Theorem aannenlem1 23284

Description: Lemma for aannen 23287. (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 4406	. . . . . . 7 |- ( a = A -> ( (deg ` d ) <_ a <-> (deg ` d ) <_ A ) )
2		breq2 4406	. . . . . . . 8 |- ( a = A -> ( ( abs ` ( (coeff ` d ) ` e ) ) <_ a <-> ( abs ` ( (coeff ` d ) ` e ) ) <_ A ) )
3	2	ralbidv 2827	. . . . . . 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 1342	. . . . . 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 3036	. . . . 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 2994	. . . 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 3036	. . 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 9620	. . . 4 |- CC e. _V
10	9	rabex 4554	. . 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 4325	. . 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 12185	. . . . . . 7 |- ( -u A ... A ) e. Fin
14		fzfi 12185	. . . . . . 7 |- ( 0 ... A ) e. Fin
15		mapfi 7870	. . . . . . 7 |- ( ( ( -u A ... A ) e. Fin /\ ( 0 ... A ) e. Fin ) -> ( ( -u A ... A ) ^m ( 0 ... A ) ) e. Fin )
16	13, 14, 15	mp2an 678	. . . . . 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 6318	. . . . . 6 |- ( ( -u A ... A ) ^m ( 0 ... A ) ) e. _V
19		neeq1 2686	. . . . . . . . . . 11 |- ( d = a -> ( d =/= 0p <-> a =/= 0p ) )
20		fveq2 5865	. . . . . . . . . . . 12 |- ( d = a -> (deg ` d ) = (deg ` a ) )
21	20	breq1d 4412	. . . . . . . . . . 11 |- ( d = a -> ( (deg ` d ) <_ A <-> (deg ` a ) <_ A ) )
22		fveq2 5865	. . . . . . . . . . . . . . 15 |- ( d = a -> (coeff ` d ) = (coeff ` a ) )
23	22	fveq1d 5867	. . . . . . . . . . . . . 14 |- ( d = a -> ( (coeff ` d ) ` e ) = ( (coeff ` a ) ` e ) )
24	23	fveq2d 5869	. . . . . . . . . . . . 13 |- ( d = a -> ( abs ` ( (coeff ` d ) ` e ) ) = ( abs ` ( (coeff ` a ) ` e ) ) )
25	24	breq1d 4412	. . . . . . . . . . . 12 |- ( d = a -> ( ( abs ` ( (coeff ` d ) ` e ) ) <_ A <-> ( abs ` ( (coeff ` a ) ` e ) ) <_ A ) )
26	25	ralbidv 2827	. . . . . . . . . . 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 1339	. . . . . . . . . 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 3196	. . . . . . . . 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 1010	. . . . . . . . . 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 573	. . . . . . . . 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 199	. . . . . . . 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 10948	. . . . . . . . . . . . . . 15 |- 0 e. ZZ
33		eqid 2451	. . . . . . . . . . . . . . . 16 |- (coeff ` a ) = (coeff ` a )
34	33	coef2 23185	. . . . . . . . . . . . . . 15 |- ( ( a e. (Poly ` ZZ ) /\ 0 e. ZZ ) -> (coeff ` a ) : NN0 --> ZZ )
35	32, 34	mpan2 677	. . . . . . . . . . . . . 14 |- ( a e. (Poly ` ZZ ) -> (coeff ` a ) : NN0 --> ZZ )
36	35	ad2antrl 734	. . . . . . . . . . . . 13 |- ( ( A e. NN0 /\ ( a e. (Poly ` ZZ ) /\ A. e e. NN0 ( abs ` ( (coeff ` a ) ` e ) ) <_ A ) ) -> (coeff ` a ) : NN0 --> ZZ )
37		ffn 5728	. . . . . . . . . . . . 13 |- ( (coeff ` a ) : NN0 --> ZZ -> (coeff ` a ) Fn NN0 )
38	36, 37	syl 17	. . . . . . . . . . . 12 |- ( ( A e. NN0 /\ ( a e. (Poly ` ZZ ) /\ A. e e. NN0 ( abs ` ( (coeff ` a ) ` e ) ) <_ A ) ) -> (coeff ` a ) Fn NN0 )
39	35	adantl 468	. . . . . . . . . . . . . . . . . . . 20 |- ( ( A e. NN0 /\ a e. (Poly ` ZZ ) ) -> (coeff ` a ) : NN0 --> ZZ )
40	39	ffvelrnda 6022	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( A e. NN0 /\ a e. (Poly ` ZZ ) ) /\ e e. NN0 ) -> ( (coeff ` a ) ` e ) e. ZZ )
41	40	zred 11040	. . . . . . . . . . . . . . . . . 18 |- ( ( ( A e. NN0 /\ a e. (Poly ` ZZ ) ) /\ e e. NN0 ) -> ( (coeff ` a ) ` e ) e. RR )
42		nn0re 10878	. . . . . . . . . . . . . . . . . . 19 |- ( A e. NN0 -> A e. RR )
43	42	ad2antrr 732	. . . . . . . . . . . . . . . . . 18 |- ( ( ( A e. NN0 /\ a e. (Poly ` ZZ ) ) /\ e e. NN0 ) -> A e. RR )
44	41, 43	absled 13492	. . . . . . . . . . . . . . . . 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 10960	. . . . . . . . . . . . . . . . . . . 20 |- ( A e. NN0 -> A e. ZZ )
46	45	ad2antrr 732	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( A e. NN0 /\ a e. (Poly ` ZZ ) ) /\ e e. NN0 ) -> A e. ZZ )
47	46	znegcld 11042	. . . . . . . . . . . . . . . . . 18 |- ( ( ( A e. NN0 /\ a e. (Poly ` ZZ ) ) /\ e e. NN0 ) -> -u A e. ZZ )
48		elfz 11790	. . . . . . . . . . . . . . . . . 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 1268	. . . . . . . . . . . . . . . . 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 260	. . . . . . . . . . . . . . . 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 211	. . . . . . . . . . . . . . 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 2796	. . . . . . . . . . . . . 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 625	. . . . . . . . . . . . 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 6051	. . . . . . . . . . . . 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 667	. . . . . . . . . . . 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 5586	. . . . . . . . . . . 12 |- ( (coeff ` a ) : NN0 --> ( -u A ... A ) <-> ( (coeff ` a ) Fn NN0 /\ ran (coeff ` a ) C_ ( -u A ... A ) ) )
57	38, 55, 56	sylanbrc 670	. . . . . . . . . . 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 11887	. . . . . . . . . . . 12 |- ( a e. ( 0 ... A ) -> a e. NN0 )
59	58	ssriv 3436	. . . . . . . . . . 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 668	. . . . . . . . . 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 6318	. . . . . . . . . . 11 |- ( -u A ... A ) e. _V
63		ovex 6318	. . . . . . . . . . 11 |- ( 0 ... A ) e. _V
64	62, 63	elmap 7500	. . . . . . . . . 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 216	. . . . . . . . 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 436	. . . . . . . 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 33	. . . . . . 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 1009	. . . . . . . . . 10 |- ( ( a =/= 0p /\ (deg ` a ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` a ) ` e ) ) <_ A ) -> (deg ` a ) <_ A )
69	68	anim2i 573	. . . . . . . . 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 199	. . . . . . . 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 2686	. . . . . . . . . . 11 |- ( d = b -> ( d =/= 0p <-> b =/= 0p ) )
72		fveq2 5865	. . . . . . . . . . . 12 |- ( d = b -> (deg ` d ) = (deg ` b ) )
73	72	breq1d 4412	. . . . . . . . . . 11 |- ( d = b -> ( (deg ` d ) <_ A <-> (deg ` b ) <_ A ) )
74		fveq2 5865	. . . . . . . . . . . . . . 15 |- ( d = b -> (coeff ` d ) = (coeff ` b ) )
75	74	fveq1d 5867	. . . . . . . . . . . . . 14 |- ( d = b -> ( (coeff ` d ) ` e ) = ( (coeff ` b ) ` e ) )
76	75	fveq2d 5869	. . . . . . . . . . . . 13 |- ( d = b -> ( abs ` ( (coeff ` d ) ` e ) ) = ( abs ` ( (coeff ` b ) ` e ) ) )
77	76	breq1d 4412	. . . . . . . . . . . 12 |- ( d = b -> ( ( abs ` ( (coeff ` d ) ` e ) ) <_ A <-> ( abs ` ( (coeff ` b ) ` e ) ) <_ A ) )
78	77	ralbidv 2827	. . . . . . . . . . 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 1339	. . . . . . . . . 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 3196	. . . . . . . . 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 1009	. . . . . . . . . 10 |- ( ( b =/= 0p /\ (deg ` b ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` b ) ` e ) ) <_ A ) -> (deg ` b ) <_ A )
82	81	anim2i 573	. . . . . . . . 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 199	. . . . . . . 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 768	. . . . . . . . . . . . 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 23152	. . . . . . . . . . . . 13 |- ( a e. (Poly ` ZZ ) -> a : CC --> CC )
86		ffn 5728	. . . . . . . . . . . . 13 |- ( a : CC --> CC -> a Fn CC )
87	84, 85, 86	3syl 18	. . . . . . . . . . . 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 770	. . . . . . . . . . . . 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 23152	. . . . . . . . . . . . 13 |- ( b e. (Poly ` ZZ ) -> b : CC --> CC )
90		ffn 5728	. . . . . . . . . . . . 13 |- ( b : CC --> CC -> b Fn CC )
91	88, 89, 90	3syl 18	. . . . . . . . . . . 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 771	. . . . . . . . . . . . . . . . . 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 467	. . . . . . . . . . . . . . . . 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 5867	. . . . . . . . . . . . . . . 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 5879	. . . . . . . . . . . . . . . . 17 |- ( d e. ( 0 ... A ) -> ( ( (coeff ` a ) |` ( 0 ... A ) ) ` d ) = ( (coeff ` a ) ` d ) )
96	95	adantl 468	. . . . . . . . . . . . . . . 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 5879	. . . . . . . . . . . . . . . . 17 |- ( d e. ( 0 ... A ) -> ( ( (coeff ` b ) |` ( 0 ... A ) ) ` d ) = ( (coeff ` b ) ` d ) )
98	97	adantl 468	. . . . . . . . . . . . . . . 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 2493	. . . . . . . . . . . . . . 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 6305	. . . . . . . . . . . . . 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 13769	. . . . . . . . . . . . 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 776	. . . . . . . . . . . . . 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 777	. . . . . . . . . . . . . . 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 23187	. . . . . . . . . . . . . . . . 17 |- ( a e. (Poly ` ZZ ) -> (deg ` a ) e. NN0 )
105		nn0z 10960	. . . . . . . . . . . . . . . . 17 |- ( (deg ` a ) e. NN0 -> (deg ` a ) e. ZZ )
106	102, 104, 105	3syl 18	. . . . . . . . . . . . . . . 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 770	. . . . . . . . . . . . . . . . 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 11038	. . . . . . . . . . . . . . . 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 11172	. . . . . . . . . . . . . . . 16 |- ( ( (deg ` a ) e. ZZ /\ A e. ZZ ) -> ( A e. ( ZZ>= ` (deg ` a ) ) <-> (deg ` a ) <_ A ) )
110	106, 108, 109	syl2anc 667	. . . . . . . . . . . . . . 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 236	. . . . . . . . . . . . . 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 463	. . . . . . . . . . . . . 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 2451	. . . . . . . . . . . . . . 15 |- (deg ` a ) = (deg ` a )
114	33, 113	coeid3 23194	. . . . . . . . . . . . . 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 1268	. . . . . . . . . . . . 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 1073	. . . . . . . . . . . . . . 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 1208	. . . . . . . . . . . . . 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 771	. . . . . . . . . . . . . . . 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 467	. . . . . . . . . . . . . . 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 23187	. . . . . . . . . . . . . . . . 17 |- ( b e. (Poly ` ZZ ) -> (deg ` b ) e. NN0 )
121		nn0z 10960	. . . . . . . . . . . . . . . . 17 |- ( (deg ` b ) e. NN0 -> (deg ` b ) e. ZZ )
122	117, 120, 121	3syl 18	. . . . . . . . . . . . . . . 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 11172	. . . . . . . . . . . . . . . 16 |- ( ( (deg ` b ) e. ZZ /\ A e. ZZ ) -> ( A e. ( ZZ>= ` (deg ` b ) ) <-> (deg ` b ) <_ A ) )
124	122, 108, 123	syl2anc 667	. . . . . . . . . . . . . . 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 236	. . . . . . . . . . . . . 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 2451	. . . . . . . . . . . . . . 15 |- (coeff ` b ) = (coeff ` b )
127		eqid 2451	. . . . . . . . . . . . . . 15 |- (deg ` b ) = (deg ` b )
128	126, 127	coeid3 23194	. . . . . . . . . . . . . 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 1268	. . . . . . . . . . . . 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 2495	. . . . . . . . . . . 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 5979	. . . . . . . . . . 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 620	. . . . . . . . . 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 5865	. . . . . . . . . . 11 |- ( a = b -> (coeff ` a ) = (coeff ` b ) )
134	133	reseq1d 5104	. . . . . . . . . 10 |- ( a = b -> ( (coeff ` a ) |` ( 0 ... A ) ) = ( (coeff ` b ) |` ( 0 ... A ) ) )
135	132, 134	impbid1 207	. . . . . . . . 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 437	. . . . . . . 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 662	. . . . . . 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 7610	. . . . . 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 20	. . . . 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 7793	. . . . 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 667	. . . 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 2686	. . . . . . . . 9 |- ( d = c -> ( d =/= 0p <-> c =/= 0p ) )
143		fveq2 5865	. . . . . . . . . 10 |- ( d = c -> (deg ` d ) = (deg ` c ) )
144	143	breq1d 4412	. . . . . . . . 9 |- ( d = c -> ( (deg ` d ) <_ A <-> (deg ` c ) <_ A ) )
145		fveq2 5865	. . . . . . . . . . . . 13 |- ( d = c -> (coeff ` d ) = (coeff ` c ) )
146	145	fveq1d 5867	. . . . . . . . . . . 12 |- ( d = c -> ( (coeff ` d ) ` e ) = ( (coeff ` c ) ` e ) )
147	146	fveq2d 5869	. . . . . . . . . . 11 |- ( d = c -> ( abs ` ( (coeff ` d ) ` e ) ) = ( abs ` ( (coeff ` c ) ` e ) ) )
148	147	breq1d 4412	. . . . . . . . . 10 |- ( d = c -> ( ( abs ` ( (coeff ` d ) ` e ) ) <_ A <-> ( abs ` ( (coeff ` c ) ` e ) ) <_ A ) )
149	148	ralbidv 2827	. . . . . . . . 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 1339	. . . . . . . 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 3196	. . . . . . 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 1008	. . . . . . . 8 |- ( ( c =/= 0p /\ (deg ` c ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` c ) ` e ) ) <_ A ) -> c =/= 0p )
153	152	anim2i 573	. . . . . . 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 199	. . . . . 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 23152	. . . . . . . . . . . . 13 |- ( c e. (Poly ` ZZ ) -> c : CC --> CC )
156		ffn 5728	. . . . . . . . . . . . 13 |- ( c : CC --> CC -> c Fn CC )
157	155, 156	syl 17	. . . . . . . . . . . 12 |- ( c e. (Poly ` ZZ ) -> c Fn CC )
158	157	adantr 467	. . . . . . . . . . 11 |- ( ( c e. (Poly ` ZZ ) /\ c =/= 0p ) -> c Fn CC )
159		fniniseg 6003	. . . . . . . . . . 11 |- ( c Fn CC -> ( a e. ( `' c " { 0 } ) <-> ( a e. CC /\ ( c ` a ) = 0 ) ) )
160	158, 159	syl 17	. . . . . . . . . 10 |- ( ( c e. (Poly ` ZZ ) /\ c =/= 0p ) -> ( a e. ( `' c " { 0 } ) <-> ( a e. CC /\ ( c ` a ) = 0 ) ) )
161		fveq2 5865	. . . . . . . . . . . 12 |- ( b = a -> ( c ` b ) = ( c ` a ) )
162	161	eqeq1d 2453	. . . . . . . . . . 11 |- ( b = a -> ( ( c ` b ) = 0 <-> ( c ` a ) = 0 ) )
163	162	elrab 3196	. . . . . . . . . 10 |- ( a e. { b e. CC | ( c ` b ) = 0 } <-> ( a e. CC /\ ( c ` a ) = 0 ) )
164	160, 163	syl6rbbr 268	. . . . . . . . 9 |- ( ( c e. (Poly ` ZZ ) /\ c =/= 0p ) -> ( a e. { b e. CC | ( c ` b ) = 0 } <-> a e. ( `' c " { 0 } ) ) )
165	164	eqrdv 2449	. . . . . . . 8 |- ( ( c e. (Poly ` ZZ ) /\ c =/= 0p ) -> { b e. CC | ( c ` b ) = 0 } = ( `' c " { 0 } ) )
166		eqid 2451	. . . . . . . . . 10 |- ( `' c " { 0 } ) = ( `' c " { 0 } )
167	166	fta1 23261	. . . . . . . . 9 |- ( ( c e. (Poly ` ZZ ) /\ c =/= 0p ) -> ( ( `' c " { 0 } ) e. Fin /\ ( # ` ( `' c " { 0 } ) ) <_ (deg ` c ) ) )
168	167	simpld 461	. . . . . . . 8 |- ( ( c e. (Poly ` ZZ ) /\ c =/= 0p ) -> ( `' c " { 0 } ) e. Fin )
169	165, 168	eqeltrd 2529	. . . . . . 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 33	. . . . 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 2800	. . . 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 7862	. . . 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 667	. . 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 2534	. 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 2529	1 |- ( A e. NN0 -> ( H ` A ) e. Fin )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   /\ w3a 985   = wceq 1444   e. wcel 1887   =/= wne 2622   A.wral 2737   E.wrex 2738   {crab 2741   _Vcvv 3045   C_ wss 3404   {csn 3968   U_ciun 4278   class class class wbr 4402   |-> cmpt 4461   `'ccnv 4833   ran crn 4835   |` cres 4836   "cima 4837   Fn wfn 5577   -->wf 5578   ` cfv 5582  (class class class)co 6290   ^m cmap 7472   ~<_ cdom 7567   Fincfn 7569   CCcc 9537   RRcr 9538   0cc0 9539   x. cmul 9544   <_ cle 9676   -ucneg 9861   NN0cn0 10869   ZZcz 10937   ZZ>=cuz 11159   ...cfz 11784   ^cexp 12272   #chash 12515   abscabs 13297   sum_csu 13752   0pc0p 22627  Polycply 23138  coeffccoe 23140  degcdgr 23141

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617  ax-addf 9618

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-of 6531  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-2o 7183  df-oadd 7186  df-er 7363  df-map 7474  df-pm 7475  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-sup 7956  df-inf 7957  df-oi 8025  df-card 8373  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-fz 11785  df-fzo 11916  df-fl 12028  df-seq 12214  df-exp 12273  df-hash 12516  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-clim 13552  df-rlim 13553  df-sum 13753  df-0p 22628  df-ply 23142  df-idp 23143  df-coe 23144  df-dgr 23145  df-quot 23244

This theorem is referenced by:  aannenlem3  23286