Theorem mpaaeu 29460

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 29455	. . . 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 10956	. . . . . . . 8 |- QQ C_ CC
4		eldifi 3473	. . . . . . . . . . . 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 10952	. . . . . . . . . . . 12 |- ZZ C_ QQ
7		0z 10649	. . . . . . . . . . . 12 |- 0 e. ZZ
8	6, 7	sselii 3348	. . . . . . . . . . 11 |- 0 e. QQ
9		eqid 2438	. . . . . . . . . . . 12 |- (coeff ` a ) = (coeff ` a )
10	9	coef2 21674	. . . . . . . . . . 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 21676	. . . . . . . . . . 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 5839	. . . . . . . . 9 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> ( (coeff ` a ) ` (deg ` a ) ) e. QQ )
15		eldifsni 3996	. . . . . . . . . . 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 2438	. . . . . . . . . . . . 13 |- (deg ` a ) = (deg ` a )
18	17, 9	dgreq0 21707	. . . . . . . . . . . 12 |- ( a e. (Poly ` QQ ) -> ( a = 0p <-> ( (coeff ` a ) ` (deg ` a ) ) = 0 ) )
19	18	necon3bid 2638	. . . . . . . . . . 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 10965	. . . . . . . . 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 21649	. . . . . . . 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 10961	. . . . . . . . 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 10963	. . . . . . . . 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 21661	. . . . . . 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 21675	. . . . . . . . . . 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 5839	. . . . . . . . 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 10092	. . . . . . . 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 10093	. . . . . . . 8 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) =/= 0 )
39		dgrmulc 21713	. . . . . . . 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 1218	. . . . . . 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 2470	. . . . . 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 21758	. . . . . . . . 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 6111	. . . . . . . . . 10 |- ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) e. _V
46		fnconstg 5593	. . . . . . . . . 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 21641	. . . . . . . . . 10 |- ( a e. (Poly ` QQ ) -> a : CC --> CC )
49		ffn 5554	. . . . . . . . . 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 9355	. . . . . . . . . 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 3554	. . . . . . . . 9 |- ( CC i^i CC ) = CC
54	45	fvconst2 5928	. . . . . . . . . 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 6324	. . . . . . . 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 9560	. . . . . . 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 2470	. . . . . 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 21697	. . . . . . . . 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 5688	. . . . . . 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 29456	. . . . . . . . . 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 10628	. . . . . . . 8 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> (degAA ` A ) e. NN0 )
67		fnconstg 5593	. . . . . . . . . 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 5554	. . . . . . . . . 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 10577	. . . . . . . . . 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 3554	. . . . . . . . 9 |- ( NN0 i^i NN0 ) = NN0
74	45	fvconst2 5928	. . . . . . . . . 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 2443	. . . . . . . . . 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 5690	. . . . . . . . 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 6324	. . . . . . . 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 10095	. . . . . . 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 2474	. . . . . 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 5686	. . . . . . . . 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 2446	. . . . . . . 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 5685	. . . . . . . . 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 2446	. . . . . . . 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 5686	. . . . . . . . . 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 5688	. . . . . . . . 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 2446	. . . . . . . 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 1289	. . . . . . 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 3068	. . . . . 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 1220	. . . . 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 2836	. . 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 21641	. . . . . . . . . . . . . . . 16 |- ( p e. (Poly ` QQ ) -> p : CC --> CC )
100		ffn 5554	. . . . . . . . . . . . . . . 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 6324	. . . . . . . . . . . . 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 10423	. . . . . . . . . . . 12 |- ( 0 - 0 ) = 0
111	109, 110	syl6eq 2486	. . . . . . . . . . 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 10668	. . . . . . . . . . . 12 |- 1 e. ZZ
122		zq 10951	. . . . . . . . . . . 12 |- ( 1 e. ZZ -> 1 e. QQ )
123		qnegcl 10962	. . . . . . . . . . . 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 21662	. . . . . . . . 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 2473	. . . . . . . . . 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 2512	. . . . . . . . . 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 5690	. . . . . . . . . . 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 5690	. . . . . . . . . . . 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 2470	. . . . . . . . . . 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 2480	. . . . . . . . . 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 2438	. . . . . . . . . . 11 |- (deg ` p ) = (deg ` p )
142	141	dgrsub2 29444	. . . . . . . . . 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 1225	. . . . . . . . 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 4311	. . . . . . . 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 29459	. . . . . . . 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 1218	. . . . . . 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 21123	. . . . . 6 |- 0p = ( CC X. { 0 } )
149	147, 148	syl6eq 2486	. . . . 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 10311	. . . . . . . 8 |- ( ( CC e. _V /\ p : CC --> CC /\ a : CC --> CC ) -> ( ( p oF - a ) = ( CC X. { 0 } ) <-> p = a ) )
151	51, 150	mp3an1 1301	. . . . . . 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 2803	. 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 5686	. . . . 5 |- ( p = a -> (deg ` p ) = (deg ` a ) )
158	157	eqeq1d 2446	. . . 4 |- ( p = a -> ( (deg ` p ) = (degAA ` A ) <-> (deg ` a ) = (degAA ` A ) ) )
159		fveq1 5685	. . . . 5 |- ( p = a -> ( p ` A ) = ( a ` A ) )
160	159	eqeq1d 2446	. . . 4 |- ( p = a -> ( ( p ` A ) = 0 <-> ( a ` A ) = 0 ) )
161		fveq2 5686	. . . . . 6 |- ( p = a -> (coeff ` p ) = (coeff ` a ) )
162	161	fveq1d 5688	. . . . 5 |- ( p = a -> ( (coeff ` p ) ` (degAA ` A ) ) = ( (coeff ` a ) ` (degAA ` A ) ) )
163	162	eqeq1d 2446	. . . 4 |- ( p = a -> ( ( (coeff ` p ) ` (degAA ` A ) ) = 1 <-> ( (coeff ` a ) ` (degAA ` A ) ) = 1 ) )
164	158, 160, 163	3anbi123d 1289	. . 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 3148	. 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 1369   e. wcel 1756   =/= wne 2601   A.wral 2710   E.wrex 2711   E!wreu 2712   _Vcvv 2967   \ cdif 3320   C_ wss 3323   {csn 3872   class class class wbr 4287   X. cxp 4833   Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6086   oFcof 6313   CCcc 9272   0cc0 9274   1c1 9275   + caddc 9277   x. cmul 9279   < clt 9410   - cmin 9587   -ucneg 9588   / cdiv 9985   NNcn 10314   NN0cn0 10571   ZZcz 10638   QQcq 10945   0pc0p 21122  Polycply 21627  coeffccoe 21629  degcdgr 21630   AAcaa 21755  deg_AAcdgraa 29450

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352  ax-addf 9353

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-of 6315  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-map 7208  df-pm 7209  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-sup 7683  df-oi 7716  df-card 8101  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-n0 10572  df-z 10639  df-uz 10854  df-q 10946  df-rp 10984  df-fz 11430  df-fzo 11541  df-fl 11634  df-mod 11701  df-seq 11799  df-exp 11858  df-hash 12096  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-clim 12958  df-rlim 12959  df-sum 13156  df-0p 21123  df-ply 21631  df-coe 21633  df-dgr 21634  df-aa 21756  df-dgraa 29452

This theorem is referenced by:  mpaalem  29462