Theorem aannenlem1 20198

Description: Lemma for aannen 20201. (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 =/= 0 p /\ (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 4176	. . . . . . 7 |- ( a = A -> ( (deg ` d ) <_ a <-> (deg ` d ) <_ A ) )
2		breq2 4176	. . . . . . . 8 |- ( a = A -> ( ( abs ` ( (coeff ` d ) ` e ) ) <_ a <-> ( abs ` ( (coeff ` d ) ` e ) ) <_ A ) )
3	2	ralbidv 2686	. . . . . . 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 1257	. . . . . 6 |- ( a = A -> ( ( d =/= 0 p /\ (deg ` d ) <_ a /\ A. e e. NN0 ( abs ` ( (coeff ` d ) ` e ) ) <_ a ) <-> ( d =/= 0 p /\ (deg ` d ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` d ) ` e ) ) <_ A ) ) )
5	4	rabbidv 2908	. . . . 5 |- ( a = A -> { d e. (Poly ` ZZ ) | ( d =/= 0 p /\ (deg ` d ) <_ a /\ A. e e. NN0 ( abs ` ( (coeff ` d ) ` e ) ) <_ a ) } = { d e. (Poly ` ZZ ) | ( d =/= 0 p /\ (deg ` d ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` d ) ` e ) ) <_ A ) } )
6	5	rexeqdv 2871	. . . 4 |- ( a = A -> ( E. c e. { d e. (Poly ` ZZ ) | ( d =/= 0 p /\ (deg ` d ) <_ a /\ A. e e. NN0 ( abs ` ( (coeff ` d ) ` e ) ) <_ a ) } ( c ` b ) = 0 <-> E. c e. { d e. (Poly ` ZZ ) | ( d =/= 0 p /\ (deg ` d ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` d ) ` e ) ) <_ A ) } ( c ` b ) = 0 ) )
7	6	rabbidv 2908	. . 3 |- ( a = A -> { b e. CC | E. c e. { d e. (Poly ` ZZ ) | ( d =/= 0 p /\ (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 =/= 0 p /\ (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 =/= 0 p /\ (deg ` d ) <_ a /\ A. e e. NN0 ( abs ` ( (coeff ` d ) ` e ) ) <_ a ) } ( c ` b ) = 0 } )
9		cnex 9027	. . . 4 |- CC e. _V
10	9	rabex 4314	. . 3 |- { b e. CC | E. c e. { d e. (Poly ` ZZ ) | ( d =/= 0 p /\ (deg ` d ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` d ) ` e ) ) <_ A ) } ( c ` b ) = 0 } e. _V
11	7, 8, 10	fvmpt 5765	. 2 |- ( A e. NN0 -> ( H ` A ) = { b e. CC | E. c e. { d e. (Poly ` ZZ ) | ( d =/= 0 p /\ (deg ` d ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` d ) ` e ) ) <_ A ) } ( c ` b ) = 0 } )
12		iunrab 4098	. . 3 |- U_ c e. { d e. (Poly ` ZZ ) | ( d =/= 0 p /\ (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 =/= 0 p /\ (deg ` d ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` d ) ` e ) ) <_ A ) } ( c ` b ) = 0 }
13		fzfi 11266	. . . . . . 7 |- ( -u A ... A ) e. Fin
14		fzfi 11266	. . . . . . 7 |- ( 0 ... A ) e. Fin
15		mapfi 7361	. . . . . . 7 |- ( ( ( -u A ... A ) e. Fin /\ ( 0 ... A ) e. Fin ) -> ( ( -u A ... A ) ^m ( 0 ... A ) ) e. Fin )
16	13, 14, 15	mp2an 654	. . . . . 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 6065	. . . . . 6 |- ( ( -u A ... A ) ^m ( 0 ... A ) ) e. _V
19		neeq1 2575	. . . . . . . . . . 11 |- ( d = a -> ( d =/= 0 p <-> a =/= 0 p ) )
20		fveq2 5687	. . . . . . . . . . . 12 |- ( d = a -> (deg ` d ) = (deg ` a ) )
21	20	breq1d 4182	. . . . . . . . . . 11 |- ( d = a -> ( (deg ` d ) <_ A <-> (deg ` a ) <_ A ) )
22		fveq2 5687	. . . . . . . . . . . . . . 15 |- ( d = a -> (coeff ` d ) = (coeff ` a ) )
23	22	fveq1d 5689	. . . . . . . . . . . . . 14 |- ( d = a -> ( (coeff ` d ) ` e ) = ( (coeff ` a ) ` e ) )
24	23	fveq2d 5691	. . . . . . . . . . . . 13 |- ( d = a -> ( abs ` ( (coeff ` d ) ` e ) ) = ( abs ` ( (coeff ` a ) ` e ) ) )
25	24	breq1d 4182	. . . . . . . . . . . 12 |- ( d = a -> ( ( abs ` ( (coeff ` d ) ` e ) ) <_ A <-> ( abs ` ( (coeff ` a ) ` e ) ) <_ A ) )
26	25	ralbidv 2686	. . . . . . . . . . 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 1254	. . . . . . . . . 10 |- ( d = a -> ( ( d =/= 0 p /\ (deg ` d ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` d ) ` e ) ) <_ A ) <-> ( a =/= 0 p /\ (deg ` a ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` a ) ` e ) ) <_ A ) ) )
28	27	elrab 3052	. . . . . . . . 9 |- ( a e. { d e. (Poly ` ZZ ) | ( d =/= 0 p /\ (deg ` d ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` d ) ` e ) ) <_ A ) } <-> ( a e. (Poly ` ZZ ) /\ ( a =/= 0 p /\ (deg ` a ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` a ) ` e ) ) <_ A ) ) )
29		simp3 959	. . . . . . . . . 10 |- ( ( a =/= 0 p /\ (deg ` a ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` a ) ` e ) ) <_ A ) -> A. e e. NN0 ( abs ` ( (coeff ` a ) ` e ) ) <_ A )
30	29	anim2i 553	. . . . . . . . 9 |- ( ( a e. (Poly ` ZZ ) /\ ( a =/= 0 p /\ (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 188	. . . . . . . 8 |- ( a e. { d e. (Poly ` ZZ ) | ( d =/= 0 p /\ (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 10249	. . . . . . . . . . . . . . 15 |- 0 e. ZZ
33		eqid 2404	. . . . . . . . . . . . . . . 16 |- (coeff ` a ) = (coeff ` a )
34	33	coef2 20103	. . . . . . . . . . . . . . 15 |- ( ( a e. (Poly ` ZZ ) /\ 0 e. ZZ ) -> (coeff ` a ) : NN0 --> ZZ )
35	32, 34	mpan2 653	. . . . . . . . . . . . . 14 |- ( a e. (Poly ` ZZ ) -> (coeff ` a ) : NN0 --> ZZ )
36	35	ad2antrl 709	. . . . . . . . . . . . 13 |- ( ( A e. NN0 /\ ( a e. (Poly ` ZZ ) /\ A. e e. NN0 ( abs ` ( (coeff ` a ) ` e ) ) <_ A ) ) -> (coeff ` a ) : NN0 --> ZZ )
37		ffn 5550	. . . . . . . . . . . . 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 453	. . . . . . . . . . . . . . . . . . . 20 |- ( ( A e. NN0 /\ a e. (Poly ` ZZ ) ) -> (coeff ` a ) : NN0 --> ZZ )
40	39	ffvelrnda 5829	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( A e. NN0 /\ a e. (Poly ` ZZ ) ) /\ e e. NN0 ) -> ( (coeff ` a ) ` e ) e. ZZ )
41	40	zred 10331	. . . . . . . . . . . . . . . . . 18 |- ( ( ( A e. NN0 /\ a e. (Poly ` ZZ ) ) /\ e e. NN0 ) -> ( (coeff ` a ) ` e ) e. RR )
42		nn0re 10186	. . . . . . . . . . . . . . . . . . 19 |- ( A e. NN0 -> A e. RR )
43	42	ad2antrr 707	. . . . . . . . . . . . . . . . . 18 |- ( ( ( A e. NN0 /\ a e. (Poly ` ZZ ) ) /\ e e. NN0 ) -> A e. RR )
44	41, 43	absled 12188	. . . . . . . . . . . . . . . . 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 10260	. . . . . . . . . . . . . . . . . . . 20 |- ( A e. NN0 -> A e. ZZ )
46	45	ad2antrr 707	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( A e. NN0 /\ a e. (Poly ` ZZ ) ) /\ e e. NN0 ) -> A e. ZZ )
47	46	znegcld 10333	. . . . . . . . . . . . . . . . . 18 |- ( ( ( A e. NN0 /\ a e. (Poly ` ZZ ) ) /\ e e. NN0 ) -> -u A e. ZZ )
48		elfz 11005	. . . . . . . . . . . . . . . . . 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 1184	. . . . . . . . . . . . . . . . 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 248	. . . . . . . . . . . . . . . 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 199	. . . . . . . . . . . . . . 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 2744	. . . . . . . . . . . . . 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 603	. . . . . . . . . . . . 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 5855	. . . . . . . . . . . . 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 643	. . . . . . . . . . . 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 5417	. . . . . . . . . . . 12 |- ( (coeff ` a ) : NN0 --> ( -u A ... A ) <-> ( (coeff ` a ) Fn NN0 /\ ran (coeff ` a ) C_ ( -u A ... A ) ) )
57	38, 55, 56	sylanbrc 646	. . . . . . . . . . 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 11039	. . . . . . . . . . . 12 |- ( a e. ( 0 ... A ) -> a e. NN0 )
59	58	ssriv 3312	. . . . . . . . . . 11 |- ( 0 ... A ) C_ NN0
60		fssres 5569	. . . . . . . . . . 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 644	. . . . . . . . . 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 6065	. . . . . . . . . . 11 |- ( -u A ... A ) e. _V
63		ovex 6065	. . . . . . . . . . 11 |- ( 0 ... A ) e. _V
64	62, 63	elmap 7001	. . . . . . . . . 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 204	. . . . . . . . 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 424	. . . . . . . 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 30	. . . . . . 7 |- ( A e. NN0 -> ( a e. { d e. (Poly ` ZZ ) | ( d =/= 0 p /\ (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 958	. . . . . . . . . 10 |- ( ( a =/= 0 p /\ (deg ` a ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` a ) ` e ) ) <_ A ) -> (deg ` a ) <_ A )
69	68	anim2i 553	. . . . . . . . 9 |- ( ( a e. (Poly ` ZZ ) /\ ( a =/= 0 p /\ (deg ` a ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` a ) ` e ) ) <_ A ) ) -> ( a e. (Poly ` ZZ ) /\ (deg ` a ) <_ A ) )
70	28, 69	sylbi 188	. . . . . . . 8 |- ( a e. { d e. (Poly ` ZZ ) | ( d =/= 0 p /\ (deg ` d ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` d ) ` e ) ) <_ A ) } -> ( a e. (Poly ` ZZ ) /\ (deg ` a ) <_ A ) )
71		neeq1 2575	. . . . . . . . . . 11 |- ( d = b -> ( d =/= 0 p <-> b =/= 0 p ) )
72		fveq2 5687	. . . . . . . . . . . 12 |- ( d = b -> (deg ` d ) = (deg ` b ) )
73	72	breq1d 4182	. . . . . . . . . . 11 |- ( d = b -> ( (deg ` d ) <_ A <-> (deg ` b ) <_ A ) )
74		fveq2 5687	. . . . . . . . . . . . . . 15 |- ( d = b -> (coeff ` d ) = (coeff ` b ) )
75	74	fveq1d 5689	. . . . . . . . . . . . . 14 |- ( d = b -> ( (coeff ` d ) ` e ) = ( (coeff ` b ) ` e ) )
76	75	fveq2d 5691	. . . . . . . . . . . . 13 |- ( d = b -> ( abs ` ( (coeff ` d ) ` e ) ) = ( abs ` ( (coeff ` b ) ` e ) ) )
77	76	breq1d 4182	. . . . . . . . . . . 12 |- ( d = b -> ( ( abs ` ( (coeff ` d ) ` e ) ) <_ A <-> ( abs ` ( (coeff ` b ) ` e ) ) <_ A ) )
78	77	ralbidv 2686	. . . . . . . . . . 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 1254	. . . . . . . . . 10 |- ( d = b -> ( ( d =/= 0 p /\ (deg ` d ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` d ) ` e ) ) <_ A ) <-> ( b =/= 0 p /\ (deg ` b ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` b ) ` e ) ) <_ A ) ) )
80	79	elrab 3052	. . . . . . . . 9 |- ( b e. { d e. (Poly ` ZZ ) | ( d =/= 0 p /\ (deg ` d ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` d ) ` e ) ) <_ A ) } <-> ( b e. (Poly ` ZZ ) /\ ( b =/= 0 p /\ (deg ` b ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` b ) ` e ) ) <_ A ) ) )
81		simp2 958	. . . . . . . . . 10 |- ( ( b =/= 0 p /\ (deg ` b ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` b ) ` e ) ) <_ A ) -> (deg ` b ) <_ A )
82	81	anim2i 553	. . . . . . . . 9 |- ( ( b e. (Poly ` ZZ ) /\ ( b =/= 0 p /\ (deg ` b ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` b ) ` e ) ) <_ A ) ) -> ( b e. (Poly ` ZZ ) /\ (deg ` b ) <_ A ) )
83	80, 82	sylbi 188	. . . . . . . 8 |- ( b e. { d e. (Poly ` ZZ ) | ( d =/= 0 p /\ (deg ` d ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` d ) ` e ) ) <_ A ) } -> ( b e. (Poly ` ZZ ) /\ (deg ` b ) <_ A ) )
84		simplll 735	. . . . . . . . . . . . 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 20070	. . . . . . . . . . . . 13 |- ( a e. (Poly ` ZZ ) -> a : CC --> CC )
86		ffn 5550	. . . . . . . . . . . . 13 |- ( a : CC --> CC -> a Fn CC )
87	84, 85, 86	3syl 19	. . . . . . . . . . . 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 737	. . . . . . . . . . . . 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 20070	. . . . . . . . . . . . 13 |- ( b e. (Poly ` ZZ ) -> b : CC --> CC )
90		ffn 5550	. . . . . . . . . . . . 13 |- ( b : CC --> CC -> b Fn CC )
91	88, 89, 90	3syl 19	. . . . . . . . . . . 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 738	. . . . . . . . . . . . . . . . . 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 452	. . . . . . . . . . . . . . . . 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 5689	. . . . . . . . . . . . . . . 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 5704	. . . . . . . . . . . . . . . . 17 |- ( d e. ( 0 ... A ) -> ( ( (coeff ` a ) |` ( 0 ... A ) ) ` d ) = ( (coeff ` a ) ` d ) )
96	95	adantl 453	. . . . . . . . . . . . . . . 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 5704	. . . . . . . . . . . . . . . . 17 |- ( d e. ( 0 ... A ) -> ( ( (coeff ` b ) |` ( 0 ... A ) ) ` d ) = ( (coeff ` b ) ` d ) )
98	97	adantl 453	. . . . . . . . . . . . . . . 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 2444	. . . . . . . . . . . . . . 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 6055	. . . . . . . . . . . . . 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 12452	. . . . . . . . . . . . 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 743	. . . . . . . . . . . . . 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 744	. . . . . . . . . . . . . . 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 20105	. . . . . . . . . . . . . . . . 17 |- ( a e. (Poly ` ZZ ) -> (deg ` a ) e. NN0 )
105		nn0z 10260	. . . . . . . . . . . . . . . . 17 |- ( (deg ` a ) e. NN0 -> (deg ` a ) e. ZZ )
106	102, 104, 105	3syl 19	. . . . . . . . . . . . . . . 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 737	. . . . . . . . . . . . . . . . 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 10329	. . . . . . . . . . . . . . . 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 10455	. . . . . . . . . . . . . . . 16 |- ( ( (deg ` a ) e. ZZ /\ A e. ZZ ) -> ( A e. ( ZZ>= ` (deg ` a ) ) <-> (deg ` a ) <_ A ) )
110	106, 108, 109	syl2anc 643	. . . . . . . . . . . . . . 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 224	. . . . . . . . . . . . . 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 448	. . . . . . . . . . . . . 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 2404	. . . . . . . . . . . . . . 15 |- (deg ` a ) = (deg ` a )
114	33, 113	coeid3 20112	. . . . . . . . . . . . . 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 1184	. . . . . . . . . . . . 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 1022	. . . . . . . . . . . . . . 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 1153	. . . . . . . . . . . . . 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 738	. . . . . . . . . . . . . . . 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 452	. . . . . . . . . . . . . . 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 20105	. . . . . . . . . . . . . . . . 17 |- ( b e. (Poly ` ZZ ) -> (deg ` b ) e. NN0 )
121		nn0z 10260	. . . . . . . . . . . . . . . . 17 |- ( (deg ` b ) e. NN0 -> (deg ` b ) e. ZZ )
122	117, 120, 121	3syl 19	. . . . . . . . . . . . . . . 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 10455	. . . . . . . . . . . . . . . 16 |- ( ( (deg ` b ) e. ZZ /\ A e. ZZ ) -> ( A e. ( ZZ>= ` (deg ` b ) ) <-> (deg ` b ) <_ A ) )
124	122, 108, 123	syl2anc 643	. . . . . . . . . . . . . . 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 224	. . . . . . . . . . . . . 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 2404	. . . . . . . . . . . . . . 15 |- (coeff ` b ) = (coeff ` b )
127		eqid 2404	. . . . . . . . . . . . . . 15 |- (deg ` b ) = (deg ` b )
128	126, 127	coeid3 20112	. . . . . . . . . . . . . 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 1184	. . . . . . . . . . . . 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 2446	. . . . . . . . . . . 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 5789	. . . . . . . . . . 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 599	. . . . . . . . . 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 5687	. . . . . . . . . . 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 195	. . . . . . . . 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 425	. . . . . . . 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 638	. . . . . . 7 |- ( A e. NN0 -> ( ( a e. { d e. (Poly ` ZZ ) | ( d =/= 0 p /\ (deg ` d ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` d ) ` e ) ) <_ A ) } /\ b e. { d e. (Poly ` ZZ ) | ( d =/= 0 p /\ (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 7107	. . . . . 6 |- ( A e. NN0 -> ( ( ( -u A ... A ) ^m ( 0 ... A ) ) e. _V -> { d e. (Poly ` ZZ ) | ( d =/= 0 p /\ (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 =/= 0 p /\ (deg ` d ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` d ) ` e ) ) <_ A ) } ~<_ ( ( -u A ... A ) ^m ( 0 ... A ) ) )
140		domfi 7289	. . . . 5 |- ( ( ( ( -u A ... A ) ^m ( 0 ... A ) ) e. Fin /\ { d e. (Poly ` ZZ ) | ( d =/= 0 p /\ (deg ` d ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` d ) ` e ) ) <_ A ) } ~<_ ( ( -u A ... A ) ^m ( 0 ... A ) ) ) -> { d e. (Poly ` ZZ ) | ( d =/= 0 p /\ (deg ` d ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` d ) ` e ) ) <_ A ) } e. Fin )
141	17, 139, 140	syl2anc 643	. . . 4 |- ( A e. NN0 -> { d e. (Poly ` ZZ ) | ( d =/= 0 p /\ (deg ` d ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` d ) ` e ) ) <_ A ) } e. Fin )
142		neeq1 2575	. . . . . . . . 9 |- ( d = c -> ( d =/= 0 p <-> c =/= 0 p ) )
143		fveq2 5687	. . . . . . . . . 10 |- ( d = c -> (deg ` d ) = (deg ` c ) )
144	143	breq1d 4182	. . . . . . . . 9 |- ( d = c -> ( (deg ` d ) <_ A <-> (deg ` c ) <_ A ) )
145		fveq2 5687	. . . . . . . . . . . . 13 |- ( d = c -> (coeff ` d ) = (coeff ` c ) )
146	145	fveq1d 5689	. . . . . . . . . . . 12 |- ( d = c -> ( (coeff ` d ) ` e ) = ( (coeff ` c ) ` e ) )
147	146	fveq2d 5691	. . . . . . . . . . 11 |- ( d = c -> ( abs ` ( (coeff ` d ) ` e ) ) = ( abs ` ( (coeff ` c ) ` e ) ) )
148	147	breq1d 4182	. . . . . . . . . 10 |- ( d = c -> ( ( abs ` ( (coeff ` d ) ` e ) ) <_ A <-> ( abs ` ( (coeff ` c ) ` e ) ) <_ A ) )
149	148	ralbidv 2686	. . . . . . . . 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 1254	. . . . . . . 8 |- ( d = c -> ( ( d =/= 0 p /\ (deg ` d ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` d ) ` e ) ) <_ A ) <-> ( c =/= 0 p /\ (deg ` c ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` c ) ` e ) ) <_ A ) ) )
151	150	elrab 3052	. . . . . . 7 |- ( c e. { d e. (Poly ` ZZ ) | ( d =/= 0 p /\ (deg ` d ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` d ) ` e ) ) <_ A ) } <-> ( c e. (Poly ` ZZ ) /\ ( c =/= 0 p /\ (deg ` c ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` c ) ` e ) ) <_ A ) ) )
152		simp1 957	. . . . . . . 8 |- ( ( c =/= 0 p /\ (deg ` c ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` c ) ` e ) ) <_ A ) -> c =/= 0 p )
153	152	anim2i 553	. . . . . . 7 |- ( ( c e. (Poly ` ZZ ) /\ ( c =/= 0 p /\ (deg ` c ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` c ) ` e ) ) <_ A ) ) -> ( c e. (Poly ` ZZ ) /\ c =/= 0 p ) )
154	151, 153	sylbi 188	. . . . . 6 |- ( c e. { d e. (Poly ` ZZ ) | ( d =/= 0 p /\ (deg ` d ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` d ) ` e ) ) <_ A ) } -> ( c e. (Poly ` ZZ ) /\ c =/= 0 p ) )
155		plyf 20070	. . . . . . . . . . . . 13 |- ( c e. (Poly ` ZZ ) -> c : CC --> CC )
156		ffn 5550	. . . . . . . . . . . . 13 |- ( c : CC --> CC -> c Fn CC )
157	155, 156	syl 16	. . . . . . . . . . . 12 |- ( c e. (Poly ` ZZ ) -> c Fn CC )
158	157	adantr 452	. . . . . . . . . . 11 |- ( ( c e. (Poly ` ZZ ) /\ c =/= 0 p ) -> c Fn CC )
159		fniniseg 5810	. . . . . . . . . . 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 =/= 0 p ) -> ( a e. ( `' c " { 0 } ) <-> ( a e. CC /\ ( c ` a ) = 0 ) ) )
161		fveq2 5687	. . . . . . . . . . . 12 |- ( b = a -> ( c ` b ) = ( c ` a ) )
162	161	eqeq1d 2412	. . . . . . . . . . 11 |- ( b = a -> ( ( c ` b ) = 0 <-> ( c ` a ) = 0 ) )
163	162	elrab 3052	. . . . . . . . . 10 |- ( a e. { b e. CC | ( c ` b ) = 0 } <-> ( a e. CC /\ ( c ` a ) = 0 ) )
164	160, 163	syl6rbbr 256	. . . . . . . . 9 |- ( ( c e. (Poly ` ZZ ) /\ c =/= 0 p ) -> ( a e. { b e. CC | ( c ` b ) = 0 } <-> a e. ( `' c " { 0 } ) ) )
165	164	eqrdv 2402	. . . . . . . 8 |- ( ( c e. (Poly ` ZZ ) /\ c =/= 0 p ) -> { b e. CC | ( c ` b ) = 0 } = ( `' c " { 0 } ) )
166		eqid 2404	. . . . . . . . . 10 |- ( `' c " { 0 } ) = ( `' c " { 0 } )
167	166	fta1 20178	. . . . . . . . 9 |- ( ( c e. (Poly ` ZZ ) /\ c =/= 0 p ) -> ( ( `' c " { 0 } ) e. Fin /\ ( # ` ( `' c " { 0 } ) ) <_ (deg ` c ) ) )
168	167	simpld 446	. . . . . . . 8 |- ( ( c e. (Poly ` ZZ ) /\ c =/= 0 p ) -> ( `' c " { 0 } ) e. Fin )
169	165, 168	eqeltrd 2478	. . . . . . 7 |- ( ( c e. (Poly ` ZZ ) /\ c =/= 0 p ) -> { b e. CC | ( c ` b ) = 0 } e. Fin )
170	169	a1i 11	. . . . . 6 |- ( A e. NN0 -> ( ( c e. (Poly ` ZZ ) /\ c =/= 0 p ) -> { b e. CC | ( c ` b ) = 0 } e. Fin ) )
171	154, 170	syl5 30	. . . . 5 |- ( A e. NN0 -> ( c e. { d e. (Poly ` ZZ ) | ( d =/= 0 p /\ (deg ` d ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` d ) ` e ) ) <_ A ) } -> { b e. CC | ( c ` b ) = 0 } e. Fin ) )
172	171	ralrimiv 2748	. . . 4 |- ( A e. NN0 -> A. c e. { d e. (Poly ` ZZ ) | ( d =/= 0 p /\ (deg ` d ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` d ) ` e ) ) <_ A ) } { b e. CC | ( c ` b ) = 0 } e. Fin )
173		iunfi 7353	. . . 4 |- ( ( { d e. (Poly ` ZZ ) | ( d =/= 0 p /\ (deg ` d ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` d ) ` e ) ) <_ A ) } e. Fin /\ A. c e. { d e. (Poly ` ZZ ) | ( d =/= 0 p /\ (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 =/= 0 p /\ (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 643	. . 3 |- ( A e. NN0 -> U_ c e. { d e. (Poly ` ZZ ) | ( d =/= 0 p /\ (deg ` d ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` d ) ` e ) ) <_ A ) } { b e. CC | ( c ` b ) = 0 } e. Fin )
175	12, 174	syl5eqelr 2489	. 2 |- ( A e. NN0 -> { b e. CC | E. c e. { d e. (Poly ` ZZ ) | ( d =/= 0 p /\ (deg ` d ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` d ) ` e ) ) <_ A ) } ( c ` b ) = 0 } e. Fin )
176	11, 175	eqeltrd 2478	1 |- ( A e. NN0 -> ( H ` A ) e. Fin )

Syntax hints:   -> wi 4   <-> wb 177   /\ wa 359   /\ w3a 936   = wceq 1649   e. wcel 1721   =/= wne 2567   A.wral 2666   E.wrex 2667   {crab 2670   _Vcvv 2916   C_ wss 3280   {csn 3774   U_ciun 4053   class class class wbr 4172   e. cmpt 4226   `'ccnv 4836   ran crn 4838   |` cres 4839   "cima 4840   Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6040   ^m cmap 6977   ~<_ cdom 7066   Fincfn 7068   CCcc 8944   RRcr 8945   0cc0 8946   x. cmul 8951   <_ cle 9077   -ucneg 9248   NN0cn0 10177   ZZcz 10238   ZZ>=cuz 10444   ...cfz 10999   ^cexp 11337   #chash 11573   abscabs 11994   sum_csu 12434   0 pc0p 19514  Polycply 20056  coeffccoe 20058  degcdgr 20059

This theorem is referenced by:  aannenlem3  20200

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

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-fz 11000  df-fzo 11091  df-fl 11157  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237  df-rlim 12238  df-sum 12435  df-0p 19515  df-ply 20060  df-idp 20061  df-coe 20062  df-dgr 20063  df-quot 20161