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Theorem aannenlem1 20245
Description: Lemma for aannen 20248. (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
StepHypRef Expression
1 breq2 4216 . . . . . . 7  |-  ( a  =  A  ->  (
(deg `  d )  <_  a  <->  (deg `  d )  <_  A ) )
2 breq2 4216 . . . . . . . 8  |-  ( a  =  A  ->  (
( abs `  (
(coeff `  d ) `  e ) )  <_ 
a  <->  ( abs `  (
(coeff `  d ) `  e ) )  <_  A ) )
32ralbidv 2725 . . . . . . 7  |-  ( a  =  A  ->  ( A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  a  <->  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  A
) )
41, 33anbi23d 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
) ) )
54rabbidv 2948 . . . . 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
) } )
65rexeqdv 2911 . . . 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 ) )
76rabbidv 2948 . . 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 9071 . . . 4  |-  CC  e.  _V
109rabex 4354 . . 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
117, 8, 10fvmpt 5806 . 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 4138 . . 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 11311 . . . . . . 7  |-  ( -u A ... A )  e. 
Fin
14 fzfi 11311 . . . . . . 7  |-  ( 0 ... A )  e. 
Fin
15 mapfi 7403 . . . . . . 7  |-  ( ( ( -u A ... A )  e.  Fin  /\  ( 0 ... A
)  e.  Fin )  ->  ( ( -u A ... A )  ^m  (
0 ... A ) )  e.  Fin )
1613, 14, 15mp2an 654 . . . . . 6  |-  ( (
-u A ... A
)  ^m  ( 0 ... A ) )  e.  Fin
1716a1i 11 . . . . 5  |-  ( A  e.  NN0  ->  ( (
-u A ... A
)  ^m  ( 0 ... A ) )  e.  Fin )
18 ovex 6106 . . . . . 6  |-  ( (
-u A ... A
)  ^m  ( 0 ... A ) )  e.  _V
19 neeq1 2609 . . . . . . . . . . 11  |-  ( d  =  a  ->  (
d  =/=  0 p  <-> 
a  =/=  0 p ) )
20 fveq2 5728 . . . . . . . . . . . 12  |-  ( d  =  a  ->  (deg `  d )  =  (deg
`  a ) )
2120breq1d 4222 . . . . . . . . . . 11  |-  ( d  =  a  ->  (
(deg `  d )  <_  A  <->  (deg `  a )  <_  A ) )
22 fveq2 5728 . . . . . . . . . . . . . . 15  |-  ( d  =  a  ->  (coeff `  d )  =  (coeff `  a ) )
2322fveq1d 5730 . . . . . . . . . . . . . 14  |-  ( d  =  a  ->  (
(coeff `  d ) `  e )  =  ( (coeff `  a ) `  e ) )
2423fveq2d 5732 . . . . . . . . . . . . 13  |-  ( d  =  a  ->  ( abs `  ( (coeff `  d ) `  e
) )  =  ( abs `  ( (coeff `  a ) `  e
) ) )
2524breq1d 4222 . . . . . . . . . . . 12  |-  ( d  =  a  ->  (
( abs `  (
(coeff `  d ) `  e ) )  <_  A 
<->  ( abs `  (
(coeff `  a ) `  e ) )  <_  A ) )
2625ralbidv 2725 . . . . . . . . . . 11  |-  ( d  =  a  ->  ( A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  A  <->  A. e  e.  NN0  ( abs `  ( (coeff `  a ) `  e
) )  <_  A
) )
2719, 21, 263anbi123d 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
) ) )
2827elrab 3092 . . . . . . . . 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 )
3029anim2i 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 ) )
3128, 30sylbi 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 10293 . . . . . . . . . . . . . . 15  |-  0  e.  ZZ
33 eqid 2436 . . . . . . . . . . . . . . . 16  |-  (coeff `  a )  =  (coeff `  a )
3433coef2 20150 . . . . . . . . . . . . . . 15  |-  ( ( a  e.  (Poly `  ZZ )  /\  0  e.  ZZ )  ->  (coeff `  a ) : NN0 --> ZZ )
3532, 34mpan2 653 . . . . . . . . . . . . . 14  |-  ( a  e.  (Poly `  ZZ )  ->  (coeff `  a
) : NN0 --> ZZ )
3635ad2antrl 709 . . . . . . . . . . . . 13  |-  ( ( A  e.  NN0  /\  ( a  e.  (Poly `  ZZ )  /\  A. e  e.  NN0  ( abs `  ( (coeff `  a
) `  e )
)  <_  A )
)  ->  (coeff `  a
) : NN0 --> ZZ )
37 ffn 5591 . . . . . . . . . . . . 13  |-  ( (coeff `  a ) : NN0 --> ZZ 
->  (coeff `  a )  Fn  NN0 )
3836, 37syl 16 . . . . . . . . . . . 12  |-  ( ( A  e.  NN0  /\  ( a  e.  (Poly `  ZZ )  /\  A. e  e.  NN0  ( abs `  ( (coeff `  a
) `  e )
)  <_  A )
)  ->  (coeff `  a
)  Fn  NN0 )
3935adantl 453 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( A  e.  NN0  /\  a  e.  (Poly `  ZZ ) )  ->  (coeff `  a ) : NN0 --> ZZ )
4039ffvelrnda 5870 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( A  e.  NN0  /\  a  e.  (Poly `  ZZ ) )  /\  e  e.  NN0 )  ->  (
(coeff `  a ) `  e )  e.  ZZ )
4140zred 10375 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  e.  NN0  /\  a  e.  (Poly `  ZZ ) )  /\  e  e.  NN0 )  ->  (
(coeff `  a ) `  e )  e.  RR )
42 nn0re 10230 . . . . . . . . . . . . . . . . . . 19  |-  ( A  e.  NN0  ->  A  e.  RR )
4342ad2antrr 707 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  e.  NN0  /\  a  e.  (Poly `  ZZ ) )  /\  e  e.  NN0 )  ->  A  e.  RR )
4441, 43absled 12233 . . . . . . . . . . . . . . . . 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 10304 . . . . . . . . . . . . . . . . . . . 20  |-  ( A  e.  NN0  ->  A  e.  ZZ )
4645ad2antrr 707 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( A  e.  NN0  /\  a  e.  (Poly `  ZZ ) )  /\  e  e.  NN0 )  ->  A  e.  ZZ )
4746znegcld 10377 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  e.  NN0  /\  a  e.  (Poly `  ZZ ) )  /\  e  e.  NN0 )  ->  -u A  e.  ZZ )
48 elfz 11049 . . . . . . . . . . . . . . . . . 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 ) ) )
4940, 47, 46, 48syl3anc 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 ) ) )
5044, 49bitr4d 248 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  NN0  /\  a  e.  (Poly `  ZZ ) )  /\  e  e.  NN0 )  ->  (
( abs `  (
(coeff `  a ) `  e ) )  <_  A 
<->  ( (coeff `  a
) `  e )  e.  ( -u A ... A ) ) )
5150biimpd 199 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  NN0  /\  a  e.  (Poly `  ZZ ) )  /\  e  e.  NN0 )  ->  (
( abs `  (
(coeff `  a ) `  e ) )  <_  A  ->  ( (coeff `  a ) `  e
)  e.  ( -u A ... A ) ) )
5251ralimdva 2784 . . . . . . . . . . . . . 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
) ) )
5352impr 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 5896 . . . . . . . . . . . . 13  |-  ( ( (coeff `  a )  Fn  NN0  /\  A. e  e.  NN0  ( (coeff `  a ) `  e
)  e.  ( -u A ... A ) )  ->  ran  (coeff `  a
)  C_  ( -u A ... A ) )
5538, 53, 54syl2anc 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 5458 . . . . . . . . . . . 12  |-  ( (coeff `  a ) : NN0 --> (
-u A ... A
)  <->  ( (coeff `  a )  Fn  NN0  /\ 
ran  (coeff `  a )  C_  ( -u A ... A ) ) )
5738, 55, 56sylanbrc 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 11083 . . . . . . . . . . . 12  |-  ( a  e.  ( 0 ... A )  ->  a  e.  NN0 )
5958ssriv 3352 . . . . . . . . . . 11  |-  ( 0 ... A )  C_  NN0
60 fssres 5610 . . . . . . . . . . 11  |-  ( ( (coeff `  a ) : NN0 --> ( -u A ... A )  /\  (
0 ... A )  C_  NN0 )  ->  ( (coeff `  a )  |`  (
0 ... A ) ) : ( 0 ... A ) --> ( -u A ... A ) )
6157, 59, 60sylancl 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 6106 . . . . . . . . . . 11  |-  ( -u A ... A )  e. 
_V
63 ovex 6106 . . . . . . . . . . 11  |-  ( 0 ... A )  e. 
_V
6462, 63elmap 7042 . . . . . . . . . 10  |-  ( ( (coeff `  a )  |`  ( 0 ... A
) )  e.  ( ( -u A ... A )  ^m  (
0 ... A ) )  <-> 
( (coeff `  a
)  |`  ( 0 ... A ) ) : ( 0 ... A
) --> ( -u A ... A ) )
6561, 64sylibr 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
) ) )
6665ex 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 ) ) ) )
6731, 66syl5 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 )
6968anim2i 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
) )
7028, 69sylbi 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 2609 . . . . . . . . . . 11  |-  ( d  =  b  ->  (
d  =/=  0 p  <-> 
b  =/=  0 p ) )
72 fveq2 5728 . . . . . . . . . . . 12  |-  ( d  =  b  ->  (deg `  d )  =  (deg
`  b ) )
7372breq1d 4222 . . . . . . . . . . 11  |-  ( d  =  b  ->  (
(deg `  d )  <_  A  <->  (deg `  b )  <_  A ) )
74 fveq2 5728 . . . . . . . . . . . . . . 15  |-  ( d  =  b  ->  (coeff `  d )  =  (coeff `  b ) )
7574fveq1d 5730 . . . . . . . . . . . . . 14  |-  ( d  =  b  ->  (
(coeff `  d ) `  e )  =  ( (coeff `  b ) `  e ) )
7675fveq2d 5732 . . . . . . . . . . . . 13  |-  ( d  =  b  ->  ( abs `  ( (coeff `  d ) `  e
) )  =  ( abs `  ( (coeff `  b ) `  e
) ) )
7776breq1d 4222 . . . . . . . . . . . 12  |-  ( d  =  b  ->  (
( abs `  (
(coeff `  d ) `  e ) )  <_  A 
<->  ( abs `  (
(coeff `  b ) `  e ) )  <_  A ) )
7877ralbidv 2725 . . . . . . . . . . 11  |-  ( d  =  b  ->  ( A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  A  <->  A. e  e.  NN0  ( abs `  ( (coeff `  b ) `  e
) )  <_  A
) )
7971, 73, 783anbi123d 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
) ) )
8079elrab 3092 . . . . . . . . 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 )
8281anim2i 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
) )
8380, 82sylbi 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 20117 . . . . . . . . . . . . 13  |-  ( a  e.  (Poly `  ZZ )  ->  a : CC --> CC )
86 ffn 5591 . . . . . . . . . . . . 13  |-  ( a : CC --> CC  ->  a  Fn  CC )
8784, 85, 863syl 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 20117 . . . . . . . . . . . . 13  |-  ( b  e.  (Poly `  ZZ )  ->  b : CC --> CC )
90 ffn 5591 . . . . . . . . . . . . 13  |-  ( b : CC --> CC  ->  b  Fn  CC )
9188, 89, 903syl 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 ) ) )
9392adantr 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
) ) )
9493fveq1d 5730 . . . . . . . . . . . . . . . 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 5745 . . . . . . . . . . . . . . . . 17  |-  ( d  e.  ( 0 ... A )  ->  (
( (coeff `  a
)  |`  ( 0 ... A ) ) `  d )  =  ( (coeff `  a ) `  d ) )
9695adantl 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 5745 . . . . . . . . . . . . . . . . 17  |-  ( d  e.  ( 0 ... A )  ->  (
( (coeff `  b
)  |`  ( 0 ... A ) ) `  d )  =  ( (coeff `  b ) `  d ) )
9897adantl 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 ) )
9994, 96, 983eqtr3d 2476 . . . . . . . . . . . . . . 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 ) )
10099oveq1d 6096 . . . . . . . . . . . . . 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
) ) )
101100sumeq2dv 12497 . . . . . . . . . . . . 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 20152 . . . . . . . . . . . . . . . . 17  |-  ( a  e.  (Poly `  ZZ )  ->  (deg `  a
)  e.  NN0 )
105 nn0z 10304 . . . . . . . . . . . . . . . . 17  |-  ( (deg
`  a )  e. 
NN0  ->  (deg `  a
)  e.  ZZ )
106102, 104, 1053syl 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 )
108107nn0zd 10373 . . . . . . . . . . . . . . . 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 10499 . . . . . . . . . . . . . . . 16  |-  ( ( (deg `  a )  e.  ZZ  /\  A  e.  ZZ )  ->  ( A  e.  ( ZZ>= `  (deg `  a ) )  <-> 
(deg `  a )  <_  A ) )
110106, 108, 109syl2anc 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 )
)
111103, 110mpbird 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 2436 . . . . . . . . . . . . . . 15  |-  (deg `  a )  =  (deg
`  a )
11433, 113coeid3 20159 . . . . . . . . . . . . . 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 ) ) )
115102, 111, 112, 114syl3anc 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 ) )
1171163expa 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 )
119118adantr 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 20152 . . . . . . . . . . . . . . . . 17  |-  ( b  e.  (Poly `  ZZ )  ->  (deg `  b
)  e.  NN0 )
121 nn0z 10304 . . . . . . . . . . . . . . . . 17  |-  ( (deg
`  b )  e. 
NN0  ->  (deg `  b
)  e.  ZZ )
122117, 120, 1213syl 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 10499 . . . . . . . . . . . . . . . 16  |-  ( ( (deg `  b )  e.  ZZ  /\  A  e.  ZZ )  ->  ( A  e.  ( ZZ>= `  (deg `  b ) )  <-> 
(deg `  b )  <_  A ) )
124122, 108, 123syl2anc 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 )
)
125119, 124mpbird 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 2436 . . . . . . . . . . . . . . 15  |-  (coeff `  b )  =  (coeff `  b )
127 eqid 2436 . . . . . . . . . . . . . . 15  |-  (deg `  b )  =  (deg
`  b )
128126, 127coeid3 20159 . . . . . . . . . . . . . 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 ) ) )
129117, 125, 112, 128syl3anc 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
) ) )
130101, 115, 1293eqtr4d 2478 . . . . . . . . . . . 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 ) )
13187, 91, 130eqfnfvd 5830 . . . . . . . . . . 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 )
132131expr 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 5728 . . . . . . . . . . 11  |-  ( a  =  b  ->  (coeff `  a )  =  (coeff `  b ) )
134133reseq1d 5145 . . . . . . . . . 10  |-  ( a  =  b  ->  (
(coeff `  a )  |`  ( 0 ... A
) )  =  ( (coeff `  b )  |`  ( 0 ... A
) ) )
135132, 134impbid1 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 ) )
136135expcom 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 ) ) )
13770, 83, 136syl2ani 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 ) ) )
13867, 137dom2d 7148 . . . . . 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 ) ) ) )
13918, 138mpi 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 7330 . . . . 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 )
14117, 139, 140syl2anc 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 2609 . . . . . . . . 9  |-  ( d  =  c  ->  (
d  =/=  0 p  <-> 
c  =/=  0 p ) )
143 fveq2 5728 . . . . . . . . . 10  |-  ( d  =  c  ->  (deg `  d )  =  (deg
`  c ) )
144143breq1d 4222 . . . . . . . . 9  |-  ( d  =  c  ->  (
(deg `  d )  <_  A  <->  (deg `  c )  <_  A ) )
145 fveq2 5728 . . . . . . . . . . . . 13  |-  ( d  =  c  ->  (coeff `  d )  =  (coeff `  c ) )
146145fveq1d 5730 . . . . . . . . . . . 12  |-  ( d  =  c  ->  (
(coeff `  d ) `  e )  =  ( (coeff `  c ) `  e ) )
147146fveq2d 5732 . . . . . . . . . . 11  |-  ( d  =  c  ->  ( abs `  ( (coeff `  d ) `  e
) )  =  ( abs `  ( (coeff `  c ) `  e
) ) )
148147breq1d 4222 . . . . . . . . . 10  |-  ( d  =  c  ->  (
( abs `  (
(coeff `  d ) `  e ) )  <_  A 
<->  ( abs `  (
(coeff `  c ) `  e ) )  <_  A ) )
149148ralbidv 2725 . . . . . . . . 9  |-  ( d  =  c  ->  ( A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  A  <->  A. e  e.  NN0  ( abs `  ( (coeff `  c ) `  e
) )  <_  A
) )
150142, 144, 1493anbi123d 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
) ) )
151150elrab 3092 . . . . . . 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 )
153152anim2i 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 ) )
154151, 153sylbi 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 20117 . . . . . . . . . . . . 13  |-  ( c  e.  (Poly `  ZZ )  ->  c : CC --> CC )
156 ffn 5591 . . . . . . . . . . . . 13  |-  ( c : CC --> CC  ->  c  Fn  CC )
157155, 156syl 16 . . . . . . . . . . . 12  |-  ( c  e.  (Poly `  ZZ )  ->  c  Fn  CC )
158157adantr 452 . . . . . . . . . . 11  |-  ( ( c  e.  (Poly `  ZZ )  /\  c  =/=  0 p )  -> 
c  Fn  CC )
159 fniniseg 5851 . . . . . . . . . . 11  |-  ( c  Fn  CC  ->  (
a  e.  ( `' c " { 0 } )  <->  ( a  e.  CC  /\  ( c `
 a )  =  0 ) ) )
160158, 159syl 16 . . . . . . . . . 10  |-  ( ( c  e.  (Poly `  ZZ )  /\  c  =/=  0 p )  -> 
( a  e.  ( `' c " {
0 } )  <->  ( a  e.  CC  /\  ( c `
 a )  =  0 ) ) )
161 fveq2 5728 . . . . . . . . . . . 12  |-  ( b  =  a  ->  (
c `  b )  =  ( c `  a ) )
162161eqeq1d 2444 . . . . . . . . . . 11  |-  ( b  =  a  ->  (
( c `  b
)  =  0  <->  (
c `  a )  =  0 ) )
163162elrab 3092 . . . . . . . . . 10  |-  ( a  e.  { b  e.  CC  |  ( c `
 b )  =  0 }  <->  ( a  e.  CC  /\  ( c `
 a )  =  0 ) )
164160, 163syl6rbbr 256 . . . . . . . . 9  |-  ( ( c  e.  (Poly `  ZZ )  /\  c  =/=  0 p )  -> 
( a  e.  {
b  e.  CC  | 
( c `  b
)  =  0 }  <-> 
a  e.  ( `' c " { 0 } ) ) )
165164eqrdv 2434 . . . . . . . 8  |-  ( ( c  e.  (Poly `  ZZ )  /\  c  =/=  0 p )  ->  { b  e.  CC  |  ( c `  b )  =  0 }  =  ( `' c " { 0 } ) )
166 eqid 2436 . . . . . . . . . 10  |-  ( `' c " { 0 } )  =  ( `' c " {
0 } )
167166fta1 20225 . . . . . . . . 9  |-  ( ( c  e.  (Poly `  ZZ )  /\  c  =/=  0 p )  -> 
( ( `' c
" { 0 } )  e.  Fin  /\  ( # `  ( `' c " { 0 } ) )  <_ 
(deg `  c )
) )
168167simpld 446 . . . . . . . 8  |-  ( ( c  e.  (Poly `  ZZ )  /\  c  =/=  0 p )  -> 
( `' c " { 0 } )  e.  Fin )
169165, 168eqeltrd 2510 . . . . . . 7  |-  ( ( c  e.  (Poly `  ZZ )  /\  c  =/=  0 p )  ->  { b  e.  CC  |  ( c `  b )  =  0 }  e.  Fin )
170169a1i 11 . . . . . 6  |-  ( A  e.  NN0  ->  ( ( c  e.  (Poly `  ZZ )  /\  c  =/=  0 p )  ->  { b  e.  CC  |  ( c `  b )  =  0 }  e.  Fin )
)
171154, 170syl5 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 ) )
172171ralrimiv 2788 . . . 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 7394 . . . 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 )
174141, 172, 173syl2anc 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 )
17512, 174syl5eqelr 2521 . 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 )
17611, 175eqeltrd 2510 1  |-  ( A  e.  NN0  ->  ( H `
 A )  e. 
Fin )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599   A.wral 2705   E.wrex 2706   {crab 2709   _Vcvv 2956    C_ wss 3320   {csn 3814   U_ciun 4093   class class class wbr 4212    e. cmpt 4266   `'ccnv 4877   ran crn 4879    |` cres 4880   "cima 4881    Fn wfn 5449   -->wf 5450   ` cfv 5454  (class class class)co 6081    ^m cmap 7018    ~<_ cdom 7107   Fincfn 7109   CCcc 8988   RRcr 8989   0cc0 8990    x. cmul 8995    <_ cle 9121   -ucneg 9292   NN0cn0 10221   ZZcz 10282   ZZ>=cuz 10488   ...cfz 11043   ^cexp 11382   #chash 11618   abscabs 12039   sum_csu 12479   0 pc0p 19561  Polycply 20103  coeffccoe 20105  degcdgr 20106
This theorem is referenced by:  aannenlem3  20247
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068  ax-addf 9069
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-of 6305  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-2o 6725  df-oadd 6728  df-er 6905  df-map 7020  df-pm 7021  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-oi 7479  df-card 7826  df-cda 8048  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-rp 10613  df-fz 11044  df-fzo 11136  df-fl 11202  df-seq 11324  df-exp 11383  df-hash 11619  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-clim 12282  df-rlim 12283  df-sum 12480  df-0p 19562  df-ply 20107  df-idp 20108  df-coe 20109  df-dgr 20110  df-quot 20208
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