Theorem aannenlem1 24083

Description: Lemma for aannen 24086. (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 4657	. . . . . . 7 |- ( a = A -> ( (deg ` d ) <_ a <-> (deg ` d ) <_ A ) )
2		breq2 4657	. . . . . . . 8 |- ( a = A -> ( ( abs ` ( (coeff ` d ) ` e ) ) <_ a <-> ( abs ` ( (coeff ` d ) ` e ) ) <_ A ) )
3	2	ralbidv 2986	. . . . . . 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 1402	. . . . . 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 3189	. . . . 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 3145	. . . 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 3189	. . 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 10017	. . . 4 |- CC e. _V
10	9	rabex 4813	. . 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 6282	. 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 4567	. . 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 12771	. . . . . . 7 |- ( -u A ... A ) e. Fin
14		fzfi 12771	. . . . . . 7 |- ( 0 ... A ) e. Fin
15		mapfi 8262	. . . . . . 7 |- ( ( ( -u A ... A ) e. Fin /\ ( 0 ... A ) e. Fin ) -> ( ( -u A ... A ) ^m ( 0 ... A ) ) e. Fin )
16	13, 14, 15	mp2an 708	. . . . . 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 6678	. . . . . 6 |- ( ( -u A ... A ) ^m ( 0 ... A ) ) e. _V
19		neeq1 2856	. . . . . . . . . . 11 |- ( d = a -> ( d =/= 0p <-> a =/= 0p ) )
20		fveq2 6191	. . . . . . . . . . . 12 |- ( d = a -> (deg ` d ) = (deg ` a ) )
21	20	breq1d 4663	. . . . . . . . . . 11 |- ( d = a -> ( (deg ` d ) <_ A <-> (deg ` a ) <_ A ) )
22		fveq2 6191	. . . . . . . . . . . . . . 15 |- ( d = a -> (coeff ` d ) = (coeff ` a ) )
23	22	fveq1d 6193	. . . . . . . . . . . . . 14 |- ( d = a -> ( (coeff ` d ) ` e ) = ( (coeff ` a ) ` e ) )
24	23	fveq2d 6195	. . . . . . . . . . . . 13 |- ( d = a -> ( abs ` ( (coeff ` d ) ` e ) ) = ( abs ` ( (coeff ` a ) ` e ) ) )
25	24	breq1d 4663	. . . . . . . . . . . 12 |- ( d = a -> ( ( abs ` ( (coeff ` d ) ` e ) ) <_ A <-> ( abs ` ( (coeff ` a ) ` e ) ) <_ A ) )
26	25	ralbidv 2986	. . . . . . . . . . 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 1399	. . . . . . . . . 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 3363	. . . . . . . . 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 1063	. . . . . . . . . 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 593	. . . . . . . . 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 207	. . . . . . . 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 11388	. . . . . . . . . . . . . . 15 |- 0 e. ZZ
33		eqid 2622	. . . . . . . . . . . . . . . 16 |- (coeff ` a ) = (coeff ` a )
34	33	coef2 23987	. . . . . . . . . . . . . . 15 |- ( ( a e. (Poly ` ZZ ) /\ 0 e. ZZ ) -> (coeff ` a ) : NN0 --> ZZ )
35	32, 34	mpan2 707	. . . . . . . . . . . . . 14 |- ( a e. (Poly ` ZZ ) -> (coeff ` a ) : NN0 --> ZZ )
36	35	ad2antrl 764	. . . . . . . . . . . . 13 |- ( ( A e. NN0 /\ ( a e. (Poly ` ZZ ) /\ A. e e. NN0 ( abs ` ( (coeff ` a ) ` e ) ) <_ A ) ) -> (coeff ` a ) : NN0 --> ZZ )
37	36	ffnd 6046	. . . . . . . . . . . 12 |- ( ( A e. NN0 /\ ( a e. (Poly ` ZZ ) /\ A. e e. NN0 ( abs ` ( (coeff ` a ) ` e ) ) <_ A ) ) -> (coeff ` a ) Fn NN0 )
38	35	adantl 482	. . . . . . . . . . . . . . . . . . . 20 |- ( ( A e. NN0 /\ a e. (Poly ` ZZ ) ) -> (coeff ` a ) : NN0 --> ZZ )
39	38	ffvelrnda 6359	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( A e. NN0 /\ a e. (Poly ` ZZ ) ) /\ e e. NN0 ) -> ( (coeff ` a ) ` e ) e. ZZ )
40	39	zred 11482	. . . . . . . . . . . . . . . . . 18 |- ( ( ( A e. NN0 /\ a e. (Poly ` ZZ ) ) /\ e e. NN0 ) -> ( (coeff ` a ) ` e ) e. RR )
41		nn0re 11301	. . . . . . . . . . . . . . . . . . 19 |- ( A e. NN0 -> A e. RR )
42	41	ad2antrr 762	. . . . . . . . . . . . . . . . . 18 |- ( ( ( A e. NN0 /\ a e. (Poly ` ZZ ) ) /\ e e. NN0 ) -> A e. RR )
43	40, 42	absled 14169	. . . . . . . . . . . . . . . . 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 ) ) )
44		nn0z 11400	. . . . . . . . . . . . . . . . . . . 20 |- ( A e. NN0 -> A e. ZZ )
45	44	ad2antrr 762	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( A e. NN0 /\ a e. (Poly ` ZZ ) ) /\ e e. NN0 ) -> A e. ZZ )
46	45	znegcld 11484	. . . . . . . . . . . . . . . . . 18 |- ( ( ( A e. NN0 /\ a e. (Poly ` ZZ ) ) /\ e e. NN0 ) -> -u A e. ZZ )
47		elfz 12332	. . . . . . . . . . . . . . . . . 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 ) ) )
48	39, 46, 45, 47	syl3anc 1326	. . . . . . . . . . . . . . . . 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 ) ) )
49	43, 48	bitr4d 271	. . . . . . . . . . . . . . . 16 |- ( ( ( A e. NN0 /\ a e. (Poly ` ZZ ) ) /\ e e. NN0 ) -> ( ( abs ` ( (coeff ` a ) ` e ) ) <_ A <-> ( (coeff ` a ) ` e ) e. ( -u A ... A ) ) )
50	49	biimpd 219	. . . . . . . . . . . . . . 15 |- ( ( ( A e. NN0 /\ a e. (Poly ` ZZ ) ) /\ e e. NN0 ) -> ( ( abs ` ( (coeff ` a ) ` e ) ) <_ A -> ( (coeff ` a ) ` e ) e. ( -u A ... A ) ) )
51	50	ralimdva 2962	. . . . . . . . . . . . . 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 ) ) )
52	51	impr 649	. . . . . . . . . . . . 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 ) )
53		fnfvrnss 6390	. . . . . . . . . . . . 13 |- ( ( (coeff ` a ) Fn NN0 /\ A. e e. NN0 ( (coeff ` a ) ` e ) e. ( -u A ... A ) ) -> ran (coeff ` a ) C_ ( -u A ... A ) )
54	37, 52, 53	syl2anc 693	. . . . . . . . . . . 12 |- ( ( A e. NN0 /\ ( a e. (Poly ` ZZ ) /\ A. e e. NN0 ( abs ` ( (coeff ` a ) ` e ) ) <_ A ) ) -> ran (coeff ` a ) C_ ( -u A ... A ) )
55		df-f 5892	. . . . . . . . . . . 12 |- ( (coeff ` a ) : NN0 --> ( -u A ... A ) <-> ( (coeff ` a ) Fn NN0 /\ ran (coeff ` a ) C_ ( -u A ... A ) ) )
56	37, 54, 55	sylanbrc 698	. . . . . . . . . . 11 |- ( ( A e. NN0 /\ ( a e. (Poly ` ZZ ) /\ A. e e. NN0 ( abs ` ( (coeff ` a ) ` e ) ) <_ A ) ) -> (coeff ` a ) : NN0 --> ( -u A ... A ) )
57		fz0ssnn0 12435	. . . . . . . . . . 11 |- ( 0 ... A ) C_ NN0
58		fssres 6070	. . . . . . . . . . 11 |- ( ( (coeff ` a ) : NN0 --> ( -u A ... A ) /\ ( 0 ... A ) C_ NN0 ) -> ( (coeff ` a ) |` ( 0 ... A ) ) : ( 0 ... A ) --> ( -u A ... A ) )
59	56, 57, 58	sylancl 694	. . . . . . . . . 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 ) )
60		ovex 6678	. . . . . . . . . . 11 |- ( -u A ... A ) e. _V
61		ovex 6678	. . . . . . . . . . 11 |- ( 0 ... A ) e. _V
62	60, 61	elmap 7886	. . . . . . . . . 10 |- ( ( (coeff ` a ) |` ( 0 ... A ) ) e. ( ( -u A ... A ) ^m ( 0 ... A ) ) <-> ( (coeff ` a ) |` ( 0 ... A ) ) : ( 0 ... A ) --> ( -u A ... A ) )
63	59, 62	sylibr 224	. . . . . . . . 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 ) ) )
64	63	ex 450	. . . . . . . 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 ) ) ) )
65	31, 64	syl5 34	. . . . . . 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 ) ) ) )
66		simp2 1062	. . . . . . . . . 10 |- ( ( a =/= 0p /\ (deg ` a ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` a ) ` e ) ) <_ A ) -> (deg ` a ) <_ A )
67	66	anim2i 593	. . . . . . . . 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 ) )
68	28, 67	sylbi 207	. . . . . . . 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 ) )
69		neeq1 2856	. . . . . . . . . . 11 |- ( d = b -> ( d =/= 0p <-> b =/= 0p ) )
70		fveq2 6191	. . . . . . . . . . . 12 |- ( d = b -> (deg ` d ) = (deg ` b ) )
71	70	breq1d 4663	. . . . . . . . . . 11 |- ( d = b -> ( (deg ` d ) <_ A <-> (deg ` b ) <_ A ) )
72		fveq2 6191	. . . . . . . . . . . . . . 15 |- ( d = b -> (coeff ` d ) = (coeff ` b ) )
73	72	fveq1d 6193	. . . . . . . . . . . . . 14 |- ( d = b -> ( (coeff ` d ) ` e ) = ( (coeff ` b ) ` e ) )
74	73	fveq2d 6195	. . . . . . . . . . . . 13 |- ( d = b -> ( abs ` ( (coeff ` d ) ` e ) ) = ( abs ` ( (coeff ` b ) ` e ) ) )
75	74	breq1d 4663	. . . . . . . . . . . 12 |- ( d = b -> ( ( abs ` ( (coeff ` d ) ` e ) ) <_ A <-> ( abs ` ( (coeff ` b ) ` e ) ) <_ A ) )
76	75	ralbidv 2986	. . . . . . . . . . 11 |- ( d = b -> ( A. e e. NN0 ( abs ` ( (coeff ` d ) ` e ) ) <_ A <-> A. e e. NN0 ( abs ` ( (coeff ` b ) ` e ) ) <_ A ) )
77	69, 71, 76	3anbi123d 1399	. . . . . . . . . 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 ) ) )
78	77	elrab 3363	. . . . . . . . 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 ) ) )
79		simp2 1062	. . . . . . . . . 10 |- ( ( b =/= 0p /\ (deg ` b ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` b ) ` e ) ) <_ A ) -> (deg ` b ) <_ A )
80	79	anim2i 593	. . . . . . . . 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 ) )
81	78, 80	sylbi 207	. . . . . . . 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 ) )
82		simplll 798	. . . . . . . . . . . . 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 ) )
83		plyf 23954	. . . . . . . . . . . . 13 |- ( a e. (Poly ` ZZ ) -> a : CC --> CC )
84		ffn 6045	. . . . . . . . . . . . 13 |- ( a : CC --> CC -> a Fn CC )
85	82, 83, 84	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 )
86		simplrl 800	. . . . . . . . . . . . 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 ) )
87		plyf 23954	. . . . . . . . . . . . 13 |- ( b e. (Poly ` ZZ ) -> b : CC --> CC )
88		ffn 6045	. . . . . . . . . . . . 13 |- ( b : CC --> CC -> b Fn CC )
89	86, 87, 88	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 )
90		simplrr 801	. . . . . . . . . . . . . . . . . 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 ) ) )
91	90	adantr 481	. . . . . . . . . . . . . . . . 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 ) ) )
92	91	fveq1d 6193	. . . . . . . . . . . . . . . 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 ) )
93		fvres 6207	. . . . . . . . . . . . . . . . 17 |- ( d e. ( 0 ... A ) -> ( ( (coeff ` a ) |` ( 0 ... A ) ) ` d ) = ( (coeff ` a ) ` d ) )
94	93	adantl 482	. . . . . . . . . . . . . . . 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 ) )
95		fvres 6207	. . . . . . . . . . . . . . . . 17 |- ( d e. ( 0 ... A ) -> ( ( (coeff ` b ) |` ( 0 ... A ) ) ` d ) = ( (coeff ` b ) ` d ) )
96	95	adantl 482	. . . . . . . . . . . . . . . 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 ) )
97	92, 94, 96	3eqtr3d 2664	. . . . . . . . . . . . . . 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 ) )
98	97	oveq1d 6665	. . . . . . . . . . . . . 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 ) ) )
99	98	sumeq2dv 14433	. . . . . . . . . . . . 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 ) ) )
100		simp-4l 806	. . . . . . . . . . . . . 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 ) )
101		simp-4r 807	. . . . . . . . . . . . . . 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 )
102		dgrcl 23989	. . . . . . . . . . . . . . . . 17 |- ( a e. (Poly ` ZZ ) -> (deg ` a ) e. NN0 )
103		nn0z 11400	. . . . . . . . . . . . . . . . 17 |- ( (deg ` a ) e. NN0 -> (deg ` a ) e. ZZ )
104	100, 102, 103	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 )
105		simplrl 800	. . . . . . . . . . . . . . . . 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 )
106	105	nn0zd 11480	. . . . . . . . . . . . . . . 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 )
107		eluz 11701	. . . . . . . . . . . . . . . 16 |- ( ( (deg ` a ) e. ZZ /\ A e. ZZ ) -> ( A e. ( ZZ>= ` (deg ` a ) ) <-> (deg ` a ) <_ A ) )
108	104, 106, 107	syl2anc 693	. . . . . . . . . . . . . . 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 ) )
109	101, 108	mpbird 247	. . . . . . . . . . . . . 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 ) ) )
110		simpr 477	. . . . . . . . . . . . . 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 )
111		eqid 2622	. . . . . . . . . . . . . . 15 |- (deg ` a ) = (deg ` a )
112	33, 111	coeid3 23996	. . . . . . . . . . . . . 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 ) ) )
113	100, 109, 110, 112	syl3anc 1326	. . . . . . . . . . . . 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 ) ) )
114		simp1rl 1126	. . . . . . . . . . . . . . 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 ) )
115	114	3expa 1265	. . . . . . . . . . . . . 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 ) )
116		simplrr 801	. . . . . . . . . . . . . . . 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 )
117	116	adantr 481	. . . . . . . . . . . . . . 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 )
118		dgrcl 23989	. . . . . . . . . . . . . . . . 17 |- ( b e. (Poly ` ZZ ) -> (deg ` b ) e. NN0 )
119		nn0z 11400	. . . . . . . . . . . . . . . . 17 |- ( (deg ` b ) e. NN0 -> (deg ` b ) e. ZZ )
120	115, 118, 119	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 )
121		eluz 11701	. . . . . . . . . . . . . . . 16 |- ( ( (deg ` b ) e. ZZ /\ A e. ZZ ) -> ( A e. ( ZZ>= ` (deg ` b ) ) <-> (deg ` b ) <_ A ) )
122	120, 106, 121	syl2anc 693	. . . . . . . . . . . . . . 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 ) )
123	117, 122	mpbird 247	. . . . . . . . . . . . . 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 ) ) )
124		eqid 2622	. . . . . . . . . . . . . . 15 |- (coeff ` b ) = (coeff ` b )
125		eqid 2622	. . . . . . . . . . . . . . 15 |- (deg ` b ) = (deg ` b )
126	124, 125	coeid3 23996	. . . . . . . . . . . . . 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 ) ) )
127	115, 123, 110, 126	syl3anc 1326	. . . . . . . . . . . . 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 ) ) )
128	99, 113, 127	3eqtr4d 2666	. . . . . . . . . . . 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 ) )
129	85, 89, 128	eqfnfvd 6314	. . . . . . . . . . 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 )
130	129	expr 643	. . . . . . . . . 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 ) )
131		fveq2 6191	. . . . . . . . . . 11 |- ( a = b -> (coeff ` a ) = (coeff ` b ) )
132	131	reseq1d 5395	. . . . . . . . . 10 |- ( a = b -> ( (coeff ` a ) |` ( 0 ... A ) ) = ( (coeff ` b ) |` ( 0 ... A ) ) )
133	130, 132	impbid1 215	. . . . . . . . 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 ) )
134	133	expcom 451	. . . . . . . 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 ) ) )
135	68, 81, 134	syl2ani 688	. . . . . . 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 ) ) )
136	65, 135	dom2d 7996	. . . . . 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 ) ) ) )
137	18, 136	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 ) ) )
138		domfi 8181	. . . . 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 )
139	17, 137, 138	syl2anc 693	. . . 4 |- ( A e. NN0 -> { d e. (Poly ` ZZ ) | ( d =/= 0p /\ (deg ` d ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` d ) ` e ) ) <_ A ) } e. Fin )
140		neeq1 2856	. . . . . . . . 9 |- ( d = c -> ( d =/= 0p <-> c =/= 0p ) )
141		fveq2 6191	. . . . . . . . . 10 |- ( d = c -> (deg ` d ) = (deg ` c ) )
142	141	breq1d 4663	. . . . . . . . 9 |- ( d = c -> ( (deg ` d ) <_ A <-> (deg ` c ) <_ A ) )
143		fveq2 6191	. . . . . . . . . . . . 13 |- ( d = c -> (coeff ` d ) = (coeff ` c ) )
144	143	fveq1d 6193	. . . . . . . . . . . 12 |- ( d = c -> ( (coeff ` d ) ` e ) = ( (coeff ` c ) ` e ) )
145	144	fveq2d 6195	. . . . . . . . . . 11 |- ( d = c -> ( abs ` ( (coeff ` d ) ` e ) ) = ( abs ` ( (coeff ` c ) ` e ) ) )
146	145	breq1d 4663	. . . . . . . . . 10 |- ( d = c -> ( ( abs ` ( (coeff ` d ) ` e ) ) <_ A <-> ( abs ` ( (coeff ` c ) ` e ) ) <_ A ) )
147	146	ralbidv 2986	. . . . . . . . 9 |- ( d = c -> ( A. e e. NN0 ( abs ` ( (coeff ` d ) ` e ) ) <_ A <-> A. e e. NN0 ( abs ` ( (coeff ` c ) ` e ) ) <_ A ) )
148	140, 142, 147	3anbi123d 1399	. . . . . . . 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 ) ) )
149	148	elrab 3363	. . . . . . 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 ) ) )
150		simp1 1061	. . . . . . . 8 |- ( ( c =/= 0p /\ (deg ` c ) <_ A /\ A. e e. NN0 ( abs ` ( (coeff ` c ) ` e ) ) <_ A ) -> c =/= 0p )
151	150	anim2i 593	. . . . . . 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 ) )
152	149, 151	sylbi 207	. . . . . 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 ) )
153		plyf 23954	. . . . . . . . . . . . 13 |- ( c e. (Poly ` ZZ ) -> c : CC --> CC )
154	153	ffnd 6046	. . . . . . . . . . . 12 |- ( c e. (Poly ` ZZ ) -> c Fn CC )
155	154	adantr 481	. . . . . . . . . . 11 |- ( ( c e. (Poly ` ZZ ) /\ c =/= 0p ) -> c Fn CC )
156		fniniseg 6338	. . . . . . . . . . 11 |- ( c Fn CC -> ( a e. ( `' c " { 0 } ) <-> ( a e. CC /\ ( c ` a ) = 0 ) ) )
157	155, 156	syl 17	. . . . . . . . . 10 |- ( ( c e. (Poly ` ZZ ) /\ c =/= 0p ) -> ( a e. ( `' c " { 0 } ) <-> ( a e. CC /\ ( c ` a ) = 0 ) ) )
158		fveq2 6191	. . . . . . . . . . . 12 |- ( b = a -> ( c ` b ) = ( c ` a ) )
159	158	eqeq1d 2624	. . . . . . . . . . 11 |- ( b = a -> ( ( c ` b ) = 0 <-> ( c ` a ) = 0 ) )
160	159	elrab 3363	. . . . . . . . . 10 |- ( a e. { b e. CC | ( c ` b ) = 0 } <-> ( a e. CC /\ ( c ` a ) = 0 ) )
161	157, 160	syl6rbbr 279	. . . . . . . . 9 |- ( ( c e. (Poly ` ZZ ) /\ c =/= 0p ) -> ( a e. { b e. CC | ( c ` b ) = 0 } <-> a e. ( `' c " { 0 } ) ) )
162	161	eqrdv 2620	. . . . . . . 8 |- ( ( c e. (Poly ` ZZ ) /\ c =/= 0p ) -> { b e. CC | ( c ` b ) = 0 } = ( `' c " { 0 } ) )
163		eqid 2622	. . . . . . . . . 10 |- ( `' c " { 0 } ) = ( `' c " { 0 } )
164	163	fta1 24063	. . . . . . . . 9 |- ( ( c e. (Poly ` ZZ ) /\ c =/= 0p ) -> ( ( `' c " { 0 } ) e. Fin /\ ( # ` ( `' c " { 0 } ) ) <_ (deg ` c ) ) )
165	164	simpld 475	. . . . . . . 8 |- ( ( c e. (Poly ` ZZ ) /\ c =/= 0p ) -> ( `' c " { 0 } ) e. Fin )
166	162, 165	eqeltrd 2701	. . . . . . 7 |- ( ( c e. (Poly ` ZZ ) /\ c =/= 0p ) -> { b e. CC | ( c ` b ) = 0 } e. Fin )
167	166	a1i 11	. . . . . 6 |- ( A e. NN0 -> ( ( c e. (Poly ` ZZ ) /\ c =/= 0p ) -> { b e. CC | ( c ` b ) = 0 } e. Fin ) )
168	152, 167	syl5 34	. . . . 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 ) )
169	168	ralrimiv 2965	. . . 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 )
170		iunfi 8254	. . . 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 )
171	139, 169, 170	syl2anc 693	. . 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 )
172	12, 171	syl5eqelr 2706	. 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 )
173	11, 172	eqeltrd 2701	1 |- ( A e. NN0 -> ( H ` A ) e. Fin )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   {crab 2916   _Vcvv 3200   C_ wss 3574   {csn 4177   U_ciun 4520   class class class wbr 4653   |-> cmpt 4729   `'ccnv 5113   ran crn 5115   |` cres 5116   "cima 5117   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   ~<_ cdom 7953   Fincfn 7955   CCcc 9934   RRcr 9935   0cc0 9936   x. cmul 9941   <_ cle 10075   -ucneg 10267   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   ...cfz 12326   ^cexp 12860   #chash 13117   abscabs 13974   sum_csu 14416   0pc0p 23436  Polycply 23940  coeffccoe 23942  degcdgr 23943

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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-rlim 14220  df-sum 14417  df-0p 23437  df-ply 23944  df-idp 23945  df-coe 23946  df-dgr 23947  df-quot 24046

This theorem is referenced by:  aannenlem3  24085