )
8	4, 5, 6, 7	plydivlem1 24048	. . . . . . . . . . . 12 |- ( ph -> 0 e. S )
9		plydiv.a	. . . . . . . . . . . . 13 |- A = (coeff ` F )
10	9	coef2 23987	. . . . . . . . . . . 12 |- ( ( F e. (Poly ` S ) /\ 0 e. S ) -> A : NN0 --> S )
11	1, 8, 10	syl2anc 693	. . . . . . . . . . 11 |- ( ph -> A : NN0 --> S )
12		plydiv.m	. . . . . . . . . . . 12 |- M = (deg ` F )
13		dgrcl 23989	. . . . . . . . . . . . 13 |- ( F e. (Poly ` S ) -> (deg ` F ) e. NN0 )
14	1, 13	syl 17	. . . . . . . . . . . 12 |- ( ph -> (deg ` F ) e. NN0 )
15	12, 14	syl5eqel 2705	. . . . . . . . . . 11 |- ( ph -> M e. NN0 )
16	11, 15	ffvelrnd 6360	. . . . . . . . . 10 |- ( ph -> ( A ` M ) e. S )
17	3, 16	sseldd 3604	. . . . . . . . 9 |- ( ph -> ( A ` M ) e. CC )
18		plydiv.g	. . . . . . . . . . . 12 |- ( ph -> G e. (Poly ` S ) )
19		plydiv.b	. . . . . . . . . . . . 13 |- B = (coeff ` G )
20	19	coef2 23987	. . . . . . . . . . . 12 |- ( ( G e. (Poly ` S ) /\ 0 e. S ) -> B : NN0 --> S )
21	18, 8, 20	syl2anc 693	. . . . . . . . . . 11 |- ( ph -> B : NN0 --> S )
22		plydiv.n	. . . . . . . . . . . 12 |- N = (deg ` G )
23		dgrcl 23989	. . . . . . . . . . . . 13 |- ( G e. (Poly ` S ) -> (deg ` G ) e. NN0 )
24	18, 23	syl 17	. . . . . . . . . . . 12 |- ( ph -> (deg ` G ) e. NN0 )
25	22, 24	syl5eqel 2705	. . . . . . . . . . 11 |- ( ph -> N e. NN0 )
26	21, 25	ffvelrnd 6360	. . . . . . . . . 10 |- ( ph -> ( B ` N ) e. S )
27	3, 26	sseldd 3604	. . . . . . . . 9 |- ( ph -> ( B ` N ) e. CC )
28		plydiv.z	. . . . . . . . . 10 |- ( ph -> G =/= 0p )
29	22, 19	dgreq0 24021	. . . . . . . . . . . 12 |- ( G e. (Poly ` S ) -> ( G = 0p <-> ( B ` N ) = 0 ) )
30	18, 29	syl 17	. . . . . . . . . . 11 |- ( ph -> ( G = 0p <-> ( B ` N ) = 0 ) )
31	30	necon3bid 2838	. . . . . . . . . 10 |- ( ph -> ( G =/= 0p <-> ( B ` N ) =/= 0 ) )
32	28, 31	mpbid 222	. . . . . . . . 9 |- ( ph -> ( B ` N ) =/= 0 )
33	17, 27, 32	divrecd 10804	. . . . . . . 8 |- ( ph -> ( ( A ` M ) / ( B ` N ) ) = ( ( A ` M ) x. ( 1 / ( B ` N ) ) ) )
34		fvex 6201	. . . . . . . . . . . 12 |- ( B ` N ) e. _V
35		eleq1 2689	. . . . . . . . . . . . . . 15 |- ( x = ( B ` N ) -> ( x e. S <-> ( B ` N ) e. S ) )
36		neeq1 2856	. . . . . . . . . . . . . . 15 |- ( x = ( B ` N ) -> ( x =/= 0 <-> ( B ` N ) =/= 0 ) )
37	35, 36	anbi12d 747	. . . . . . . . . . . . . 14 |- ( x = ( B ` N ) -> ( ( x e. S /\ x =/= 0 ) <-> ( ( B ` N ) e. S /\ ( B ` N ) =/= 0 ) ) )
38	37	anbi2d 740	. . . . . . . . . . . . 13 |- ( x = ( B ` N ) -> ( ( ph /\ ( x e. S /\ x =/= 0 ) ) <-> ( ph /\ ( ( B ` N ) e. S /\ ( B ` N ) =/= 0 ) ) ) )
39		oveq2 6658	. . . . . . . . . . . . . 14 |- ( x = ( B ` N ) -> ( 1 / x ) = ( 1 / ( B ` N ) ) )
40	39	eleq1d 2686	. . . . . . . . . . . . 13 |- ( x = ( B ` N ) -> ( ( 1 / x ) e. S <-> ( 1 / ( B ` N ) ) e. S ) )
41	38, 40	imbi12d 334	. . . . . . . . . . . 12 |- ( x = ( B ` N ) -> ( ( ( ph /\ ( x e. S /\ x =/= 0 ) ) -> ( 1 / x ) e. S ) <-> ( ( ph /\ ( ( B ` N ) e. S /\ ( B ` N ) =/= 0 ) ) -> ( 1 / ( B ` N ) ) e. S ) ) )
42	34, 41, 6	vtocl 3259	. . . . . . . . . . 11 |- ( ( ph /\ ( ( B ` N ) e. S /\ ( B ` N ) =/= 0 ) ) -> ( 1 / ( B ` N ) ) e. S )
43	42	ex 450	. . . . . . . . . 10 |- ( ph -> ( ( ( B ` N ) e. S /\ ( B ` N ) =/= 0 ) -> ( 1 / ( B ` N ) ) e. S ) )
44	26, 32, 43	mp2and 715	. . . . . . . . 9 |- ( ph -> ( 1 / ( B ` N ) ) e. S )
45	5, 16, 44	caovcld 6827	. . . . . . . 8 |- ( ph -> ( ( A ` M ) x. ( 1 / ( B ` N ) ) ) e. S )
46	33, 45	eqeltrd 2701	. . . . . . 7 |- ( ph -> ( ( A ` M ) / ( B ` N ) ) e. S )
47		plydiv.d	. . . . . . 7 |- ( ph -> D e. NN0 )
48		plydiv.h	. . . . . . . 8 |- H = ( z e. CC |-> ( ( ( A ` M ) / ( B ` N ) ) x. ( z ^ D ) ) )
49	48	ply1term 23960	. . . . . . 7 |- ( ( S C_ CC /\ ( ( A ` M ) / ( B ` N ) ) e. S /\ D e. NN0 ) -> H e. (Poly ` S ) )
50	3, 46, 47, 49	syl3anc 1326	. . . . . 6 |- ( ph -> H e. (Poly ` S ) )
51	50	adantr 481	. . . . 5 |- ( ( ph /\ p e. (Poly ` S ) ) -> H e. (Poly ` S ) )
52		simpr 477	. . . . 5 |- ( ( ph /\ p e. (Poly ` S ) ) -> p e. (Poly ` S ) )
53	4	adantlr 751	. . . . 5 |- ( ( ( ph /\ p e. (Poly ` S ) ) /\ ( x e. S /\ y e. S ) ) -> ( x + y ) e. S )
54	51, 52, 53	plyadd 23973	. . . 4 |- ( ( ph /\ p e. (Poly ` S ) ) -> ( H oF + p ) e. (Poly ` S ) )
55	54	adantr 481	. . 3 |- ( ( ( ph /\ p e. (Poly ` S ) ) /\ ( ( ( F oF - ( H oF x. G ) ) oF - ( G oF x. p ) ) = 0p \/ (deg ` ( ( F oF - ( H oF x. G ) ) oF - ( G oF x. p ) ) ) < N ) ) -> ( H oF + p ) e. (Poly ` S ) )
56		cnex 10017	. . . . . . . . 9 |- CC e. _V
57	56	a1i 11	. . . . . . . 8 |- ( ( ph /\ p e. (Poly ` S ) ) -> CC e. _V )
58	1	adantr 481	. . . . . . . . 9 |- ( ( ph /\ p e. (Poly ` S ) ) -> F e. (Poly ` S ) )
59		plyf 23954	. . . . . . . . 9 |- ( F e. (Poly ` S ) -> F : CC --> CC )
60	58, 59	syl 17	. . . . . . . 8 |- ( ( ph /\ p e. (Poly ` S ) ) -> F : CC --> CC )
61		mulcl 10020	. . . . . . . . . 10 |- ( ( x e. CC /\ y e. CC ) -> ( x x. y ) e. CC )
62	61	adantl 482	. . . . . . . . 9 |- ( ( ( ph /\ p e. (Poly ` S ) ) /\ ( x e. CC /\ y e. CC ) ) -> ( x x. y ) e. CC )
63		plyf 23954	. . . . . . . . . 10 |- ( H e. (Poly ` S ) -> H : CC --> CC )
64	51, 63	syl 17	. . . . . . . . 9 |- ( ( ph /\ p e. (Poly ` S ) ) -> H : CC --> CC )
65	18	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ p e. (Poly ` S ) ) -> G e. (Poly ` S ) )
66		plyf 23954	. . . . . . . . . 10 |- ( G e. (Poly ` S ) -> G : CC --> CC )
67	65, 66	syl 17	. . . . . . . . 9 |- ( ( ph /\ p e. (Poly ` S ) ) -> G : CC --> CC )
68		inidm 3822	. . . . . . . . 9 |- ( CC i^i CC ) = CC
69	62, 64, 67, 57, 57, 68	off 6912	. . . . . . . 8 |- ( ( ph /\ p e. (Poly ` S ) ) -> ( H oF x. G ) : CC --> CC )
70		plyf 23954	. . . . . . . . . 10 |- ( p e. (Poly ` S ) -> p : CC --> CC )
71	70	adantl 482	. . . . . . . . 9 |- ( ( ph /\ p e. (Poly ` S ) ) -> p : CC --> CC )
72	62, 67, 71, 57, 57, 68	off 6912	. . . . . . . 8 |- ( ( ph /\ p e. (Poly ` S ) ) -> ( G oF x. p ) : CC --> CC )
73		subsub4 10314	. . . . . . . . 9 |- ( ( x e. CC /\ y e. CC /\ z e. CC ) -> ( ( x - y ) - z ) = ( x - ( y + z ) ) )
74	73	adantl 482	. . . . . . . 8 |- ( ( ( ph /\ p e. (Poly ` S ) ) /\ ( x e. CC /\ y e. CC /\ z e. CC ) ) -> ( ( x - y ) - z ) = ( x - ( y + z ) ) )
75	57, 60, 69, 72, 74	caofass 6931	. . . . . . 7 |- ( ( ph /\ p e. (Poly ` S ) ) -> ( ( F oF - ( H oF x. G ) ) oF - ( G oF x. p ) ) = ( F oF - ( ( H oF x. G ) oF + ( G oF x. p ) ) ) )
76		mulcom 10022	. . . . . . . . . . . 12 |- ( ( x e. CC /\ y e. CC ) -> ( x x. y ) = ( y x. x ) )
77	76	adantl 482	. . . . . . . . . . 11 |- ( ( ( ph /\ p e. (Poly ` S ) ) /\ ( x e. CC /\ y e. CC ) ) -> ( x x. y ) = ( y x. x ) )
78	57, 64, 67, 77	caofcom 6929	. . . . . . . . . 10 |- ( ( ph /\ p e. (Poly ` S ) ) -> ( H oF x. G ) = ( G oF x. H ) )
79	78	oveq1d 6665	. . . . . . . . 9 |- ( ( ph /\ p e. (Poly ` S ) ) -> ( ( H oF x. G ) oF + ( G oF x. p ) ) = ( ( G oF x. H ) oF + ( G oF x. p ) ) )
80		adddi 10025	. . . . . . . . . . 11 |- ( ( x e. CC /\ y e. CC /\ z e. CC ) -> ( x x. ( y + z ) ) = ( ( x x. y ) + ( x x. z ) ) )
81	80	adantl 482	. . . . . . . . . 10 |- ( ( ( ph /\ p e. (Poly ` S ) ) /\ ( x e. CC /\ y e. CC /\ z e. CC ) ) -> ( x x. ( y + z ) ) = ( ( x x. y ) + ( x x. z ) ) )
82	57, 67, 64, 71, 81	caofdi 6933	. . . . . . . . 9 |- ( ( ph /\ p e. (Poly ` S ) ) -> ( G oF x. ( H oF + p ) ) = ( ( G oF x. H ) oF + ( G oF x. p ) ) )
83	79, 82	eqtr4d 2659	. . . . . . . 8 |- ( ( ph /\ p e. (Poly ` S ) ) -> ( ( H oF x. G ) oF + ( G oF x. p ) ) = ( G oF x. ( H oF + p ) ) )
84	83	oveq2d 6666	. . . . . . 7 |- ( ( ph /\ p e. (Poly ` S ) ) -> ( F oF - ( ( H oF x. G ) oF + ( G oF x. p ) ) ) = ( F oF - ( G oF x. ( H oF + p ) ) ) )
85	75, 84	eqtrd 2656	. . . . . 6 |- ( ( ph /\ p e. (Poly ` S ) ) -> ( ( F oF - ( H oF x. G ) ) oF - ( G oF x. p ) ) = ( F oF - ( G oF x. ( H oF + p ) ) ) )
86	85	eqeq1d 2624	. . . . 5 |- ( ( ph /\ p e. (Poly ` S ) ) -> ( ( ( F oF - ( H oF x. G ) ) oF - ( G oF x. p ) ) = 0p <-> ( F oF - ( G oF x. ( H oF + p ) ) ) = 0p ) )
87	85	fveq2d 6195	. . . . . 6 |- ( ( ph /\ p e. (Poly ` S ) ) -> (deg ` ( ( F oF - ( H oF x. G ) ) oF - ( G oF x. p ) ) ) = (deg ` ( F oF - ( G oF x. ( H oF + p ) ) ) ) )
88	87	breq1d 4663	. . . . 5 |- ( ( ph /\ p e. (Poly ` S ) ) -> ( (deg ` ( ( F oF - ( H oF x. G ) ) oF - ( G oF x. p ) ) ) < N <-> (deg ` ( F oF - ( G oF x. ( H oF + p ) ) ) ) < N ) )
89	86, 88	orbi12d 746	. . . 4 |- ( ( ph /\ p e. (Poly ` S ) ) -> ( ( ( ( F oF - ( H oF x. G ) ) oF - ( G oF x. p ) ) = 0p \/ (deg ` ( ( F oF - ( H oF x. G ) ) oF - ( G oF x. p ) ) ) < N ) <-> ( ( F oF - ( G oF x. ( H oF + p ) ) ) = 0p \/ (deg ` ( F oF - ( G oF x. ( H oF + p ) ) ) ) < N ) ) )
90	89	biimpa 501	. . 3 |- ( ( ( ph /\ p e. (Poly ` S ) ) /\ ( ( ( F oF - ( H oF x. G ) ) oF - ( G oF x. p ) ) = 0p \/ (deg ` ( ( F oF - ( H oF x. G ) ) oF - ( G oF x. p ) ) ) < N ) ) -> ( ( F oF - ( G oF x. ( H oF + p ) ) ) = 0p \/ (deg ` ( F oF - ( G oF x. ( H oF + p ) ) ) ) < N ) )
91		plydiv.r	. . . . . . 7 |- R = ( F oF - ( G oF x. q ) )
92		oveq2 6658	. . . . . . . 8 |- ( q = ( H oF + p ) -> ( G oF x. q ) = ( G oF x. ( H oF + p ) ) )
93	92	oveq2d 6666	. . . . . . 7 |- ( q = ( H oF + p ) -> ( F oF - ( G oF x. q ) ) = ( F oF - ( G oF x. ( H oF + p ) ) ) )
94	91, 93	syl5eq 2668	. . . . . 6 |- ( q = ( H oF + p ) -> R = ( F oF - ( G oF x. ( H oF + p ) ) ) )
95	94	eqeq1d 2624	. . . . 5 |- ( q = ( H oF + p ) -> ( R = 0p <-> ( F oF - ( G oF x. ( H oF + p ) ) ) = 0p ) )
96	94	fveq2d 6195	. . . . . 6 |- ( q = ( H oF + p ) -> (deg ` R ) = (deg ` ( F oF - ( G oF x. ( H oF + p ) ) ) ) )
97	96	breq1d 4663	. . . . 5 |- ( q = ( H oF + p ) -> ( (deg ` R ) < N <-> (deg ` ( F oF - ( G oF x. ( H oF + p ) ) ) ) < N ) )
98	95, 97	orbi12d 746	. . . 4 |- ( q = ( H oF + p ) -> ( ( R = 0p \/ (deg ` R ) < N ) <-> ( ( F oF - ( G oF x. ( H oF + p ) ) ) = 0p \/ (deg ` ( F oF - ( G oF x. ( H oF + p ) ) ) ) < N ) ) )
99	98	rspcev 3309	. . 3 |- ( ( ( H oF + p ) e. (Poly ` S ) /\ ( ( F oF - ( G oF x. ( H oF + p ) ) ) = 0p \/ (deg ` ( F oF - ( G oF x. ( H oF + p ) ) ) ) < N ) ) -> E. q e. (Poly ` S ) ( R = 0p \/ (deg ` R ) < N ) )
100	55, 90, 99	syl2anc 693	. 2 |- ( ( ( ph /\ p e. (Poly ` S ) ) /\ ( ( ( F oF - ( H oF x. G ) ) oF - ( G oF x. p ) ) = 0p \/ (deg ` ( ( F oF - ( H oF x. G ) ) oF - ( G oF x. p ) ) ) < N ) ) -> E. q e. (Poly ` S ) ( R = 0p \/ (deg ` R ) < N ) )
101	50, 18, 4, 5	plymul 23974	. . . 4 |- ( ph -> ( H oF x. G ) e. (Poly ` S ) )
102	1, 101, 4, 5, 7	plysub 23975	. . 3 |- ( ph -> ( F oF - ( H oF x. G ) ) e. (Poly ` S ) )
103		plydiv.al	. . 3 |- ( ph -> A. f e. (Poly ` S ) ( ( f = 0p \/ ( (deg ` f ) - N ) < D ) -> E. p e. (Poly ` S ) ( U = 0p \/ (deg ` U ) < N ) ) )
104		eqid 2622	. . . . . . 7 |- (deg ` ( H oF x. G ) ) = (deg ` ( H oF x. G ) )
105	12, 104	dgrsub 24028	. . . . . 6 |- ( ( F e. (Poly ` S ) /\ ( H oF x. G ) e. (Poly ` S ) ) -> (deg ` ( F oF - ( H oF x. G ) ) ) <_ if ( M <_ (deg ` ( H oF x. G ) ) , (deg ` ( H oF x. G ) ) , M ) )
106	1, 101, 105	syl2anc 693	. . . . 5 |- ( ph -> (deg ` ( F oF - ( H oF x. G ) ) ) <_ if ( M <_ (deg ` ( H oF x. G ) ) , (deg ` ( H oF x. G ) ) , M ) )
107		plydiv.fz	. . . . . . . . . . . . 13 |- ( ph -> F =/= 0p )
108	12, 9	dgreq0 24021	. . . . . . . . . . . . . . 15 |- ( F e. (Poly ` S ) -> ( F = 0p <-> ( A ` M ) = 0 ) )
109	1, 108	syl 17	. . . . . . . . . . . . . 14 |- ( ph -> ( F = 0p <-> ( A ` M ) = 0 ) )
110	109	necon3bid 2838	. . . . . . . . . . . . 13 |- ( ph -> ( F =/= 0p <-> ( A ` M ) =/= 0 ) )
111	107, 110	mpbid 222	. . . . . . . . . . . 12 |- ( ph -> ( A ` M ) =/= 0 )
112	17, 27, 111, 32	divne0d 10817	. . . . . . . . . . 11 |- ( ph -> ( ( A ` M ) / ( B ` N ) ) =/= 0 )
113	3, 46	sseldd 3604	. . . . . . . . . . . . 13 |- ( ph -> ( ( A ` M ) / ( B ` N ) ) e. CC )
114	48	coe1term 24015	. . . . . . . . . . . . 13 |- ( ( ( ( A ` M ) / ( B ` N ) ) e. CC /\ D e. NN0 /\ D e. NN0 ) -> ( (coeff ` H ) ` D ) = if ( D = D , ( ( A ` M ) / ( B ` N ) ) , 0 ) )
115	113, 47, 47, 114	syl3anc 1326	. . . . . . . . . . . 12 |- ( ph -> ( (coeff ` H ) ` D ) = if ( D = D , ( ( A ` M ) / ( B ` N ) ) , 0 ) )
116		eqid 2622	. . . . . . . . . . . . 13 |- D = D
117	116	iftruei 4093	. . . . . . . . . . . 12 |- if ( D = D , ( ( A ` M ) / ( B ` N ) ) , 0 ) = ( ( A ` M ) / ( B ` N ) )
118	115, 117	syl6eq 2672	. . . . . . . . . . 11 |- ( ph -> ( (coeff ` H ) ` D ) = ( ( A ` M ) / ( B ` N ) ) )
119		c0ex 10034	. . . . . . . . . . . . 13 |- 0 e. _V
120	119	fvconst2 6469	. . . . . . . . . . . 12 |- ( D e. NN0 -> ( ( NN0 X. { 0 } ) ` D ) = 0 )
121	47, 120	syl 17	. . . . . . . . . . 11 |- ( ph -> ( ( NN0 X. { 0 } ) ` D ) = 0 )
122	112, 118, 121	3netr4d 2871	. . . . . . . . . 10 |- ( ph -> ( (coeff ` H ) ` D ) =/= ( ( NN0 X. { 0 } ) ` D ) )
123		fveq2 6191	. . . . . . . . . . . . 13 |- ( H = 0p -> (coeff ` H ) = (coeff ` 0p ) )
124		coe0 24012	. . . . . . . . . . . . 13 |- (coeff ` 0p ) = ( NN0 X. { 0 } )
125	123, 124	syl6eq 2672	. . . . . . . . . . . 12 |- ( H = 0p -> (coeff ` H ) = ( NN0 X. { 0 } ) )
126	125	fveq1d 6193	. . . . . . . . . . 11 |- ( H = 0p -> ( (coeff ` H ) ` D ) = ( ( NN0 X. { 0 } ) ` D ) )
127	126	necon3i 2826	. . . . . . . . . 10 |- ( ( (coeff ` H ) ` D ) =/= ( ( NN0 X. { 0 } ) ` D ) -> H =/= 0p )
128	122, 127	syl 17	. . . . . . . . 9 |- ( ph -> H =/= 0p )
129		eqid 2622	. . . . . . . . . 10 |- (deg ` H ) = (deg ` H )
130	129, 22	dgrmul 24026	. . . . . . . . 9 |- ( ( ( H e. (Poly ` S ) /\ H =/= 0p ) /\ ( G e. (Poly ` S ) /\ G =/= 0p ) ) -> (deg ` ( H oF x. G ) ) = ( (deg ` H ) + N ) )
131	50, 128, 18, 28, 130	syl22anc 1327	. . . . . . . 8 |- ( ph -> (deg ` ( H oF x. G ) ) = ( (deg ` H ) + N ) )
132	48	dgr1term 24016	. . . . . . . . . . . 12 |- ( ( ( ( A ` M ) / ( B ` N ) ) e. CC /\ ( ( A ` M ) / ( B ` N ) ) =/= 0 /\ D e. NN0 ) -> (deg ` H ) = D )
133	113, 112, 47, 132	syl3anc 1326	. . . . . . . . . . 11 |- ( ph -> (deg ` H ) = D )
134		plydiv.e	. . . . . . . . . . 11 |- ( ph -> ( M - N ) = D )
135	133, 134	eqtr4d 2659	. . . . . . . . . 10 |- ( ph -> (deg ` H ) = ( M - N ) )
136	135	oveq1d 6665	. . . . . . . . 9 |- ( ph -> ( (deg ` H ) + N ) = ( ( M - N ) + N ) )
137	15	nn0cnd 11353	. . . . . . . . . 10 |- ( ph -> M e. CC )
138	25	nn0cnd 11353	. . . . . . . . . 10 |- ( ph -> N e. CC )
139	137, 138	npcand 10396	. . . . . . . . 9 |- ( ph -> ( ( M - N ) + N ) = M )
140	136, 139	eqtrd 2656	. . . . . . . 8 |- ( ph -> ( (deg ` H ) + N ) = M )
141	131, 140	eqtrd 2656	. . . . . . 7 |- ( ph -> (deg ` ( H oF x. G ) ) = M )
142	141	ifeq1d 4104	. . . . . 6 |- ( ph -> if ( M <_ (deg ` ( H oF x. G ) ) , (deg ` ( H oF x. G ) ) , M ) = if ( M <_ (deg ` ( H oF x. G ) ) , M , M ) )
143		ifid 4125	. . . . . 6 |- if ( M <_ (deg ` ( H oF x. G ) ) , M , M ) = M
144	142, 143	syl6eq 2672	. . . . 5 |- ( ph -> if ( M <_ (deg ` ( H oF x. G ) ) , (deg ` ( H oF x. G ) ) , M ) = M )
145	106, 144	breqtrd 4679	. . . 4 |- ( ph -> (deg ` ( F oF - ( H oF x. G ) ) ) <_ M )
146		eqid 2622	. . . . . . . 8 |- (coeff ` ( H oF x. G ) ) = (coeff ` ( H oF x. G ) )
147	9, 146	coesub 24013	. . . . . . 7 |- ( ( F e. (Poly ` S ) /\ ( H oF x. G ) e. (Poly ` S ) ) -> (coeff ` ( F oF - ( H oF x. G ) ) ) = ( A oF - (coeff ` ( H oF x. G ) ) ) )
148	1, 101, 147	syl2anc 693	. . . . . 6 |- ( ph -> (coeff ` ( F oF - ( H oF x. G ) ) ) = ( A oF - (coeff ` ( H oF x. G ) ) ) )
149	148	fveq1d 6193	. . . . 5 |- ( ph -> ( (coeff ` ( F oF - ( H oF x. G ) ) ) ` M ) = ( ( A oF - (coeff ` ( H oF x. G ) ) ) ` M ) )
150	9	coef3 23988	. . . . . . . 8 |- ( F e. (Poly ` S ) -> A : NN0 --> CC )
151		ffn 6045	. . . . . . . 8 |- ( A : NN0 --> CC -> A Fn NN0 )
152	1, 150, 151	3syl 18	. . . . . . 7 |- ( ph -> A Fn NN0 )
153	146	coef3 23988	. . . . . . . 8 |- ( ( H oF x. G ) e. (Poly ` S ) -> (coeff ` ( H oF x. G ) ) : NN0 --> CC )
154		ffn 6045	. . . . . . . 8 |- ( (coeff ` ( H oF x. G ) ) : NN0 --> CC -> (coeff ` ( H oF x. G ) ) Fn NN0 )
155	101, 153, 154	3syl 18	. . . . . . 7 |- ( ph -> (coeff ` ( H oF x. G ) ) Fn NN0 )
156		nn0ex 11298	. . . . . . . 8 |- NN0 e. _V
157	156	a1i 11	. . . . . . 7 |- ( ph -> NN0 e. _V )
158		inidm 3822	. . . . . . 7 |- ( NN0 i^i NN0 ) = NN0
159		eqidd 2623	. . . . . . 7 |- ( ( ph /\ M e. NN0 ) -> ( A ` M ) = ( A ` M ) )
160		eqid 2622	. . . . . . . . . . 11 |- (coeff ` H ) = (coeff ` H )
161	160, 19, 129, 22	coemulhi 24010	. . . . . . . . . 10 |- ( ( H e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> ( (coeff ` ( H oF x. G ) ) ` ( (deg ` H ) + N ) ) = ( ( (coeff ` H ) ` (deg ` H ) ) x. ( B ` N ) ) )
162	50, 18, 161	syl2anc 693	. . . . . . . . 9 |- ( ph -> ( (coeff ` ( H oF x. G ) ) ` ( (deg ` H ) + N ) ) = ( ( (coeff ` H ) ` (deg ` H ) ) x. ( B ` N ) ) )
163	140	fveq2d 6195	. . . . . . . . 9 |- ( ph -> ( (coeff ` ( H oF x. G ) ) ` ( (deg ` H ) + N ) ) = ( (coeff ` ( H oF x. G ) ) ` M ) )
164	133	fveq2d 6195	. . . . . . . . . . . 12 |- ( ph -> ( (coeff ` H ) ` (deg ` H ) ) = ( (coeff ` H ) ` D ) )
165	164, 118	eqtrd 2656	. . . . . . . . . . 11 |- ( ph -> ( (coeff ` H ) ` (deg ` H ) ) = ( ( A ` M ) / ( B ` N ) ) )
166	165	oveq1d 6665	. . . . . . . . . 10 |- ( ph -> ( ( (coeff ` H ) ` (deg ` H ) ) x. ( B ` N ) ) = ( ( ( A ` M ) / ( B ` N ) ) x. ( B ` N ) ) )
167	17, 27, 32	divcan1d 10802	. . . . . . . . . 10 |- ( ph -> ( ( ( A ` M ) / ( B ` N ) ) x. ( B ` N ) ) = ( A ` M ) )
168	166, 167	eqtrd 2656	. . . . . . . . 9 |- ( ph -> ( ( (coeff ` H ) ` (deg ` H ) ) x. ( B ` N ) ) = ( A ` M ) )
169	162, 163, 168	3eqtr3d 2664	. . . . . . . 8 |- ( ph -> ( (coeff ` ( H oF x. G ) ) ` M ) = ( A ` M ) )
170	169	adantr 481	. . . . . . 7 |- ( ( ph /\ M e. NN0 ) -> ( (coeff ` ( H oF x. G ) ) ` M ) = ( A ` M ) )
171	152, 155, 157, 157, 158, 159, 170	ofval 6906	. . . . . 6 |- ( ( ph /\ M e. NN0 ) -> ( ( A oF - (coeff ` ( H oF x. G ) ) ) ` M ) = ( ( A ` M ) - ( A ` M ) ) )
172	15, 171	mpdan 702	. . . . 5 |- ( ph -> ( ( A oF - (coeff ` ( H oF x. G ) ) ) ` M ) = ( ( A ` M ) - ( A ` M ) ) )
173	17	subidd 10380	. . . . 5 |- ( ph -> ( ( A ` M ) - ( A ` M ) ) = 0 )
174	149, 172, 173	3eqtrd 2660	. . . 4 |- ( ph -> ( (coeff ` ( F oF - ( H oF x. G ) ) ) ` M ) = 0 )
175		dgrcl 23989	. . . . . . . . . 10 |- ( ( F oF - ( H oF x. G ) ) e. (Poly ` S ) -> (deg ` ( F oF - ( H oF x. G ) ) ) e. NN0 )
176	102, 175	syl 17	. . . . . . . . 9 |- ( ph -> (deg ` ( F oF - ( H oF x. G ) ) ) e. NN0 )
177	176	nn0red 11352	. . . . . . . 8 |- ( ph -> (deg ` ( F oF - ( H oF x. G ) ) ) e. RR )
178	15	nn0red 11352	. . . . . . . 8 |- ( ph -> M e. RR )
179	25	nn0red 11352	. . . . . . . 8 |- ( ph -> N e. RR )
180	177, 178, 179	ltsub1d 10636	. . . . . . 7 |- ( ph -> ( (deg ` ( F oF - ( H oF x. G ) ) ) < M <-> ( (deg ` ( F oF - ( H oF x. G ) ) ) - N ) < ( M - N ) ) )
181	134	breq2d 4665	. . . . . . 7 |- ( ph -> ( ( (deg ` ( F oF - ( H oF x. G ) ) ) - N ) < ( M - N ) <-> ( (deg ` ( F oF - ( H oF x. G ) ) ) - N ) < D ) )
182	180, 181	bitrd 268	. . . . . 6 |- ( ph -> ( (deg ` ( F oF - ( H oF x. G ) ) ) < M <-> ( (deg ` ( F oF - ( H oF x. G ) ) ) - N ) < D ) )
183	182	orbi2d 738	. . . . 5 |- ( ph -> ( ( ( F oF - ( H oF x. G ) ) = 0p \/ (deg ` ( F oF - ( H oF x. G ) ) ) < M ) <-> ( ( F oF - ( H oF x. G ) ) = 0p \/ ( (deg ` ( F oF - ( H oF x. G ) ) ) - N ) < D ) ) )
184		eqid 2622	. . . . . . 7 |- (deg ` ( F oF - ( H oF x. G ) ) ) = (deg ` ( F oF - ( H oF x. G ) ) )
185		eqid 2622	. . . . . . 7 |- (coeff ` ( F oF - ( H oF x. G ) ) ) = (coeff ` ( F oF - ( H oF x. G ) ) )
186	184, 185	dgrlt 24022	. . . . . 6 |- ( ( ( F oF - ( H oF x. G ) ) e. (Poly ` S ) /\ M e. NN0 ) -> ( ( ( F oF - ( H oF x. G ) ) = 0p \/ (deg ` ( F oF - ( H oF x. G ) ) ) < M ) <-> ( (deg ` ( F oF - ( H oF x. G ) ) ) <_ M /\ ( (coeff ` ( F oF - ( H oF x. G ) ) ) ` M ) = 0 ) ) )
187	102, 15, 186	syl2anc 693	. . . . 5 |- ( ph -> ( ( ( F oF - ( H oF x. G ) ) = 0p \/ (deg ` ( F oF - ( H oF x. G ) ) ) < M ) <-> ( (deg ` ( F oF - ( H oF x. G ) ) ) <_ M /\ ( (coeff ` ( F oF - ( H oF x. G ) ) ) ` M ) = 0 ) ) )
188	183, 187	bitr3d 270	. . . 4 |- ( ph -> ( ( ( F oF - ( H oF x. G ) ) = 0p \/ ( (deg ` ( F oF - ( H oF x. G ) ) ) - N ) < D ) <-> ( (deg ` ( F oF - ( H oF x. G ) ) ) <_ M /\ ( (coeff ` ( F oF - ( H oF x. G ) ) ) ` M ) = 0 ) ) )
189	145, 174, 188	mpbir2and 957	. . 3 |- ( ph -> ( ( F oF - ( H oF x. G ) ) = 0p \/ ( (deg ` ( F oF - ( H oF x. G ) ) ) - N ) < D ) )
190		eqeq1 2626	. . . . . 6 |- ( f = ( F oF - ( H oF x. G ) ) -> ( f = 0p <-> ( F oF - ( H oF x. G ) ) = 0p ) )
191		fveq2 6191	. . . . . . . 8 |- ( f = ( F oF - ( H oF x. G ) ) -> (deg ` f ) = (deg ` ( F oF - ( H oF x. G ) ) ) )
192	191	oveq1d 6665	. . . . . . 7 |- ( f = ( F oF - ( H oF x. G ) ) -> ( (deg ` f ) - N ) = ( (deg ` ( F oF - ( H oF x. G ) ) ) - N ) )
193	192	breq1d 4663	. . . . . 6 |- ( f = ( F oF - ( H oF x. G ) ) -> ( ( (deg ` f ) - N ) < D <-> ( (deg ` ( F oF - ( H oF x. G ) ) ) - N ) < D ) )
194	190, 193	orbi12d 746	. . . . 5 |- ( f = ( F oF - ( H oF x. G ) ) -> ( ( f = 0p \/ ( (deg ` f ) - N ) < D ) <-> ( ( F oF - ( H oF x. G ) ) = 0p \/ ( (deg ` ( F oF - ( H oF x. G ) ) ) - N ) < D ) ) )
195		plydiv.u	. . . . . . . . 9 |- U = ( f oF - ( G oF x. p ) )
196		oveq1 6657	. . . . . . . . 9 |- ( f = ( F oF - ( H oF x. G ) ) -> ( f oF - ( G oF x. p ) ) = ( ( F oF - ( H oF x. G ) ) oF - ( G oF x. p ) ) )
197	195, 196	syl5eq 2668	. . . . . . . 8 |- ( f = ( F oF - ( H oF x. G ) ) -> U = ( ( F oF - ( H oF x. G ) ) oF - ( G oF x. p ) ) )
198	197	eqeq1d 2624	. . . . . . 7 |- ( f = ( F oF - ( H oF x. G ) ) -> ( U = 0p <-> ( ( F oF - ( H oF x. G ) ) oF - ( G oF x. p ) ) = 0p ) )
199	197	fveq2d 6195	. . . . . . . 8 |- ( f = ( F oF - ( H oF x. G ) ) -> (deg ` U ) = (deg ` ( ( F oF - ( H oF x. G ) ) oF - ( G oF x. p ) ) ) )
200	199	breq1d 4663	. . . . . . 7 |- ( f = ( F oF - ( H oF x. G ) ) -> ( (deg ` U ) < N <-> (deg ` ( ( F oF - ( H oF x. G ) ) oF - ( G oF x. p ) ) ) < N ) )
201	198, 200	orbi12d 746	. . . . . 6 |- ( f = ( F oF - ( H oF x. G ) ) -> ( ( U = 0p \/ (deg ` U ) < N ) <-> ( ( ( F oF - ( H oF x. G ) ) oF - ( G oF x. p ) ) = 0p \/ (deg ` ( ( F oF - ( H oF x. G ) ) oF - ( G oF x. p ) ) ) < N ) ) )
202	201	rexbidv 3052	. . . . 5 |- ( f = ( F oF - ( H oF x. G ) ) -> ( E. p e. (Poly ` S ) ( U = 0p \/ (deg ` U ) < N ) <-> E. p e. (Poly ` S ) ( ( ( F oF - ( H oF x. G ) ) oF - ( G oF x. p ) ) = 0p \/ (deg ` ( ( F oF - ( H oF x. G ) ) oF - ( G oF x. p ) ) ) < N ) ) )
203	194, 202	imbi12d 334	. . . 4 |- ( f = ( F oF - ( H oF x. G ) ) -> ( ( ( f = 0p \/ ( (deg ` f ) - N ) < D ) -> E. p e. (Poly ` S ) ( U = 0p \/ (deg ` U ) < N ) ) <-> ( ( ( F oF - ( H oF x. G ) ) = 0p \/ ( (deg ` ( F oF - ( H oF x. G ) ) ) - N ) < D ) -> E. p e. (Poly ` S ) ( ( ( F oF - ( H oF x. G ) ) oF - ( G oF x. p ) ) = 0p \/ (deg ` ( ( F oF - ( H oF x. G ) ) oF - ( G oF x. p ) ) ) < N ) ) ) )
204	203	rspcv 3305	. . 3 |- ( ( F oF - ( H oF x. G ) ) e. (Poly ` S ) -> ( A. f e. (Poly ` S ) ( ( f = 0p \/ ( (deg ` f ) - N ) < D ) -> E. p e. (Poly ` S ) ( U = 0p \/ (deg ` U ) < N ) ) -> ( ( ( F oF - ( H oF x. G ) ) = 0p \/ ( (deg ` ( F oF - ( H oF x. G ) ) ) - N ) < D ) -> E. p e. (Poly ` S ) ( ( ( F oF - ( H oF x. G ) ) oF - ( G oF x. p ) ) = 0p \/ (deg ` ( ( F oF - ( H oF x. G ) ) oF - ( G oF x. p ) ) ) < N ) ) ) )
205	102, 103, 189, 204	syl3c 66	. 2 |- ( ph -> E. p e. (Poly ` S ) ( ( ( F oF - ( H oF x. G ) ) oF - ( G oF x. p ) ) = 0p \/ (deg ` ( ( F oF - ( H oF x. G ) ) oF - ( G oF x. p ) ) ) < N ) )
206	100, 205	r19.29a 3078	1 |- ( ph -> E. q e. (Poly ` S ) ( R = 0p \/ (deg ` R ) < N ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   _Vcvv 3200   C_ wss 3574   ifcif 4086   {csn 4177   class class class wbr 4653   |-> cmpt 4729   X. cxp 5112   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   < clt 10074   <_ cle 10075   - cmin 10266   -ucneg 10267   / cdiv 10684   NN0cn0 11292   ^cexp 12860   0pc0p 23436  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:  plydivex  24052