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Theorem aannenlem1 19708
Description: Lemma for aannen 19711. (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 4027 . . . . . . 7  |-  ( a  =  A  ->  (
(deg `  d )  <_  a  <->  (deg `  d )  <_  A ) )
2 breq2 4027 . . . . . . . 8  |-  ( a  =  A  ->  (
( abs `  (
(coeff `  d ) `  e ) )  <_ 
a  <->  ( abs `  (
(coeff `  d ) `  e ) )  <_  A ) )
32ralbidv 2563 . . . . . . 7  |-  ( a  =  A  ->  ( A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  a  <->  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  A
) )
41, 33anbi23d 1255 . . . . . 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 2780 . . . . 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 2743 . . . 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 2780 . . 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 8818 . . . 4  |-  CC  e.  _V
109rabex 4165 . . 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 5602 . 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 3949 . . 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 11034 . . . . . . 7  |-  ( -u A ... A )  e. 
Fin
14 fzfi 11034 . . . . . . 7  |-  ( 0 ... A )  e. 
Fin
15 mapfi 7152 . . . . . . 7  |-  ( ( ( -u A ... A )  e.  Fin  /\  ( 0 ... A
)  e.  Fin )  ->  ( ( -u A ... A )  ^m  (
0 ... A ) )  e.  Fin )
1613, 14, 15mp2an 653 . . . . . 6  |-  ( (
-u A ... A
)  ^m  ( 0 ... A ) )  e.  Fin
1716a1i 10 . . . . 5  |-  ( A  e.  NN0  ->  ( (
-u A ... A
)  ^m  ( 0 ... A ) )  e.  Fin )
18 ovex 5883 . . . . . 6  |-  ( (
-u A ... A
)  ^m  ( 0 ... A ) )  e.  _V
19 neeq1 2454 . . . . . . . . . . 11  |-  ( d  =  a  ->  (
d  =/=  0 p  <-> 
a  =/=  0 p ) )
20 fveq2 5525 . . . . . . . . . . . 12  |-  ( d  =  a  ->  (deg `  d )  =  (deg
`  a ) )
2120breq1d 4033 . . . . . . . . . . 11  |-  ( d  =  a  ->  (
(deg `  d )  <_  A  <->  (deg `  a )  <_  A ) )
22 fveq2 5525 . . . . . . . . . . . . . . 15  |-  ( d  =  a  ->  (coeff `  d )  =  (coeff `  a ) )
2322fveq1d 5527 . . . . . . . . . . . . . 14  |-  ( d  =  a  ->  (
(coeff `  d ) `  e )  =  ( (coeff `  a ) `  e ) )
2423fveq2d 5529 . . . . . . . . . . . . 13  |-  ( d  =  a  ->  ( abs `  ( (coeff `  d ) `  e
) )  =  ( abs `  ( (coeff `  a ) `  e
) ) )
2524breq1d 4033 . . . . . . . . . . . 12  |-  ( d  =  a  ->  (
( abs `  (
(coeff `  d ) `  e ) )  <_  A 
<->  ( abs `  (
(coeff `  a ) `  e ) )  <_  A ) )
2625ralbidv 2563 . . . . . . . . . . 11  |-  ( d  =  a  ->  ( A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  A  <->  A. e  e.  NN0  ( abs `  ( (coeff `  a ) `  e
) )  <_  A
) )
2719, 21, 263anbi123d 1252 . . . . . . . . . 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 2923 . . . . . . . . 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 957 . . . . . . . . . 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 552 . . . . . . . . 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 187 . . . . . . . 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 10035 . . . . . . . . . . . . . . 15  |-  0  e.  ZZ
33 eqid 2283 . . . . . . . . . . . . . . . 16  |-  (coeff `  a )  =  (coeff `  a )
3433coef2 19613 . . . . . . . . . . . . . . 15  |-  ( ( a  e.  (Poly `  ZZ )  /\  0  e.  ZZ )  ->  (coeff `  a ) : NN0 --> ZZ )
3532, 34mpan2 652 . . . . . . . . . . . . . 14  |-  ( a  e.  (Poly `  ZZ )  ->  (coeff `  a
) : NN0 --> ZZ )
3635ad2antrl 708 . . . . . . . . . . . . 13  |-  ( ( A  e.  NN0  /\  ( a  e.  (Poly `  ZZ )  /\  A. e  e.  NN0  ( abs `  ( (coeff `  a
) `  e )
)  <_  A )
)  ->  (coeff `  a
) : NN0 --> ZZ )
37 ffn 5389 . . . . . . . . . . . . 13  |-  ( (coeff `  a ) : NN0 --> ZZ 
->  (coeff `  a )  Fn  NN0 )
3836, 37syl 15 . . . . . . . . . . . 12  |-  ( ( A  e.  NN0  /\  ( a  e.  (Poly `  ZZ )  /\  A. e  e.  NN0  ( abs `  ( (coeff `  a
) `  e )
)  <_  A )
)  ->  (coeff `  a
)  Fn  NN0 )
3935adantl 452 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( A  e.  NN0  /\  a  e.  (Poly `  ZZ ) )  ->  (coeff `  a ) : NN0 --> ZZ )
4039ffvelrnda 5665 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( A  e.  NN0  /\  a  e.  (Poly `  ZZ ) )  /\  e  e.  NN0 )  ->  (
(coeff `  a ) `  e )  e.  ZZ )
4140zred 10117 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  e.  NN0  /\  a  e.  (Poly `  ZZ ) )  /\  e  e.  NN0 )  ->  (
(coeff `  a ) `  e )  e.  RR )
42 nn0re 9974 . . . . . . . . . . . . . . . . . . 19  |-  ( A  e.  NN0  ->  A  e.  RR )
4342ad2antrr 706 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  e.  NN0  /\  a  e.  (Poly `  ZZ ) )  /\  e  e.  NN0 )  ->  A  e.  RR )
4441, 43absled 11913 . . . . . . . . . . . . . . . . 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 10046 . . . . . . . . . . . . . . . . . . . 20  |-  ( A  e.  NN0  ->  A  e.  ZZ )
4645ad2antrr 706 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( A  e.  NN0  /\  a  e.  (Poly `  ZZ ) )  /\  e  e.  NN0 )  ->  A  e.  ZZ )
4746znegcld 10119 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  e.  NN0  /\  a  e.  (Poly `  ZZ ) )  /\  e  e.  NN0 )  ->  -u A  e.  ZZ )
48 elfz 10788 . . . . . . . . . . . . . . . . . 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 1182 . . . . . . . . . . . . . . . . 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 247 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  NN0  /\  a  e.  (Poly `  ZZ ) )  /\  e  e.  NN0 )  ->  (
( abs `  (
(coeff `  a ) `  e ) )  <_  A 
<->  ( (coeff `  a
) `  e )  e.  ( -u A ... A ) ) )
5150biimpd 198 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  NN0  /\  a  e.  (Poly `  ZZ ) )  /\  e  e.  NN0 )  ->  (
( abs `  (
(coeff `  a ) `  e ) )  <_  A  ->  ( (coeff `  a ) `  e
)  e.  ( -u A ... A ) ) )
5251ralimdva 2621 . . . . . . . . . . . . . 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 602 . . . . . . . . . . . . 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 5687 . . . . . . . . . . . . 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 642 . . . . . . . . . . . 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 5259 . . . . . . . . . . . 12  |-  ( (coeff `  a ) : NN0 --> (
-u A ... A
)  <->  ( (coeff `  a )  Fn  NN0  /\ 
ran  (coeff `  a )  C_  ( -u A ... A ) ) )
5738, 55, 56sylanbrc 645 . . . . . . . . . . 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 10822 . . . . . . . . . . . 12  |-  ( a  e.  ( 0 ... A )  ->  a  e.  NN0 )
5958ssriv 3184 . . . . . . . . . . 11  |-  ( 0 ... A )  C_  NN0
60 fssres 5408 . . . . . . . . . . 11  |-  ( ( (coeff `  a ) : NN0 --> ( -u A ... A )  /\  (
0 ... A )  C_  NN0 )  ->  ( (coeff `  a )  |`  (
0 ... A ) ) : ( 0 ... A ) --> ( -u A ... A ) )
6157, 59, 60sylancl 643 . . . . . . . . . 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 5883 . . . . . . . . . . 11  |-  ( -u A ... A )  e. 
_V
63 ovex 5883 . . . . . . . . . . 11  |-  ( 0 ... A )  e. 
_V
6462, 63elmap 6796 . . . . . . . . . 10  |-  ( ( (coeff `  a )  |`  ( 0 ... A
) )  e.  ( ( -u A ... A )  ^m  (
0 ... A ) )  <-> 
( (coeff `  a
)  |`  ( 0 ... A ) ) : ( 0 ... A
) --> ( -u A ... A ) )
6561, 64sylibr 203 . . . . . . . . 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 423 . . . . . . . 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 28 . . . . . . 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 956 . . . . . . . . . 10  |-  ( ( a  =/=  0 p  /\  (deg `  a
)  <_  A  /\  A. e  e.  NN0  ( abs `  ( (coeff `  a ) `  e
) )  <_  A
)  ->  (deg `  a
)  <_  A )
6968anim2i 552 . . . . . . . . 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 187 . . . . . . . 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 2454 . . . . . . . . . . 11  |-  ( d  =  b  ->  (
d  =/=  0 p  <-> 
b  =/=  0 p ) )
72 fveq2 5525 . . . . . . . . . . . 12  |-  ( d  =  b  ->  (deg `  d )  =  (deg
`  b ) )
7372breq1d 4033 . . . . . . . . . . 11  |-  ( d  =  b  ->  (
(deg `  d )  <_  A  <->  (deg `  b )  <_  A ) )
74 fveq2 5525 . . . . . . . . . . . . . . 15  |-  ( d  =  b  ->  (coeff `  d )  =  (coeff `  b ) )
7574fveq1d 5527 . . . . . . . . . . . . . 14  |-  ( d  =  b  ->  (
(coeff `  d ) `  e )  =  ( (coeff `  b ) `  e ) )
7675fveq2d 5529 . . . . . . . . . . . . 13  |-  ( d  =  b  ->  ( abs `  ( (coeff `  d ) `  e
) )  =  ( abs `  ( (coeff `  b ) `  e
) ) )
7776breq1d 4033 . . . . . . . . . . . 12  |-  ( d  =  b  ->  (
( abs `  (
(coeff `  d ) `  e ) )  <_  A 
<->  ( abs `  (
(coeff `  b ) `  e ) )  <_  A ) )
7877ralbidv 2563 . . . . . . . . . . 11  |-  ( d  =  b  ->  ( A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  A  <->  A. e  e.  NN0  ( abs `  ( (coeff `  b ) `  e
) )  <_  A
) )
7971, 73, 783anbi123d 1252 . . . . . . . . . 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 2923 . . . . . . . . 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 956 . . . . . . . . . 10  |-  ( ( b  =/=  0 p  /\  (deg `  b
)  <_  A  /\  A. e  e.  NN0  ( abs `  ( (coeff `  b ) `  e
) )  <_  A
)  ->  (deg `  b
)  <_  A )
8281anim2i 552 . . . . . . . . 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 187 . . . . . . . 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 734 . . . . . . . . . . . . 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 19580 . . . . . . . . . . . . 13  |-  ( a  e.  (Poly `  ZZ )  ->  a : CC --> CC )
86 ffn 5389 . . . . . . . . . . . . 13  |-  ( a : CC --> CC  ->  a  Fn  CC )
8784, 85, 863syl 18 . . . . . . . . . . . 12  |-  ( ( ( ( a  e.  (Poly `  ZZ )  /\  (deg `  a )  <_  A )  /\  (
b  e.  (Poly `  ZZ )  /\  (deg `  b )  <_  A
) )  /\  ( A  e.  NN0  /\  (
(coeff `  a )  |`  ( 0 ... A
) )  =  ( (coeff `  b )  |`  ( 0 ... A
) ) ) )  ->  a  Fn  CC )
88 simplrl 736 . . . . . . . . . . . . 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 19580 . . . . . . . . . . . . 13  |-  ( b  e.  (Poly `  ZZ )  ->  b : CC --> CC )
90 ffn 5389 . . . . . . . . . . . . 13  |-  ( b : CC --> CC  ->  b  Fn  CC )
9188, 89, 903syl 18 . . . . . . . . . . . 12  |-  ( ( ( ( a  e.  (Poly `  ZZ )  /\  (deg `  a )  <_  A )  /\  (
b  e.  (Poly `  ZZ )  /\  (deg `  b )  <_  A
) )  /\  ( A  e.  NN0  /\  (
(coeff `  a )  |`  ( 0 ... A
) )  =  ( (coeff `  b )  |`  ( 0 ... A
) ) ) )  ->  b  Fn  CC )
92 simplrr 737 . . . . . . . . . . . . . . . . . 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 451 . . . . . . . . . . . . . . . . 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 5527 . . . . . . . . . . . . . . . 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 5542 . . . . . . . . . . . . . . . . 17  |-  ( d  e.  ( 0 ... A )  ->  (
( (coeff `  a
)  |`  ( 0 ... A ) ) `  d )  =  ( (coeff `  a ) `  d ) )
9695adantl 452 . . . . . . . . . . . . . . . 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 5542 . . . . . . . . . . . . . . . . 17  |-  ( d  e.  ( 0 ... A )  ->  (
( (coeff `  b
)  |`  ( 0 ... A ) ) `  d )  =  ( (coeff `  b ) `  d ) )
9897adantl 452 . . . . . . . . . . . . . . . 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 2323 . . . . . . . . . . . . . . 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 5873 . . . . . . . . . . . . . 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 12176 . . . . . . . . . . . . 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 simp1ll 1018 . . . . . . . . . . . . . . 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.  (Poly `  ZZ ) )
1031023expa 1151 . . . . . . . . . . . . . 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 ) )
104 simpllr 735 . . . . . . . . . . . . . . . 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 `  a
)  <_  A )
105104adantr 451 . . . . . . . . . . . . . . 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 )
106 dgrcl 19615 . . . . . . . . . . . . . . . . 17  |-  ( a  e.  (Poly `  ZZ )  ->  (deg `  a
)  e.  NN0 )
107 nn0z 10046 . . . . . . . . . . . . . . . . 17  |-  ( (deg
`  a )  e. 
NN0  ->  (deg `  a
)  e.  ZZ )
108103, 106, 1073syl 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 )
109 simplrl 736 . . . . . . . . . . . . . . . . 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 )
110109nn0zd 10115 . . . . . . . . . . . . . . . 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 )
111 eluz 10241 . . . . . . . . . . . . . . . 16  |-  ( ( (deg `  a )  e.  ZZ  /\  A  e.  ZZ )  ->  ( A  e.  ( ZZ>= `  (deg `  a ) )  <-> 
(deg `  a )  <_  A ) )
112108, 110, 111syl2anc 642 . . . . . . . . . . . . . . 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 )
)
113105, 112mpbird 223 . . . . . . . . . . . . . 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
) ) )
114 simpr 447 . . . . . . . . . . . . . 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 )
115 eqid 2283 . . . . . . . . . . . . . . 15  |-  (deg `  a )  =  (deg
`  a )
11633, 115coeid3 19622 . . . . . . . . . . . . . 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 ) ) )
117103, 113, 114, 116syl3anc 1182 . . . . . . . . . . . . 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
) ) )
118 simp1rl 1020 . . . . . . . . . . . . . . 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 ) )
1191183expa 1151 . . . . . . . . . . . . . 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 ) )
120 simplrr 737 . . . . . . . . . . . . . . . 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 )
121120adantr 451 . . . . . . . . . . . . . . 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 )
122 dgrcl 19615 . . . . . . . . . . . . . . . . 17  |-  ( b  e.  (Poly `  ZZ )  ->  (deg `  b
)  e.  NN0 )
123 nn0z 10046 . . . . . . . . . . . . . . . . 17  |-  ( (deg
`  b )  e. 
NN0  ->  (deg `  b
)  e.  ZZ )
124119, 122, 1233syl 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 )
125 eluz 10241 . . . . . . . . . . . . . . . 16  |-  ( ( (deg `  b )  e.  ZZ  /\  A  e.  ZZ )  ->  ( A  e.  ( ZZ>= `  (deg `  b ) )  <-> 
(deg `  b )  <_  A ) )
126124, 110, 125syl2anc 642 . . . . . . . . . . . . . . 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 )
)
127121, 126mpbird 223 . . . . . . . . . . . . . 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
) ) )
128 eqid 2283 . . . . . . . . . . . . . . 15  |-  (coeff `  b )  =  (coeff `  b )
129 eqid 2283 . . . . . . . . . . . . . . 15  |-  (deg `  b )  =  (deg
`  b )
130128, 129coeid3 19622 . . . . . . . . . . . . . 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 ) ) )
131119, 127, 114, 130syl3anc 1182 . . . . . . . . . . . . 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
) ) )
132101, 117, 1313eqtr4d 2325 . . . . . . . . . . . 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 ) )
13387, 91, 132eqfnfvd 5625 . . . . . . . . . . 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 )
134133expr 598 . . . . . . . . . 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 ) )
135 fveq2 5525 . . . . . . . . . . 11  |-  ( a  =  b  ->  (coeff `  a )  =  (coeff `  b ) )
136135reseq1d 4954 . . . . . . . . . 10  |-  ( a  =  b  ->  (
(coeff `  a )  |`  ( 0 ... A
) )  =  ( (coeff `  b )  |`  ( 0 ... A
) ) )
137134, 136impbid1 194 . . . . . . . . 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 ) )
138137expcom 424 . . . . . . . 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 ) ) )
13970, 83, 138syl2ani 637 . . . . . . 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 ) ) )
14067, 139dom2d 6902 . . . . . 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 ) ) ) )
14118, 140mpi 16 . . . . 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 ) ) )
142 domfi 7084 . . . . 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 )
14317, 141, 142syl2anc 642 . . . 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 )
144 neeq1 2454 . . . . . . . . 9  |-  ( d  =  c  ->  (
d  =/=  0 p  <-> 
c  =/=  0 p ) )
145 fveq2 5525 . . . . . . . . . 10  |-  ( d  =  c  ->  (deg `  d )  =  (deg
`  c ) )
146145breq1d 4033 . . . . . . . . 9  |-  ( d  =  c  ->  (
(deg `  d )  <_  A  <->  (deg `  c )  <_  A ) )
147 fveq2 5525 . . . . . . . . . . . . 13  |-  ( d  =  c  ->  (coeff `  d )  =  (coeff `  c ) )
148147fveq1d 5527 . . . . . . . . . . . 12  |-  ( d  =  c  ->  (
(coeff `  d ) `  e )  =  ( (coeff `  c ) `  e ) )
149148fveq2d 5529 . . . . . . . . . . 11  |-  ( d  =  c  ->  ( abs `  ( (coeff `  d ) `  e
) )  =  ( abs `  ( (coeff `  c ) `  e
) ) )
150149breq1d 4033 . . . . . . . . . 10  |-  ( d  =  c  ->  (
( abs `  (
(coeff `  d ) `  e ) )  <_  A 
<->  ( abs `  (
(coeff `  c ) `  e ) )  <_  A ) )
151150ralbidv 2563 . . . . . . . . 9  |-  ( d  =  c  ->  ( A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  A  <->  A. e  e.  NN0  ( abs `  ( (coeff `  c ) `  e
) )  <_  A
) )
152144, 146, 1513anbi123d 1252 . . . . . . . 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
) ) )
153152elrab 2923 . . . . . . 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
) ) )
154 simp1 955 . . . . . . . 8  |-  ( ( c  =/=  0 p  /\  (deg `  c
)  <_  A  /\  A. e  e.  NN0  ( abs `  ( (coeff `  c ) `  e
) )  <_  A
)  ->  c  =/=  0 p )
155154anim2i 552 . . . . . . 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 ) )
156153, 155sylbi 187 . . . . . 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 ) )
157 plyf 19580 . . . . . . . . . . . . 13  |-  ( c  e.  (Poly `  ZZ )  ->  c : CC --> CC )
158 ffn 5389 . . . . . . . . . . . . 13  |-  ( c : CC --> CC  ->  c  Fn  CC )
159157, 158syl 15 . . . . . . . . . . . 12  |-  ( c  e.  (Poly `  ZZ )  ->  c  Fn  CC )
160159adantr 451 . . . . . . . . . . 11  |-  ( ( c  e.  (Poly `  ZZ )  /\  c  =/=  0 p )  -> 
c  Fn  CC )
161 fniniseg 5646 . . . . . . . . . . 11  |-  ( c  Fn  CC  ->  (
a  e.  ( `' c " { 0 } )  <->  ( a  e.  CC  /\  ( c `
 a )  =  0 ) ) )
162160, 161syl 15 . . . . . . . . . 10  |-  ( ( c  e.  (Poly `  ZZ )  /\  c  =/=  0 p )  -> 
( a  e.  ( `' c " {
0 } )  <->  ( a  e.  CC  /\  ( c `
 a )  =  0 ) ) )
163 fveq2 5525 . . . . . . . . . . . 12  |-  ( b  =  a  ->  (
c `  b )  =  ( c `  a ) )
164163eqeq1d 2291 . . . . . . . . . . 11  |-  ( b  =  a  ->  (
( c `  b
)  =  0  <->  (
c `  a )  =  0 ) )
165164elrab 2923 . . . . . . . . . 10  |-  ( a  e.  { b  e.  CC  |  ( c `
 b )  =  0 }  <->  ( a  e.  CC  /\  ( c `
 a )  =  0 ) )
166162, 165syl6rbbr 255 . . . . . . . . 9  |-  ( ( c  e.  (Poly `  ZZ )  /\  c  =/=  0 p )  -> 
( a  e.  {
b  e.  CC  | 
( c `  b
)  =  0 }  <-> 
a  e.  ( `' c " { 0 } ) ) )
167166eqrdv 2281 . . . . . . . 8  |-  ( ( c  e.  (Poly `  ZZ )  /\  c  =/=  0 p )  ->  { b  e.  CC  |  ( c `  b )  =  0 }  =  ( `' c " { 0 } ) )
168 eqid 2283 . . . . . . . . . 10  |-  ( `' c " { 0 } )  =  ( `' c " {
0 } )
169168fta1 19688 . . . . . . . . 9  |-  ( ( c  e.  (Poly `  ZZ )  /\  c  =/=  0 p )  -> 
( ( `' c
" { 0 } )  e.  Fin  /\  ( # `  ( `' c " { 0 } ) )  <_ 
(deg `  c )
) )
170169simpld 445 . . . . . . . 8  |-  ( ( c  e.  (Poly `  ZZ )  /\  c  =/=  0 p )  -> 
( `' c " { 0 } )  e.  Fin )
171167, 170eqeltrd 2357 . . . . . . 7  |-  ( ( c  e.  (Poly `  ZZ )  /\  c  =/=  0 p )  ->  { b  e.  CC  |  ( c `  b )  =  0 }  e.  Fin )
172171a1i 10 . . . . . 6  |-  ( A  e.  NN0  ->  ( ( c  e.  (Poly `  ZZ )  /\  c  =/=  0 p )  ->  { b  e.  CC  |  ( c `  b )  =  0 }  e.  Fin )
)
173156, 172syl5 28 . . . . 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 ) )
174173ralrimiv 2625 . . . 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 )
175 iunfi 7144 . . . 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 )
176143, 174, 175syl2anc 642 . . 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 )
17712, 176syl5eqelr 2368 . 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 )
17811, 177eqeltrd 2357 1  |-  ( A  e.  NN0  ->  ( H `
 A )  e. 
Fin )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544   {crab 2547   _Vcvv 2788    C_ wss 3152   {csn 3640   U_ciun 3905   class class class wbr 4023    e. cmpt 4077   `'ccnv 4688   ran crn 4690    |` cres 4691   "cima 4692    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858    ^m cmap 6772    ~<_ cdom 6861   Fincfn 6863   CCcc 8735   RRcr 8736   0cc0 8737    x. cmul 8742    <_ cle 8868   -ucneg 9038   NN0cn0 9965   ZZcz 10024   ZZ>=cuz 10230   ...cfz 10782   ^cexp 11104   #chash 11337   abscabs 11719   sum_csu 12158   0 pc0p 19024  Polycply 19566  coeffccoe 19568  degcdgr 19569
This theorem is referenced by:  aannenlem3  19710
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-rlim 11963  df-sum 12159  df-0p 19025  df-ply 19570  df-idp 19571  df-coe 19572  df-dgr 19573  df-quot 19671
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