Theorem aannenlem1 22458

Description: Lemma for aannen 22461. (Contributed by Stefan O'Rear, 16-Nov-2014.)

Hypothesis

Ref	Expression
aannenlem.a	|- H = ( a e. NN0 |-> { b e. CC | E. c e. { d e. (Poly ` ZZ ) | ( d =/= 0p /\ (deg ` d ) <_ a /\ A. e e. NN0 ( abs ` ( (coeff ` d ) ` e ) ) <_ a ) } ( c ` b ) = 0 } )

Assertion

Ref	Expression
aannenlem1	|- ( A e. NN0 -> ( H ` A ) e. Fin )

Distinct variable group:   A, a, b, c, d, e

Allowed substitution hints:   H( e, a, b, c, d)

Proof of Theorem aannenlem1

Step	Hyp	Ref	Expression
1		breq2 4451	. . . . . . 7 |- ( a = A -> ( (deg ` d ) <_ a <-> (deg ` d ) <_ A ) )
2		breq2 4451	. . . . . . . 8 |- ( a = A -> ( ( abs ` ( (coeff ` d ) ` e ) ) <_ a <-> ( abs ` ( (coeff ` d ) ` e ) ) <_ A ) )
3	2	ralbidv 2903	. . . . . . 7 |- ( a = A -> ( A. e e. NN0 ( abs ` ( (coeff ` d ) ` e ) ) <_ a <-> A. e e. NN0 ( abs ` ( (coeff ` d ) ` e ) ) <_ A ) )
4	1, 3	3anbi23d 1302	. . . . . 6 |- ( a = A -> ( ( d =/= 0p /\ (deg ` d ) <_ a /\ A. e e. NN0 ( abs ` ( (coeff ` d ) ` e ) ) <_ a ) <-> ( d =/= 0p /\ (deg ` d ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` d ) ` e ) ) <_ A ) ) )
5	4	rabbidv 3105	. . . . 5 |- ( a = A -> { d e. (Poly ` ZZ ) | ( d =/= 0p /\ (deg ` d ) <_ a /\ A. e e. NN0 ( abs ` ( (coeff ` d ) ` e ) ) <_ a ) } = { d e. (Poly ` ZZ ) | ( d =/= 0p /\ (deg ` d ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` d ) ` e ) ) <_ A ) } )
6	5	rexeqdv 3065	. . . 4 |- ( a = A -> ( E. c e. { d e. (Poly ` ZZ ) | ( d =/= 0p /\ (deg ` d ) <_ a /\ A. e e. NN0 ( abs ` ( (coeff ` d ) ` e ) ) <_ a ) } ( c ` b ) = 0 <-> E. c e. { d e. (Poly ` ZZ ) | ( d =/= 0p /\ (deg ` d ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` d ) ` e ) ) <_ A ) } ( c ` b ) = 0 ) )
7	6	rabbidv 3105	. . 3 |- ( a = A -> { b e. CC | E. c e. { d e. (Poly ` ZZ ) | ( d =/= 0p /\ (deg ` d ) <_ a /\ A. e e. NN0 ( abs ` ( (coeff ` d ) ` e ) ) <_ a ) } ( c ` b ) = 0 } = { b e. CC | E. c e. { d e. (Poly ` ZZ ) | ( d =/= 0p /\ (deg ` d ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` d ) ` e ) ) <_ A ) } ( c ` b ) = 0 } )
8		aannenlem.a	. . 3 |- H = ( a e. NN0 |-> { b e. CC | E. c e. { d e. (Poly ` ZZ ) | ( d =/= 0p /\ (deg ` d ) <_ a /\ A. e e. NN0 ( abs ` ( (coeff ` d ) ` e ) ) <_ a ) } ( c ` b ) = 0 } )
9		cnex 9569	. . . 4 |- CC e. _V
10	9	rabex 4598	. . 3 |- { b e. CC | E. c e. { d e. (Poly ` ZZ ) | ( d =/= 0p /\ (deg ` d ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` d ) ` e ) ) <_ A ) } ( c ` b ) = 0 } e. _V
11	7, 8, 10	fvmpt 5948	. 2 |- ( A e. NN0 -> ( H ` A ) = { b e. CC | E. c e. { d e. (Poly ` ZZ ) | ( d =/= 0p /\ (deg ` d ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` d ) ` e ) ) <_ A ) } ( c ` b ) = 0 } )
12		iunrab 4372	. . 3 |- U_ c e. { d e. (Poly ` ZZ ) | ( d =/= 0p /\ (deg ` d ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` d ) ` e ) ) <_ A ) } { b e. CC | ( c ` b ) = 0 } = { b e. CC | E. c e. { d e. (Poly ` ZZ ) | ( d =/= 0p /\ (deg ` d ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` d ) ` e ) ) <_ A ) } ( c ` b ) = 0 }
13		fzfi 12046	. . . . . . 7 |- ( -u A ... A ) e. Fin
14		fzfi 12046	. . . . . . 7 |- ( 0 ... A ) e. Fin
15		mapfi 7812	. . . . . . 7 |- ( ( ( -u A ... A ) e. Fin /\ ( 0 ... A ) e. Fin ) -> ( ( -u A ... A ) ^m ( 0 ... A ) ) e. Fin )
16	13, 14, 15	mp2an 672	. . . . . 6 |- ( ( -u A ... A ) ^m ( 0 ... A ) ) e. Fin
17	16	a1i 11	. . . . 5 |- ( A e. NN0 -> ( ( -u A ... A ) ^m ( 0 ... A ) ) e. Fin )
18		ovex 6307	. . . . . 6 |- ( ( -u A ... A ) ^m ( 0 ... A ) ) e. _V
19		neeq1 2748	. . . . . . . . . . 11 |- ( d = a -> ( d =/= 0p <-> a =/= 0p ) )
20		fveq2 5864	. . . . . . . . . . . 12 |- ( d = a -> (deg ` d ) = (deg ` a ) )
21	20	breq1d 4457	. . . . . . . . . . 11 |- ( d = a -> ( (deg ` d ) <_ A <-> (deg ` a ) <_ A ) )
22		fveq2 5864	. . . . . . . . . . . . . . 15 |- ( d = a -> (coeff ` d ) = (coeff ` a ) )
23	22	fveq1d 5866	. . . . . . . . . . . . . 14 |- ( d = a -> ( (coeff ` d ) ` e ) = ( (coeff ` a ) ` e ) )
24	23	fveq2d 5868	. . . . . . . . . . . . 13 |- ( d = a -> ( abs ` ( (coeff ` d ) ` e ) ) = ( abs ` ( (coeff ` a ) ` e ) ) )
25	24	breq1d 4457	. . . . . . . . . . . 12 |- ( d = a -> ( ( abs ` ( (coeff ` d ) ` e ) ) <_ A <-> ( abs ` ( (coeff ` a ) ` e ) ) <_ A ) )
26	25	ralbidv 2903	. . . . . . . . . . 11 |- ( d = a -> ( A. e e. NN0 ( abs ` ( (coeff ` d ) ` e ) ) <_ A <-> A. e e. NN0 ( abs ` ( (coeff ` a ) ` e ) ) <_ A ) )
27	19, 21, 26	3anbi123d 1299	. . . . . . . . . 10 |- ( d = a -> ( ( d =/= 0p /\ (deg ` d ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` d ) ` e ) ) <_ A ) <-> ( a =/= 0p /\ (deg ` a ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` a ) ` e ) ) <_ A ) ) )
28	27	elrab 3261	. . . . . . . . 9 |- ( a e. { d e. (Poly ` ZZ ) | ( d =/= 0p /\ (deg ` d ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` d ) ` e ) ) <_ A ) } <-> ( a e. (Poly ` ZZ ) /\ ( a =/= 0p /\ (deg ` a ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` a ) ` e ) ) <_ A ) ) )
29		simp3 998	. . . . . . . . . 10 |- ( ( a =/= 0p /\ (deg ` a ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` a ) ` e ) ) <_ A ) -> A. e e. NN0 ( abs ` ( (coeff ` a ) ` e ) ) <_ A )
30	29	anim2i 569	. . . . . . . . 9 |- ( ( a e. (Poly ` ZZ ) /\ ( a =/= 0p /\ (deg ` a ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` a ) ` e ) ) <_ A ) ) -> ( a e. (Poly ` ZZ ) /\ A. e e. NN0 ( abs ` ( (coeff ` a ) ` e ) ) <_ A ) )
31	28, 30	sylbi 195	. . . . . . . 8 |- ( a e. { d e. (Poly ` ZZ ) | ( d =/= 0p /\ (deg ` d ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` d ) ` e ) ) <_ A ) } -> ( a e. (Poly ` ZZ ) /\ A. e e. NN0 ( abs ` ( (coeff ` a ) ` e ) ) <_ A ) )
32		0z 10871	. . . . . . . . . . . . . . 15 |- 0 e. ZZ
33		eqid 2467	. . . . . . . . . . . . . . . 16 |- (coeff ` a ) = (coeff ` a )
34	33	coef2 22363	. . . . . . . . . . . . . . 15 |- ( ( a e. (Poly ` ZZ ) /\ 0 e. ZZ ) -> (coeff ` a ) : NN0 --> ZZ )
35	32, 34	mpan2 671	. . . . . . . . . . . . . 14 |- ( a e. (Poly ` ZZ ) -> (coeff ` a ) : NN0 --> ZZ )
36	35	ad2antrl 727	. . . . . . . . . . . . 13 |- ( ( A e. NN0 /\ ( a e. (Poly ` ZZ ) /\ A. e e. NN0 ( abs ` ( (coeff ` a ) ` e ) ) <_ A ) ) -> (coeff ` a ) : NN0 --> ZZ )
37		ffn 5729	. . . . . . . . . . . . 13 |- ( (coeff ` a ) : NN0 --> ZZ -> (coeff ` a ) Fn NN0 )
38	36, 37	syl 16	. . . . . . . . . . . 12 |- ( ( A e. NN0 /\ ( a e. (Poly ` ZZ ) /\ A. e e. NN0 ( abs ` ( (coeff ` a ) ` e ) ) <_ A ) ) -> (coeff ` a ) Fn NN0 )
39	35	adantl 466	. . . . . . . . . . . . . . . . . . . 20 |- ( ( A e. NN0 /\ a e. (Poly ` ZZ ) ) -> (coeff ` a ) : NN0 --> ZZ )
40	39	ffvelrnda 6019	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( A e. NN0 /\ a e. (Poly ` ZZ ) ) /\ e e. NN0 ) -> ( (coeff ` a ) ` e ) e. ZZ )
41	40	zred 10962	. . . . . . . . . . . . . . . . . 18 |- ( ( ( A e. NN0 /\ a e. (Poly ` ZZ ) ) /\ e e. NN0 ) -> ( (coeff ` a ) ` e ) e. RR )
42		nn0re 10800	. . . . . . . . . . . . . . . . . . 19 |- ( A e. NN0 -> A e. RR )
43	42	ad2antrr 725	. . . . . . . . . . . . . . . . . 18 |- ( ( ( A e. NN0 /\ a e. (Poly ` ZZ ) ) /\ e e. NN0 ) -> A e. RR )
44	41, 43	absled 13221	. . . . . . . . . . . . . . . . 17 |- ( ( ( A e. NN0 /\ a e. (Poly ` ZZ ) ) /\ e e. NN0 ) -> ( ( abs ` ( (coeff ` a ) ` e ) ) <_ A <-> ( -u A <_ ( (coeff ` a ) ` e ) /\ ( (coeff ` a ) ` e ) <_ A ) ) )
45		nn0z 10883	. . . . . . . . . . . . . . . . . . . 20 |- ( A e. NN0 -> A e. ZZ )
46	45	ad2antrr 725	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( A e. NN0 /\ a e. (Poly ` ZZ ) ) /\ e e. NN0 ) -> A e. ZZ )
47	46	znegcld 10964	. . . . . . . . . . . . . . . . . 18 |- ( ( ( A e. NN0 /\ a e. (Poly ` ZZ ) ) /\ e e. NN0 ) -> -u A e. ZZ )
48		elfz 11674	. . . . . . . . . . . . . . . . . 18 |- ( ( ( (coeff ` a ) ` e ) e. ZZ /\ -u A e. ZZ /\ A e. ZZ ) -> ( ( (coeff ` a ) ` e ) e. ( -u A ... A ) <-> ( -u A <_ ( (coeff ` a ) ` e ) /\ ( (coeff ` a ) ` e ) <_ A ) ) )
49	40, 47, 46, 48	syl3anc 1228	. . . . . . . . . . . . . . . . 17 |- ( ( ( A e. NN0 /\ a e. (Poly ` ZZ ) ) /\ e e. NN0 ) -> ( ( (coeff ` a ) ` e ) e. ( -u A ... A ) <-> ( -u A <_ ( (coeff ` a ) ` e ) /\ ( (coeff ` a ) ` e ) <_ A ) ) )
50	44, 49	bitr4d 256	. . . . . . . . . . . . . . . 16 |- ( ( ( A e. NN0 /\ a e. (Poly ` ZZ ) ) /\ e e. NN0 ) -> ( ( abs ` ( (coeff ` a ) ` e ) ) <_ A <-> ( (coeff ` a ) ` e ) e. ( -u A ... A ) ) )
51	50	biimpd 207	. . . . . . . . . . . . . . 15 |- ( ( ( A e. NN0 /\ a e. (Poly ` ZZ ) ) /\ e e. NN0 ) -> ( ( abs ` ( (coeff ` a ) ` e ) ) <_ A -> ( (coeff ` a ) ` e ) e. ( -u A ... A ) ) )
52	51	ralimdva 2872	. . . . . . . . . . . . . 14 |- ( ( A e. NN0 /\ a e. (Poly ` ZZ ) ) -> ( A. e e. NN0 ( abs ` ( (coeff ` a ) ` e ) ) <_ A -> A. e e. NN0 ( (coeff ` a ) ` e ) e. ( -u A ... A ) ) )
53	52	impr 619	. . . . . . . . . . . . 13 |- ( ( A e. NN0 /\ ( a e. (Poly ` ZZ ) /\ A. e e. NN0 ( abs ` ( (coeff ` a ) ` e ) ) <_ A ) ) -> A. e e. NN0 ( (coeff ` a ) ` e ) e. ( -u A ... A ) )
54		fnfvrnss 6047	. . . . . . . . . . . . 13 |- ( ( (coeff ` a ) Fn NN0 /\ A. e e. NN0 ( (coeff ` a ) ` e ) e. ( -u A ... A ) ) -> ran (coeff ` a ) C_ ( -u A ... A ) )
55	38, 53, 54	syl2anc 661	. . . . . . . . . . . 12 |- ( ( A e. NN0 /\ ( a e. (Poly ` ZZ ) /\ A. e e. NN0 ( abs ` ( (coeff ` a ) ` e ) ) <_ A ) ) -> ran (coeff ` a ) C_ ( -u A ... A ) )
56		df-f 5590	. . . . . . . . . . . 12 |- ( (coeff ` a ) : NN0 --> ( -u A ... A ) <-> ( (coeff ` a ) Fn NN0 /\ ran (coeff ` a ) C_ ( -u A ... A ) ) )
57	38, 55, 56	sylanbrc 664	. . . . . . . . . . 11 |- ( ( A e. NN0 /\ ( a e. (Poly ` ZZ ) /\ A. e e. NN0 ( abs ` ( (coeff ` a ) ` e ) ) <_ A ) ) -> (coeff ` a ) : NN0 --> ( -u A ... A ) )
58		elfznn0 11766	. . . . . . . . . . . 12 |- ( a e. ( 0 ... A ) -> a e. NN0 )
59	58	ssriv 3508	. . . . . . . . . . 11 |- ( 0 ... A ) C_ NN0
60		fssres 5749	. . . . . . . . . . 11 |- ( ( (coeff ` a ) : NN0 --> ( -u A ... A ) /\ ( 0 ... A ) C_ NN0 ) -> ( (coeff ` a ) |` ( 0 ... A ) ) : ( 0 ... A ) --> ( -u A ... A ) )
61	57, 59, 60	sylancl 662	. . . . . . . . . 10 |- ( ( A e. NN0 /\ ( a e. (Poly ` ZZ ) /\ A. e e. NN0 ( abs ` ( (coeff ` a ) ` e ) ) <_ A ) ) -> ( (coeff ` a ) |` ( 0 ... A ) ) : ( 0 ... A ) --> ( -u A ... A ) )
62		ovex 6307	. . . . . . . . . . 11 |- ( -u A ... A ) e. _V
63		ovex 6307	. . . . . . . . . . 11 |- ( 0 ... A ) e. _V
64	62, 63	elmap 7444	. . . . . . . . . 10 |- ( ( (coeff ` a ) |` ( 0 ... A ) ) e. ( ( -u A ... A ) ^m ( 0 ... A ) ) <-> ( (coeff ` a ) |` ( 0 ... A ) ) : ( 0 ... A ) --> ( -u A ... A ) )
65	61, 64	sylibr 212	. . . . . . . . 9 |- ( ( A e. NN0 /\ ( a e. (Poly ` ZZ ) /\ A. e e. NN0 ( abs ` ( (coeff ` a ) ` e ) ) <_ A ) ) -> ( (coeff ` a ) |` ( 0 ... A ) ) e. ( ( -u A ... A ) ^m ( 0 ... A ) ) )
66	65	ex 434	. . . . . . . 8 |- ( A e. NN0 -> ( ( a e. (Poly ` ZZ ) /\ A. e e. NN0 ( abs ` ( (coeff ` a ) ` e ) ) <_ A ) -> ( (coeff ` a ) |` ( 0 ... A ) ) e. ( ( -u A ... A ) ^m ( 0 ... A ) ) ) )
67	31, 66	syl5 32	. . . . . . 7 |- ( A e. NN0 -> ( a e. { d e. (Poly ` ZZ ) | ( d =/= 0p /\ (deg ` d ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` d ) ` e ) ) <_ A ) } -> ( (coeff ` a ) |` ( 0 ... A ) ) e. ( ( -u A ... A ) ^m ( 0 ... A ) ) ) )
68		simp2 997	. . . . . . . . . 10 |- ( ( a =/= 0p /\ (deg ` a ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` a ) ` e ) ) <_ A ) -> (deg ` a ) <_ A )
69	68	anim2i 569	. . . . . . . . 9 |- ( ( a e. (Poly ` ZZ ) /\ ( a =/= 0p /\ (deg ` a ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` a ) ` e ) ) <_ A ) ) -> ( a e. (Poly ` ZZ ) /\ (deg ` a ) <_ A ) )
70	28, 69	sylbi 195	. . . . . . . 8 |- ( a e. { d e. (Poly ` ZZ ) | ( d =/= 0p /\ (deg ` d ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` d ) ` e ) ) <_ A ) } -> ( a e. (Poly ` ZZ ) /\ (deg ` a ) <_ A ) )
71		neeq1 2748	. . . . . . . . . . 11 |- ( d = b -> ( d =/= 0p <-> b =/= 0p ) )
72		fveq2 5864	. . . . . . . . . . . 12 |- ( d = b -> (deg ` d ) = (deg ` b ) )
73	72	breq1d 4457	. . . . . . . . . . 11 |- ( d = b -> ( (deg ` d ) <_ A <-> (deg ` b ) <_ A ) )
74		fveq2 5864	. . . . . . . . . . . . . . 15 |- ( d = b -> (coeff ` d ) = (coeff ` b ) )
75	74	fveq1d 5866	. . . . . . . . . . . . . 14 |- ( d = b -> ( (coeff ` d ) ` e ) = ( (coeff ` b ) ` e ) )
76	75	fveq2d 5868	. . . . . . . . . . . . 13 |- ( d = b -> ( abs ` ( (coeff ` d ) ` e ) ) = ( abs ` ( (coeff ` b ) ` e ) ) )
77	76	breq1d 4457	. . . . . . . . . . . 12 |- ( d = b -> ( ( abs ` ( (coeff ` d ) ` e ) ) <_ A <-> ( abs ` ( (coeff ` b ) ` e ) ) <_ A ) )
78	77	ralbidv 2903	. . . . . . . . . . 11 |- ( d = b -> ( A. e e. NN0 ( abs ` ( (coeff ` d ) ` e ) ) <_ A <-> A. e e. NN0 ( abs ` ( (coeff ` b ) ` e ) ) <_ A ) )
79	71, 73, 78	3anbi123d 1299	. . . . . . . . . 10 |- ( d = b -> ( ( d =/= 0p /\ (deg ` d ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` d ) ` e ) ) <_ A ) <-> ( b =/= 0p /\ (deg ` b ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` b ) ` e ) ) <_ A ) ) )
80	79	elrab 3261	. . . . . . . . 9 |- ( b e. { d e. (Poly ` ZZ ) | ( d =/= 0p /\ (deg ` d ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` d ) ` e ) ) <_ A ) } <-> ( b e. (Poly ` ZZ ) /\ ( b =/= 0p /\ (deg ` b ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` b ) ` e ) ) <_ A ) ) )
81		simp2 997	. . . . . . . . . 10 |- ( ( b =/= 0p /\ (deg ` b ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` b ) ` e ) ) <_ A ) -> (deg ` b ) <_ A )
82	81	anim2i 569	. . . . . . . . 9 |- ( ( b e. (Poly ` ZZ ) /\ ( b =/= 0p /\ (deg ` b ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` b ) ` e ) ) <_ A ) ) -> ( b e. (Poly ` ZZ ) /\ (deg ` b ) <_ A ) )
83	80, 82	sylbi 195	. . . . . . . 8 |- ( b e. { d e. (Poly ` ZZ ) | ( d =/= 0p /\ (deg ` d ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` d ) ` e ) ) <_ A ) } -> ( b e. (Poly ` ZZ ) /\ (deg ` b ) <_ A ) )
84		simplll 757	. . . . . . . . . . . . 13 |- ( ( ( ( a e. (Poly ` ZZ ) /\ (deg ` a ) <_ A ) /\ ( b e. (Poly ` ZZ ) /\ (deg ` b ) <_ A ) ) /\ ( A e. NN0 /\ ( (coeff ` a ) |` ( 0 ... A ) ) = ( (coeff ` b ) |` ( 0 ... A ) ) ) ) -> a e. (Poly ` ZZ ) )
85		plyf 22330	. . . . . . . . . . . . 13 |- ( a e. (Poly ` ZZ ) -> a : CC --> CC )
86		ffn 5729	. . . . . . . . . . . . 13 |- ( a : CC --> CC -> a Fn CC )
87	84, 85, 86	3syl 20	. . . . . . . . . . . 12 |- ( ( ( ( a e. (Poly ` ZZ ) /\ (deg ` a ) <_ A ) /\ ( b e. (Poly ` ZZ ) /\ (deg ` b ) <_ A ) ) /\ ( A e. NN0 /\ ( (coeff ` a ) |` ( 0 ... A ) ) = ( (coeff ` b ) |` ( 0 ... A ) ) ) ) -> a Fn CC )
88		simplrl 759	. . . . . . . . . . . . 13 |- ( ( ( ( a e. (Poly ` ZZ ) /\ (deg ` a ) <_ A ) /\ ( b e. (Poly ` ZZ ) /\ (deg ` b ) <_ A ) ) /\ ( A e. NN0 /\ ( (coeff ` a ) |` ( 0 ... A ) ) = ( (coeff ` b ) |` ( 0 ... A ) ) ) ) -> b e. (Poly ` ZZ ) )
89		plyf 22330	. . . . . . . . . . . . 13 |- ( b e. (Poly ` ZZ ) -> b : CC --> CC )
90		ffn 5729	. . . . . . . . . . . . 13 |- ( b : CC --> CC -> b Fn CC )
91	88, 89, 90	3syl 20	. . . . . . . . . . . 12 |- ( ( ( ( a e. (Poly ` ZZ ) /\ (deg ` a ) <_ A ) /\ ( b e. (Poly ` ZZ ) /\ (deg ` b ) <_ A ) ) /\ ( A e. NN0 /\ ( (coeff ` a ) |` ( 0 ... A ) ) = ( (coeff ` b ) |` ( 0 ... A ) ) ) ) -> b Fn CC )
92		simplrr 760	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( a e. (Poly ` ZZ ) /\ (deg ` a ) <_ A ) /\ ( b e. (Poly ` ZZ ) /\ (deg ` b ) <_ A ) ) /\ ( A e. NN0 /\ ( (coeff ` a ) |` ( 0 ... A ) ) = ( (coeff ` b ) |` ( 0 ... A ) ) ) ) /\ c e. CC ) -> ( (coeff ` a ) |` ( 0 ... A ) ) = ( (coeff ` b ) |` ( 0 ... A ) ) )
93	92	adantr 465	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ( a e. (Poly ` ZZ ) /\ (deg ` a ) <_ A ) /\ ( b e. (Poly ` ZZ ) /\ (deg ` b ) <_ A ) ) /\ ( A e. NN0 /\ ( (coeff ` a ) |` ( 0 ... A ) ) = ( (coeff ` b ) |` ( 0 ... A ) ) ) ) /\ c e. CC ) /\ d e. ( 0 ... A ) ) -> ( (coeff ` a ) |` ( 0 ... A ) ) = ( (coeff ` b ) |` ( 0 ... A ) ) )
94	93	fveq1d 5866	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( a e. (Poly ` ZZ ) /\ (deg ` a ) <_ A ) /\ ( b e. (Poly ` ZZ ) /\ (deg ` b ) <_ A ) ) /\ ( A e. NN0 /\ ( (coeff ` a ) |` ( 0 ... A ) ) = ( (coeff ` b ) |` ( 0 ... A ) ) ) ) /\ c e. CC ) /\ d e. ( 0 ... A ) ) -> ( ( (coeff ` a ) |` ( 0 ... A ) ) ` d ) = ( ( (coeff ` b ) |` ( 0 ... A ) ) ` d ) )
95		fvres 5878	. . . . . . . . . . . . . . . . 17 |- ( d e. ( 0 ... A ) -> ( ( (coeff ` a ) |` ( 0 ... A ) ) ` d ) = ( (coeff ` a ) ` d ) )
96	95	adantl 466	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( a e. (Poly ` ZZ ) /\ (deg ` a ) <_ A ) /\ ( b e. (Poly ` ZZ ) /\ (deg ` b ) <_ A ) ) /\ ( A e. NN0 /\ ( (coeff ` a ) |` ( 0 ... A ) ) = ( (coeff ` b ) |` ( 0 ... A ) ) ) ) /\ c e. CC ) /\ d e. ( 0 ... A ) ) -> ( ( (coeff ` a ) |` ( 0 ... A ) ) ` d ) = ( (coeff ` a ) ` d ) )
97		fvres 5878	. . . . . . . . . . . . . . . . 17 |- ( d e. ( 0 ... A ) -> ( ( (coeff ` b ) |` ( 0 ... A ) ) ` d ) = ( (coeff ` b ) ` d ) )
98	97	adantl 466	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( a e. (Poly ` ZZ ) /\ (deg ` a ) <_ A ) /\ ( b e. (Poly ` ZZ ) /\ (deg ` b ) <_ A ) ) /\ ( A e. NN0 /\ ( (coeff ` a ) |` ( 0 ... A ) ) = ( (coeff ` b ) |` ( 0 ... A ) ) ) ) /\ c e. CC ) /\ d e. ( 0 ... A ) ) -> ( ( (coeff ` b ) |` ( 0 ... A ) ) ` d ) = ( (coeff ` b ) ` d ) )
99	94, 96, 98	3eqtr3d 2516	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( a e. (Poly ` ZZ ) /\ (deg ` a ) <_ A ) /\ ( b e. (Poly ` ZZ ) /\ (deg ` b ) <_ A ) ) /\ ( A e. NN0 /\ ( (coeff ` a ) |` ( 0 ... A ) ) = ( (coeff ` b ) |` ( 0 ... A ) ) ) ) /\ c e. CC ) /\ d e. ( 0 ... A ) ) -> ( (coeff ` a ) ` d ) = ( (coeff ` b ) ` d ) )
100	99	oveq1d 6297	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( a e. (Poly ` ZZ ) /\ (deg ` a ) <_ A ) /\ ( b e. (Poly ` ZZ ) /\ (deg ` b ) <_ A ) ) /\ ( A e. NN0 /\ ( (coeff ` a ) |` ( 0 ... A ) ) = ( (coeff ` b ) |` ( 0 ... A ) ) ) ) /\ c e. CC ) /\ d e. ( 0 ... A ) ) -> ( ( (coeff ` a ) ` d ) x. ( c ^ d ) ) = ( ( (coeff ` b ) ` d ) x. ( c ^ d ) ) )
101	100	sumeq2dv 13484	. . . . . . . . . . . . 13 |- ( ( ( ( ( a e. (Poly ` ZZ ) /\ (deg ` a ) <_ A ) /\ ( b e. (Poly ` ZZ ) /\ (deg ` b ) <_ A ) ) /\ ( A e. NN0 /\ ( (coeff ` a ) |` ( 0 ... A ) ) = ( (coeff ` b ) |` ( 0 ... A ) ) ) ) /\ c e. CC ) -> sum_ d e. ( 0 ... A ) ( ( (coeff ` a ) ` d ) x. ( c ^ d ) ) = sum_ d e. ( 0 ... A ) ( ( (coeff ` b ) ` d ) x. ( c ^ d ) ) )
102		simp-4l 765	. . . . . . . . . . . . . 14 |- ( ( ( ( ( a e. (Poly ` ZZ ) /\ (deg ` a ) <_ A ) /\ ( b e. (Poly ` ZZ ) /\ (deg ` b ) <_ A ) ) /\ ( A e. NN0 /\ ( (coeff ` a ) |` ( 0 ... A ) ) = ( (coeff ` b ) |` ( 0 ... A ) ) ) ) /\ c e. CC ) -> a e. (Poly ` ZZ ) )
103		simp-4r 766	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( a e. (Poly ` ZZ ) /\ (deg ` a ) <_ A ) /\ ( b e. (Poly ` ZZ ) /\ (deg ` b ) <_ A ) ) /\ ( A e. NN0 /\ ( (coeff ` a ) |` ( 0 ... A ) ) = ( (coeff ` b ) |` ( 0 ... A ) ) ) ) /\ c e. CC ) -> (deg ` a ) <_ A )
104		dgrcl 22365	. . . . . . . . . . . . . . . . 17 |- ( a e. (Poly ` ZZ ) -> (deg ` a ) e. NN0 )
105		nn0z 10883	. . . . . . . . . . . . . . . . 17 |- ( (deg ` a ) e. NN0 -> (deg ` a ) e. ZZ )
106	102, 104, 105	3syl 20	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( a e. (Poly ` ZZ ) /\ (deg ` a ) <_ A ) /\ ( b e. (Poly ` ZZ ) /\ (deg ` b ) <_ A ) ) /\ ( A e. NN0 /\ ( (coeff ` a ) |` ( 0 ... A ) ) = ( (coeff ` b ) |` ( 0 ... A ) ) ) ) /\ c e. CC ) -> (deg ` a ) e. ZZ )
107		simplrl 759	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( a e. (Poly ` ZZ ) /\ (deg ` a ) <_ A ) /\ ( b e. (Poly ` ZZ ) /\ (deg ` b ) <_ A ) ) /\ ( A e. NN0 /\ ( (coeff ` a ) |` ( 0 ... A ) ) = ( (coeff ` b ) |` ( 0 ... A ) ) ) ) /\ c e. CC ) -> A e. NN0 )
108	107	nn0zd 10960	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( a e. (Poly ` ZZ ) /\ (deg ` a ) <_ A ) /\ ( b e. (Poly ` ZZ ) /\ (deg ` b ) <_ A ) ) /\ ( A e. NN0 /\ ( (coeff ` a ) |` ( 0 ... A ) ) = ( (coeff ` b ) |` ( 0 ... A ) ) ) ) /\ c e. CC ) -> A e. ZZ )
109		eluz 11091	. . . . . . . . . . . . . . . 16 |- ( ( (deg ` a ) e. ZZ /\ A e. ZZ ) -> ( A e. ( ZZ>= ` (deg ` a ) ) <-> (deg ` a ) <_ A ) )
110	106, 108, 109	syl2anc 661	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( a e. (Poly ` ZZ ) /\ (deg ` a ) <_ A ) /\ ( b e. (Poly ` ZZ ) /\ (deg ` b ) <_ A ) ) /\ ( A e. NN0 /\ ( (coeff ` a ) |` ( 0 ... A ) ) = ( (coeff ` b ) |` ( 0 ... A ) ) ) ) /\ c e. CC ) -> ( A e. ( ZZ>= ` (deg ` a ) ) <-> (deg ` a ) <_ A ) )
111	103, 110	mpbird 232	. . . . . . . . . . . . . 14 |- ( ( ( ( ( a e. (Poly ` ZZ ) /\ (deg ` a ) <_ A ) /\ ( b e. (Poly ` ZZ ) /\ (deg ` b ) <_ A ) ) /\ ( A e. NN0 /\ ( (coeff ` a ) |` ( 0 ... A ) ) = ( (coeff ` b ) |` ( 0 ... A ) ) ) ) /\ c e. CC ) -> A e. ( ZZ>= ` (deg ` a ) ) )
112		simpr 461	. . . . . . . . . . . . . 14 |- ( ( ( ( ( a e. (Poly ` ZZ ) /\ (deg ` a ) <_ A ) /\ ( b e. (Poly ` ZZ ) /\ (deg ` b ) <_ A ) ) /\ ( A e. NN0 /\ ( (coeff ` a ) |` ( 0 ... A ) ) = ( (coeff ` b ) |` ( 0 ... A ) ) ) ) /\ c e. CC ) -> c e. CC )
113		eqid 2467	. . . . . . . . . . . . . . 15 |- (deg ` a ) = (deg ` a )
114	33, 113	coeid3 22372	. . . . . . . . . . . . . 14 |- ( ( a e. (Poly ` ZZ ) /\ A e. ( ZZ>= ` (deg ` a ) ) /\ c e. CC ) -> ( a ` c ) = sum_ d e. ( 0 ... A ) ( ( (coeff ` a ) ` d ) x. ( c ^ d ) ) )
115	102, 111, 112, 114	syl3anc 1228	. . . . . . . . . . . . 13 |- ( ( ( ( ( a e. (Poly ` ZZ ) /\ (deg ` a ) <_ A ) /\ ( b e. (Poly ` ZZ ) /\ (deg ` b ) <_ A ) ) /\ ( A e. NN0 /\ ( (coeff ` a ) |` ( 0 ... A ) ) = ( (coeff ` b ) |` ( 0 ... A ) ) ) ) /\ c e. CC ) -> ( a ` c ) = sum_ d e. ( 0 ... A ) ( ( (coeff ` a ) ` d ) x. ( c ^ d ) ) )
116		simp1rl 1061	. . . . . . . . . . . . . . 15 |- ( ( ( ( a e. (Poly ` ZZ ) /\ (deg ` a ) <_ A ) /\ ( b e. (Poly ` ZZ ) /\ (deg ` b ) <_ A ) ) /\ ( A e. NN0 /\ ( (coeff ` a ) |` ( 0 ... A ) ) = ( (coeff ` b ) |` ( 0 ... A ) ) ) /\ c e. CC ) -> b e. (Poly ` ZZ ) )
117	116	3expa 1196	. . . . . . . . . . . . . 14 |- ( ( ( ( ( a e. (Poly ` ZZ ) /\ (deg ` a ) <_ A ) /\ ( b e. (Poly ` ZZ ) /\ (deg ` b ) <_ A ) ) /\ ( A e. NN0 /\ ( (coeff ` a ) |` ( 0 ... A ) ) = ( (coeff ` b ) |` ( 0 ... A ) ) ) ) /\ c e. CC ) -> b e. (Poly ` ZZ ) )
118		simplrr 760	. . . . . . . . . . . . . . . 16 |- ( ( ( ( a e. (Poly ` ZZ ) /\ (deg ` a ) <_ A ) /\ ( b e. (Poly ` ZZ ) /\ (deg ` b ) <_ A ) ) /\ ( A e. NN0 /\ ( (coeff ` a ) |` ( 0 ... A ) ) = ( (coeff ` b ) |` ( 0 ... A ) ) ) ) -> (deg ` b ) <_ A )
119	118	adantr 465	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( a e. (Poly ` ZZ ) /\ (deg ` a ) <_ A ) /\ ( b e. (Poly ` ZZ ) /\ (deg ` b ) <_ A ) ) /\ ( A e. NN0 /\ ( (coeff ` a ) |` ( 0 ... A ) ) = ( (coeff ` b ) |` ( 0 ... A ) ) ) ) /\ c e. CC ) -> (deg ` b ) <_ A )
120		dgrcl 22365	. . . . . . . . . . . . . . . . 17 |- ( b e. (Poly ` ZZ ) -> (deg ` b ) e. NN0 )
121		nn0z 10883	. . . . . . . . . . . . . . . . 17 |- ( (deg ` b ) e. NN0 -> (deg ` b ) e. ZZ )
122	117, 120, 121	3syl 20	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( a e. (Poly ` ZZ ) /\ (deg ` a ) <_ A ) /\ ( b e. (Poly ` ZZ ) /\ (deg ` b ) <_ A ) ) /\ ( A e. NN0 /\ ( (coeff ` a ) |` ( 0 ... A ) ) = ( (coeff ` b ) |` ( 0 ... A ) ) ) ) /\ c e. CC ) -> (deg ` b ) e. ZZ )
123		eluz 11091	. . . . . . . . . . . . . . . 16 |- ( ( (deg ` b ) e. ZZ /\ A e. ZZ ) -> ( A e. ( ZZ>= ` (deg ` b ) ) <-> (deg ` b ) <_ A ) )
124	122, 108, 123	syl2anc 661	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( a e. (Poly ` ZZ ) /\ (deg ` a ) <_ A ) /\ ( b e. (Poly ` ZZ ) /\ (deg ` b ) <_ A ) ) /\ ( A e. NN0 /\ ( (coeff ` a ) |` ( 0 ... A ) ) = ( (coeff ` b ) |` ( 0 ... A ) ) ) ) /\ c e. CC ) -> ( A e. ( ZZ>= ` (deg ` b ) ) <-> (deg ` b ) <_ A ) )
125	119, 124	mpbird 232	. . . . . . . . . . . . . 14 |- ( ( ( ( ( a e. (Poly ` ZZ ) /\ (deg ` a ) <_ A ) /\ ( b e. (Poly ` ZZ ) /\ (deg ` b ) <_ A ) ) /\ ( A e. NN0 /\ ( (coeff ` a ) |` ( 0 ... A ) ) = ( (coeff ` b ) |` ( 0 ... A ) ) ) ) /\ c e. CC ) -> A e. ( ZZ>= ` (deg ` b ) ) )
126		eqid 2467	. . . . . . . . . . . . . . 15 |- (coeff ` b ) = (coeff ` b )
127		eqid 2467	. . . . . . . . . . . . . . 15 |- (deg ` b ) = (deg ` b )
128	126, 127	coeid3 22372	. . . . . . . . . . . . . 14 |- ( ( b e. (Poly ` ZZ ) /\ A e. ( ZZ>= ` (deg ` b ) ) /\ c e. CC ) -> ( b ` c ) = sum_ d e. ( 0 ... A ) ( ( (coeff ` b ) ` d ) x. ( c ^ d ) ) )
129	117, 125, 112, 128	syl3anc 1228	. . . . . . . . . . . . 13 |- ( ( ( ( ( a e. (Poly ` ZZ ) /\ (deg ` a ) <_ A ) /\ ( b e. (Poly ` ZZ ) /\ (deg ` b ) <_ A ) ) /\ ( A e. NN0 /\ ( (coeff ` a ) |` ( 0 ... A ) ) = ( (coeff ` b ) |` ( 0 ... A ) ) ) ) /\ c e. CC ) -> ( b ` c ) = sum_ d e. ( 0 ... A ) ( ( (coeff ` b ) ` d ) x. ( c ^ d ) ) )
130	101, 115, 129	3eqtr4d 2518	. . . . . . . . . . . 12 |- ( ( ( ( ( a e. (Poly ` ZZ ) /\ (deg ` a ) <_ A ) /\ ( b e. (Poly ` ZZ ) /\ (deg ` b ) <_ A ) ) /\ ( A e. NN0 /\ ( (coeff ` a ) |` ( 0 ... A ) ) = ( (coeff ` b ) |` ( 0 ... A ) ) ) ) /\ c e. CC ) -> ( a ` c ) = ( b ` c ) )
131	87, 91, 130	eqfnfvd 5976	. . . . . . . . . . 11 |- ( ( ( ( a e. (Poly ` ZZ ) /\ (deg ` a ) <_ A ) /\ ( b e. (Poly ` ZZ ) /\ (deg ` b ) <_ A ) ) /\ ( A e. NN0 /\ ( (coeff ` a ) |` ( 0 ... A ) ) = ( (coeff ` b ) |` ( 0 ... A ) ) ) ) -> a = b )
132	131	expr 615	. . . . . . . . . 10 |- ( ( ( ( a e. (Poly ` ZZ ) /\ (deg ` a ) <_ A ) /\ ( b e. (Poly ` ZZ ) /\ (deg ` b ) <_ A ) ) /\ A e. NN0 ) -> ( ( (coeff ` a ) |` ( 0 ... A ) ) = ( (coeff ` b ) |` ( 0 ... A ) ) -> a = b ) )
133		fveq2 5864	. . . . . . . . . . 11 |- ( a = b -> (coeff ` a ) = (coeff ` b ) )
134	133	reseq1d 5270	. . . . . . . . . 10 |- ( a = b -> ( (coeff ` a ) |` ( 0 ... A ) ) = ( (coeff ` b ) |` ( 0 ... A ) ) )
135	132, 134	impbid1 203	. . . . . . . . 9 |- ( ( ( ( a e. (Poly ` ZZ ) /\ (deg ` a ) <_ A ) /\ ( b e. (Poly ` ZZ ) /\ (deg ` b ) <_ A ) ) /\ A e. NN0 ) -> ( ( (coeff ` a ) |` ( 0 ... A ) ) = ( (coeff ` b ) |` ( 0 ... A ) ) <-> a = b ) )
136	135	expcom 435	. . . . . . . 8 |- ( A e. NN0 -> ( ( ( a e. (Poly ` ZZ ) /\ (deg ` a ) <_ A ) /\ ( b e. (Poly ` ZZ ) /\ (deg ` b ) <_ A ) ) -> ( ( (coeff ` a ) |` ( 0 ... A ) ) = ( (coeff ` b ) |` ( 0 ... A ) ) <-> a = b ) ) )
137	70, 83, 136	syl2ani 656	. . . . . . 7 |- ( A e. NN0 -> ( ( a e. { d e. (Poly ` ZZ ) | ( d =/= 0p /\ (deg ` d ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` d ) ` e ) ) <_ A ) } /\ b e. { d e. (Poly ` ZZ ) | ( d =/= 0p /\ (deg ` d ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` d ) ` e ) ) <_ A ) } ) -> ( ( (coeff ` a ) |` ( 0 ... A ) ) = ( (coeff ` b ) |` ( 0 ... A ) ) <-> a = b ) ) )
138	67, 137	dom2d 7553	. . . . . 6 |- ( A e. NN0 -> ( ( ( -u A ... A ) ^m ( 0 ... A ) ) e. _V -> { d e. (Poly ` ZZ ) | ( d =/= 0p /\ (deg ` d ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` d ) ` e ) ) <_ A ) } ~<_ ( ( -u A ... A ) ^m ( 0 ... A ) ) ) )
139	18, 138	mpi 17	. . . . 5 |- ( A e. NN0 -> { d e. (Poly ` ZZ ) | ( d =/= 0p /\ (deg ` d ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` d ) ` e ) ) <_ A ) } ~<_ ( ( -u A ... A ) ^m ( 0 ... A ) ) )
140		domfi 7738	. . . . 5 |- ( ( ( ( -u A ... A ) ^m ( 0 ... A ) ) e. Fin /\ { d e. (Poly ` ZZ ) | ( d =/= 0p /\ (deg ` d ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` d ) ` e ) ) <_ A ) } ~<_ ( ( -u A ... A ) ^m ( 0 ... A ) ) ) -> { d e. (Poly ` ZZ ) | ( d =/= 0p /\ (deg ` d ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` d ) ` e ) ) <_ A ) } e. Fin )
141	17, 139, 140	syl2anc 661	. . . 4 |- ( A e. NN0 -> { d e. (Poly ` ZZ ) | ( d =/= 0p /\ (deg ` d ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` d ) ` e ) ) <_ A ) } e. Fin )
142		neeq1 2748	. . . . . . . . 9 |- ( d = c -> ( d =/= 0p <-> c =/= 0p ) )
143		fveq2 5864	. . . . . . . . . 10 |- ( d = c -> (deg ` d ) = (deg ` c ) )
144	143	breq1d 4457	. . . . . . . . 9 |- ( d = c -> ( (deg ` d ) <_ A <-> (deg ` c ) <_ A ) )
145		fveq2 5864	. . . . . . . . . . . . 13 |- ( d = c -> (coeff ` d ) = (coeff ` c ) )
146	145	fveq1d 5866	. . . . . . . . . . . 12 |- ( d = c -> ( (coeff ` d ) ` e ) = ( (coeff ` c ) ` e ) )
147	146	fveq2d 5868	. . . . . . . . . . 11 |- ( d = c -> ( abs ` ( (coeff ` d ) ` e ) ) = ( abs ` ( (coeff ` c ) ` e ) ) )
148	147	breq1d 4457	. . . . . . . . . 10 |- ( d = c -> ( ( abs ` ( (coeff ` d ) ` e ) ) <_ A <-> ( abs ` ( (coeff ` c ) ` e ) ) <_ A ) )
149	148	ralbidv 2903	. . . . . . . . 9 |- ( d = c -> ( A. e e. NN0 ( abs ` ( (coeff ` d ) ` e ) ) <_ A <-> A. e e. NN0 ( abs ` ( (coeff ` c ) ` e ) ) <_ A ) )
150	142, 144, 149	3anbi123d 1299	. . . . . . . 8 |- ( d = c -> ( ( d =/= 0p /\ (deg ` d ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` d ) ` e ) ) <_ A ) <-> ( c =/= 0p /\ (deg ` c ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` c ) ` e ) ) <_ A ) ) )
151	150	elrab 3261	. . . . . . 7 |- ( c e. { d e. (Poly ` ZZ ) | ( d =/= 0p /\ (deg ` d ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` d ) ` e ) ) <_ A ) } <-> ( c e. (Poly ` ZZ ) /\ ( c =/= 0p /\ (deg ` c ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` c ) ` e ) ) <_ A ) ) )
152		simp1 996	. . . . . . . 8 |- ( ( c =/= 0p /\ (deg ` c ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` c ) ` e ) ) <_ A ) -> c =/= 0p )
153	152	anim2i 569	. . . . . . 7 |- ( ( c e. (Poly ` ZZ ) /\ ( c =/= 0p /\ (deg ` c ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` c ) ` e ) ) <_ A ) ) -> ( c e. (Poly ` ZZ ) /\ c =/= 0p ) )
154	151, 153	sylbi 195	. . . . . 6 |- ( c e. { d e. (Poly ` ZZ ) | ( d =/= 0p /\ (deg ` d ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` d ) ` e ) ) <_ A ) } -> ( c e. (Poly ` ZZ ) /\ c =/= 0p ) )
155		plyf 22330	. . . . . . . . . . . . 13 |- ( c e. (Poly ` ZZ ) -> c : CC --> CC )
156		ffn 5729	. . . . . . . . . . . . 13 |- ( c : CC --> CC -> c Fn CC )
157	155, 156	syl 16	. . . . . . . . . . . 12 |- ( c e. (Poly ` ZZ ) -> c Fn CC )
158	157	adantr 465	. . . . . . . . . . 11 |- ( ( c e. (Poly ` ZZ ) /\ c =/= 0p ) -> c Fn CC )
159		fniniseg 6000	. . . . . . . . . . 11 |- ( c Fn CC -> ( a e. ( `' c " { 0 } ) <-> ( a e. CC /\ ( c ` a ) = 0 ) ) )
160	158, 159	syl 16	. . . . . . . . . 10 |- ( ( c e. (Poly ` ZZ ) /\ c =/= 0p ) -> ( a e. ( `' c " { 0 } ) <-> ( a e. CC /\ ( c ` a ) = 0 ) ) )
161		fveq2 5864	. . . . . . . . . . . 12 |- ( b = a -> ( c ` b ) = ( c ` a ) )
162	161	eqeq1d 2469	. . . . . . . . . . 11 |- ( b = a -> ( ( c ` b ) = 0 <-> ( c ` a ) = 0 ) )
163	162	elrab 3261	. . . . . . . . . 10 |- ( a e. { b e. CC | ( c ` b ) = 0 } <-> ( a e. CC /\ ( c ` a ) = 0 ) )
164	160, 163	syl6rbbr 264	. . . . . . . . 9 |- ( ( c e. (Poly ` ZZ ) /\ c =/= 0p ) -> ( a e. { b e. CC | ( c ` b ) = 0 } <-> a e. ( `' c " { 0 } ) ) )
165	164	eqrdv 2464	. . . . . . . 8 |- ( ( c e. (Poly ` ZZ ) /\ c =/= 0p ) -> { b e. CC | ( c ` b ) = 0 } = ( `' c " { 0 } ) )
166		eqid 2467	. . . . . . . . . 10 |- ( `' c " { 0 } ) = ( `' c " { 0 } )
167	166	fta1 22438	. . . . . . . . 9 |- ( ( c e. (Poly ` ZZ ) /\ c =/= 0p ) -> ( ( `' c " { 0 } ) e. Fin /\ ( # ` ( `' c " { 0 } ) ) <_ (deg ` c ) ) )
168	167	simpld 459	. . . . . . . 8 |- ( ( c e. (Poly ` ZZ ) /\ c =/= 0p ) -> ( `' c " { 0 } ) e. Fin )
169	165, 168	eqeltrd 2555	. . . . . . 7 |- ( ( c e. (Poly ` ZZ ) /\ c =/= 0p ) -> { b e. CC | ( c ` b ) = 0 } e. Fin )
170	169	a1i 11	. . . . . 6 |- ( A e. NN0 -> ( ( c e. (Poly ` ZZ ) /\ c =/= 0p ) -> { b e. CC | ( c ` b ) = 0 } e. Fin ) )
171	154, 170	syl5 32	. . . . 5 |- ( A e. NN0 -> ( c e. { d e. (Poly ` ZZ ) | ( d =/= 0p /\ (deg ` d ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` d ) ` e ) ) <_ A ) } -> { b e. CC | ( c ` b ) = 0 } e. Fin ) )
172	171	ralrimiv 2876	. . . 4 |- ( A e. NN0 -> A. c e. { d e. (Poly ` ZZ ) | ( d =/= 0p /\ (deg ` d ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` d ) ` e ) ) <_ A ) } { b e. CC | ( c ` b ) = 0 } e. Fin )
173		iunfi 7804	. . . 4 |- ( ( { d e. (Poly ` ZZ ) | ( d =/= 0p /\ (deg ` d ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` d ) ` e ) ) <_ A ) } e. Fin /\ A. c e. { d e. (Poly ` ZZ ) | ( d =/= 0p /\ (deg ` d ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` d ) ` e ) ) <_ A ) } { b e. CC | ( c ` b ) = 0 } e. Fin ) -> U_ c e. { d e. (Poly ` ZZ ) | ( d =/= 0p /\ (deg ` d ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` d ) ` e ) ) <_ A ) } { b e. CC | ( c ` b ) = 0 } e. Fin )
174	141, 172, 173	syl2anc 661	. . 3 |- ( A e. NN0 -> U_ c e. { d e. (Poly ` ZZ ) | ( d =/= 0p /\ (deg ` d ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` d ) ` e ) ) <_ A ) } { b e. CC | ( c ` b ) = 0 } e. Fin )
175	12, 174	syl5eqelr 2560	. 2 |- ( A e. NN0 -> { b e. CC | E. c e. { d e. (Poly ` ZZ ) | ( d =/= 0p /\ (deg ` d ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` d ) ` e ) ) <_ A ) } ( c ` b ) = 0 } e. Fin )
176	11, 175	eqeltrd 2555	1 |- ( A e. NN0 -> ( H ` A ) e. Fin )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 973   = wceq 1379   e. wcel 1767   =/= wne 2662   A.wral 2814   E.wrex 2815   {crab 2818   _Vcvv 3113   C_ wss 3476   {csn 4027   U_ciun 4325   class class class wbr 4447   |-> cmpt 4505   `'ccnv 4998   ran crn 5000   |` cres 5001   "cima 5002   Fn wfn 5581   -->wf 5582   ` cfv 5586  (class class class)co 6282   ^m cmap 7417   ~<_ cdom 7511   Fincfn 7513   CCcc 9486   RRcr 9487   0cc0 9488   x. cmul 9493   <_ cle 9625   -ucneg 9802   NN0cn0 10791   ZZcz 10860   ZZ>=cuz 11078   ...cfz 11668   ^cexp 12130   #chash 12369   abscabs 13026   sum_csu 13467   0pc0p 21811  Polycply 22316  coeffccoe 22318  degcdgr 22319

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566  ax-addf 9567

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-of 6522  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-2o 7128  df-oadd 7131  df-er 7308  df-map 7419  df-pm 7420  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-sup 7897  df-oi 7931  df-card 8316  df-cda 8544  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11079  df-rp 11217  df-fz 11669  df-fzo 11789  df-fl 11893  df-seq 12072  df-exp 12131  df-hash 12370  df-cj 12891  df-re 12892  df-im 12893  df-sqrt 13027  df-abs 13028  df-clim 13270  df-rlim 13271  df-sum 13468  df-0p 21812  df-ply 22320  df-idp 22321  df-coe 22322  df-dgr 22323  df-quot 22421

This theorem is referenced by:  aannenlem3  22460