Theorem mpaaeu 37720

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		qsscn 11799	. . . . . 6 |- QQ C_ CC
2		eldifi 3732	. . . . . . . . . 10 |- ( a e. ( (Poly ` QQ ) \ { 0p } ) -> a e. (Poly ` QQ ) )
3	2	ad2antlr 763	. . . . . . . . 9 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> a e. (Poly ` QQ ) )
4		zssq 11795	. . . . . . . . . 10 |- ZZ C_ QQ
5		0z 11388	. . . . . . . . . 10 |- 0 e. ZZ
6	4, 5	sselii 3600	. . . . . . . . 9 |- 0 e. QQ
7		eqid 2622	. . . . . . . . . 10 |- (coeff ` a ) = (coeff ` a )
8	7	coef2 23987	. . . . . . . . 9 |- ( ( a e. (Poly ` QQ ) /\ 0 e. QQ ) -> (coeff ` a ) : NN0 --> QQ )
9	3, 6, 8	sylancl 694	. . . . . . . 8 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> (coeff ` a ) : NN0 --> QQ )
10		dgrcl 23989	. . . . . . . . 9 |- ( a e. (Poly ` QQ ) -> (deg ` a ) e. NN0 )
11	3, 10	syl 17	. . . . . . . 8 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> (deg ` a ) e. NN0 )
12	9, 11	ffvelrnd 6360	. . . . . . 7 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> ( (coeff ` a ) ` (deg ` a ) ) e. QQ )
13		eldifsni 4320	. . . . . . . . 9 |- ( a e. ( (Poly ` QQ ) \ { 0p } ) -> a =/= 0p )
14	13	ad2antlr 763	. . . . . . . 8 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> a =/= 0p )
15		eqid 2622	. . . . . . . . . . 11 |- (deg ` a ) = (deg ` a )
16	15, 7	dgreq0 24021	. . . . . . . . . 10 |- ( a e. (Poly ` QQ ) -> ( a = 0p <-> ( (coeff ` a ) ` (deg ` a ) ) = 0 ) )
17	16	necon3bid 2838	. . . . . . . . 9 |- ( a e. (Poly ` QQ ) -> ( a =/= 0p <-> ( (coeff ` a ) ` (deg ` a ) ) =/= 0 ) )
18	3, 17	syl 17	. . . . . . . 8 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> ( a =/= 0p <-> ( (coeff ` a ) ` (deg ` a ) ) =/= 0 ) )
19	14, 18	mpbid 222	. . . . . . 7 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> ( (coeff ` a ) ` (deg ` a ) ) =/= 0 )
20		qreccl 11808	. . . . . . 7 |- ( ( ( (coeff ` a ) ` (deg ` a ) ) e. QQ /\ ( (coeff ` a ) ` (deg ` a ) ) =/= 0 ) -> ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) e. QQ )
21	12, 19, 20	syl2anc 693	. . . . . 6 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) e. QQ )
22		plyconst 23962	. . . . . 6 |- ( ( QQ C_ CC /\ ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) e. QQ ) -> ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) e. (Poly ` QQ ) )
23	1, 21, 22	sylancr 695	. . . . 5 |- ( ( ( 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 ) )
24		simpl 473	. . . . . 6 |- ( ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) e. (Poly ` QQ ) /\ a e. (Poly ` QQ ) ) -> ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) e. (Poly ` QQ ) )
25		simpr 477	. . . . . 6 |- ( ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) e. (Poly ` QQ ) /\ a e. (Poly ` QQ ) ) -> a e. (Poly ` QQ ) )
26		qaddcl 11804	. . . . . . 7 |- ( ( b e. QQ /\ c e. QQ ) -> ( b + c ) e. QQ )
27	26	adantl 482	. . . . . 6 |- ( ( ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) e. (Poly ` QQ ) /\ a e. (Poly ` QQ ) ) /\ ( b e. QQ /\ c e. QQ ) ) -> ( b + c ) e. QQ )
28		qmulcl 11806	. . . . . . 7 |- ( ( b e. QQ /\ c e. QQ ) -> ( b x. c ) e. QQ )
29	28	adantl 482	. . . . . 6 |- ( ( ( ( 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 )
30	24, 25, 27, 29	plymul 23974	. . . . 5 |- ( ( ( 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 ) )
31	23, 3, 30	syl2anc 693	. . . 4 |- ( ( ( 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 ) )
32	7	coef3 23988	. . . . . . . . 9 |- ( a e. (Poly ` QQ ) -> (coeff ` a ) : NN0 --> CC )
33	3, 32	syl 17	. . . . . . . 8 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> (coeff ` a ) : NN0 --> CC )
34	33, 11	ffvelrnd 6360	. . . . . . 7 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> ( (coeff ` a ) ` (deg ` a ) ) e. CC )
35	34, 19	reccld 10794	. . . . . 6 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) e. CC )
36	34, 19	recne0d 10795	. . . . . 6 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) =/= 0 )
37		dgrmulc 24027	. . . . . 6 |- ( ( ( 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 ) )
38	35, 36, 3, 37	syl3anc 1326	. . . . 5 |- ( ( ( 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 ) )
39		simprl 794	. . . . 5 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> (deg ` a ) = (degAA ` A ) )
40	38, 39	eqtrd 2656	. . . 4 |- ( ( ( 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 ) )
41		aacn 24072	. . . . . . 7 |- ( A e. AA -> A e. CC )
42	41	ad2antrr 762	. . . . . 6 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> A e. CC )
43		ovex 6678	. . . . . . . 8 |- ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) e. _V
44		fnconstg 6093	. . . . . . . 8 |- ( ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) e. _V -> ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) Fn CC )
45	43, 44	mp1i 13	. . . . . . 7 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) Fn CC )
46		plyf 23954	. . . . . . . 8 |- ( a e. (Poly ` QQ ) -> a : CC --> CC )
47		ffn 6045	. . . . . . . 8 |- ( a : CC --> CC -> a Fn CC )
48	3, 46, 47	3syl 18	. . . . . . 7 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> a Fn CC )
49		cnex 10017	. . . . . . . 8 |- CC e. _V
50	49	a1i 11	. . . . . . 7 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> CC e. _V )
51		inidm 3822	. . . . . . 7 |- ( CC i^i CC ) = CC
52	43	fvconst2 6469	. . . . . . . 8 |- ( A e. CC -> ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) ` A ) = ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) )
53	52	adantl 482	. . . . . . 7 |- ( ( ( ( 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 ) ) ) )
54		simplrr 801	. . . . . . 7 |- ( ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) /\ A e. CC ) -> ( a ` A ) = 0 )
55	45, 48, 50, 50, 51, 53, 54	ofval 6906	. . . . . 6 |- ( ( ( ( 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 ) )
56	42, 55	mpdan 702	. . . . 5 |- ( ( ( 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 ) )
57	35	mul01d 10235	. . . . 5 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> ( ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) x. 0 ) = 0 )
58	56, 57	eqtrd 2656	. . . 4 |- ( ( ( 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 )
59		coemulc 24011	. . . . . . 7 |- ( ( ( 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 ) ) )
60	35, 3, 59	syl2anc 693	. . . . . 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 ) ) = ( ( NN0 X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) oF x. (coeff ` a ) ) )
61	60	fveq1d 6193	. . . . 5 |- ( ( ( 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 ) ) )
62		dgraacl 37716	. . . . . . . 8 |- ( A e. AA -> (degAA ` A ) e. NN )
63	62	ad2antrr 762	. . . . . . 7 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> (degAA ` A ) e. NN )
64	63	nnnn0d 11351	. . . . . 6 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> (degAA ` A ) e. NN0 )
65		fnconstg 6093	. . . . . . . 8 |- ( ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) e. _V -> ( NN0 X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) Fn NN0 )
66	43, 65	mp1i 13	. . . . . . 7 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> ( NN0 X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) Fn NN0 )
67		ffn 6045	. . . . . . . 8 |- ( (coeff ` a ) : NN0 --> CC -> (coeff ` a ) Fn NN0 )
68	33, 67	syl 17	. . . . . . 7 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> (coeff ` a ) Fn NN0 )
69		nn0ex 11298	. . . . . . . 8 |- NN0 e. _V
70	69	a1i 11	. . . . . . 7 |- ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) -> NN0 e. _V )
71		inidm 3822	. . . . . . 7 |- ( NN0 i^i NN0 ) = NN0
72	43	fvconst2 6469	. . . . . . . 8 |- ( (degAA ` A ) e. NN0 -> ( ( NN0 X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) ` (degAA ` A ) ) = ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) )
73	72	adantl 482	. . . . . . 7 |- ( ( ( ( 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 ) ) ) )
74		simplrl 800	. . . . . . . . 9 |- ( ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) /\ (degAA ` A ) e. NN0 ) -> (deg ` a ) = (degAA ` A ) )
75	74	eqcomd 2628	. . . . . . . 8 |- ( ( ( ( A e. AA /\ a e. ( (Poly ` QQ ) \ { 0p } ) ) /\ ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) /\ (degAA ` A ) e. NN0 ) -> (degAA ` A ) = (deg ` a ) )
76	75	fveq2d 6195	. . . . . . 7 |- ( ( ( ( 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 ) ) )
77	66, 68, 70, 70, 71, 73, 76	ofval 6906	. . . . . 6 |- ( ( ( ( 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 ) ) ) )
78	64, 77	mpdan 702	. . . . 5 |- ( ( ( 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 ) ) ) )
79	34, 19	recid2d 10797	. . . . 5 |- ( ( ( 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 )
80	61, 78, 79	3eqtrd 2660	. . . 4 |- ( ( ( 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 )
81		fveq2 6191	. . . . . . 7 |- ( p = ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) oF x. a ) -> (deg ` p ) = (deg ` ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) oF x. a ) ) )
82	81	eqeq1d 2624	. . . . . 6 |- ( 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 ) ) )
83		fveq1 6190	. . . . . . 7 |- ( p = ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) oF x. a ) -> ( p ` A ) = ( ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) oF x. a ) ` A ) )
84	83	eqeq1d 2624	. . . . . 6 |- ( 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 ) )
85		fveq2 6191	. . . . . . . 8 |- ( p = ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) oF x. a ) -> (coeff ` p ) = (coeff ` ( ( CC X. { ( 1 / ( (coeff ` a ) ` (deg ` a ) ) ) } ) oF x. a ) ) )
86	85	fveq1d 6193	. . . . . . 7 |- ( 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 ) ) )
87	86	eqeq1d 2624	. . . . . 6 |- ( 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 ) )
88	82, 84, 87	3anbi123d 1399	. . . . 5 |- ( 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 ) ) )
89	88	rspcev 3309	. . . 4 |- ( ( ( ( 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 ) )
90	31, 40, 58, 80, 89	syl13anc 1328	. . 3 |- ( ( ( 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 ) )
91		dgraalem 37715	. . . 4 |- ( A e. AA -> ( (degAA ` A ) e. NN /\ E. a e. ( (Poly ` QQ ) \ { 0p } ) ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) ) )
92	91	simprd 479	. . 3 |- ( A e. AA -> E. a e. ( (Poly ` QQ ) \ { 0p } ) ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 ) )
93	90, 92	r19.29a 3078	. 2 |- ( A e. AA -> E. p e. (Poly ` QQ ) ( (deg ` p ) = (degAA ` A ) /\ ( p ` A ) = 0 /\ ( (coeff ` p ) ` (degAA ` A ) ) = 1 ) )
94		simp2 1062	. . . . . . . . . . 11 |- ( ( (deg ` p ) = (degAA ` A ) /\ ( p ` A ) = 0 /\ ( (coeff ` p ) ` (degAA ` A ) ) = 1 ) -> ( p ` A ) = 0 )
95		simp2 1062	. . . . . . . . . . 11 |- ( ( (deg ` a ) = (degAA ` A ) /\ ( a ` A ) = 0 /\ ( (coeff ` a ) ` (degAA ` A ) ) = 1 ) -> ( a ` A ) = 0 )
96	94, 95	anim12i 590	. . . . . . . . . 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 ) )
97		plyf 23954	. . . . . . . . . . . . . . . 16 |- ( p e. (Poly ` QQ ) -> p : CC --> CC )
98		ffn 6045	. . . . . . . . . . . . . . . 16 |- ( p : CC --> CC -> p Fn CC )
99	97, 98	syl 17	. . . . . . . . . . . . . . 15 |- ( p e. (Poly ` QQ ) -> p Fn CC )
100	99	ad2antrr 762	. . . . . . . . . . . . . 14 |- ( ( ( p e. (Poly ` QQ ) /\ a e. (Poly ` QQ ) ) /\ ( ( p ` A ) = 0 /\ ( a ` A ) = 0 ) ) -> p Fn CC )
101	46, 47	syl 17	. . . . . . . . . . . . . . 15 |- ( a e. (Poly ` QQ ) -> a Fn CC )
102	101	ad2antlr 763	. . . . . . . . . . . . . 14 |- ( ( ( p e. (Poly ` QQ ) /\ a e. (Poly ` QQ ) ) /\ ( ( p ` A ) = 0 /\ ( a ` A ) = 0 ) ) -> a Fn CC )
103	49	a1i 11	. . . . . . . . . . . . . 14 |- ( ( ( p e. (Poly ` QQ ) /\ a e. (Poly ` QQ ) ) /\ ( ( p ` A ) = 0 /\ ( a ` A ) = 0 ) ) -> CC e. _V )
104		simplrl 800	. . . . . . . . . . . . . 14 |- ( ( ( ( p e. (Poly ` QQ ) /\ a e. (Poly ` QQ ) ) /\ ( ( p ` A ) = 0 /\ ( a ` A ) = 0 ) ) /\ A e. CC ) -> ( p ` A ) = 0 )
105		simplrr 801	. . . . . . . . . . . . . 14 |- ( ( ( ( p e. (Poly ` QQ ) /\ a e. (Poly ` QQ ) ) /\ ( ( p ` A ) = 0 /\ ( a ` A ) = 0 ) ) /\ A e. CC ) -> ( a ` A ) = 0 )
106	100, 102, 103, 103, 51, 104, 105	ofval 6906	. . . . . . . . . . . . 13 |- ( ( ( ( p e. (Poly ` QQ ) /\ a e. (Poly ` QQ ) ) /\ ( ( p ` A ) = 0 /\ ( a ` A ) = 0 ) ) /\ A e. CC ) -> ( ( p oF - a ) ` A ) = ( 0 - 0 ) )
107	41, 106	sylan2 491	. . . . . . . . . . . 12 |- ( ( ( ( p e. (Poly ` QQ ) /\ a e. (Poly ` QQ ) ) /\ ( ( p ` A ) = 0 /\ ( a ` A ) = 0 ) ) /\ A e. AA ) -> ( ( p oF - a ) ` A ) = ( 0 - 0 ) )
108		0m0e0 11130	. . . . . . . . . . . 12 |- ( 0 - 0 ) = 0
109	107, 108	syl6eq 2672	. . . . . . . . . . 11 |- ( ( ( ( p e. (Poly ` QQ ) /\ a e. (Poly ` QQ ) ) /\ ( ( p ` A ) = 0 /\ ( a ` A ) = 0 ) ) /\ A e. AA ) -> ( ( p oF - a ) ` A ) = 0 )
110	109	ex 450	. . . . . . . . . 10 |- ( ( ( p e. (Poly ` QQ ) /\ a e. (Poly ` QQ ) ) /\ ( ( p ` A ) = 0 /\ ( a ` A ) = 0 ) ) -> ( A e. AA -> ( ( p oF - a ) ` A ) = 0 ) )
111	96, 110	sylan2 491	. . . . . . . . 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 ) )
112	111	com12 32	. . . . . . . 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 ) )
113	112	impl 650	. . . . . . 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 )
114		simpll 790	. . . . . . . 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 )
115		simpl 473	. . . . . . . . . 10 |- ( ( p e. (Poly ` QQ ) /\ a e. (Poly ` QQ ) ) -> p e. (Poly ` QQ ) )
116		simpr 477	. . . . . . . . . 10 |- ( ( p e. (Poly ` QQ ) /\ a e. (Poly ` QQ ) ) -> a e. (Poly ` QQ ) )
117	26	adantl 482	. . . . . . . . . 10 |- ( ( ( p e. (Poly ` QQ ) /\ a e. (Poly ` QQ ) ) /\ ( b e. QQ /\ c e. QQ ) ) -> ( b + c ) e. QQ )
118	28	adantl 482	. . . . . . . . . 10 |- ( ( ( p e. (Poly ` QQ ) /\ a e. (Poly ` QQ ) ) /\ ( b e. QQ /\ c e. QQ ) ) -> ( b x. c ) e. QQ )
119		1z 11407	. . . . . . . . . . . 12 |- 1 e. ZZ
120		zq 11794	. . . . . . . . . . . 12 |- ( 1 e. ZZ -> 1 e. QQ )
121		qnegcl 11805	. . . . . . . . . . . 12 |- ( 1 e. QQ -> -u 1 e. QQ )
122	119, 120, 121	mp2b 10	. . . . . . . . . . 11 |- -u 1 e. QQ
123	122	a1i 11	. . . . . . . . . 10 |- ( ( p e. (Poly ` QQ ) /\ a e. (Poly ` QQ ) ) -> -u 1 e. QQ )
124	115, 116, 117, 118, 123	plysub 23975	. . . . . . . . 9 |- ( ( p e. (Poly ` QQ ) /\ a e. (Poly ` QQ ) ) -> ( p oF - a ) e. (Poly ` QQ ) )
125	124	ad2antlr 763	. . . . . . . 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 ) )
126		simplrl 800	. . . . . . . . . 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 ) )
127		simplrr 801	. . . . . . . . . 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 ) )
128		simprr1 1109	. . . . . . . . . . 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 ) )
129		simprl1 1106	. . . . . . . . . . 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 ) )
130	128, 129	eqtr4d 2659	. . . . . . . . . 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 ) )
131	62	ad2antrr 762	. . . . . . . . . . 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 )
132	129, 131	eqeltrd 2701	. . . . . . . . . 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 )
133		simprl3 1108	. . . . . . . . . . 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 )
134	129	fveq2d 6195	. . . . . . . . . . 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 ) ) )
135	129	fveq2d 6195	. . . . . . . . . . . 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 ) ) )
136		simprr3 1111	. . . . . . . . . . . 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 )
137	135, 136	eqtrd 2656	. . . . . . . . . . 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 )
138	133, 134, 137	3eqtr4d 2666	. . . . . . . . . 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 ) ) )
139		eqid 2622	. . . . . . . . . . 11 |- (deg ` p ) = (deg ` p )
140	139	dgrsub2 37705	. . . . . . . . . 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 ) )
141	126, 127, 130, 132, 138, 140	syl23anc 1333	. . . . . . . . 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 ) )
142	141, 129	breqtrd 4679	. . . . . . . 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 ) )
143		dgraa0p 37719	. . . . . . . 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 ) )
144	114, 125, 142, 143	syl3anc 1326	. . . . . . 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 ) )
145	113, 144	mpbid 222	. . . . . 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 )
146		df-0p 23437	. . . . . 6 |- 0p = ( CC X. { 0 } )
147	145, 146	syl6eq 2672	. . . . 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 } ) )
148		ofsubeq0 11017	. . . . . . . 8 |- ( ( CC e. _V /\ p : CC --> CC /\ a : CC --> CC ) -> ( ( p oF - a ) = ( CC X. { 0 } ) <-> p = a ) )
149	49, 148	mp3an1 1411	. . . . . . 7 |- ( ( p : CC --> CC /\ a : CC --> CC ) -> ( ( p oF - a ) = ( CC X. { 0 } ) <-> p = a ) )
150	97, 46, 149	syl2an 494	. . . . . 6 |- ( ( p e. (Poly ` QQ ) /\ a e. (Poly ` QQ ) ) -> ( ( p oF - a ) = ( CC X. { 0 } ) <-> p = a ) )
151	150	ad2antlr 763	. . . . 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 ) )
152	147, 151	mpbid 222	. . . 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 )
153	152	ex 450	. . 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 ) )
154	153	ralrimivva 2971	. 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 ) )
155		fveq2 6191	. . . . 5 |- ( p = a -> (deg ` p ) = (deg ` a ) )
156	155	eqeq1d 2624	. . . 4 |- ( p = a -> ( (deg ` p ) = (degAA ` A ) <-> (deg ` a ) = (degAA ` A ) ) )
157		fveq1 6190	. . . . 5 |- ( p = a -> ( p ` A ) = ( a ` A ) )
158	157	eqeq1d 2624	. . . 4 |- ( p = a -> ( ( p ` A ) = 0 <-> ( a ` A ) = 0 ) )
159		fveq2 6191	. . . . . 6 |- ( p = a -> (coeff ` p ) = (coeff ` a ) )
160	159	fveq1d 6193	. . . . 5 |- ( p = a -> ( (coeff ` p ) ` (degAA ` A ) ) = ( (coeff ` a ) ` (degAA ` A ) ) )
161	160	eqeq1d 2624	. . . 4 |- ( p = a -> ( ( (coeff ` p ) ` (degAA ` A ) ) = 1 <-> ( (coeff ` a ) ` (degAA ` A ) ) = 1 ) )
162	156, 158, 161	3anbi123d 1399	. . 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 ) ) )
163	162	reu4 3400	. 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 ) ) )
164	93, 154, 163	sylanbrc 698	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 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   E!wreu 2914   _Vcvv 3200   \ cdif 3571   C_ wss 3574   {csn 4177   class class class wbr 4653   X. cxp 5112   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   < clt 10074   - cmin 10266   -ucneg 10267   / cdiv 10684   NNcn 11020   NN0cn0 11292   ZZcz 11377   QQcq 11788   0pc0p 23436  Polycply 23940  coeffccoe 23942  degcdgr 23943   AAcaa 24069  deg_AAcdgraa 37710

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-rlim 14220  df-sum 14417  df-0p 23437  df-ply 23944  df-coe 23946  df-dgr 23947  df-aa 24070  df-dgraa 37712

This theorem is referenced by:  mpaalem  37722