Theorem mpaaeu 30704

Description: An algebraic number has exactly one monic polynomial of the least degree. (Contributed by Stefan O'Rear, 25-Nov-2014.)

Assertion

Ref	Expression
mpaaeu	|- ( A e. AA -> E! p e. (Poly ` QQ ) ( (deg ` p ) = (degAA ` A ) /\ ( p ` A ) = 0 /\ ( (coeff ` p ) ` (degAA ` A ) ) = 1 ) )

Distinct variable group:   A, p

Proof of Theorem mpaaeu

Dummy variables a b c are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dgraalem 30699	. . . 4 |- ( A e. AA -> ( (degAA ` A ) e. NN /\ E. a e. ( (Poly ` QQ ) \ { 0p } ) ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) )
2	1	simprd 463	. . 3 |- ( A e. AA -> E. a e. ( (Poly ` QQ ) \ { 0p } ) ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) )
3		qsscn 11189	. . . . . . . 8 |- QQ C_ CC
4		eldifi 3626	. . . . . . . . . . . 12 |- ( a e. ( (Poly ` QQ ) \ { 0p } ) -> a e. (Poly ` QQ ) )
5	4	ad2antlr 726	. . . . . . . . . . 11 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> a e. (Poly ` QQ ) )
6		zssq 11185	. . . . . . . . . . . 12 |- ZZ C_ QQ
7		0z 10871	. . . . . . . . . . . 12 |- 0 e. ZZ
8	6, 7	sselii 3501	. . . . . . . . . . 11 |- 0 e. QQ
9		eqid 2467	. . . . . . . . . . . 12 |- (coeff ` a ) = (coeff ` a )
10	9	coef2 22360	. . . . . . . . . . 11 |- ( ( a e. (Poly ` QQ ) /\ 0 e. QQ ) -> (coeff ` a ) : NN0 --> QQ )
11	5, 8, 10	sylancl 662	. . . . . . . . . 10 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> (coeff ` a ) : NN0 --> QQ )
12		dgrcl 22362	. . . . . . . . . . 11 |- ( a e. (Poly ` QQ ) -> (deg ` a ) e. NN0 )
13	5, 12	syl 16	. . . . . . . . . 10 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> (deg ` a ) e. NN0 )
14	11, 13	ffvelrnd 6020	. . . . . . . . 9 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> ( (coeff ` a ) ` (deg ` a ) ) e. QQ )
15		eldifsni 4153	. . . . . . . . . . 11 |- ( a e. ( (Poly ` QQ ) \ { 0p } ) -> a =/= 0p )
16	15	ad2antlr 726	. . . . . . . . . 10 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> a =/= 0p )
17		eqid 2467	. . . . . . . . . . . . 13 |- (deg ` a ) = (deg ` a )
18	17, 9	dgreq0 22393	. . . . . . . . . . . 12 |- ( a e. (Poly ` QQ ) -> ( a = 0p <-> ( (coeff ` a ) ` (deg ` a ) ) = 0 ) )
19	18	necon3bid 2725	. . . . . . . . . . 11 |- ( a e. (Poly ` QQ ) -> ( a =/= 0p <-> ( (coeff ` a ) ` (deg ` a ) ) =/= 0 ) )
20	5, 19	syl 16	. . . . . . . . . 10 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> ( a =/= 0p <-> ( (coeff ` a ) ` (deg ` a ) ) =/= 0 ) )
21	16, 20	mpbid 210	. . . . . . . . 9 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> ( (coeff ` a ) ` (deg ` a ) ) =/= 0 )
22		qreccl 11198	. . . . . . . . 9 |- ( ( ( (coeff ` a ) ` (deg ` a ) ) e. QQ /\ ( (coeff ` a ) ` (deg ` a ) ) =/= 0 ) -> ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) e. QQ )
23	14, 21, 22	syl2anc 661	. . . . . . . 8 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) e. QQ )
24		plyconst 22335	. . . . . . . 8 |- ( ( QQ C_ CC /\ ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) e. QQ ) -> ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) e. (Poly ` QQ ) )
25	3, 23, 24	sylancr 663	. . . . . . 7 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) e. (Poly ` QQ ) )
26		simpl 457	. . . . . . . 8 |- ( ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) e. (Poly ` QQ ) /\ a e. (Poly ` QQ ) ) -> ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) e. (Poly ` QQ ) )
27		simpr 461	. . . . . . . 8 |- ( ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) e. (Poly ` QQ ) /\ a e. (Poly ` QQ ) ) -> a e. (Poly ` QQ ) )
28		qaddcl 11194	. . . . . . . . 9 |- ( ( b e. QQ /\ c e. QQ ) -> ( b + c ) e. QQ )
29	28	adantl 466	. . . . . . . 8 |- ( ( ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) e. (Poly ` QQ ) /\ a e. (Poly ` QQ ) ) /\ ( b e. QQ /\ c e. QQ ) ) -> ( b + c ) e. QQ )
30		qmulcl 11196	. . . . . . . . 9 |- ( ( b e. QQ /\ c e. QQ ) -> ( b x. c ) e. QQ )
31	30	adantl 466	. . . . . . . 8 |- ( ( ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) e. (Poly ` QQ ) /\ a e. (Poly ` QQ ) ) /\ ( b e. QQ /\ c e. QQ ) ) -> ( b x. c ) e. QQ )
32	26, 27, 29, 31	plymul 22347	. . . . . . 7 |- ( ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) e. (Poly ` QQ ) /\ a e. (Poly ` QQ ) ) -> ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) oF x. a ) e. (Poly ` QQ ) )
33	25, 5, 32	syl2anc 661	. . . . . 6 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) oF x. a ) e. (Poly ` QQ ) )
34	9	coef3 22361	. . . . . . . . . . 11 |- ( a e. (Poly ` QQ ) -> (coeff ` a ) : NN0 --> CC )
35	5, 34	syl 16	. . . . . . . . . 10 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> (coeff ` a ) : NN0 --> CC )
36	35, 13	ffvelrnd 6020	. . . . . . . . 9 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> ( (coeff ` a ) ` (deg ` a ) ) e. CC )
37	36, 21	reccld 10309	. . . . . . . 8 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) e. CC )
38	36, 21	recne0d 10310	. . . . . . . 8 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) =/= 0 )
39		dgrmulc 22399	. . . . . . . 8 |- ( ( ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) e. CC /\ ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) =/= 0 /\ a e. (Poly ` QQ ) ) -> (deg ` ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) oF x. a ) ) = (deg ` a ) )
40	37, 38, 5, 39	syl3anc 1228	. . . . . . 7 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> (deg ` ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) oF x. a ) ) = (deg ` a ) )
41		simprl 755	. . . . . . 7 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> (deg ` a ) = (degAA ` A ) )
42	40, 41	eqtrd 2508	. . . . . 6 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> (deg ` ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) oF x. a ) ) = (degAA ` A ) )
43		aacn 22444	. . . . . . . . 9 |- ( A e. AA -> A e. CC )
44	43	ad2antrr 725	. . . . . . . 8 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> A e. CC )
45		ovex 6307	. . . . . . . . . 10 |- ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) e. _V
46		fnconstg 5771	. . . . . . . . . 10 |- ( ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) e. _V -> ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) Fn CC )
47	45, 46	mp1i 12	. . . . . . . . 9 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) Fn CC )
48		plyf 22327	. . . . . . . . . 10 |- ( a e. (Poly ` QQ ) -> a : CC --> CC )
49		ffn 5729	. . . . . . . . . 10 |- ( a : CC --> CC -> a Fn CC )
50	5, 48, 49	3syl 20	. . . . . . . . 9 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> a Fn CC )
51		cnex 9569	. . . . . . . . . 10 |- CC e. _V
52	51	a1i 11	. . . . . . . . 9 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> CC e. _V )
53		inidm 3707	. . . . . . . . 9 |- ( CC i^i CC ) = CC
54	45	fvconst2 6114	. . . . . . . . . 10 |- ( A e. CC -> ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) ` A ) = ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) )
55	54	adantl 466	. . . . . . . . 9 |- ( ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) /\ A e. CC ) -> ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) ` A ) = ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) )
56		simplrr 760	. . . . . . . . 9 |- ( ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) /\ A e. CC ) -> ( a ` A ) = 0 )
57	47, 50, 52, 52, 53, 55, 56	ofval 6531	. . . . . . . 8 |- ( ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) /\ A e. CC ) -> ( ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) oF x. a ) ` A ) = ( ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) x. 0 ) )
58	44, 57	mpdan 668	. . . . . . 7 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> ( ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) oF x. a ) ` A ) = ( ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) x. 0 ) )
59	37	mul01d 9774	. . . . . . 7 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> ( ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) x. 0 ) = 0 )
60	58, 59	eqtrd 2508	. . . . . 6 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> ( ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) oF x. a ) ` A ) = 0 )
61		coemulc 22383	. . . . . . . . 9 |- ( ( ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) e. CC /\ a e. (Poly ` QQ ) ) -> (coeff ` ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) oF x. a ) ) = ( ( NN0 X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) oF x. (coeff ` a ) ) )
62	37, 5, 61	syl2anc 661	. . . . . . . 8 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> (coeff ` ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) oF x. a ) ) = ( ( NN0 X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) oF x. (coeff ` a ) ) )
63	62	fveq1d 5866	. . . . . . 7 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> ( (coeff ` ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) oF x. a ) ) ` (degAA ` A ) ) = ( ( ( NN0 X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) oF x. (coeff ` a ) ) ` (degAA ` A ) ) )
64		dgraacl 30700	. . . . . . . . . 10 |- ( A e. AA -> (degAA ` A ) e. NN )
65	64	ad2antrr 725	. . . . . . . . 9 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> (degAA ` A ) e. NN )
66	65	nnnn0d 10848	. . . . . . . 8 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> (degAA ` A ) e. NN0 )
67		fnconstg 5771	. . . . . . . . . 10 |- ( ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) e. _V -> ( NN0 X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) Fn NN0 )
68	45, 67	mp1i 12	. . . . . . . . 9 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> ( NN0 X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) Fn NN0 )
69		ffn 5729	. . . . . . . . . 10 |- ( (coeff ` a ) : NN0 --> CC -> (coeff ` a ) Fn NN0 )
70	35, 69	syl 16	. . . . . . . . 9 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> (coeff ` a ) Fn NN0 )
71		nn0ex 10797	. . . . . . . . . 10 |- NN0 e. _V
72	71	a1i 11	. . . . . . . . 9 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> NN0 e. _V )
73		inidm 3707	. . . . . . . . 9 |- ( NN0 i^i NN0 ) = NN0
74	45	fvconst2 6114	. . . . . . . . . 10 |- ( (degAA ` A ) e. NN0 -> ( ( NN0 X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) ` (degAA ` A ) ) = ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) )
75	74	adantl 466	. . . . . . . . 9 |- ( ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) /\ (degAA ` A ) e. NN0 ) -> ( ( NN0 X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) ` (degAA ` A ) ) = ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) )
76		simplrl 759	. . . . . . . . . . 11 |- ( ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) /\ (degAA ` A ) e. NN0 ) -> (deg ` a ) = (degAA ` A ) )
77	76	eqcomd 2475	. . . . . . . . . 10 |- ( ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) /\ (degAA ` A ) e. NN0 ) -> (degAA ` A ) = (deg ` a ) )
78	77	fveq2d 5868	. . . . . . . . 9 |- ( ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) /\ (degAA ` A ) e. NN0 ) -> ( (coeff ` a ) ` (degAA ` A ) ) = ( (coeff ` a ) ` (deg ` a ) ) )
79	68, 70, 72, 72, 73, 75, 78	ofval 6531	. . . . . . . 8 |- ( ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) /\ (degAA ` A ) e. NN0 ) -> ( ( ( NN0 X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) oF x. (coeff ` a ) ) ` (degAA ` A ) ) = ( ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) x. ( (coeff ` a ) ` (deg ` a ) ) ) )
80	66, 79	mpdan 668	. . . . . . 7 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> ( ( ( NN0 X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) oF x. (coeff ` a ) ) ` (degAA ` A ) ) = ( ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) x. ( (coeff ` a ) ` (deg ` a ) ) ) )
81	36, 21	recid2d 10312	. . . . . . 7 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> ( ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) x. ( (coeff ` a ) ` (deg ` a ) ) ) = 1 )
82	63, 80, 81	3eqtrd 2512	. . . . . 6 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> ( (coeff ` ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) oF x. a ) ) ` (degAA ` A ) ) = 1 )
83		fveq2 5864	. . . . . . . . 9 |- ( p = ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) oF x. a ) -> (deg ` p ) = (deg ` ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) oF x. a ) ) )
84	83	eqeq1d 2469	. . . . . . . 8 |- ( p = ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) oF x. a ) -> ( (deg ` p ) = (degAA ` A ) <-> (deg ` ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) oF x. a ) ) = (degAA ` A ) ) )
85		fveq1 5863	. . . . . . . . 9 |- ( p = ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) oF x. a ) -> ( p ` A ) = ( ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) oF x. a ) ` A ) )
86	85	eqeq1d 2469	. . . . . . . 8 |- ( p = ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) oF x. a ) -> ( ( p ` A ) = 0 <-> ( ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) oF x. a ) ` A ) = 0 ) )
87		fveq2 5864	. . . . . . . . . 10 |- ( p = ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) oF x. a ) -> (coeff ` p ) = (coeff ` ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) oF x. a ) ) )
88	87	fveq1d 5866	. . . . . . . . 9 |- ( p = ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) oF x. a ) -> ( (coeff ` p ) ` (degAA ` A ) ) = ( (coeff ` ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) oF x. a ) ) ` (degAA ` A ) ) )
89	88	eqeq1d 2469	. . . . . . . 8 |- ( p = ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) oF x. a ) -> ( ( (coeff ` p ) ` (degAA ` A ) ) = 1 <-> ( (coeff ` ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) oF x. a ) ) ` (degAA ` A ) ) = 1 ) )
90	84, 86, 89	3anbi123d 1299	. . . . . . 7 |- ( p = ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) oF x. a ) -> ( ( (deg ` p ) = (degAA ` A ) /\ ( p ` A ) = 0 /\ ( (coeff ` p ) ` (degAA ` A ) ) = 1 ) <-> ( (deg ` ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) oF x. a ) ) = (degAA ` A ) /\ ( ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) oF x. a ) ` A ) = 0 /\ ( (coeff ` ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) oF x. a ) ) ` (degAA ` A ) ) = 1 ) ) )
91	90	rspcev 3214	. . . . . 6 |- ( ( ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) oF x. a ) e. (Poly ` QQ ) /\ ( (deg ` ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) oF x. a ) ) = (degAA ` A ) /\ ( ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) oF x. a ) ` A ) = 0 /\ ( (coeff ` ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) oF x. a ) ) ` (degAA ` A ) ) = 1 ) ) -> E. p e. (Poly ` QQ ) ( (deg ` p ) = (degAA ` A ) /\ ( p ` A ) = 0 /\ ( (coeff ` p ) ` (degAA ` A ) ) = 1 ) )
92	33, 42, 60, 82, 91	syl13anc 1230	. . . . 5 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> E. p e. (Poly ` QQ ) ( (deg ` p ) = (degAA ` A ) /\ ( p ` A ) = 0 /\ ( (coeff ` p ) ` (degAA ` A ) ) = 1 ) )
93	92	ex 434	. . . 4 |- ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0p } ) ) -> ( ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) -> E. p e. (Poly ` QQ ) ( (deg ` p ) = (degAA ` A ) /\ ( p ` A ) = 0 /\ ( (coeff ` p ) ` (degAA ` A ) ) = 1 ) ) )
94	93	rexlimdva 2955	. . 3 |- ( A e. AA -> ( E. a e. ( (Poly ` QQ ) \ { 0p } ) ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) -> E. p e. (Poly ` QQ ) ( (deg ` p ) = (degAA ` A ) /\ ( p ` A ) = 0 /\ ( (coeff ` p ) ` (degAA ` A ) ) = 1 ) ) )
95	2, 94	mpd 15	. 2 |- ( A e. AA -> E. p e. (Poly ` QQ ) ( (deg ` p ) = (degAA ` A ) /\ ( p ` A ) = 0 /\ ( (coeff ` p ) ` (degAA ` A ) ) = 1 ) )
96		simp2 997	. . . . . . . . . . 11 |- ( ( (deg ` p ) = (degAA ` A ) /\ ( p ` A ) = 0 /\ ( (coeff ` p ) ` (degAA ` A ) ) = 1 ) -> ( p ` A ) = 0 )
97		simp2 997	. . . . . . . . . . 11 |- ( ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 /\ ( (coeff ` a ) ` (degAA ` A ) ) = 1 ) -> ( a ` A ) = 0 )
98	96, 97	anim12i 566	. . . . . . . . . 10 |- ( ( ( (deg ` p ) = (degAA ` A ) /\ ( p ` A ) = 0 /\ ( (coeff ` p ) ` (degAA ` A ) ) = 1 ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 /\ ( (coeff ` a ) ` (degAA ` A ) ) = 1 ) ) -> ( ( p ` A ) = 0 /\ ( a ` A ) = 0 ) )
99		plyf 22327	. . . . . . . . . . . . . . . 16 |- ( p e. (Poly ` QQ ) -> p : CC --> CC )
100		ffn 5729	. . . . . . . . . . . . . . . 16 |- ( p : CC --> CC -> p Fn CC )
101	99, 100	syl 16	. . . . . . . . . . . . . . 15 |- ( p e. (Poly ` QQ ) -> p Fn CC )
102	101	ad2antrr 725	. . . . . . . . . . . . . 14 |- ( ( ( p e. (Poly ` QQ ) /\ a e. (Poly ` QQ ) ) /\ ( ( p ` A ) = 0 /\ ( a ` A ) = 0 ) ) -> p Fn CC )
103	48, 49	syl 16	. . . . . . . . . . . . . . 15 |- ( a e. (Poly ` QQ ) -> a Fn CC )
104	103	ad2antlr 726	. . . . . . . . . . . . . 14 |- ( ( ( p e. (Poly ` QQ ) /\ a e. (Poly ` QQ ) ) /\ ( ( p ` A ) = 0 /\ ( a ` A ) = 0 ) ) -> a Fn CC )
105	51	a1i 11	. . . . . . . . . . . . . 14 |- ( ( ( p e. (Poly ` QQ ) /\ a e. (Poly ` QQ ) ) /\ ( ( p ` A ) = 0 /\ ( a ` A ) = 0 ) ) -> CC e. _V )
106		simplrl 759	. . . . . . . . . . . . . 14 |- ( ( ( ( p e. (Poly ` QQ ) /\ a e. (Poly ` QQ ) ) /\ ( ( p ` A ) = 0 /\ ( a ` A ) = 0 ) ) /\ A e. CC ) -> ( p ` A ) = 0 )
107		simplrr 760	. . . . . . . . . . . . . 14 |- ( ( ( ( p e. (Poly ` QQ ) /\ a e. (Poly ` QQ ) ) /\ ( ( p ` A ) = 0 /\ ( a ` A ) = 0 ) ) /\ A e. CC ) -> ( a ` A ) = 0 )
108	102, 104, 105, 105, 53, 106, 107	ofval 6531	. . . . . . . . . . . . 13 |- ( ( ( ( p e. (Poly ` QQ ) /\ a e. (Poly ` QQ ) ) /\ ( ( p ` A ) = 0 /\ ( a ` A ) = 0 ) ) /\ A e. CC ) -> ( ( p oF - a ) ` A ) = ( 0 - 0 ) )
109	43, 108	sylan2 474	. . . . . . . . . . . 12 |- ( ( ( ( p e. (Poly ` QQ ) /\ a e. (Poly ` QQ ) ) /\ ( ( p ` A ) = 0 /\ ( a ` A ) = 0 ) ) /\ A e. AA ) -> ( ( p oF - a ) ` A ) = ( 0 - 0 ) )
110		0m0e0 10641	. . . . . . . . . . . 12 |- ( 0 - 0 ) = 0
111	109, 110	syl6eq 2524	. . . . . . . . . . 11 |- ( ( ( ( p e. (Poly ` QQ ) /\ a e. (Poly ` QQ ) ) /\ ( ( p ` A ) = 0 /\ ( a ` A ) = 0 ) ) /\ A e. AA ) -> ( ( p oF - a ) ` A ) = 0 )
112	111	ex 434	. . . . . . . . . 10 |- ( ( ( p e. (Poly ` QQ ) /\ a e. (Poly ` QQ ) ) /\ ( ( p ` A ) = 0 /\ ( a ` A ) = 0 ) ) -> ( A e. AA -> ( ( p oF - a ) ` A ) = 0 ) )
113	98, 112	sylan2 474	. . . . . . . . 9 |- ( ( ( p e. (Poly ` QQ ) /\ a e. (Poly ` QQ ) ) /\ ( ( (deg ` p ) = (degAA ` A ) /\ ( p ` A ) = 0 /\ ( (coeff ` p ) ` (degAA ` A ) ) = 1 ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 /\ ( (coeff ` a ) ` (degAA ` A ) ) = 1 ) ) ) -> ( A e. AA -> ( ( p oF - a ) ` A ) = 0 ) )
114	113	com12 31	. . . . . . . 8 |- ( A e. AA -> ( ( ( p e. (Poly ` QQ ) /\ a e. (Poly ` QQ ) ) /\ ( ( (deg ` p ) = (degAA ` A ) /\ ( p ` A ) = 0 /\ ( (coeff ` p ) ` (degAA ` A ) ) = 1 ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 /\ ( (coeff ` a ) ` (degAA ` A ) ) = 1 ) ) ) -> ( ( p oF - a ) ` A ) = 0 ) )
115	114	impl 620	. . . . . . 7 |- ( ( ( A e. AA /\ ( p e. (Poly ` QQ ) /\ a e. (Poly ` QQ ) ) ) /\ ( ( (deg ` p ) = (degAA ` A ) /\ ( p ` A ) = 0 /\ ( (coeff ` p ) ` (degAA ` A ) ) = 1 ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 /\ ( (coeff ` a ) ` (degAA ` A ) ) = 1 ) ) ) -> ( ( p oF - a ) ` A ) = 0 )
116		simpll 753	. . . . . . . 8 |- ( ( ( A e. AA /\ ( p e. (Poly ` QQ ) /\ a e. (Poly ` QQ ) ) ) /\ ( ( (deg ` p ) = (degAA ` A ) /\ ( p ` A ) = 0 /\ ( (coeff ` p ) ` (degAA ` A ) ) = 1 ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 /\ ( (coeff ` a ) ` (degAA ` A ) ) = 1 ) ) ) -> A e. AA )
117		simpl 457	. . . . . . . . . 10 |- ( ( p e. (Poly ` QQ ) /\ a e. (Poly ` QQ ) ) -> p e. (Poly ` QQ ) )
118		simpr 461	. . . . . . . . . 10 |- ( ( p e. (Poly ` QQ ) /\ a e. (Poly ` QQ ) ) -> a e. (Poly ` QQ ) )
119	28	adantl 466	. . . . . . . . . 10 |- ( ( ( p e. (Poly ` QQ ) /\ a e. (Poly ` QQ ) ) /\ ( b e. QQ /\ c e. QQ ) ) -> ( b + c ) e. QQ )
120	30	adantl 466	. . . . . . . . . 10 |- ( ( ( p e. (Poly ` QQ ) /\ a e. (Poly ` QQ ) ) /\ ( b e. QQ /\ c e. QQ ) ) -> ( b x. c ) e. QQ )
121		1z 10890	. . . . . . . . . . . 12 |- 1 e. ZZ
122		zq 11184	. . . . . . . . . . . 12 |- ( 1 e. ZZ -> 1 e. QQ )
123		qnegcl 11195	. . . . . . . . . . . 12 |- ( 1 e. QQ -> -u 1 e. QQ )
124	121, 122, 123	mp2b 10	. . . . . . . . . . 11 |- -u 1 e. QQ
125	124	a1i 11	. . . . . . . . . 10 |- ( ( p e. (Poly ` QQ ) /\ a e. (Poly ` QQ ) ) -> -u 1 e. QQ )
126	117, 118, 119, 120, 125	plysub 22348	. . . . . . . . 9 |- ( ( p e. (Poly ` QQ ) /\ a e. (Poly ` QQ ) ) -> ( p oF - a ) e. (Poly ` QQ ) )
127	126	ad2antlr 726	. . . . . . . 8 |- ( ( ( A e. AA /\ ( p e. (Poly ` QQ ) /\ a e. (Poly ` QQ ) ) ) /\ ( ( (deg ` p ) = (degAA ` A ) /\ ( p ` A ) = 0 /\ ( (coeff ` p ) ` (degAA ` A ) ) = 1 ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 /\ ( (coeff ` a ) ` (degAA ` A ) ) = 1 ) ) ) -> ( p oF - a ) e. (Poly ` QQ ) )
128		simplrl 759	. . . . . . . . . 10 |- ( ( ( A e. AA /\ ( p e. (Poly ` QQ ) /\ a e. (Poly ` QQ ) ) ) /\ ( ( (deg ` p ) = (degAA ` A ) /\ ( p ` A ) = 0 /\ ( (coeff ` p ) ` (degAA ` A ) ) = 1 ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 /\ ( (coeff ` a ) ` (degAA ` A ) ) = 1 ) ) ) -> p e. (Poly ` QQ ) )
129		simplrr 760	. . . . . . . . . 10 |- ( ( ( A e. AA /\ ( p e. (Poly ` QQ ) /\ a e. (Poly ` QQ ) ) ) /\ ( ( (deg ` p ) = (degAA ` A ) /\ ( p ` A ) = 0 /\ ( (coeff ` p ) ` (degAA ` A ) ) = 1 ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 /\ ( (coeff ` a ) ` (degAA ` A ) ) = 1 ) ) ) -> a e. (Poly ` QQ ) )
130		simprr1 1044	. . . . . . . . . . 11 |- ( ( ( A e. AA /\ ( p e. (Poly ` QQ ) /\ a e. (Poly ` QQ ) ) ) /\ ( ( (deg ` p ) = (degAA ` A ) /\ ( p ` A ) = 0 /\ ( (coeff ` p ) ` (degAA ` A ) ) = 1 ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 /\ ( (coeff ` a ) ` (degAA ` A ) ) = 1 ) ) ) -> (deg ` a ) = (degAA ` A ) )
131		simprl1 1041	. . . . . . . . . . 11 |- ( ( ( A e. AA /\ ( p e. (Poly ` QQ ) /\ a e. (Poly ` QQ ) ) ) /\ ( ( (deg ` p ) = (degAA ` A ) /\ ( p ` A ) = 0 /\ ( (coeff ` p ) ` (degAA ` A ) ) = 1 ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 /\ ( (coeff ` a ) ` (degAA ` A ) ) = 1 ) ) ) -> (deg ` p ) = (degAA ` A ) )
132	130, 131	eqtr4d 2511	. . . . . . . . . 10 |- ( ( ( A e. AA /\ ( p e. (Poly ` QQ ) /\ a e. (Poly ` QQ ) ) ) /\ ( ( (deg ` p ) = (degAA ` A ) /\ ( p ` A ) = 0 /\ ( (coeff ` p ) ` (degAA ` A ) ) = 1 ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 /\ ( (coeff ` a ) ` (degAA ` A ) ) = 1 ) ) ) -> (deg ` a ) = (deg ` p ) )
133	64	ad2antrr 725	. . . . . . . . . . 11 |- ( ( ( A e. AA /\ ( p e. (Poly ` QQ ) /\ a e. (Poly ` QQ ) ) ) /\ ( ( (deg ` p ) = (degAA ` A ) /\ ( p ` A ) = 0 /\ ( (coeff ` p ) ` (degAA ` A ) ) = 1 ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 /\ ( (coeff ` a ) ` (degAA ` A ) ) = 1 ) ) ) -> (degAA ` A ) e. NN )
134	131, 133	eqeltrd 2555	. . . . . . . . . 10 |- ( ( ( A e. AA /\ ( p e. (Poly ` QQ ) /\ a e. (Poly ` QQ ) ) ) /\ ( ( (deg ` p ) = (degAA ` A ) /\ ( p ` A ) = 0 /\ ( (coeff ` p ) ` (degAA ` A ) ) = 1 ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 /\ ( (coeff ` a ) ` (degAA ` A ) ) = 1 ) ) ) -> (deg ` p ) e. NN )
135		simprl3 1043	. . . . . . . . . . 11 |- ( ( ( A e. AA /\ ( p e. (Poly ` QQ ) /\ a e. (Poly ` QQ ) ) ) /\ ( ( (deg ` p ) = (degAA ` A ) /\ ( p ` A ) = 0 /\ ( (coeff ` p ) ` (degAA ` A ) ) = 1 ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 /\ ( (coeff ` a ) ` (degAA ` A ) ) = 1 ) ) ) -> ( (coeff ` p ) ` (degAA ` A ) ) = 1 )
136	131	fveq2d 5868	. . . . . . . . . . 11 |- ( ( ( A e. AA /\ ( p e. (Poly ` QQ ) /\ a e. (Poly ` QQ ) ) ) /\ ( ( (deg ` p ) = (degAA ` A ) /\ ( p ` A ) = 0 /\ ( (coeff ` p ) ` (degAA ` A ) ) = 1 ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 /\ ( (coeff ` a ) ` (degAA ` A ) ) = 1 ) ) ) -> ( (coeff ` p ) ` (deg ` p ) ) = ( (coeff ` p ) ` (degAA ` A ) ) )
137	131	fveq2d 5868	. . . . . . . . . . . 12 |- ( ( ( A e. AA /\ ( p e. (Poly ` QQ ) /\ a e. (Poly ` QQ ) ) ) /\ ( ( (deg ` p ) = (degAA ` A ) /\ ( p ` A ) = 0 /\ ( (coeff ` p ) ` (degAA ` A ) ) = 1 ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 /\ ( (coeff ` a ) ` (degAA ` A ) ) = 1 ) ) ) -> ( (coeff ` a ) ` (deg ` p ) ) = ( (coeff ` a ) ` (degAA ` A ) ) )
138		simprr3 1046	. . . . . . . . . . . 12 |- ( ( ( A e. AA /\ ( p e. (Poly ` QQ ) /\ a e. (Poly ` QQ ) ) ) /\ ( ( (deg ` p ) = (degAA ` A ) /\ ( p ` A ) = 0 /\ ( (coeff ` p ) ` (degAA ` A ) ) = 1 ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 /\ ( (coeff ` a ) ` (degAA ` A ) ) = 1 ) ) ) -> ( (coeff ` a ) ` (degAA ` A ) ) = 1 )
139	137, 138	eqtrd 2508	. . . . . . . . . . 11 |- ( ( ( A e. AA /\ ( p e. (Poly ` QQ ) /\ a e. (Poly ` QQ ) ) ) /\ ( ( (deg ` p ) = (degAA ` A ) /\ ( p ` A ) = 0 /\ ( (coeff ` p ) ` (degAA ` A ) ) = 1 ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 /\ ( (coeff ` a ) ` (degAA ` A ) ) = 1 ) ) ) -> ( (coeff ` a ) ` (deg ` p ) ) = 1 )
140	135, 136, 139	3eqtr4d 2518	. . . . . . . . . 10 |- ( ( ( A e. AA /\ ( p e. (Poly ` QQ ) /\ a e. (Poly ` QQ ) ) ) /\ ( ( (deg ` p ) = (degAA ` A ) /\ ( p ` A ) = 0 /\ ( (coeff ` p ) ` (degAA ` A ) ) = 1 ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 /\ ( (coeff ` a ) ` (degAA ` A ) ) = 1 ) ) ) -> ( (coeff ` p ) ` (deg ` p ) ) = ( (coeff ` a ) ` (deg ` p ) ) )
141		eqid 2467	. . . . . . . . . . 11 |- (deg ` p ) = (deg ` p )
142	141	dgrsub2 30688	. . . . . . . . . 10 |- ( ( ( p e. (Poly ` QQ ) /\ a e. (Poly ` QQ ) ) /\ ( (deg ` a ) = (deg ` p ) /\ (deg ` p ) e. NN /\ ( (coeff ` p ) ` (deg ` p ) ) = ( (coeff ` a ) ` (deg ` p ) ) ) ) -> (deg ` ( p oF - a ) ) < (deg ` p ) )
143	128, 129, 132, 134, 140, 142	syl23anc 1235	. . . . . . . . 9 |- ( ( ( A e. AA /\ ( p e. (Poly ` QQ ) /\ a e. (Poly ` QQ ) ) ) /\ ( ( (deg ` p ) = (degAA ` A ) /\ ( p ` A ) = 0 /\ ( (coeff ` p ) ` (degAA ` A ) ) = 1 ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 /\ ( (coeff ` a ) ` (degAA ` A ) ) = 1 ) ) ) -> (deg ` ( p oF - a ) ) < (deg ` p ) )
144	143, 131	breqtrd 4471	. . . . . . . 8 |- ( ( ( A e. AA /\ ( p e. (Poly ` QQ ) /\ a e. (Poly ` QQ ) ) ) /\ ( ( (deg ` p ) = (degAA ` A ) /\ ( p ` A ) = 0 /\ ( (coeff ` p ) ` (degAA ` A ) ) = 1 ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 /\ ( (coeff ` a ) ` (degAA ` A ) ) = 1 ) ) ) -> (deg ` ( p oF - a ) ) < (degAA ` A ) )
145		dgraa0p 30703	. . . . . . . 8 |- ( ( A e. AA /\ ( p oF - a ) e. (Poly ` QQ ) /\ (deg ` ( p oF - a ) ) < (degAA ` A ) ) -> ( ( ( p oF - a ) ` A ) = 0 <-> ( p oF - a ) = 0p ) )
146	116, 127, 144, 145	syl3anc 1228	. . . . . . 7 |- ( ( ( A e. AA /\ ( p e. (Poly ` QQ ) /\ a e. (Poly ` QQ ) ) ) /\ ( ( (deg ` p ) = (degAA ` A ) /\ ( p ` A ) = 0 /\ ( (coeff ` p ) ` (degAA ` A ) ) = 1 ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 /\ ( (coeff ` a ) ` (degAA ` A ) ) = 1 ) ) ) -> ( ( ( p oF - a ) ` A ) = 0 <-> ( p oF - a ) = 0p ) )
147	115, 146	mpbid 210	. . . . . 6 |- ( ( ( A e. AA /\ ( p e. (Poly ` QQ ) /\ a e. (Poly ` QQ ) ) ) /\ ( ( (deg ` p ) = (degAA ` A ) /\ ( p ` A ) = 0 /\ ( (coeff ` p ) ` (degAA ` A ) ) = 1 ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 /\ ( (coeff ` a ) ` (degAA ` A ) ) = 1 ) ) ) -> ( p oF - a ) = 0p )
148		df-0p 21809	. . . . . 6 |- 0p = ( CC X. { 0 } )
149	147, 148	syl6eq 2524	. . . . 5 |- ( ( ( A e. AA /\ ( p e. (Poly ` QQ ) /\ a e. (Poly ` QQ ) ) ) /\ ( ( (deg ` p ) = (degAA ` A ) /\ ( p ` A ) = 0 /\ ( (coeff ` p ) ` (degAA ` A ) ) = 1 ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 /\ ( (coeff ` a ) ` (degAA ` A ) ) = 1 ) ) ) -> ( p oF - a ) = ( CC X. { 0 } ) )
150		ofsubeq0 10529	. . . . . . . 8 |- ( ( CC e. _V /\ p : CC --> CC /\ a : CC --> CC ) -> ( ( p oF - a ) = ( CC X. { 0 } ) <-> p = a ) )
151	51, 150	mp3an1 1311	. . . . . . 7 |- ( ( p : CC --> CC /\ a : CC --> CC ) -> ( ( p oF - a ) = ( CC X. { 0 } ) <-> p = a ) )
152	99, 48, 151	syl2an 477	. . . . . 6 |- ( ( p e. (Poly ` QQ ) /\ a e. (Poly ` QQ ) ) -> ( ( p oF - a ) = ( CC X. { 0 } ) <-> p = a ) )
153	152	ad2antlr 726	. . . . 5 |- ( ( ( A e. AA /\ ( p e. (Poly ` QQ ) /\ a e. (Poly ` QQ ) ) ) /\ ( ( (deg ` p ) = (degAA ` A ) /\ ( p ` A ) = 0 /\ ( (coeff ` p ) ` (degAA ` A ) ) = 1 ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 /\ ( (coeff ` a ) ` (degAA ` A ) ) = 1 ) ) ) -> ( ( p oF - a ) = ( CC X. { 0 } ) <-> p = a ) )
154	149, 153	mpbid 210	. . . 4 |- ( ( ( A e. AA /\ ( p e. (Poly ` QQ ) /\ a e. (Poly ` QQ ) ) ) /\ ( ( (deg ` p ) = (degAA ` A ) /\ ( p ` A ) = 0 /\ ( (coeff ` p ) ` (degAA ` A ) ) = 1 ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 /\ ( (coeff ` a ) ` (degAA ` A ) ) = 1 ) ) ) -> p = a )
155	154	ex 434	. . 3 |- ( ( A e. AA /\ ( p e. (Poly ` QQ ) /\ a e. (Poly ` QQ ) ) ) -> ( ( ( (deg ` p ) = (degAA ` A ) /\ ( p ` A ) = 0 /\ ( (coeff ` p ) ` (degAA ` A ) ) = 1 ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 /\ ( (coeff ` a ) ` (degAA ` A ) ) = 1 ) ) -> p = a ) )
156	155	ralrimivva 2885	. 2 |- ( A e. AA -> A. p e. (Poly ` QQ ) A. a e. (Poly ` QQ ) ( ( ( (deg ` p ) = (degAA ` A ) /\ ( p ` A ) = 0 /\ ( (coeff ` p ) ` (degAA ` A ) ) = 1 ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 /\ ( (coeff ` a ) ` (degAA ` A ) ) = 1 ) ) -> p = a ) )
157		fveq2 5864	. . . . 5 |- ( p = a -> (deg ` p ) = (deg ` a ) )
158	157	eqeq1d 2469	. . . 4 |- ( p = a -> ( (deg ` p ) = (degAA ` A ) <-> (deg ` a ) = (degAA ` A ) ) )
159		fveq1 5863	. . . . 5 |- ( p = a -> ( p ` A ) = ( a ` A ) )
160	159	eqeq1d 2469	. . . 4 |- ( p = a -> ( ( p ` A ) = 0 <-> ( a ` A ) = 0 ) )
161		fveq2 5864	. . . . . 6 |- ( p = a -> (coeff ` p ) = (coeff ` a ) )
162	161	fveq1d 5866	. . . . 5 |- ( p = a -> ( (coeff ` p ) ` (degAA ` A ) ) = ( (coeff ` a ) ` (degAA ` A ) ) )
163	162	eqeq1d 2469	. . . 4 |- ( p = a -> ( ( (coeff ` p ) ` (degAA ` A ) ) = 1 <-> ( (coeff ` a ) ` (degAA ` A ) ) = 1 ) )
164	158, 160, 163	3anbi123d 1299	. . 3 |- ( p = a -> ( ( (deg ` p ) = (degAA ` A ) /\ ( p ` A ) = 0 /\ ( (coeff ` p ) ` (degAA ` A ) ) = 1 ) <-> ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 /\ ( (coeff ` a ) ` (degAA ` A ) ) = 1 ) ) )
165	164	reu4 3297	. 2 |- ( E! p e. (Poly ` QQ ) ( (deg ` p ) = (degAA ` A ) /\ ( p ` A ) = 0 /\ ( (coeff ` p ) ` (degAA ` A ) ) = 1 ) <-> ( E. p e. (Poly ` QQ ) ( (deg ` p ) = (degAA ` A ) /\ ( p ` A ) = 0 /\ ( (coeff ` p ) ` (degAA ` A ) ) = 1 ) /\ A. p e. (Poly ` QQ ) A. a e. (Poly ` QQ ) ( ( ( (deg ` p ) = (degAA ` A ) /\ ( p ` A ) = 0 /\ ( (coeff ` p ) ` (degAA ` A ) ) = 1 ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 /\ ( (coeff ` a ) ` (degAA ` A ) ) = 1 ) ) -> p = a ) ) )
166	95, 156, 165	sylanbrc 664	1 |- ( A e. AA -> E! p e. (Poly ` QQ ) ( (deg ` p ) = (degAA ` A ) /\ ( p ` A ) = 0 /\ ( (coeff ` p ) ` (degAA ` A ) ) = 1 ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 973   = wceq 1379   e. wcel 1767   =/= wne 2662   A.wral 2814   E.wrex 2815   E!wreu 2816   _Vcvv 3113   \ cdif 3473   C_ wss 3476   {csn 4027   class class class wbr 4447   X. cxp 4997   Fn wfn 5581   -->wf 5582   ` cfv 5586  (class class class)co 6282   oFcof 6520   CCcc 9486   0cc0 9488   1c1 9489   + caddc 9491   x. cmul 9493   < clt 9624   - cmin 9801   -ucneg 9802   / cdiv 10202   NNcn 10532   NN0cn0 10791   ZZcz 10860   QQcq 11178   0pc0p 21808  Polycply 22313  coeffccoe 22315  degcdgr 22316   AAcaa 22441  deg_AAcdgraa 30694

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566  ax-addf 9567

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-of 6522  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-map 7419  df-pm 7420  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-sup 7897  df-oi 7931  df-card 8316  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11079  df-q 11179  df-rp 11217  df-fz 11669  df-fzo 11789  df-fl 11893  df-mod 11960  df-seq 12071  df-exp 12130  df-hash 12368  df-cj 12889  df-re 12890  df-im 12891  df-sqrt 13025  df-abs 13026  df-clim 13267  df-rlim 13268  df-sum 13465  df-0p 21809  df-ply 22317  df-coe 22319  df-dgr 22320  df-aa 22442  df-dgraa 30696

This theorem is referenced by:  mpaalem  30706