Theorem plydiveu 24053

Description: Lemma for plydivalg 24054. (Contributed by Mario Carneiro, 24-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 ) )
plydiveu.q	|- ( ph -> q e. (Poly ` S ) )
plydiveu.qd	|- ( ph -> ( R = 0p \/ (deg ` R ) < (deg ` G ) ) )
plydiveu.t	|- T = ( F oF - ( G oF x. p ) )
plydiveu.p	|- ( ph -> p e. (Poly ` S ) )
plydiveu.pd	|- ( ph -> ( T = 0p \/ (deg ` T ) < (deg ` G ) ) )

Assertion

Ref	Expression
plydiveu	|- ( ph -> p = q )

Distinct variable groups:   x, y   q, p, x, y, F   ph, x, y   x, T, y   G, p, q, x, y   R, p, x, y   S, p, q, x, y

Allowed substitution hints:   ph( q, p)   R( q)   T( q, p)

Proof of Theorem plydiveu

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		idd 24	. . . 4 |- ( ph -> ( ( p oF - q ) = 0p -> ( p oF - q ) = 0p ) )
2		plydiveu.q	. . . . . . . . . . . . . . . 16 |- ( ph -> q e. (Poly ` S ) )
3		plydiv.pl	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x + y ) e. S )
4		plydiv.tm	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x x. y ) e. S )
5		plydiv.rc	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ ( x e. S /\ x =/= 0 ) ) -> ( 1 / x ) e. S )
6		plydiv.m1	. . . . . . . . . . . . . . . . 17 |- ( ph -> -u 1 e. S )
7		plydiv.f	. . . . . . . . . . . . . . . . 17 |- ( ph -> F e. (Poly ` S ) )
8		plydiv.g	. . . . . . . . . . . . . . . . 17 |- ( ph -> G e. (Poly ` S ) )
9		plydiv.z	. . . . . . . . . . . . . . . . 17 |- ( ph -> G =/= 0p )
10		plydiv.r	. . . . . . . . . . . . . . . . 17 |- R = ( F oF - ( G oF x. q ) )
11	3, 4, 5, 6, 7, 8, 9, 10	plydivlem2 24049	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ q e. (Poly ` S ) ) -> R e. (Poly ` S ) )
12	2, 11	mpdan 702	. . . . . . . . . . . . . . 15 |- ( ph -> R e. (Poly ` S ) )
13		plydiveu.p	. . . . . . . . . . . . . . . 16 |- ( ph -> p e. (Poly ` S ) )
14		plydiveu.t	. . . . . . . . . . . . . . . . 17 |- T = ( F oF - ( G oF x. p ) )
15	3, 4, 5, 6, 7, 8, 9, 14	plydivlem2 24049	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ p e. (Poly ` S ) ) -> T e. (Poly ` S ) )
16	13, 15	mpdan 702	. . . . . . . . . . . . . . 15 |- ( ph -> T e. (Poly ` S ) )
17	12, 16, 3, 4, 6	plysub 23975	. . . . . . . . . . . . . 14 |- ( ph -> ( R oF - T ) e. (Poly ` S ) )
18		dgrcl 23989	. . . . . . . . . . . . . 14 |- ( ( R oF - T ) e. (Poly ` S ) -> (deg ` ( R oF - T ) ) e. NN0 )
19	17, 18	syl 17	. . . . . . . . . . . . 13 |- ( ph -> (deg ` ( R oF - T ) ) e. NN0 )
20	19	nn0red 11352	. . . . . . . . . . . 12 |- ( ph -> (deg ` ( R oF - T ) ) e. RR )
21		dgrcl 23989	. . . . . . . . . . . . . . 15 |- ( T e. (Poly ` S ) -> (deg ` T ) e. NN0 )
22	16, 21	syl 17	. . . . . . . . . . . . . 14 |- ( ph -> (deg ` T ) e. NN0 )
23	22	nn0red 11352	. . . . . . . . . . . . 13 |- ( ph -> (deg ` T ) e. RR )
24		dgrcl 23989	. . . . . . . . . . . . . . 15 |- ( R e. (Poly ` S ) -> (deg ` R ) e. NN0 )
25	12, 24	syl 17	. . . . . . . . . . . . . 14 |- ( ph -> (deg ` R ) e. NN0 )
26	25	nn0red 11352	. . . . . . . . . . . . 13 |- ( ph -> (deg ` R ) e. RR )
27	23, 26	ifcld 4131	. . . . . . . . . . . 12 |- ( ph -> if ( (deg ` R ) <_ (deg ` T ) , (deg ` T ) , (deg ` R ) ) e. RR )
28		dgrcl 23989	. . . . . . . . . . . . . 14 |- ( G e. (Poly ` S ) -> (deg ` G ) e. NN0 )
29	8, 28	syl 17	. . . . . . . . . . . . 13 |- ( ph -> (deg ` G ) e. NN0 )
30	29	nn0red 11352	. . . . . . . . . . . 12 |- ( ph -> (deg ` G ) e. RR )
31		eqid 2622	. . . . . . . . . . . . . 14 |- (deg ` R ) = (deg ` R )
32		eqid 2622	. . . . . . . . . . . . . 14 |- (deg ` T ) = (deg ` T )
33	31, 32	dgrsub 24028	. . . . . . . . . . . . 13 |- ( ( R e. (Poly ` S ) /\ T e. (Poly ` S ) ) -> (deg ` ( R oF - T ) ) <_ if ( (deg ` R ) <_ (deg ` T ) , (deg ` T ) , (deg ` R ) ) )
34	12, 16, 33	syl2anc 693	. . . . . . . . . . . 12 |- ( ph -> (deg ` ( R oF - T ) ) <_ if ( (deg ` R ) <_ (deg ` T ) , (deg ` T ) , (deg ` R ) ) )
35		plydiveu.pd	. . . . . . . . . . . . . . 15 |- ( ph -> ( T = 0p \/ (deg ` T ) < (deg ` G ) ) )
36		eqid 2622	. . . . . . . . . . . . . . . . 17 |- (coeff ` T ) = (coeff ` T )
37	32, 36	dgrlt 24022	. . . . . . . . . . . . . . . 16 |- ( ( T e. (Poly ` S ) /\ (deg ` G ) e. NN0 ) -> ( ( T = 0p \/ (deg ` T ) < (deg ` G ) ) <-> ( (deg ` T ) <_ (deg ` G ) /\ ( (coeff ` T ) ` (deg ` G ) ) = 0 ) ) )
38	16, 29, 37	syl2anc 693	. . . . . . . . . . . . . . 15 |- ( ph -> ( ( T = 0p \/ (deg ` T ) < (deg ` G ) ) <-> ( (deg ` T ) <_ (deg ` G ) /\ ( (coeff ` T ) ` (deg ` G ) ) = 0 ) ) )
39	35, 38	mpbid 222	. . . . . . . . . . . . . 14 |- ( ph -> ( (deg ` T ) <_ (deg ` G ) /\ ( (coeff ` T ) ` (deg ` G ) ) = 0 ) )
40	39	simpld 475	. . . . . . . . . . . . 13 |- ( ph -> (deg ` T ) <_ (deg ` G ) )
41		plydiveu.qd	. . . . . . . . . . . . . . 15 |- ( ph -> ( R = 0p \/ (deg ` R ) < (deg ` G ) ) )
42		eqid 2622	. . . . . . . . . . . . . . . . 17 |- (coeff ` R ) = (coeff ` R )
43	31, 42	dgrlt 24022	. . . . . . . . . . . . . . . 16 |- ( ( R e. (Poly ` S ) /\ (deg ` G ) e. NN0 ) -> ( ( R = 0p \/ (deg ` R ) < (deg ` G ) ) <-> ( (deg ` R ) <_ (deg ` G ) /\ ( (coeff ` R ) ` (deg ` G ) ) = 0 ) ) )
44	12, 29, 43	syl2anc 693	. . . . . . . . . . . . . . 15 |- ( ph -> ( ( R = 0p \/ (deg ` R ) < (deg ` G ) ) <-> ( (deg ` R ) <_ (deg ` G ) /\ ( (coeff ` R ) ` (deg ` G ) ) = 0 ) ) )
45	41, 44	mpbid 222	. . . . . . . . . . . . . 14 |- ( ph -> ( (deg ` R ) <_ (deg ` G ) /\ ( (coeff ` R ) ` (deg ` G ) ) = 0 ) )
46	45	simpld 475	. . . . . . . . . . . . 13 |- ( ph -> (deg ` R ) <_ (deg ` G ) )
47		breq1 4656	. . . . . . . . . . . . . 14 |- ( (deg ` T ) = if ( (deg ` R ) <_ (deg ` T ) , (deg ` T ) , (deg ` R ) ) -> ( (deg ` T ) <_ (deg ` G ) <-> if ( (deg ` R ) <_ (deg ` T ) , (deg ` T ) , (deg ` R ) ) <_ (deg ` G ) ) )
48		breq1 4656	. . . . . . . . . . . . . 14 |- ( (deg ` R ) = if ( (deg ` R ) <_ (deg ` T ) , (deg ` T ) , (deg ` R ) ) -> ( (deg ` R ) <_ (deg ` G ) <-> if ( (deg ` R ) <_ (deg ` T ) , (deg ` T ) , (deg ` R ) ) <_ (deg ` G ) ) )
49	47, 48	ifboth 4124	. . . . . . . . . . . . 13 |- ( ( (deg ` T ) <_ (deg ` G ) /\ (deg ` R ) <_ (deg ` G ) ) -> if ( (deg ` R ) <_ (deg ` T ) , (deg ` T ) , (deg ` R ) ) <_ (deg ` G ) )
50	40, 46, 49	syl2anc 693	. . . . . . . . . . . 12 |- ( ph -> if ( (deg ` R ) <_ (deg ` T ) , (deg ` T ) , (deg ` R ) ) <_ (deg ` G ) )
51	20, 27, 30, 34, 50	letrd 10194	. . . . . . . . . . 11 |- ( ph -> (deg ` ( R oF - T ) ) <_ (deg ` G ) )
52	51	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ ( p oF - q ) =/= 0p ) -> (deg ` ( R oF - T ) ) <_ (deg ` G ) )
53	13, 2, 3, 4, 6	plysub 23975	. . . . . . . . . . . . . 14 |- ( ph -> ( p oF - q ) e. (Poly ` S ) )
54		dgrcl 23989	. . . . . . . . . . . . . 14 |- ( ( p oF - q ) e. (Poly ` S ) -> (deg ` ( p oF - q ) ) e. NN0 )
55	53, 54	syl 17	. . . . . . . . . . . . 13 |- ( ph -> (deg ` ( p oF - q ) ) e. NN0 )
56		nn0addge1 11339	. . . . . . . . . . . . 13 |- ( ( (deg ` G ) e. RR /\ (deg ` ( p oF - q ) ) e. NN0 ) -> (deg ` G ) <_ ( (deg ` G ) + (deg ` ( p oF - q ) ) ) )
57	30, 55, 56	syl2anc 693	. . . . . . . . . . . 12 |- ( ph -> (deg ` G ) <_ ( (deg ` G ) + (deg ` ( p oF - q ) ) ) )
58	57	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ ( p oF - q ) =/= 0p ) -> (deg ` G ) <_ ( (deg ` G ) + (deg ` ( p oF - q ) ) ) )
59		plyf 23954	. . . . . . . . . . . . . . . . . . . 20 |- ( F e. (Poly ` S ) -> F : CC --> CC )
60	7, 59	syl 17	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> F : CC --> CC )
61	60	ffvelrnda 6359	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ z e. CC ) -> ( F ` z ) e. CC )
62	8, 2, 3, 4	plymul 23974	. . . . . . . . . . . . . . . . . . . 20 |- ( ph -> ( G oF x. q ) e. (Poly ` S ) )
63		plyf 23954	. . . . . . . . . . . . . . . . . . . 20 |- ( ( G oF x. q ) e. (Poly ` S ) -> ( G oF x. q ) : CC --> CC )
64	62, 63	syl 17	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> ( G oF x. q ) : CC --> CC )
65	64	ffvelrnda 6359	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ z e. CC ) -> ( ( G oF x. q ) ` z ) e. CC )
66	8, 13, 3, 4	plymul 23974	. . . . . . . . . . . . . . . . . . . 20 |- ( ph -> ( G oF x. p ) e. (Poly ` S ) )
67		plyf 23954	. . . . . . . . . . . . . . . . . . . 20 |- ( ( G oF x. p ) e. (Poly ` S ) -> ( G oF x. p ) : CC --> CC )
68	66, 67	syl 17	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> ( G oF x. p ) : CC --> CC )
69	68	ffvelrnda 6359	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ z e. CC ) -> ( ( G oF x. p ) ` z ) e. CC )
70	61, 65, 69	nnncan1d 10426	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ z e. CC ) -> ( ( ( F ` z ) - ( ( G oF x. q ) ` z ) ) - ( ( F ` z ) - ( ( G oF x. p ) ` z ) ) ) = ( ( ( G oF x. p ) ` z ) - ( ( G oF x. q ) ` z ) ) )
71	70	mpteq2dva 4744	. . . . . . . . . . . . . . . 16 |- ( ph -> ( z e. CC |-> ( ( ( F ` z ) - ( ( G oF x. q ) ` z ) ) - ( ( F ` z ) - ( ( G oF x. p ) ` z ) ) ) ) = ( z e. CC |-> ( ( ( G oF x. p ) ` z ) - ( ( G oF x. q ) ` z ) ) ) )
72		cnex 10017	. . . . . . . . . . . . . . . . . 18 |- CC e. _V
73	72	a1i 11	. . . . . . . . . . . . . . . . 17 |- ( ph -> CC e. _V )
74	61, 65	subcld 10392	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ z e. CC ) -> ( ( F ` z ) - ( ( G oF x. q ) ` z ) ) e. CC )
75	61, 69	subcld 10392	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ z e. CC ) -> ( ( F ` z ) - ( ( G oF x. p ) ` z ) ) e. CC )
76	60	feqmptd 6249	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> F = ( z e. CC |-> ( F ` z ) ) )
77	64	feqmptd 6249	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> ( G oF x. q ) = ( z e. CC |-> ( ( G oF x. q ) ` z ) ) )
78	73, 61, 65, 76, 77	offval2 6914	. . . . . . . . . . . . . . . . . 18 |- ( ph -> ( F oF - ( G oF x. q ) ) = ( z e. CC |-> ( ( F ` z ) - ( ( G oF x. q ) ` z ) ) ) )
79	10, 78	syl5eq 2668	. . . . . . . . . . . . . . . . 17 |- ( ph -> R = ( z e. CC |-> ( ( F ` z ) - ( ( G oF x. q ) ` z ) ) ) )
80	68	feqmptd 6249	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> ( G oF x. p ) = ( z e. CC |-> ( ( G oF x. p ) ` z ) ) )
81	73, 61, 69, 76, 80	offval2 6914	. . . . . . . . . . . . . . . . . 18 |- ( ph -> ( F oF - ( G oF x. p ) ) = ( z e. CC |-> ( ( F ` z ) - ( ( G oF x. p ) ` z ) ) ) )
82	14, 81	syl5eq 2668	. . . . . . . . . . . . . . . . 17 |- ( ph -> T = ( z e. CC |-> ( ( F ` z ) - ( ( G oF x. p ) ` z ) ) ) )
83	73, 74, 75, 79, 82	offval2 6914	. . . . . . . . . . . . . . . 16 |- ( ph -> ( R oF - T ) = ( z e. CC |-> ( ( ( F ` z ) - ( ( G oF x. q ) ` z ) ) - ( ( F ` z ) - ( ( G oF x. p ) ` z ) ) ) ) )
84	73, 69, 65, 80, 77	offval2 6914	. . . . . . . . . . . . . . . 16 |- ( ph -> ( ( G oF x. p ) oF - ( G oF x. q ) ) = ( z e. CC |-> ( ( ( G oF x. p ) ` z ) - ( ( G oF x. q ) ` z ) ) ) )
85	71, 83, 84	3eqtr4d 2666	. . . . . . . . . . . . . . 15 |- ( ph -> ( R oF - T ) = ( ( G oF x. p ) oF - ( G oF x. q ) ) )
86		plyf 23954	. . . . . . . . . . . . . . . . 17 |- ( G e. (Poly ` S ) -> G : CC --> CC )
87	8, 86	syl 17	. . . . . . . . . . . . . . . 16 |- ( ph -> G : CC --> CC )
88		plyf 23954	. . . . . . . . . . . . . . . . 17 |- ( p e. (Poly ` S ) -> p : CC --> CC )
89	13, 88	syl 17	. . . . . . . . . . . . . . . 16 |- ( ph -> p : CC --> CC )
90		plyf 23954	. . . . . . . . . . . . . . . . 17 |- ( q e. (Poly ` S ) -> q : CC --> CC )
91	2, 90	syl 17	. . . . . . . . . . . . . . . 16 |- ( ph -> q : CC --> CC )
92		subdi 10463	. . . . . . . . . . . . . . . . 17 |- ( ( x e. CC /\ y e. CC /\ z e. CC ) -> ( x x. ( y - z ) ) = ( ( x x. y ) - ( x x. z ) ) )
93	92	adantl 482	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ ( x e. CC /\ y e. CC /\ z e. CC ) ) -> ( x x. ( y - z ) ) = ( ( x x. y ) - ( x x. z ) ) )
94	73, 87, 89, 91, 93	caofdi 6933	. . . . . . . . . . . . . . 15 |- ( ph -> ( G oF x. ( p oF - q ) ) = ( ( G oF x. p ) oF - ( G oF x. q ) ) )
95	85, 94	eqtr4d 2659	. . . . . . . . . . . . . 14 |- ( ph -> ( R oF - T ) = ( G oF x. ( p oF - q ) ) )
96	95	fveq2d 6195	. . . . . . . . . . . . 13 |- ( ph -> (deg ` ( R oF - T ) ) = (deg ` ( G oF x. ( p oF - q ) ) ) )
97	96	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ ( p oF - q ) =/= 0p ) -> (deg ` ( R oF - T ) ) = (deg ` ( G oF x. ( p oF - q ) ) ) )
98	8	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ ( p oF - q ) =/= 0p ) -> G e. (Poly ` S ) )
99	9	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ ( p oF - q ) =/= 0p ) -> G =/= 0p )
100	53	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ ( p oF - q ) =/= 0p ) -> ( p oF - q ) e. (Poly ` S ) )
101		simpr 477	. . . . . . . . . . . . 13 |- ( ( ph /\ ( p oF - q ) =/= 0p ) -> ( p oF - q ) =/= 0p )
102		eqid 2622	. . . . . . . . . . . . . 14 |- (deg ` G ) = (deg ` G )
103		eqid 2622	. . . . . . . . . . . . . 14 |- (deg ` ( p oF - q ) ) = (deg ` ( p oF - q ) )
104	102, 103	dgrmul 24026	. . . . . . . . . . . . 13 |- ( ( ( G e. (Poly ` S ) /\ G =/= 0p ) /\ ( ( p oF - q ) e. (Poly ` S ) /\ ( p oF - q ) =/= 0p ) ) -> (deg ` ( G oF x. ( p oF - q ) ) ) = ( (deg ` G ) + (deg ` ( p oF - q ) ) ) )
105	98, 99, 100, 101, 104	syl22anc 1327	. . . . . . . . . . . 12 |- ( ( ph /\ ( p oF - q ) =/= 0p ) -> (deg ` ( G oF x. ( p oF - q ) ) ) = ( (deg ` G ) + (deg ` ( p oF - q ) ) ) )
106	97, 105	eqtrd 2656	. . . . . . . . . . 11 |- ( ( ph /\ ( p oF - q ) =/= 0p ) -> (deg ` ( R oF - T ) ) = ( (deg ` G ) + (deg ` ( p oF - q ) ) ) )
107	58, 106	breqtrrd 4681	. . . . . . . . . 10 |- ( ( ph /\ ( p oF - q ) =/= 0p ) -> (deg ` G ) <_ (deg ` ( R oF - T ) ) )
108	20, 30	letri3d 10179	. . . . . . . . . . 11 |- ( ph -> ( (deg ` ( R oF - T ) ) = (deg ` G ) <-> ( (deg ` ( R oF - T ) ) <_ (deg ` G ) /\ (deg ` G ) <_ (deg ` ( R oF - T ) ) ) ) )
109	108	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ ( p oF - q ) =/= 0p ) -> ( (deg ` ( R oF - T ) ) = (deg ` G ) <-> ( (deg ` ( R oF - T ) ) <_ (deg ` G ) /\ (deg ` G ) <_ (deg ` ( R oF - T ) ) ) ) )
110	52, 107, 109	mpbir2and 957	. . . . . . . . 9 |- ( ( ph /\ ( p oF - q ) =/= 0p ) -> (deg ` ( R oF - T ) ) = (deg ` G ) )
111	110	fveq2d 6195	. . . . . . . 8 |- ( ( ph /\ ( p oF - q ) =/= 0p ) -> ( (coeff ` ( R oF - T ) ) ` (deg ` ( R oF - T ) ) ) = ( (coeff ` ( R oF - T ) ) ` (deg ` G ) ) )
112	42, 36	coesub 24013	. . . . . . . . . . . . 13 |- ( ( R e. (Poly ` S ) /\ T e. (Poly ` S ) ) -> (coeff ` ( R oF - T ) ) = ( (coeff ` R ) oF - (coeff ` T ) ) )
113	12, 16, 112	syl2anc 693	. . . . . . . . . . . 12 |- ( ph -> (coeff ` ( R oF - T ) ) = ( (coeff ` R ) oF - (coeff ` T ) ) )
114	113	fveq1d 6193	. . . . . . . . . . 11 |- ( ph -> ( (coeff ` ( R oF - T ) ) ` (deg ` G ) ) = ( ( (coeff ` R ) oF - (coeff ` T ) ) ` (deg ` G ) ) )
115	42	coef3 23988	. . . . . . . . . . . . . 14 |- ( R e. (Poly ` S ) -> (coeff ` R ) : NN0 --> CC )
116		ffn 6045	. . . . . . . . . . . . . 14 |- ( (coeff ` R ) : NN0 --> CC -> (coeff ` R ) Fn NN0 )
117	12, 115, 116	3syl 18	. . . . . . . . . . . . 13 |- ( ph -> (coeff ` R ) Fn NN0 )
118	36	coef3 23988	. . . . . . . . . . . . . 14 |- ( T e. (Poly ` S ) -> (coeff ` T ) : NN0 --> CC )
119		ffn 6045	. . . . . . . . . . . . . 14 |- ( (coeff ` T ) : NN0 --> CC -> (coeff ` T ) Fn NN0 )
120	16, 118, 119	3syl 18	. . . . . . . . . . . . 13 |- ( ph -> (coeff ` T ) Fn NN0 )
121		nn0ex 11298	. . . . . . . . . . . . . 14 |- NN0 e. _V
122	121	a1i 11	. . . . . . . . . . . . 13 |- ( ph -> NN0 e. _V )
123		inidm 3822	. . . . . . . . . . . . 13 |- ( NN0 i^i NN0 ) = NN0
124	45	simprd 479	. . . . . . . . . . . . . 14 |- ( ph -> ( (coeff ` R ) ` (deg ` G ) ) = 0 )
125	124	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ (deg ` G ) e. NN0 ) -> ( (coeff ` R ) ` (deg ` G ) ) = 0 )
126	39	simprd 479	. . . . . . . . . . . . . 14 |- ( ph -> ( (coeff ` T ) ` (deg ` G ) ) = 0 )
127	126	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ (deg ` G ) e. NN0 ) -> ( (coeff ` T ) ` (deg ` G ) ) = 0 )
128	117, 120, 122, 122, 123, 125, 127	ofval 6906	. . . . . . . . . . . 12 |- ( ( ph /\ (deg ` G ) e. NN0 ) -> ( ( (coeff ` R ) oF - (coeff ` T ) ) ` (deg ` G ) ) = ( 0 - 0 ) )
129	29, 128	mpdan 702	. . . . . . . . . . 11 |- ( ph -> ( ( (coeff ` R ) oF - (coeff ` T ) ) ` (deg ` G ) ) = ( 0 - 0 ) )
130	114, 129	eqtrd 2656	. . . . . . . . . 10 |- ( ph -> ( (coeff ` ( R oF - T ) ) ` (deg ` G ) ) = ( 0 - 0 ) )
131		0m0e0 11130	. . . . . . . . . 10 |- ( 0 - 0 ) = 0
132	130, 131	syl6eq 2672	. . . . . . . . 9 |- ( ph -> ( (coeff ` ( R oF - T ) ) ` (deg ` G ) ) = 0 )
133	132	adantr 481	. . . . . . . 8 |- ( ( ph /\ ( p oF - q ) =/= 0p ) -> ( (coeff ` ( R oF - T ) ) ` (deg ` G ) ) = 0 )
134	111, 133	eqtrd 2656	. . . . . . 7 |- ( ( ph /\ ( p oF - q ) =/= 0p ) -> ( (coeff ` ( R oF - T ) ) ` (deg ` ( R oF - T ) ) ) = 0 )
135		eqid 2622	. . . . . . . . . 10 |- (deg ` ( R oF - T ) ) = (deg ` ( R oF - T ) )
136		eqid 2622	. . . . . . . . . 10 |- (coeff ` ( R oF - T ) ) = (coeff ` ( R oF - T ) )
137	135, 136	dgreq0 24021	. . . . . . . . 9 |- ( ( R oF - T ) e. (Poly ` S ) -> ( ( R oF - T ) = 0p <-> ( (coeff ` ( R oF - T ) ) ` (deg ` ( R oF - T ) ) ) = 0 ) )
138	17, 137	syl 17	. . . . . . . 8 |- ( ph -> ( ( R oF - T ) = 0p <-> ( (coeff ` ( R oF - T ) ) ` (deg ` ( R oF - T ) ) ) = 0 ) )
139	138	biimpar 502	. . . . . . 7 |- ( ( ph /\ ( (coeff ` ( R oF - T ) ) ` (deg ` ( R oF - T ) ) ) = 0 ) -> ( R oF - T ) = 0p )
140	134, 139	syldan 487	. . . . . 6 |- ( ( ph /\ ( p oF - q ) =/= 0p ) -> ( R oF - T ) = 0p )
141	140	ex 450	. . . . 5 |- ( ph -> ( ( p oF - q ) =/= 0p -> ( R oF - T ) = 0p ) )
142		plymul0or 24036	. . . . . . 7 |- ( ( G e. (Poly ` S ) /\ ( p oF - q ) e. (Poly ` S ) ) -> ( ( G oF x. ( p oF - q ) ) = 0p <-> ( G = 0p \/ ( p oF - q ) = 0p ) ) )
143	8, 53, 142	syl2anc 693	. . . . . 6 |- ( ph -> ( ( G oF x. ( p oF - q ) ) = 0p <-> ( G = 0p \/ ( p oF - q ) = 0p ) ) )
144	95	eqeq1d 2624	. . . . . 6 |- ( ph -> ( ( R oF - T ) = 0p <-> ( G oF x. ( p oF - q ) ) = 0p ) )
145	9	neneqd 2799	. . . . . . 7 |- ( ph -> -. G = 0p )
146		biorf 420	. . . . . . 7 |- ( -. G = 0p -> ( ( p oF - q ) = 0p <-> ( G = 0p \/ ( p oF - q ) = 0p ) ) )
147	145, 146	syl 17	. . . . . 6 |- ( ph -> ( ( p oF - q ) = 0p <-> ( G = 0p \/ ( p oF - q ) = 0p ) ) )
148	143, 144, 147	3bitr4d 300	. . . . 5 |- ( ph -> ( ( R oF - T ) = 0p <-> ( p oF - q ) = 0p ) )
149	141, 148	sylibd 229	. . . 4 |- ( ph -> ( ( p oF - q ) =/= 0p -> ( p oF - q ) = 0p ) )
150	1, 149	pm2.61dne 2880	. . 3 |- ( ph -> ( p oF - q ) = 0p )
151		df-0p 23437	. . 3 |- 0p = ( CC X. { 0 } )
152	150, 151	syl6eq 2672	. 2 |- ( ph -> ( p oF - q ) = ( CC X. { 0 } ) )
153		ofsubeq0 11017	. . 3 |- ( ( CC e. _V /\ p : CC --> CC /\ q : CC --> CC ) -> ( ( p oF - q ) = ( CC X. { 0 } ) <-> p = q ) )
154	73, 89, 91, 153	syl3anc 1326	. 2 |- ( ph -> ( ( p oF - q ) = ( CC X. { 0 } ) <-> p = q ) )
155	152, 154	mpbid 222	1 |- ( ph -> p = q )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   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   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   < clt 10074   <_ cle 10075   - cmin 10266   -ucneg 10267   / cdiv 10684   NN0cn0 11292   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:  plydivalg  24054