Theorem mpaaeu 27223

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 27218	. . . 4 |- ( A e. AA -> ( (degAA ` A ) e. NN /\ E. a e. ( (Poly ` QQ ) \ { 0 p } ) ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) )
2	1	simprd 450	. . 3 |- ( A e. AA -> E. a e. ( (Poly ` QQ ) \ { 0 p } ) ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) )
3		qsscn 10541	. . . . . . . 8 |- QQ C_ CC
4		eldifi 3429	. . . . . . . . . . . 12 |- ( a e. ( (Poly ` QQ ) \ { 0 p } ) -> a e. (Poly ` QQ ) )
5	4	ad2antlr 708	. . . . . . . . . . 11 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0 p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> a e. (Poly ` QQ ) )
6		zssq 10537	. . . . . . . . . . . 12 |- ZZ C_ QQ
7		0z 10249	. . . . . . . . . . . 12 |- 0 e. ZZ
8	6, 7	sselii 3305	. . . . . . . . . . 11 |- 0 e. QQ
9		eqid 2404	. . . . . . . . . . . 12 |- (coeff ` a ) = (coeff ` a )
10	9	coef2 20103	. . . . . . . . . . 11 |- ( ( a e. (Poly ` QQ ) /\ 0 e. QQ ) -> (coeff ` a ) : NN0 --> QQ )
11	5, 8, 10	sylancl 644	. . . . . . . . . 10 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0 p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> (coeff ` a ) : NN0 --> QQ )
12		dgrcl 20105	. . . . . . . . . . 11 |- ( a e. (Poly ` QQ ) -> (deg ` a ) e. NN0 )
13	5, 12	syl 16	. . . . . . . . . 10 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0 p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> (deg ` a ) e. NN0 )
14	11, 13	ffvelrnd 5830	. . . . . . . . 9 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0 p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> ( (coeff ` a ) ` (deg ` a ) ) e. QQ )
15		eldifsni 3888	. . . . . . . . . . 11 |- ( a e. ( (Poly ` QQ ) \ { 0 p } ) -> a =/= 0 p )
16	15	ad2antlr 708	. . . . . . . . . 10 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0 p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> a =/= 0 p )
17		eqid 2404	. . . . . . . . . . . . 13 |- (deg ` a ) = (deg ` a )
18	17, 9	dgreq0 20136	. . . . . . . . . . . 12 |- ( a e. (Poly ` QQ ) -> ( a = 0 p <-> ( (coeff ` a ) ` (deg ` a ) ) = 0 ) )
19	18	necon3bid 2602	. . . . . . . . . . 11 |- ( a e. (Poly ` QQ ) -> ( a =/= 0 p <-> ( (coeff ` a ) ` (deg ` a ) ) =/= 0 ) )
20	5, 19	syl 16	. . . . . . . . . 10 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0 p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> ( a =/= 0 p <-> ( (coeff ` a ) ` (deg ` a ) ) =/= 0 ) )
21	16, 20	mpbid 202	. . . . . . . . 9 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0 p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> ( (coeff ` a ) ` (deg ` a ) ) =/= 0 )
22		qreccl 10550	. . . . . . . . 9 |- ( ( ( (coeff ` a ) ` (deg ` a ) ) e. QQ /\ ( (coeff ` a ) ` (deg ` a ) ) =/= 0 ) -> ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) e. QQ )
23	14, 21, 22	syl2anc 643	. . . . . . . 8 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0 p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) e. QQ )
24		plyconst 20078	. . . . . . . 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 645	. . . . . . 7 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0 p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) e. (Poly ` QQ ) )
26		simpl 444	. . . . . . . 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 448	. . . . . . . 8 |- ( ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) e. (Poly ` QQ ) /\ a e. (Poly ` QQ ) ) -> a e. (Poly ` QQ ) )
28		qaddcl 10546	. . . . . . . . 9 |- ( ( b e. QQ /\ c e. QQ ) -> ( b + c ) e. QQ )
29	28	adantl 453	. . . . . . . 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 10548	. . . . . . . . 9 |- ( ( b e. QQ /\ c e. QQ ) -> ( b x. c ) e. QQ )
31	30	adantl 453	. . . . . . . 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 20090	. . . . . . 7 |- ( ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) e. (Poly ` QQ ) /\ a e. (Poly ` QQ ) ) -> ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) o F x. a ) e. (Poly ` QQ ) )
33	25, 5, 32	syl2anc 643	. . . . . 6 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0 p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) o F x. a ) e. (Poly ` QQ ) )
34	9	coef3 20104	. . . . . . . . . . 11 |- ( a e. (Poly ` QQ ) -> (coeff ` a ) : NN0 --> CC )
35	5, 34	syl 16	. . . . . . . . . 10 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0 p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> (coeff ` a ) : NN0 --> CC )
36	35, 13	ffvelrnd 5830	. . . . . . . . 9 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0 p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> ( (coeff ` a ) ` (deg ` a ) ) e. CC )
37	36, 21	reccld 9739	. . . . . . . 8 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0 p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) e. CC )
38	36, 21	recne0d 9740	. . . . . . . 8 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0 p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) =/= 0 )
39		dgrmulc 20142	. . . . . . . 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 ) ) ) } ) o F x. a ) ) = (deg ` a ) )
40	37, 38, 5, 39	syl3anc 1184	. . . . . . 7 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0 p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> (deg ` ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) o F x. a ) ) = (deg ` a ) )
41		simprl 733	. . . . . . 7 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0 p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> (deg ` a ) = (degAA ` A ) )
42	40, 41	eqtrd 2436	. . . . . 6 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0 p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> (deg ` ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) o F x. a ) ) = (degAA ` A ) )
43		aacn 20187	. . . . . . . . 9 |- ( A e. AA -> A e. CC )
44	43	ad2antrr 707	. . . . . . . 8 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0 p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> A e. CC )
45		ovex 6065	. . . . . . . . . 10 |- ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) e. _V
46		fnconstg 5590	. . . . . . . . . 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 ) \ { 0 p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) Fn CC )
48		plyf 20070	. . . . . . . . . 10 |- ( a e. (Poly ` QQ ) -> a : CC --> CC )
49		ffn 5550	. . . . . . . . . 10 |- ( a : CC --> CC -> a Fn CC )
50	5, 48, 49	3syl 19	. . . . . . . . 9 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0 p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> a Fn CC )
51		cnex 9027	. . . . . . . . . 10 |- CC e. _V
52	51	a1i 11	. . . . . . . . 9 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0 p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> CC e. _V )
53		inidm 3510	. . . . . . . . 9 |- ( CC i^i CC ) = CC
54	45	fvconst2 5906	. . . . . . . . . 10 |- ( A e. CC -> ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) ` A ) = ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) )
55	54	adantl 453	. . . . . . . . 9 |- ( ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0 p } ) ) /\ ( (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 738	. . . . . . . . 9 |- ( ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0 p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) /\ A e. CC ) -> ( a ` A ) = 0 )
57	47, 50, 52, 52, 53, 55, 56	ofval 6273	. . . . . . . 8 |- ( ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0 p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) /\ A e. CC ) -> ( ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) o F x. a ) ` A ) = ( ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) x. 0 ) )
58	44, 57	mpdan 650	. . . . . . 7 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0 p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> ( ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) o F x. a ) ` A ) = ( ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) x. 0 ) )
59	37	mul01d 9221	. . . . . . 7 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0 p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> ( ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) x. 0 ) = 0 )
60	58, 59	eqtrd 2436	. . . . . 6 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0 p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> ( ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) o F x. a ) ` A ) = 0 )
61		coemulc 20126	. . . . . . . . 9 |- ( ( ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) e. CC /\ a e. (Poly ` QQ ) ) -> (coeff ` ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) o F x. a ) ) = ( ( NN0 X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) o F x. (coeff ` a ) ) )
62	37, 5, 61	syl2anc 643	. . . . . . . 8 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0 p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> (coeff ` ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) o F x. a ) ) = ( ( NN0 X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) o F x. (coeff ` a ) ) )
63	62	fveq1d 5689	. . . . . . 7 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0 p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> ( (coeff ` ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) o F x. a ) ) ` (degAA ` A ) ) = ( ( ( NN0 X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) o F x. (coeff ` a ) ) ` (degAA ` A ) ) )
64		dgraacl 27219	. . . . . . . . . 10 |- ( A e. AA -> (degAA ` A ) e. NN )
65	64	ad2antrr 707	. . . . . . . . 9 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0 p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> (degAA ` A ) e. NN )
66	65	nnnn0d 10230	. . . . . . . 8 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0 p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> (degAA ` A ) e. NN0 )
67		fnconstg 5590	. . . . . . . . . 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 ) \ { 0 p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> ( NN0 X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) Fn NN0 )
69		ffn 5550	. . . . . . . . . 10 |- ( (coeff ` a ) : NN0 --> CC -> (coeff ` a ) Fn NN0 )
70	35, 69	syl 16	. . . . . . . . 9 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0 p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> (coeff ` a ) Fn NN0 )
71		nn0ex 10183	. . . . . . . . . 10 |- NN0 e. _V
72	71	a1i 11	. . . . . . . . 9 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0 p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> NN0 e. _V )
73		inidm 3510	. . . . . . . . 9 |- ( NN0 i^i NN0 ) = NN0
74	45	fvconst2 5906	. . . . . . . . . 10 |- ( (degAA ` A ) e. NN0 -> ( ( NN0 X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) ` (degAA ` A ) ) = ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) )
75	74	adantl 453	. . . . . . . . 9 |- ( ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0 p } ) ) /\ ( (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 737	. . . . . . . . . . 11 |- ( ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0 p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) /\ (degAA ` A ) e. NN0 ) -> (deg ` a ) = (degAA ` A ) )
77	76	eqcomd 2409	. . . . . . . . . 10 |- ( ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0 p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) /\ (degAA ` A ) e. NN0 ) -> (degAA ` A ) = (deg ` a ) )
78	77	fveq2d 5691	. . . . . . . . 9 |- ( ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0 p } ) ) /\ ( (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 6273	. . . . . . . 8 |- ( ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0 p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) /\ (degAA ` A ) e. NN0 ) -> ( ( ( NN0 X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) o F x. (coeff ` a ) ) ` (degAA ` A ) ) = ( ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) x. ( (coeff ` a ) ` (deg ` a ) ) ) )
80	66, 79	mpdan 650	. . . . . . 7 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0 p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> ( ( ( NN0 X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) o F x. (coeff ` a ) ) ` (degAA ` A ) ) = ( ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) x. ( (coeff ` a ) ` (deg ` a ) ) ) )
81	36, 21	recid2d 9742	. . . . . . 7 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0 p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> ( ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) x. ( (coeff ` a ) ` (deg ` a ) ) ) = 1 )
82	63, 80, 81	3eqtrd 2440	. . . . . 6 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0 p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> ( (coeff ` ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) o F x. a ) ) ` (degAA ` A ) ) = 1 )
83		fveq2 5687	. . . . . . . . 9 |- ( p = ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) o F x. a ) -> (deg ` p ) = (deg ` ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) o F x. a ) ) )
84	83	eqeq1d 2412	. . . . . . . 8 |- ( p = ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) o F x. a ) -> ( (deg ` p ) = (degAA ` A ) <-> (deg ` ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) o F x. a ) ) = (degAA ` A ) ) )
85		fveq1 5686	. . . . . . . . 9 |- ( p = ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) o F x. a ) -> ( p ` A ) = ( ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) o F x. a ) ` A ) )
86	85	eqeq1d 2412	. . . . . . . 8 |- ( p = ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) o F x. a ) -> ( ( p ` A ) = 0 <-> ( ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) o F x. a ) ` A ) = 0 ) )
87		fveq2 5687	. . . . . . . . . 10 |- ( p = ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) o F x. a ) -> (coeff ` p ) = (coeff ` ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) o F x. a ) ) )
88	87	fveq1d 5689	. . . . . . . . 9 |- ( p = ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) o F x. a ) -> ( (coeff ` p ) ` (degAA ` A ) ) = ( (coeff ` ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) o F x. a ) ) ` (degAA ` A ) ) )
89	88	eqeq1d 2412	. . . . . . . 8 |- ( p = ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) o F x. a ) -> ( ( (coeff ` p ) ` (degAA ` A ) ) = 1 <-> ( (coeff ` ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) o F x. a ) ) ` (degAA ` A ) ) = 1 ) )
90	84, 86, 89	3anbi123d 1254	. . . . . . 7 |- ( p = ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) o F x. a ) -> ( ( (deg ` p ) = (degAA ` A ) /\ ( p ` A ) = 0 /\ ( (coeff ` p ) ` (degAA ` A ) ) = 1 ) <-> ( (deg ` ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) o F x. a ) ) = (degAA ` A ) /\ ( ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) o F x. a ) ` A ) = 0 /\ ( (coeff ` ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) o F x. a ) ) ` (degAA ` A ) ) = 1 ) ) )
91	90	rspcev 3012	. . . . . 6 |- ( ( ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) o F x. a ) e. (Poly ` QQ ) /\ ( (deg ` ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) o F x. a ) ) = (degAA ` A ) /\ ( ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) o F x. a ) ` A ) = 0 /\ ( (coeff ` ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) o F 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 1186	. . . . 5 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0 p } ) ) /\ ( (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 424	. . . 4 |- ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0 p } ) ) -> ( ( (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 2790	. . 3 |- ( A e. AA -> ( E. a e. ( (Poly ` QQ ) \ { 0 p } ) ( (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 958	. . . . . . . . . . 11 |- ( ( (deg ` p ) = (degAA ` A ) /\ ( p ` A ) = 0 /\ ( (coeff ` p ) ` (degAA ` A ) ) = 1 ) -> ( p ` A ) = 0 )
97		simp2 958	. . . . . . . . . . 11 |- ( ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 /\ ( (coeff ` a ) ` (degAA ` A ) ) = 1 ) -> ( a ` A ) = 0 )
98	96, 97	anim12i 550	. . . . . . . . . 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 20070	. . . . . . . . . . . . . . . 16 |- ( p e. (Poly ` QQ ) -> p : CC --> CC )
100		ffn 5550	. . . . . . . . . . . . . . . 16 |- ( p : CC --> CC -> p Fn CC )
101	99, 100	syl 16	. . . . . . . . . . . . . . 15 |- ( p e. (Poly ` QQ ) -> p Fn CC )
102	101	ad2antrr 707	. . . . . . . . . . . . . 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 708	. . . . . . . . . . . . . 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 737	. . . . . . . . . . . . . 14 |- ( ( ( ( p e. (Poly ` QQ ) /\ a e. (Poly ` QQ ) ) /\ ( ( p ` A ) = 0 /\ ( a ` A ) = 0 ) ) /\ A e. CC ) -> ( p ` A ) = 0 )
107		simplrr 738	. . . . . . . . . . . . . 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 6273	. . . . . . . . . . . . 13 |- ( ( ( ( p e. (Poly ` QQ ) /\ a e. (Poly ` QQ ) ) /\ ( ( p ` A ) = 0 /\ ( a ` A ) = 0 ) ) /\ A e. CC ) -> ( ( p o F - a ) ` A ) = ( 0 - 0 ) )
109	43, 108	sylan2 461	. . . . . . . . . . . 12 |- ( ( ( ( p e. (Poly ` QQ ) /\ a e. (Poly ` QQ ) ) /\ ( ( p ` A ) = 0 /\ ( a ` A ) = 0 ) ) /\ A e. AA ) -> ( ( p o F - a ) ` A ) = ( 0 - 0 ) )
110		0cn 9040	. . . . . . . . . . . . 13 |- 0 e. CC
111	110	subid1i 9328	. . . . . . . . . . . 12 |- ( 0 - 0 ) = 0
112	109, 111	syl6eq 2452	. . . . . . . . . . 11 |- ( ( ( ( p e. (Poly ` QQ ) /\ a e. (Poly ` QQ ) ) /\ ( ( p ` A ) = 0 /\ ( a ` A ) = 0 ) ) /\ A e. AA ) -> ( ( p o F - a ) ` A ) = 0 )
113	112	ex 424	. . . . . . . . . 10 |- ( ( ( p e. (Poly ` QQ ) /\ a e. (Poly ` QQ ) ) /\ ( ( p ` A ) = 0 /\ ( a ` A ) = 0 ) ) -> ( A e. AA -> ( ( p o F - a ) ` A ) = 0 ) )
114	98, 113	sylan2 461	. . . . . . . . 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 o F - a ) ` A ) = 0 ) )
115	114	com12 29	. . . . . . . 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 o F - a ) ` A ) = 0 ) )
116	115	impl 604	. . . . . . 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 o F - a ) ` A ) = 0 )
117		simpll 731	. . . . . . . 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 )
118		simpl 444	. . . . . . . . . 10 |- ( ( p e. (Poly ` QQ ) /\ a e. (Poly ` QQ ) ) -> p e. (Poly ` QQ ) )
119		simpr 448	. . . . . . . . . 10 |- ( ( p e. (Poly ` QQ ) /\ a e. (Poly ` QQ ) ) -> a e. (Poly ` QQ ) )
120	28	adantl 453	. . . . . . . . . 10 |- ( ( ( p e. (Poly ` QQ ) /\ a e. (Poly ` QQ ) ) /\ ( b e. QQ /\ c e. QQ ) ) -> ( b + c ) e. QQ )
121	30	adantl 453	. . . . . . . . . 10 |- ( ( ( p e. (Poly ` QQ ) /\ a e. (Poly ` QQ ) ) /\ ( b e. QQ /\ c e. QQ ) ) -> ( b x. c ) e. QQ )
122		1z 10267	. . . . . . . . . . . 12 |- 1 e. ZZ
123		zq 10536	. . . . . . . . . . . 12 |- ( 1 e. ZZ -> 1 e. QQ )
124		qnegcl 10547	. . . . . . . . . . . 12 |- ( 1 e. QQ -> -u 1 e. QQ )
125	122, 123, 124	mp2b 10	. . . . . . . . . . 11 |- -u 1 e. QQ
126	125	a1i 11	. . . . . . . . . 10 |- ( ( p e. (Poly ` QQ ) /\ a e. (Poly ` QQ ) ) -> -u 1 e. QQ )
127	118, 119, 120, 121, 126	plysub 20091	. . . . . . . . 9 |- ( ( p e. (Poly ` QQ ) /\ a e. (Poly ` QQ ) ) -> ( p o F - a ) e. (Poly ` QQ ) )
128	127	ad2antlr 708	. . . . . . . 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 o F - a ) e. (Poly ` QQ ) )
129		simplrl 737	. . . . . . . . . 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 ) )
130		simplrr 738	. . . . . . . . . 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 ) )
131		simprr1 1005	. . . . . . . . . . 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 ) )
132		simprl1 1002	. . . . . . . . . . 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 ) )
133	131, 132	eqtr4d 2439	. . . . . . . . . 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 ) )
134	64	ad2antrr 707	. . . . . . . . . . 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 )
135	132, 134	eqeltrd 2478	. . . . . . . . . 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 )
136		simprl3 1004	. . . . . . . . . . 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 )
137	132	fveq2d 5691	. . . . . . . . . . 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 ) ) )
138	132	fveq2d 5691	. . . . . . . . . . . 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 ) ) )
139		simprr3 1007	. . . . . . . . . . . 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 )
140	138, 139	eqtrd 2436	. . . . . . . . . . 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 )
141	136, 137, 140	3eqtr4d 2446	. . . . . . . . . 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 ) ) )
142		eqid 2404	. . . . . . . . . . 11 |- (deg ` p ) = (deg ` p )
143	142	dgrsub2 27207	. . . . . . . . . 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 o F - a ) ) < (deg ` p ) )
144	129, 130, 133, 135, 141, 143	syl23anc 1191	. . . . . . . . 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 o F - a ) ) < (deg ` p ) )
145	144, 132	breqtrd 4196	. . . . . . . 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 o F - a ) ) < (degAA ` A ) )
146		dgraa0p 27222	. . . . . . . 8 |- ( ( A e. AA /\ ( p o F - a ) e. (Poly ` QQ ) /\ (deg ` ( p o F - a ) ) < (degAA ` A ) ) -> ( ( ( p o F - a ) ` A ) = 0 <-> ( p o F - a ) = 0 p ) )
147	117, 128, 145, 146	syl3anc 1184	. . . . . . 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 o F - a ) ` A ) = 0 <-> ( p o F - a ) = 0 p ) )
148	116, 147	mpbid 202	. . . . . 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 o F - a ) = 0 p )
149		df-0p 19515	. . . . . 6 |- 0 p = ( CC X. { 0 } )
150	148, 149	syl6eq 2452	. . . . 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 o F - a ) = ( CC X. { 0 } ) )
151		ofsubeq0 9953	. . . . . . . 8 |- ( ( CC e. _V /\ p : CC --> CC /\ a : CC --> CC ) -> ( ( p o F - a ) = ( CC X. { 0 } ) <-> p = a ) )
152	51, 151	mp3an1 1266	. . . . . . 7 |- ( ( p : CC --> CC /\ a : CC --> CC ) -> ( ( p o F - a ) = ( CC X. { 0 } ) <-> p = a ) )
153	99, 48, 152	syl2an 464	. . . . . 6 |- ( ( p e. (Poly ` QQ ) /\ a e. (Poly ` QQ ) ) -> ( ( p o F - a ) = ( CC X. { 0 } ) <-> p = a ) )
154	153	ad2antlr 708	. . . . 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 o F - a ) = ( CC X. { 0 } ) <-> p = a ) )
155	150, 154	mpbid 202	. . . 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 )
156	155	ex 424	. . 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 ) )
157	156	ralrimivva 2758	. 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 ) )
158		fveq2 5687	. . . . 5 |- ( p = a -> (deg ` p ) = (deg ` a ) )
159	158	eqeq1d 2412	. . . 4 |- ( p = a -> ( (deg ` p ) = (degAA ` A ) <-> (deg ` a ) = (degAA ` A ) ) )
160		fveq1 5686	. . . . 5 |- ( p = a -> ( p ` A ) = ( a ` A ) )
161	160	eqeq1d 2412	. . . 4 |- ( p = a -> ( ( p ` A ) = 0 <-> ( a ` A ) = 0 ) )
162		fveq2 5687	. . . . . 6 |- ( p = a -> (coeff ` p ) = (coeff ` a ) )
163	162	fveq1d 5689	. . . . 5 |- ( p = a -> ( (coeff ` p ) ` (degAA ` A ) ) = ( (coeff ` a ) ` (degAA ` A ) ) )
164	163	eqeq1d 2412	. . . 4 |- ( p = a -> ( ( (coeff ` p ) ` (degAA ` A ) ) = 1 <-> ( (coeff ` a ) ` (degAA ` A ) ) = 1 ) )
165	159, 161, 164	3anbi123d 1254	. . 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 ) ) )
166	165	reu4 3088	. 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 ) ) )
167	95, 157, 166	sylanbrc 646	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 177   /\ wa 359   /\ w3a 936   = wceq 1649   e. wcel 1721   =/= wne 2567   A.wral 2666   E.wrex 2667   E!wreu 2668   _Vcvv 2916   \ cdif 3277   C_ wss 3280   {csn 3774   class class class wbr 4172   X. cxp 4835   Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6040   o Fcof 6262   CCcc 8944   0cc0 8946   1c1 8947   + caddc 8949   x. cmul 8951   < clt 9076   - cmin 9247   -ucneg 9248   / cdiv 9633   NNcn 9956   NN0cn0 10177   ZZcz 10238   QQcq 10530   0 pc0p 19514  Polycply 20056  coeffccoe 20058  degcdgr 20059   AAcaa 20184  deg_AAcdgraa 27213

This theorem is referenced by:  mpaalem  27225

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  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-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-oi 7435  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-q 10531  df-rp 10569  df-fz 11000  df-fzo 11091  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237  df-rlim 12238  df-sum 12435  df-0p 19515  df-ply 20060  df-coe 20062  df-dgr 20063  df-aa 20185  df-dgraa 27215