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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
StepHypRef 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 ) )
32ralbidv 2686 . . . . . . 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 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
) } )
65rexeqdv 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 ) )
76rabbidv 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
109rabex 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
117, 8, 10fvmpt 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 )
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 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 ) )
2120breq1d 4182 . . . . . . . . . . 11  |-  ( d  =  a  ->  (
(deg `  d )  <_  A  <->  (deg `  a )  <_  A ) )
22 fveq2 5687 . . . . . . . . . . . . . . 15  |-  ( d  =  a  ->  (coeff `  d )  =  (coeff `  a ) )
2322fveq1d 5689 . . . . . . . . . . . . . 14  |-  ( d  =  a  ->  (
(coeff `  d ) `  e )  =  ( (coeff `  a ) `  e ) )
2423fveq2d 5691 . . . . . . . . . . . . 13  |-  ( d  =  a  ->  ( abs `  ( (coeff `  d ) `  e
) )  =  ( abs `  ( (coeff `  a ) `  e
) ) )
2524breq1d 4182 . . . . . . . . . . . 12  |-  ( d  =  a  ->  (
( abs `  (
(coeff `  d ) `  e ) )  <_  A 
<->  ( abs `  (
(coeff `  a ) `  e ) )  <_  A ) )
2625ralbidv 2686 . . . . . . . . . . 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 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 )
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 10249 . . . . . . . . . . . . . . 15  |-  0  e.  ZZ
33 eqid 2404 . . . . . . . . . . . . . . . 16  |-  (coeff `  a )  =  (coeff `  a )
3433coef2 20103 . . . . . . . . . . . . . . 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 5550 . . . . . . . . . . . . 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 5829 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( A  e.  NN0  /\  a  e.  (Poly `  ZZ ) )  /\  e  e.  NN0 )  ->  (
(coeff `  a ) `  e )  e.  ZZ )
4140zred 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 )
4342ad2antrr 707 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  e.  NN0  /\  a  e.  (Poly `  ZZ ) )  /\  e  e.  NN0 )  ->  A  e.  RR )
4441, 43absled 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 )
4645ad2antrr 707 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( A  e.  NN0  /\  a  e.  (Poly `  ZZ ) )  /\  e  e.  NN0 )  ->  A  e.  ZZ )
4746znegcld 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 ) ) )
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 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
) ) )
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 5855 . . . . . . . . . . . . 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 5417 . . . . . . . . . . . 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 11039 . . . . . . . . . . . 12  |-  ( a  e.  ( 0 ... A )  ->  a  e.  NN0 )
5958ssriv 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 ) )
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 6065 . . . . . . . . . . 11  |-  ( -u A ... A )  e. 
_V
63 ovex 6065 . . . . . . . . . . 11  |-  ( 0 ... A )  e. 
_V
6462, 63elmap 7001 . . . . . . . . . 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 2575 . . . . . . . . . . 11  |-  ( d  =  b  ->  (
d  =/=  0 p  <-> 
b  =/=  0 p ) )
72 fveq2 5687 . . . . . . . . . . . 12  |-  ( d  =  b  ->  (deg `  d )  =  (deg
`  b ) )
7372breq1d 4182 . . . . . . . . . . 11  |-  ( d  =  b  ->  (
(deg `  d )  <_  A  <->  (deg `  b )  <_  A ) )
74 fveq2 5687 . . . . . . . . . . . . . . 15  |-  ( d  =  b  ->  (coeff `  d )  =  (coeff `  b ) )
7574fveq1d 5689 . . . . . . . . . . . . . 14  |-  ( d  =  b  ->  (
(coeff `  d ) `  e )  =  ( (coeff `  b ) `  e ) )
7675fveq2d 5691 . . . . . . . . . . . . 13  |-  ( d  =  b  ->  ( abs `  ( (coeff `  d ) `  e
) )  =  ( abs `  ( (coeff `  b ) `  e
) ) )
7776breq1d 4182 . . . . . . . . . . . 12  |-  ( d  =  b  ->  (
( abs `  (
(coeff `  d ) `  e ) )  <_  A 
<->  ( abs `  (
(coeff `  b ) `  e ) )  <_  A ) )
7877ralbidv 2686 . . . . . . . . . . 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 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 )
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 20070 . . . . . . . . . . . . 13  |-  ( a  e.  (Poly `  ZZ )  ->  a : CC --> CC )
86 ffn 5550 . . . . . . . . . . . . 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 20070 . . . . . . . . . . . . 13  |-  ( b  e.  (Poly `  ZZ )  ->  b : CC --> CC )
90 ffn 5550 . . . . . . . . . . . . 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 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 ) )
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 5704 . . . . . . . . . . . . . . . . 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 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 ) )
10099oveq1d 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
) ) )
101100sumeq2dv 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 )
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 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 ) )
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 2404 . . . . . . . . . . . . . . 15  |-  (deg `  a )  =  (deg
`  a )
11433, 113coeid3 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 ) ) )
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 20105 . . . . . . . . . . . . . . . . 17  |-  ( b  e.  (Poly `  ZZ )  ->  (deg `  b
)  e.  NN0 )
121 nn0z 10260 . . . . . . . . . . . . . . . . 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 10455 . . . . . . . . . . . . . . . 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 2404 . . . . . . . . . . . . . . 15  |-  (coeff `  b )  =  (coeff `  b )
127 eqid 2404 . . . . . . . . . . . . . . 15  |-  (deg `  b )  =  (deg
`  b )
128126, 127coeid3 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 ) ) )
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 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 ) )
13187, 91, 130eqfnfvd 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 )
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 5687 . . . . . . . . . . 11  |-  ( a  =  b  ->  (coeff `  a )  =  (coeff `  b ) )
134133reseq1d 5104 . . . . . . . . . 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 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 ) ) ) )
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 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 )
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 2575 . . . . . . . . 9  |-  ( d  =  c  ->  (
d  =/=  0 p  <-> 
c  =/=  0 p ) )
143 fveq2 5687 . . . . . . . . . 10  |-  ( d  =  c  ->  (deg `  d )  =  (deg
`  c ) )
144143breq1d 4182 . . . . . . . . 9  |-  ( d  =  c  ->  (
(deg `  d )  <_  A  <->  (deg `  c )  <_  A ) )
145 fveq2 5687 . . . . . . . . . . . . 13  |-  ( d  =  c  ->  (coeff `  d )  =  (coeff `  c ) )
146145fveq1d 5689 . . . . . . . . . . . 12  |-  ( d  =  c  ->  (
(coeff `  d ) `  e )  =  ( (coeff `  c ) `  e ) )
147146fveq2d 5691 . . . . . . . . . . 11  |-  ( d  =  c  ->  ( abs `  ( (coeff `  d ) `  e
) )  =  ( abs `  ( (coeff `  c ) `  e
) ) )
148147breq1d 4182 . . . . . . . . . 10  |-  ( d  =  c  ->  (
( abs `  (
(coeff `  d ) `  e ) )  <_  A 
<->  ( abs `  (
(coeff `  c ) `  e ) )  <_  A ) )
149148ralbidv 2686 . . . . . . . . 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 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 )
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 20070 . . . . . . . . . . . . 13  |-  ( c  e.  (Poly `  ZZ )  ->  c : CC --> CC )
156 ffn 5550 . . . . . . . . . . . . 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 5810 . . . . . . . . . . 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 5687 . . . . . . . . . . . 12  |-  ( b  =  a  ->  (
c `  b )  =  ( c `  a ) )
162161eqeq1d 2412 . . . . . . . . . . 11  |-  ( b  =  a  ->  (
( c `  b
)  =  0  <->  (
c `  a )  =  0 ) )
163162elrab 3052 . . . . . . . . . 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 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 } )
167166fta1 20178 . . . . . . . . 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 2478 . . . . . . 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 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 )
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 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 )
17611, 175eqeltrd 2478 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 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
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