Theorem mpaaeu 29645

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 29640	. . . 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 11065	. . . . . . . 8 |- QQ C_ CC
4		eldifi 3576	. . . . . . . . . . . 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 11061	. . . . . . . . . . . 12 |- ZZ C_ QQ
7		0z 10758	. . . . . . . . . . . 12 |- 0 e. ZZ
8	6, 7	sselii 3451	. . . . . . . . . . 11 |- 0 e. QQ
9		eqid 2451	. . . . . . . . . . . 12 |- (coeff ` a ) = (coeff ` a )
10	9	coef2 21815	. . . . . . . . . . 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 21817	. . . . . . . . . . 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 5943	. . . . . . . . 9 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> ( (coeff ` a ) ` (deg ` a ) ) e. QQ )
15		eldifsni 4099	. . . . . . . . . . 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 2451	. . . . . . . . . . . . 13 |- (deg ` a ) = (deg ` a )
18	17, 9	dgreq0 21848	. . . . . . . . . . . 12 |- ( a e. (Poly ` QQ ) -> ( a = 0p <-> ( (coeff ` a ) ` (deg ` a ) ) = 0 ) )
19	18	necon3bid 2706	. . . . . . . . . . 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 11074	. . . . . . . . 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 21790	. . . . . . . 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 11070	. . . . . . . . 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 11072	. . . . . . . . 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 21802	. . . . . . 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 21816	. . . . . . . . . . 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 5943	. . . . . . . . 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 10201	. . . . . . . 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 10202	. . . . . . . 8 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) =/= 0 )
39		dgrmulc 21854	. . . . . . . 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 1219	. . . . . . 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 2492	. . . . . 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 21899	. . . . . . . . 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 6215	. . . . . . . . . 10 |- ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) e. _V
46		fnconstg 5696	. . . . . . . . . 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 21782	. . . . . . . . . 10 |- ( a e. (Poly ` QQ ) -> a : CC --> CC )
49		ffn 5657	. . . . . . . . . 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 9464	. . . . . . . . . 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 3657	. . . . . . . . 9 |- ( CC i^i CC ) = CC
54	45	fvconst2 6032	. . . . . . . . . 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 6429	. . . . . . . 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 9669	. . . . . . 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 2492	. . . . . 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 21838	. . . . . . . . 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 5791	. . . . . . 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 29641	. . . . . . . . . 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 10737	. . . . . . . 8 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> (degAA ` A ) e. NN0 )
67		fnconstg 5696	. . . . . . . . . 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 5657	. . . . . . . . . 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 10686	. . . . . . . . . 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 3657	. . . . . . . . 9 |- ( NN0 i^i NN0 ) = NN0
74	45	fvconst2 6032	. . . . . . . . . 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 2459	. . . . . . . . . 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 5793	. . . . . . . . 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 6429	. . . . . . . 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 10204	. . . . . . 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 2496	. . . . . 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 5789	. . . . . . . . 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 2453	. . . . . . . 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 5788	. . . . . . . . 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 2453	. . . . . . . 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 5789	. . . . . . . . . 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 5791	. . . . . . . . 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 2453	. . . . . . . 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 1290	. . . . . . 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 3169	. . . . . 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 1221	. . . . 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 2937	. . 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 989	. . . . . . . . . . 11 |- ( ( (deg ` p ) = (degAA ` A ) /\ ( p ` A ) = 0 /\ ( (coeff ` p ) ` (degAA ` A ) ) = 1 ) -> ( p ` A ) = 0 )
97		simp2 989	. . . . . . . . . . 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 21782	. . . . . . . . . . . . . . . 16 |- ( p e. (Poly ` QQ ) -> p : CC --> CC )
100		ffn 5657	. . . . . . . . . . . . . . . 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 6429	. . . . . . . . . . . . 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 10532	. . . . . . . . . . . 12 |- ( 0 - 0 ) = 0
111	109, 110	syl6eq 2508	. . . . . . . . . . 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 10777	. . . . . . . . . . . 12 |- 1 e. ZZ
122		zq 11060	. . . . . . . . . . . 12 |- ( 1 e. ZZ -> 1 e. QQ )
123		qnegcl 11071	. . . . . . . . . . . 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 21803	. . . . . . . . 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 1036	. . . . . . . . . . 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 1033	. . . . . . . . . . 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 2495	. . . . . . . . . 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 2539	. . . . . . . . . 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 1035	. . . . . . . . . . 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 5793	. . . . . . . . . . 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 5793	. . . . . . . . . . . 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 1038	. . . . . . . . . . . 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 2492	. . . . . . . . . . 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 2502	. . . . . . . . . 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 2451	. . . . . . . . . . 11 |- (deg ` p ) = (deg ` p )
142	141	dgrsub2 29629	. . . . . . . . . 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 1226	. . . . . . . . 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 4414	. . . . . . . 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 29644	. . . . . . . 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 1219	. . . . . . 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 21264	. . . . . 6 |- 0p = ( CC X. { 0 } )
149	147, 148	syl6eq 2508	. . . . 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 10420	. . . . . . . 8 |- ( ( CC e. _V /\ p : CC --> CC /\ a : CC --> CC ) -> ( ( p oF - a ) = ( CC X. { 0 } ) <-> p = a ) )
151	51, 150	mp3an1 1302	. . . . . . 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 2904	. 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 5789	. . . . 5 |- ( p = a -> (deg ` p ) = (deg ` a ) )
158	157	eqeq1d 2453	. . . 4 |- ( p = a -> ( (deg ` p ) = (degAA ` A ) <-> (deg ` a ) = (degAA ` A ) ) )
159		fveq1 5788	. . . . 5 |- ( p = a -> ( p ` A ) = ( a ` A ) )
160	159	eqeq1d 2453	. . . 4 |- ( p = a -> ( ( p ` A ) = 0 <-> ( a ` A ) = 0 ) )
161		fveq2 5789	. . . . . 6 |- ( p = a -> (coeff ` p ) = (coeff ` a ) )
162	161	fveq1d 5791	. . . . 5 |- ( p = a -> ( (coeff ` p ) ` (degAA ` A ) ) = ( (coeff ` a ) ` (degAA ` A ) ) )
163	162	eqeq1d 2453	. . . 4 |- ( p = a -> ( ( (coeff ` p ) ` (degAA ` A ) ) = 1 <-> ( (coeff ` a ) ` (degAA ` A ) ) = 1 ) )
164	158, 160, 163	3anbi123d 1290	. . 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 3250	. 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 965   = wceq 1370   e. wcel 1758   =/= wne 2644   A.wral 2795   E.wrex 2796   E!wreu 2797   _Vcvv 3068   \ cdif 3423   C_ wss 3426   {csn 3975   class class class wbr 4390   X. cxp 4936   Fn wfn 5511   -->wf 5512   ` cfv 5516  (class class class)co 6190   oFcof 6418   CCcc 9381   0cc0 9383   1c1 9384   + caddc 9386   x. cmul 9388   < clt 9519   - cmin 9696   -ucneg 9697   / cdiv 10094   NNcn 10423   NN0cn0 10680   ZZcz 10747   QQcq 11054   0pc0p 21263  Polycply 21768  coeffccoe 21770  degcdgr 21771   AAcaa 21896  deg_AAcdgraa 29635

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-inf2 7948  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460  ax-pre-sup 9461  ax-addf 9462

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-int 4227  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-se 4778  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-isom 5525  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-of 6420  df-om 6577  df-1st 6677  df-2nd 6678  df-recs 6932  df-rdg 6966  df-1o 7020  df-oadd 7024  df-er 7201  df-map 7316  df-pm 7317  df-en 7411  df-dom 7412  df-sdom 7413  df-fin 7414  df-sup 7792  df-oi 7825  df-card 8210  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-div 10095  df-nn 10424  df-2 10481  df-3 10482  df-n0 10681  df-z 10748  df-uz 10963  df-q 11055  df-rp 11093  df-fz 11539  df-fzo 11650  df-fl 11743  df-mod 11810  df-seq 11908  df-exp 11967  df-hash 12205  df-cj 12690  df-re 12691  df-im 12692  df-sqr 12826  df-abs 12827  df-clim 13068  df-rlim 13069  df-sum 13266  df-0p 21264  df-ply 21772  df-coe 21774  df-dgr 21775  df-aa 21897  df-dgraa 29637

This theorem is referenced by:  mpaalem  29647