Theorem fta1 24063

Description: The easy direction of the Fundamental Theorem of Algebra: A nonzero polynomial has at most deg ( F ) roots. (Contributed by Mario Carneiro, 26-Jul-2014.)

Hypothesis

Ref	Expression
fta1.1	|- R = ( `' F " { 0 } )

Assertion

Ref	Expression
fta1	|- ( ( F e. (Poly ` S ) /\ F =/= 0p ) -> ( R e. Fin /\ ( # ` R ) <_ (deg ` F ) ) )

Proof of Theorem fta1

Dummy variables x g f d are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. 2 |- (deg ` F ) = (deg ` F )
2		dgrcl 23989	. . . . 5 |- ( F e. (Poly ` S ) -> (deg ` F ) e. NN0 )
3	2	adantr 481	. . . 4 |- ( ( F e. (Poly ` S ) /\ F =/= 0p ) -> (deg ` F ) e. NN0 )
4		eqeq2 2633	. . . . . . 7 |- ( x = 0 -> ( (deg ` f ) = x <-> (deg ` f ) = 0 ) )
5	4	imbi1d 331	. . . . . 6 |- ( x = 0 -> ( ( (deg ` f ) = x -> ( ( `' f " { 0 } ) e. Fin /\ ( # ` ( `' f " { 0 } ) ) <_ (deg ` f ) ) ) <-> ( (deg ` f ) = 0 -> ( ( `' f " { 0 } ) e. Fin /\ ( # ` ( `' f " { 0 } ) ) <_ (deg ` f ) ) ) ) )
6	5	ralbidv 2986	. . . . 5 |- ( x = 0 -> ( A. f e. ( (Poly ` CC ) \ { 0p } ) ( (deg ` f ) = x -> ( ( `' f " { 0 } ) e. Fin /\ ( # ` ( `' f " { 0 } ) ) <_ (deg ` f ) ) ) <-> A. f e. ( (Poly ` CC ) \ { 0p } ) ( (deg ` f ) = 0 -> ( ( `' f " { 0 } ) e. Fin /\ ( # ` ( `' f " { 0 } ) ) <_ (deg ` f ) ) ) ) )
7		eqeq2 2633	. . . . . . 7 |- ( x = d -> ( (deg ` f ) = x <-> (deg ` f ) = d ) )
8	7	imbi1d 331	. . . . . 6 |- ( x = d -> ( ( (deg ` f ) = x -> ( ( `' f " { 0 } ) e. Fin /\ ( # ` ( `' f " { 0 } ) ) <_ (deg ` f ) ) ) <-> ( (deg ` f ) = d -> ( ( `' f " { 0 } ) e. Fin /\ ( # ` ( `' f " { 0 } ) ) <_ (deg ` f ) ) ) ) )
9	8	ralbidv 2986	. . . . 5 |- ( x = d -> ( A. f e. ( (Poly ` CC ) \ { 0p } ) ( (deg ` f ) = x -> ( ( `' f " { 0 } ) e. Fin /\ ( # ` ( `' f " { 0 } ) ) <_ (deg ` f ) ) ) <-> A. f e. ( (Poly ` CC ) \ { 0p } ) ( (deg ` f ) = d -> ( ( `' f " { 0 } ) e. Fin /\ ( # ` ( `' f " { 0 } ) ) <_ (deg ` f ) ) ) ) )
10		eqeq2 2633	. . . . . . 7 |- ( x = ( d + 1 ) -> ( (deg ` f ) = x <-> (deg ` f ) = ( d + 1 ) ) )
11	10	imbi1d 331	. . . . . 6 |- ( x = ( d + 1 ) -> ( ( (deg ` f ) = x -> ( ( `' f " { 0 } ) e. Fin /\ ( # ` ( `' f " { 0 } ) ) <_ (deg ` f ) ) ) <-> ( (deg ` f ) = ( d + 1 ) -> ( ( `' f " { 0 } ) e. Fin /\ ( # ` ( `' f " { 0 } ) ) <_ (deg ` f ) ) ) ) )
12	11	ralbidv 2986	. . . . 5 |- ( x = ( d + 1 ) -> ( A. f e. ( (Poly ` CC ) \ { 0p } ) ( (deg ` f ) = x -> ( ( `' f " { 0 } ) e. Fin /\ ( # ` ( `' f " { 0 } ) ) <_ (deg ` f ) ) ) <-> A. f e. ( (Poly ` CC ) \ { 0p } ) ( (deg ` f ) = ( d + 1 ) -> ( ( `' f " { 0 } ) e. Fin /\ ( # ` ( `' f " { 0 } ) ) <_ (deg ` f ) ) ) ) )
13		eqeq2 2633	. . . . . . 7 |- ( x = (deg ` F ) -> ( (deg ` f ) = x <-> (deg ` f ) = (deg ` F ) ) )
14	13	imbi1d 331	. . . . . 6 |- ( x = (deg ` F ) -> ( ( (deg ` f ) = x -> ( ( `' f " { 0 } ) e. Fin /\ ( # ` ( `' f " { 0 } ) ) <_ (deg ` f ) ) ) <-> ( (deg ` f ) = (deg ` F ) -> ( ( `' f " { 0 } ) e. Fin /\ ( # ` ( `' f " { 0 } ) ) <_ (deg ` f ) ) ) ) )
15	14	ralbidv 2986	. . . . 5 |- ( x = (deg ` F ) -> ( A. f e. ( (Poly ` CC ) \ { 0p } ) ( (deg ` f ) = x -> ( ( `' f " { 0 } ) e. Fin /\ ( # ` ( `' f " { 0 } ) ) <_ (deg ` f ) ) ) <-> A. f e. ( (Poly ` CC ) \ { 0p } ) ( (deg ` f ) = (deg ` F ) -> ( ( `' f " { 0 } ) e. Fin /\ ( # ` ( `' f " { 0 } ) ) <_ (deg ` f ) ) ) ) )
16		eldifsni 4320	. . . . . . . . . . 11 |- ( f e. ( (Poly ` CC ) \ { 0p } ) -> f =/= 0p )
17	16	adantr 481	. . . . . . . . . 10 |- ( ( f e. ( (Poly ` CC ) \ { 0p } ) /\ (deg ` f ) = 0 ) -> f =/= 0p )
18		simplr 792	. . . . . . . . . . . . . . 15 |- ( ( ( f e. ( (Poly ` CC ) \ { 0p } ) /\ (deg ` f ) = 0 ) /\ x e. ( `' f " { 0 } ) ) -> (deg ` f ) = 0 )
19		eldifi 3732	. . . . . . . . . . . . . . . . 17 |- ( f e. ( (Poly ` CC ) \ { 0p } ) -> f e. (Poly ` CC ) )
20	19	ad2antrr 762	. . . . . . . . . . . . . . . 16 |- ( ( ( f e. ( (Poly ` CC ) \ { 0p } ) /\ (deg ` f ) = 0 ) /\ x e. ( `' f " { 0 } ) ) -> f e. (Poly ` CC ) )
21		0dgrb 24002	. . . . . . . . . . . . . . . 16 |- ( f e. (Poly ` CC ) -> ( (deg ` f ) = 0 <-> f = ( CC X. { ( f ` 0 ) } ) ) )
22	20, 21	syl 17	. . . . . . . . . . . . . . 15 |- ( ( ( f e. ( (Poly ` CC ) \ { 0p } ) /\ (deg ` f ) = 0 ) /\ x e. ( `' f " { 0 } ) ) -> ( (deg ` f ) = 0 <-> f = ( CC X. { ( f ` 0 ) } ) ) )
23	18, 22	mpbid 222	. . . . . . . . . . . . . 14 |- ( ( ( f e. ( (Poly ` CC ) \ { 0p } ) /\ (deg ` f ) = 0 ) /\ x e. ( `' f " { 0 } ) ) -> f = ( CC X. { ( f ` 0 ) } ) )
24	23	fveq1d 6193	. . . . . . . . . . . . . . . . 17 |- ( ( ( f e. ( (Poly ` CC ) \ { 0p } ) /\ (deg ` f ) = 0 ) /\ x e. ( `' f " { 0 } ) ) -> ( f ` x ) = ( ( CC X. { ( f ` 0 ) } ) ` x ) )
25	19	adantr 481	. . . . . . . . . . . . . . . . . . . 20 |- ( ( f e. ( (Poly ` CC ) \ { 0p } ) /\ (deg ` f ) = 0 ) -> f e. (Poly ` CC ) )
26		plyf 23954	. . . . . . . . . . . . . . . . . . . 20 |- ( f e. (Poly ` CC ) -> f : CC --> CC )
27		ffn 6045	. . . . . . . . . . . . . . . . . . . 20 |- ( f : CC --> CC -> f Fn CC )
28		fniniseg 6338	. . . . . . . . . . . . . . . . . . . 20 |- ( f Fn CC -> ( x e. ( `' f " { 0 } ) <-> ( x e. CC /\ ( f ` x ) = 0 ) ) )
29	25, 26, 27, 28	4syl 19	. . . . . . . . . . . . . . . . . . 19 |- ( ( f e. ( (Poly ` CC ) \ { 0p } ) /\ (deg ` f ) = 0 ) -> ( x e. ( `' f " { 0 } ) <-> ( x e. CC /\ ( f ` x ) = 0 ) ) )
30	29	biimpa 501	. . . . . . . . . . . . . . . . . 18 |- ( ( ( f e. ( (Poly ` CC ) \ { 0p } ) /\ (deg ` f ) = 0 ) /\ x e. ( `' f " { 0 } ) ) -> ( x e. CC /\ ( f ` x ) = 0 ) )
31	30	simprd 479	. . . . . . . . . . . . . . . . 17 |- ( ( ( f e. ( (Poly ` CC ) \ { 0p } ) /\ (deg ` f ) = 0 ) /\ x e. ( `' f " { 0 } ) ) -> ( f ` x ) = 0 )
32	30	simpld 475	. . . . . . . . . . . . . . . . . 18 |- ( ( ( f e. ( (Poly ` CC ) \ { 0p } ) /\ (deg ` f ) = 0 ) /\ x e. ( `' f " { 0 } ) ) -> x e. CC )
33		fvex 6201	. . . . . . . . . . . . . . . . . . 19 |- ( f ` 0 ) e. _V
34	33	fvconst2 6469	. . . . . . . . . . . . . . . . . 18 |- ( x e. CC -> ( ( CC X. { ( f ` 0 ) } ) ` x ) = ( f ` 0 ) )
35	32, 34	syl 17	. . . . . . . . . . . . . . . . 17 |- ( ( ( f e. ( (Poly ` CC ) \ { 0p } ) /\ (deg ` f ) = 0 ) /\ x e. ( `' f " { 0 } ) ) -> ( ( CC X. { ( f ` 0 ) } ) ` x ) = ( f ` 0 ) )
36	24, 31, 35	3eqtr3rd 2665	. . . . . . . . . . . . . . . 16 |- ( ( ( f e. ( (Poly ` CC ) \ { 0p } ) /\ (deg ` f ) = 0 ) /\ x e. ( `' f " { 0 } ) ) -> ( f ` 0 ) = 0 )
37	36	sneqd 4189	. . . . . . . . . . . . . . 15 |- ( ( ( f e. ( (Poly ` CC ) \ { 0p } ) /\ (deg ` f ) = 0 ) /\ x e. ( `' f " { 0 } ) ) -> { ( f ` 0 ) } = { 0 } )
38	37	xpeq2d 5139	. . . . . . . . . . . . . 14 |- ( ( ( f e. ( (Poly ` CC ) \ { 0p } ) /\ (deg ` f ) = 0 ) /\ x e. ( `' f " { 0 } ) ) -> ( CC X. { ( f ` 0 ) } ) = ( CC X. { 0 } ) )
39	23, 38	eqtrd 2656	. . . . . . . . . . . . 13 |- ( ( ( f e. ( (Poly ` CC ) \ { 0p } ) /\ (deg ` f ) = 0 ) /\ x e. ( `' f " { 0 } ) ) -> f = ( CC X. { 0 } ) )
40		df-0p 23437	. . . . . . . . . . . . 13 |- 0p = ( CC X. { 0 } )
41	39, 40	syl6eqr 2674	. . . . . . . . . . . 12 |- ( ( ( f e. ( (Poly ` CC ) \ { 0p } ) /\ (deg ` f ) = 0 ) /\ x e. ( `' f " { 0 } ) ) -> f = 0p )
42	41	ex 450	. . . . . . . . . . 11 |- ( ( f e. ( (Poly ` CC ) \ { 0p } ) /\ (deg ` f ) = 0 ) -> ( x e. ( `' f " { 0 } ) -> f = 0p ) )
43	42	necon3ad 2807	. . . . . . . . . 10 |- ( ( f e. ( (Poly ` CC ) \ { 0p } ) /\ (deg ` f ) = 0 ) -> ( f =/= 0p -> -. x e. ( `' f " { 0 } ) ) )
44	17, 43	mpd 15	. . . . . . . . 9 |- ( ( f e. ( (Poly ` CC ) \ { 0p } ) /\ (deg ` f ) = 0 ) -> -. x e. ( `' f " { 0 } ) )
45	44	eq0rdv 3979	. . . . . . . 8 |- ( ( f e. ( (Poly ` CC ) \ { 0p } ) /\ (deg ` f ) = 0 ) -> ( `' f " { 0 } ) = (/) )
46	45	ex 450	. . . . . . 7 |- ( f e. ( (Poly ` CC ) \ { 0p } ) -> ( (deg ` f ) = 0 -> ( `' f " { 0 } ) = (/) ) )
47		dgrcl 23989	. . . . . . . . 9 |- ( f e. (Poly ` CC ) -> (deg ` f ) e. NN0 )
48		nn0ge0 11318	. . . . . . . . 9 |- ( (deg ` f ) e. NN0 -> 0 <_ (deg ` f ) )
49	19, 47, 48	3syl 18	. . . . . . . 8 |- ( f e. ( (Poly ` CC ) \ { 0p } ) -> 0 <_ (deg ` f ) )
50		id 22	. . . . . . . . . . 11 |- ( ( `' f " { 0 } ) = (/) -> ( `' f " { 0 } ) = (/) )
51		0fin 8188	. . . . . . . . . . 11 |- (/) e. Fin
52	50, 51	syl6eqel 2709	. . . . . . . . . 10 |- ( ( `' f " { 0 } ) = (/) -> ( `' f " { 0 } ) e. Fin )
53	52	biantrurd 529	. . . . . . . . 9 |- ( ( `' f " { 0 } ) = (/) -> ( ( # ` ( `' f " { 0 } ) ) <_ (deg ` f ) <-> ( ( `' f " { 0 } ) e. Fin /\ ( # ` ( `' f " { 0 } ) ) <_ (deg ` f ) ) ) )
54		fveq2 6191	. . . . . . . . . . 11 |- ( ( `' f " { 0 } ) = (/) -> ( # ` ( `' f " { 0 } ) ) = ( # ` (/) ) )
55		hash0 13158	. . . . . . . . . . 11 |- ( # ` (/) ) = 0
56	54, 55	syl6eq 2672	. . . . . . . . . 10 |- ( ( `' f " { 0 } ) = (/) -> ( # ` ( `' f " { 0 } ) ) = 0 )
57	56	breq1d 4663	. . . . . . . . 9 |- ( ( `' f " { 0 } ) = (/) -> ( ( # ` ( `' f " { 0 } ) ) <_ (deg ` f ) <-> 0 <_ (deg ` f ) ) )
58	53, 57	bitr3d 270	. . . . . . . 8 |- ( ( `' f " { 0 } ) = (/) -> ( ( ( `' f " { 0 } ) e. Fin /\ ( # ` ( `' f " { 0 } ) ) <_ (deg ` f ) ) <-> 0 <_ (deg ` f ) ) )
59	49, 58	syl5ibrcom 237	. . . . . . 7 |- ( f e. ( (Poly ` CC ) \ { 0p } ) -> ( ( `' f " { 0 } ) = (/) -> ( ( `' f " { 0 } ) e. Fin /\ ( # ` ( `' f " { 0 } ) ) <_ (deg ` f ) ) ) )
60	46, 59	syld 47	. . . . . 6 |- ( f e. ( (Poly ` CC ) \ { 0p } ) -> ( (deg ` f ) = 0 -> ( ( `' f " { 0 } ) e. Fin /\ ( # ` ( `' f " { 0 } ) ) <_ (deg ` f ) ) ) )
61	60	rgen 2922	. . . . 5 |- A. f e. ( (Poly ` CC ) \ { 0p } ) ( (deg ` f ) = 0 -> ( ( `' f " { 0 } ) e. Fin /\ ( # ` ( `' f " { 0 } ) ) <_ (deg ` f ) ) )
62		fveq2 6191	. . . . . . . . 9 |- ( f = g -> (deg ` f ) = (deg ` g ) )
63	62	eqeq1d 2624	. . . . . . . 8 |- ( f = g -> ( (deg ` f ) = d <-> (deg ` g ) = d ) )
64		cnveq 5296	. . . . . . . . . . 11 |- ( f = g -> `' f = `' g )
65	64	imaeq1d 5465	. . . . . . . . . 10 |- ( f = g -> ( `' f " { 0 } ) = ( `' g " { 0 } ) )
66	65	eleq1d 2686	. . . . . . . . 9 |- ( f = g -> ( ( `' f " { 0 } ) e. Fin <-> ( `' g " { 0 } ) e. Fin ) )
67	65	fveq2d 6195	. . . . . . . . . 10 |- ( f = g -> ( # ` ( `' f " { 0 } ) ) = ( # ` ( `' g " { 0 } ) ) )
68	67, 62	breq12d 4666	. . . . . . . . 9 |- ( f = g -> ( ( # ` ( `' f " { 0 } ) ) <_ (deg ` f ) <-> ( # ` ( `' g " { 0 } ) ) <_ (deg ` g ) ) )
69	66, 68	anbi12d 747	. . . . . . . 8 |- ( f = g -> ( ( ( `' f " { 0 } ) e. Fin /\ ( # ` ( `' f " { 0 } ) ) <_ (deg ` f ) ) <-> ( ( `' g " { 0 } ) e. Fin /\ ( # ` ( `' g " { 0 } ) ) <_ (deg ` g ) ) ) )
70	63, 69	imbi12d 334	. . . . . . 7 |- ( f = g -> ( ( (deg ` f ) = d -> ( ( `' f " { 0 } ) e. Fin /\ ( # ` ( `' f " { 0 } ) ) <_ (deg ` f ) ) ) <-> ( (deg ` g ) = d -> ( ( `' g " { 0 } ) e. Fin /\ ( # ` ( `' g " { 0 } ) ) <_ (deg ` g ) ) ) ) )
71	70	cbvralv 3171	. . . . . 6 |- ( A. f e. ( (Poly ` CC ) \ { 0p } ) ( (deg ` f ) = d -> ( ( `' f " { 0 } ) e. Fin /\ ( # ` ( `' f " { 0 } ) ) <_ (deg ` f ) ) ) <-> A. g e. ( (Poly ` CC ) \ { 0p } ) ( (deg ` g ) = d -> ( ( `' g " { 0 } ) e. Fin /\ ( # ` ( `' g " { 0 } ) ) <_ (deg ` g ) ) ) )
72	49	ad2antlr 763	. . . . . . . . . . . 12 |- ( ( ( d e. NN0 /\ f e. ( (Poly ` CC ) \ { 0p } ) ) /\ (deg ` f ) = ( d + 1 ) ) -> 0 <_ (deg ` f ) )
73	72, 58	syl5ibrcom 237	. . . . . . . . . . 11 |- ( ( ( d e. NN0 /\ f e. ( (Poly ` CC ) \ { 0p } ) ) /\ (deg ` f ) = ( d + 1 ) ) -> ( ( `' f " { 0 } ) = (/) -> ( ( `' f " { 0 } ) e. Fin /\ ( # ` ( `' f " { 0 } ) ) <_ (deg ` f ) ) ) )
74	73	a1dd 50	. . . . . . . . . 10 |- ( ( ( d e. NN0 /\ f e. ( (Poly ` CC ) \ { 0p } ) ) /\ (deg ` f ) = ( d + 1 ) ) -> ( ( `' f " { 0 } ) = (/) -> ( A. g e. ( (Poly ` CC ) \ { 0p } ) ( (deg ` g ) = d -> ( ( `' g " { 0 } ) e. Fin /\ ( # ` ( `' g " { 0 } ) ) <_ (deg ` g ) ) ) -> ( ( `' f " { 0 } ) e. Fin /\ ( # ` ( `' f " { 0 } ) ) <_ (deg ` f ) ) ) ) )
75		n0 3931	. . . . . . . . . . 11 |- ( ( `' f " { 0 } ) =/= (/) <-> E. x x e. ( `' f " { 0 } ) )
76		eqid 2622	. . . . . . . . . . . . . 14 |- ( `' f " { 0 } ) = ( `' f " { 0 } )
77		simplll 798	. . . . . . . . . . . . . 14 |- ( ( ( ( d e. NN0 /\ f e. ( (Poly ` CC ) \ { 0p } ) ) /\ (deg ` f ) = ( d + 1 ) ) /\ ( x e. ( `' f " { 0 } ) /\ A. g e. ( (Poly ` CC ) \ { 0p } ) ( (deg ` g ) = d -> ( ( `' g " { 0 } ) e. Fin /\ ( # ` ( `' g " { 0 } ) ) <_ (deg ` g ) ) ) ) ) -> d e. NN0 )
78		simpllr 799	. . . . . . . . . . . . . 14 |- ( ( ( ( d e. NN0 /\ f e. ( (Poly ` CC ) \ { 0p } ) ) /\ (deg ` f ) = ( d + 1 ) ) /\ ( x e. ( `' f " { 0 } ) /\ A. g e. ( (Poly ` CC ) \ { 0p } ) ( (deg ` g ) = d -> ( ( `' g " { 0 } ) e. Fin /\ ( # ` ( `' g " { 0 } ) ) <_ (deg ` g ) ) ) ) ) -> f e. ( (Poly ` CC ) \ { 0p } ) )
79		simplr 792	. . . . . . . . . . . . . 14 |- ( ( ( ( d e. NN0 /\ f e. ( (Poly ` CC ) \ { 0p } ) ) /\ (deg ` f ) = ( d + 1 ) ) /\ ( x e. ( `' f " { 0 } ) /\ A. g e. ( (Poly ` CC ) \ { 0p } ) ( (deg ` g ) = d -> ( ( `' g " { 0 } ) e. Fin /\ ( # ` ( `' g " { 0 } ) ) <_ (deg ` g ) ) ) ) ) -> (deg ` f ) = ( d + 1 ) )
80		simprl 794	. . . . . . . . . . . . . 14 |- ( ( ( ( d e. NN0 /\ f e. ( (Poly ` CC ) \ { 0p } ) ) /\ (deg ` f ) = ( d + 1 ) ) /\ ( x e. ( `' f " { 0 } ) /\ A. g e. ( (Poly ` CC ) \ { 0p } ) ( (deg ` g ) = d -> ( ( `' g " { 0 } ) e. Fin /\ ( # ` ( `' g " { 0 } ) ) <_ (deg ` g ) ) ) ) ) -> x e. ( `' f " { 0 } ) )
81		simprr 796	. . . . . . . . . . . . . 14 |- ( ( ( ( d e. NN0 /\ f e. ( (Poly ` CC ) \ { 0p } ) ) /\ (deg ` f ) = ( d + 1 ) ) /\ ( x e. ( `' f " { 0 } ) /\ A. g e. ( (Poly ` CC ) \ { 0p } ) ( (deg ` g ) = d -> ( ( `' g " { 0 } ) e. Fin /\ ( # ` ( `' g " { 0 } ) ) <_ (deg ` g ) ) ) ) ) -> A. g e. ( (Poly ` CC ) \ { 0p } ) ( (deg ` g ) = d -> ( ( `' g " { 0 } ) e. Fin /\ ( # ` ( `' g " { 0 } ) ) <_ (deg ` g ) ) ) )
82	76, 77, 78, 79, 80, 81	fta1lem 24062	. . . . . . . . . . . . 13 |- ( ( ( ( d e. NN0 /\ f e. ( (Poly ` CC ) \ { 0p } ) ) /\ (deg ` f ) = ( d + 1 ) ) /\ ( x e. ( `' f " { 0 } ) /\ A. g e. ( (Poly ` CC ) \ { 0p } ) ( (deg ` g ) = d -> ( ( `' g " { 0 } ) e. Fin /\ ( # ` ( `' g " { 0 } ) ) <_ (deg ` g ) ) ) ) ) -> ( ( `' f " { 0 } ) e. Fin /\ ( # ` ( `' f " { 0 } ) ) <_ (deg ` f ) ) )
83	82	exp32 631	. . . . . . . . . . . 12 |- ( ( ( d e. NN0 /\ f e. ( (Poly ` CC ) \ { 0p } ) ) /\ (deg ` f ) = ( d + 1 ) ) -> ( x e. ( `' f " { 0 } ) -> ( A. g e. ( (Poly ` CC ) \ { 0p } ) ( (deg ` g ) = d -> ( ( `' g " { 0 } ) e. Fin /\ ( # ` ( `' g " { 0 } ) ) <_ (deg ` g ) ) ) -> ( ( `' f " { 0 } ) e. Fin /\ ( # ` ( `' f " { 0 } ) ) <_ (deg ` f ) ) ) ) )
84	83	exlimdv 1861	. . . . . . . . . . 11 |- ( ( ( d e. NN0 /\ f e. ( (Poly ` CC ) \ { 0p } ) ) /\ (deg ` f ) = ( d + 1 ) ) -> ( E. x x e. ( `' f " { 0 } ) -> ( A. g e. ( (Poly ` CC ) \ { 0p } ) ( (deg ` g ) = d -> ( ( `' g " { 0 } ) e. Fin /\ ( # ` ( `' g " { 0 } ) ) <_ (deg ` g ) ) ) -> ( ( `' f " { 0 } ) e. Fin /\ ( # ` ( `' f " { 0 } ) ) <_ (deg ` f ) ) ) ) )
85	75, 84	syl5bi 232	. . . . . . . . . 10 |- ( ( ( d e. NN0 /\ f e. ( (Poly ` CC ) \ { 0p } ) ) /\ (deg ` f ) = ( d + 1 ) ) -> ( ( `' f " { 0 } ) =/= (/) -> ( A. g e. ( (Poly ` CC ) \ { 0p } ) ( (deg ` g ) = d -> ( ( `' g " { 0 } ) e. Fin /\ ( # ` ( `' g " { 0 } ) ) <_ (deg ` g ) ) ) -> ( ( `' f " { 0 } ) e. Fin /\ ( # ` ( `' f " { 0 } ) ) <_ (deg ` f ) ) ) ) )
86	74, 85	pm2.61dne 2880	. . . . . . . . 9 |- ( ( ( d e. NN0 /\ f e. ( (Poly ` CC ) \ { 0p } ) ) /\ (deg ` f ) = ( d + 1 ) ) -> ( A. g e. ( (Poly ` CC ) \ { 0p } ) ( (deg ` g ) = d -> ( ( `' g " { 0 } ) e. Fin /\ ( # ` ( `' g " { 0 } ) ) <_ (deg ` g ) ) ) -> ( ( `' f " { 0 } ) e. Fin /\ ( # ` ( `' f " { 0 } ) ) <_ (deg ` f ) ) ) )
87	86	ex 450	. . . . . . . 8 |- ( ( d e. NN0 /\ f e. ( (Poly ` CC ) \ { 0p } ) ) -> ( (deg ` f ) = ( d + 1 ) -> ( A. g e. ( (Poly ` CC ) \ { 0p } ) ( (deg ` g ) = d -> ( ( `' g " { 0 } ) e. Fin /\ ( # ` ( `' g " { 0 } ) ) <_ (deg ` g ) ) ) -> ( ( `' f " { 0 } ) e. Fin /\ ( # ` ( `' f " { 0 } ) ) <_ (deg ` f ) ) ) ) )
88	87	com23 86	. . . . . . 7 |- ( ( d e. NN0 /\ f e. ( (Poly ` CC ) \ { 0p } ) ) -> ( A. g e. ( (Poly ` CC ) \ { 0p } ) ( (deg ` g ) = d -> ( ( `' g " { 0 } ) e. Fin /\ ( # ` ( `' g " { 0 } ) ) <_ (deg ` g ) ) ) -> ( (deg ` f ) = ( d + 1 ) -> ( ( `' f " { 0 } ) e. Fin /\ ( # ` ( `' f " { 0 } ) ) <_ (deg ` f ) ) ) ) )
89	88	ralrimdva 2969	. . . . . 6 |- ( d e. NN0 -> ( A. g e. ( (Poly ` CC ) \ { 0p } ) ( (deg ` g ) = d -> ( ( `' g " { 0 } ) e. Fin /\ ( # ` ( `' g " { 0 } ) ) <_ (deg ` g ) ) ) -> A. f e. ( (Poly ` CC ) \ { 0p } ) ( (deg ` f ) = ( d + 1 ) -> ( ( `' f " { 0 } ) e. Fin /\ ( # ` ( `' f " { 0 } ) ) <_ (deg ` f ) ) ) ) )
90	71, 89	syl5bi 232	. . . . 5 |- ( d e. NN0 -> ( A. f e. ( (Poly ` CC ) \ { 0p } ) ( (deg ` f ) = d -> ( ( `' f " { 0 } ) e. Fin /\ ( # ` ( `' f " { 0 } ) ) <_ (deg ` f ) ) ) -> A. f e. ( (Poly ` CC ) \ { 0p } ) ( (deg ` f ) = ( d + 1 ) -> ( ( `' f " { 0 } ) e. Fin /\ ( # ` ( `' f " { 0 } ) ) <_ (deg ` f ) ) ) ) )
91	6, 9, 12, 15, 61, 90	nn0ind 11472	. . . 4 |- ( (deg ` F ) e. NN0 -> A. f e. ( (Poly ` CC ) \ { 0p } ) ( (deg ` f ) = (deg ` F ) -> ( ( `' f " { 0 } ) e. Fin /\ ( # ` ( `' f " { 0 } ) ) <_ (deg ` f ) ) ) )
92	3, 91	syl 17	. . 3 |- ( ( F e. (Poly ` S ) /\ F =/= 0p ) -> A. f e. ( (Poly ` CC ) \ { 0p } ) ( (deg ` f ) = (deg ` F ) -> ( ( `' f " { 0 } ) e. Fin /\ ( # ` ( `' f " { 0 } ) ) <_ (deg ` f ) ) ) )
93		plyssc 23956	. . . . 5 |- (Poly ` S ) C_ (Poly ` CC )
94	93	sseli 3599	. . . 4 |- ( F e. (Poly ` S ) -> F e. (Poly ` CC ) )
95		eldifsn 4317	. . . . 5 |- ( F e. ( (Poly ` CC ) \ { 0p } ) <-> ( F e. (Poly ` CC ) /\ F =/= 0p ) )
96		fveq2 6191	. . . . . . . 8 |- ( f = F -> (deg ` f ) = (deg ` F ) )
97	96	eqeq1d 2624	. . . . . . 7 |- ( f = F -> ( (deg ` f ) = (deg ` F ) <-> (deg ` F ) = (deg ` F ) ) )
98		cnveq 5296	. . . . . . . . . . 11 |- ( f = F -> `' f = `' F )
99	98	imaeq1d 5465	. . . . . . . . . 10 |- ( f = F -> ( `' f " { 0 } ) = ( `' F " { 0 } ) )
100		fta1.1	. . . . . . . . . 10 |- R = ( `' F " { 0 } )
101	99, 100	syl6eqr 2674	. . . . . . . . 9 |- ( f = F -> ( `' f " { 0 } ) = R )
102	101	eleq1d 2686	. . . . . . . 8 |- ( f = F -> ( ( `' f " { 0 } ) e. Fin <-> R e. Fin ) )
103	101	fveq2d 6195	. . . . . . . . 9 |- ( f = F -> ( # ` ( `' f " { 0 } ) ) = ( # ` R ) )
104	103, 96	breq12d 4666	. . . . . . . 8 |- ( f = F -> ( ( # ` ( `' f " { 0 } ) ) <_ (deg ` f ) <-> ( # ` R ) <_ (deg ` F ) ) )
105	102, 104	anbi12d 747	. . . . . . 7 |- ( f = F -> ( ( ( `' f " { 0 } ) e. Fin /\ ( # ` ( `' f " { 0 } ) ) <_ (deg ` f ) ) <-> ( R e. Fin /\ ( # ` R ) <_ (deg ` F ) ) ) )
106	97, 105	imbi12d 334	. . . . . 6 |- ( f = F -> ( ( (deg ` f ) = (deg ` F ) -> ( ( `' f " { 0 } ) e. Fin /\ ( # ` ( `' f " { 0 } ) ) <_ (deg ` f ) ) ) <-> ( (deg ` F ) = (deg ` F ) -> ( R e. Fin /\ ( # ` R ) <_ (deg ` F ) ) ) ) )
107	106	rspcv 3305	. . . . 5 |- ( F e. ( (Poly ` CC ) \ { 0p } ) -> ( A. f e. ( (Poly ` CC ) \ { 0p } ) ( (deg ` f ) = (deg ` F ) -> ( ( `' f " { 0 } ) e. Fin /\ ( # ` ( `' f " { 0 } ) ) <_ (deg ` f ) ) ) -> ( (deg ` F ) = (deg ` F ) -> ( R e. Fin /\ ( # ` R ) <_ (deg ` F ) ) ) ) )
108	95, 107	sylbir 225	. . . 4 |- ( ( F e. (Poly ` CC ) /\ F =/= 0p ) -> ( A. f e. ( (Poly ` CC ) \ { 0p } ) ( (deg ` f ) = (deg ` F ) -> ( ( `' f " { 0 } ) e. Fin /\ ( # ` ( `' f " { 0 } ) ) <_ (deg ` f ) ) ) -> ( (deg ` F ) = (deg ` F ) -> ( R e. Fin /\ ( # ` R ) <_ (deg ` F ) ) ) ) )
109	94, 108	sylan 488	. . 3 |- ( ( F e. (Poly ` S ) /\ F =/= 0p ) -> ( A. f e. ( (Poly ` CC ) \ { 0p } ) ( (deg ` f ) = (deg ` F ) -> ( ( `' f " { 0 } ) e. Fin /\ ( # ` ( `' f " { 0 } ) ) <_ (deg ` f ) ) ) -> ( (deg ` F ) = (deg ` F ) -> ( R e. Fin /\ ( # ` R ) <_ (deg ` F ) ) ) ) )
110	92, 109	mpd 15	. 2 |- ( ( F e. (Poly ` S ) /\ F =/= 0p ) -> ( (deg ` F ) = (deg ` F ) -> ( R e. Fin /\ ( # ` R ) <_ (deg ` F ) ) ) )
111	1, 110	mpi 20	1 |- ( ( F e. (Poly ` S ) /\ F =/= 0p ) -> ( R e. Fin /\ ( # ` R ) <_ (deg ` F ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   A.wral 2912   \ cdif 3571   (/)c0 3915   {csn 4177   class class class wbr 4653   X. cxp 5112   `'ccnv 5113   "cima 5117   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   Fincfn 7955   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   <_ cle 10075   NN0cn0 11292   #chash 13117   0pc0p 23436  Polycply 23940  degcdgr 23943

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-cda 8990  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-xnn0 11364  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  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-idp 23945  df-coe 23946  df-dgr 23947  df-quot 24046

This theorem is referenced by:  vieta1lem2  24066  vieta1  24067  plyexmo  24068  aannenlem1  24083  aalioulem2  24088  basellem4  24810  basellem5  24811  dchrfi  24980