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Theorem aannenlem1 21753
Description: Lemma for aannen 21756. (Contributed by Stefan O'Rear, 16-Nov-2014.)
Hypothesis
Ref Expression
aannenlem.a  |-  H  =  ( a  e.  NN0  |->  { b  e.  CC  |  E. c  e.  {
d  e.  (Poly `  ZZ )  |  (
d  =/=  0p  /\  (deg `  d
)  <_  a  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  a
) }  ( c `
 b )  =  0 } )
Assertion
Ref Expression
aannenlem1  |-  ( A  e.  NN0  ->  ( H `
 A )  e. 
Fin )
Distinct variable group:    A, a, b, c, d, e
Allowed substitution hints:    H( e, a, b, c, d)

Proof of Theorem aannenlem1
StepHypRef Expression
1 breq2 4293 . . . . . . 7  |-  ( a  =  A  ->  (
(deg `  d )  <_  a  <->  (deg `  d )  <_  A ) )
2 breq2 4293 . . . . . . . 8  |-  ( a  =  A  ->  (
( abs `  (
(coeff `  d ) `  e ) )  <_ 
a  <->  ( abs `  (
(coeff `  d ) `  e ) )  <_  A ) )
32ralbidv 2733 . . . . . . 7  |-  ( a  =  A  ->  ( A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  a  <->  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  A
) )
41, 33anbi23d 1287 . . . . . 6  |-  ( a  =  A  ->  (
( d  =/=  0p  /\  (deg `  d
)  <_  a  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  a
)  <->  ( d  =/=  0p  /\  (deg `  d )  <_  A  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  A
) ) )
54rabbidv 2962 . . . . 5  |-  ( a  =  A  ->  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0p  /\  (deg `  d )  <_  a  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  a
) }  =  {
d  e.  (Poly `  ZZ )  |  (
d  =/=  0p  /\  (deg `  d
)  <_  A  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  A
) } )
65rexeqdv 2922 . . . 4  |-  ( a  =  A  ->  ( E. c  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0p  /\  (deg `  d )  <_  a  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  a
) }  ( c `
 b )  =  0  <->  E. c  e.  {
d  e.  (Poly `  ZZ )  |  (
d  =/=  0p  /\  (deg `  d
)  <_  A  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  A
) }  ( c `
 b )  =  0 ) )
76rabbidv 2962 . . 3  |-  ( a  =  A  ->  { b  e.  CC  |  E. c  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0p  /\  (deg `  d )  <_  a  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  a
) }  ( c `
 b )  =  0 }  =  {
b  e.  CC  |  E. c  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0p  /\  (deg `  d )  <_  A  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  A
) }  ( c `
 b )  =  0 } )
8 aannenlem.a . . 3  |-  H  =  ( a  e.  NN0  |->  { b  e.  CC  |  E. c  e.  {
d  e.  (Poly `  ZZ )  |  (
d  =/=  0p  /\  (deg `  d
)  <_  a  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  a
) }  ( c `
 b )  =  0 } )
9 cnex 9359 . . . 4  |-  CC  e.  _V
109rabex 4440 . . 3  |-  { b  e.  CC  |  E. c  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0p  /\  (deg `  d )  <_  A  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  A
) }  ( c `
 b )  =  0 }  e.  _V
117, 8, 10fvmpt 5771 . 2  |-  ( A  e.  NN0  ->  ( H `
 A )  =  { b  e.  CC  |  E. c  e.  {
d  e.  (Poly `  ZZ )  |  (
d  =/=  0p  /\  (deg `  d
)  <_  A  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  A
) }  ( c `
 b )  =  0 } )
12 iunrab 4214 . . 3  |-  U_ c  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0p  /\  (deg `  d
)  <_  A  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  A
) }  { b  e.  CC  |  ( c `  b )  =  0 }  =  { b  e.  CC  |  E. c  e.  {
d  e.  (Poly `  ZZ )  |  (
d  =/=  0p  /\  (deg `  d
)  <_  A  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  A
) }  ( c `
 b )  =  0 }
13 fzfi 11790 . . . . . . 7  |-  ( -u A ... A )  e. 
Fin
14 fzfi 11790 . . . . . . 7  |-  ( 0 ... A )  e. 
Fin
15 mapfi 7603 . . . . . . 7  |-  ( ( ( -u A ... A )  e.  Fin  /\  ( 0 ... A
)  e.  Fin )  ->  ( ( -u A ... A )  ^m  (
0 ... A ) )  e.  Fin )
1613, 14, 15mp2an 667 . . . . . 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 6115 . . . . . 6  |-  ( (
-u A ... A
)  ^m  ( 0 ... A ) )  e.  _V
19 neeq1 2614 . . . . . . . . . . 11  |-  ( d  =  a  ->  (
d  =/=  0p  <-> 
a  =/=  0p ) )
20 fveq2 5688 . . . . . . . . . . . 12  |-  ( d  =  a  ->  (deg `  d )  =  (deg
`  a ) )
2120breq1d 4299 . . . . . . . . . . 11  |-  ( d  =  a  ->  (
(deg `  d )  <_  A  <->  (deg `  a )  <_  A ) )
22 fveq2 5688 . . . . . . . . . . . . . . 15  |-  ( d  =  a  ->  (coeff `  d )  =  (coeff `  a ) )
2322fveq1d 5690 . . . . . . . . . . . . . 14  |-  ( d  =  a  ->  (
(coeff `  d ) `  e )  =  ( (coeff `  a ) `  e ) )
2423fveq2d 5692 . . . . . . . . . . . . 13  |-  ( d  =  a  ->  ( abs `  ( (coeff `  d ) `  e
) )  =  ( abs `  ( (coeff `  a ) `  e
) ) )
2524breq1d 4299 . . . . . . . . . . . 12  |-  ( d  =  a  ->  (
( abs `  (
(coeff `  d ) `  e ) )  <_  A 
<->  ( abs `  (
(coeff `  a ) `  e ) )  <_  A ) )
2625ralbidv 2733 . . . . . . . . . . 11  |-  ( d  =  a  ->  ( A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  A  <->  A. e  e.  NN0  ( abs `  ( (coeff `  a ) `  e
) )  <_  A
) )
2719, 21, 263anbi123d 1284 . . . . . . . . . 10  |-  ( d  =  a  ->  (
( d  =/=  0p  /\  (deg `  d
)  <_  A  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  A
)  <->  ( a  =/=  0p  /\  (deg `  a )  <_  A  /\  A. e  e.  NN0  ( abs `  ( (coeff `  a ) `  e
) )  <_  A
) ) )
2827elrab 3114 . . . . . . . . 9  |-  ( a  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0p  /\  (deg `  d )  <_  A  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  A
) }  <->  ( a  e.  (Poly `  ZZ )  /\  ( a  =/=  0p  /\  (deg `  a
)  <_  A  /\  A. e  e.  NN0  ( abs `  ( (coeff `  a ) `  e
) )  <_  A
) ) )
29 simp3 985 . . . . . . . . . 10  |-  ( ( a  =/=  0p  /\  (deg `  a
)  <_  A  /\  A. e  e.  NN0  ( abs `  ( (coeff `  a ) `  e
) )  <_  A
)  ->  A. e  e.  NN0  ( abs `  (
(coeff `  a ) `  e ) )  <_  A )
3029anim2i 566 . . . . . . . . 9  |-  ( ( a  e.  (Poly `  ZZ )  /\  (
a  =/=  0p  /\  (deg `  a
)  <_  A  /\  A. e  e.  NN0  ( abs `  ( (coeff `  a ) `  e
) )  <_  A
) )  ->  (
a  e.  (Poly `  ZZ )  /\  A. e  e.  NN0  ( abs `  (
(coeff `  a ) `  e ) )  <_  A ) )
3128, 30sylbi 195 . . . . . . . 8  |-  ( a  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0p  /\  (deg `  d )  <_  A  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  A
) }  ->  (
a  e.  (Poly `  ZZ )  /\  A. e  e.  NN0  ( abs `  (
(coeff `  a ) `  e ) )  <_  A ) )
32 0z 10653 . . . . . . . . . . . . . . 15  |-  0  e.  ZZ
33 eqid 2441 . . . . . . . . . . . . . . . 16  |-  (coeff `  a )  =  (coeff `  a )
3433coef2 21658 . . . . . . . . . . . . . . 15  |-  ( ( a  e.  (Poly `  ZZ )  /\  0  e.  ZZ )  ->  (coeff `  a ) : NN0 --> ZZ )
3532, 34mpan2 666 . . . . . . . . . . . . . 14  |-  ( a  e.  (Poly `  ZZ )  ->  (coeff `  a
) : NN0 --> ZZ )
3635ad2antrl 722 . . . . . . . . . . . . 13  |-  ( ( A  e.  NN0  /\  ( a  e.  (Poly `  ZZ )  /\  A. e  e.  NN0  ( abs `  ( (coeff `  a
) `  e )
)  <_  A )
)  ->  (coeff `  a
) : NN0 --> ZZ )
37 ffn 5556 . . . . . . . . . . . . 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 463 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( A  e.  NN0  /\  a  e.  (Poly `  ZZ ) )  ->  (coeff `  a ) : NN0 --> ZZ )
4039ffvelrnda 5840 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( A  e.  NN0  /\  a  e.  (Poly `  ZZ ) )  /\  e  e.  NN0 )  ->  (
(coeff `  a ) `  e )  e.  ZZ )
4140zred 10743 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  e.  NN0  /\  a  e.  (Poly `  ZZ ) )  /\  e  e.  NN0 )  ->  (
(coeff `  a ) `  e )  e.  RR )
42 nn0re 10584 . . . . . . . . . . . . . . . . . . 19  |-  ( A  e.  NN0  ->  A  e.  RR )
4342ad2antrr 720 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  e.  NN0  /\  a  e.  (Poly `  ZZ ) )  /\  e  e.  NN0 )  ->  A  e.  RR )
4441, 43absled 12913 . . . . . . . . . . . . . . . . 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 10665 . . . . . . . . . . . . . . . . . . . 20  |-  ( A  e.  NN0  ->  A  e.  ZZ )
4645ad2antrr 720 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( A  e.  NN0  /\  a  e.  (Poly `  ZZ ) )  /\  e  e.  NN0 )  ->  A  e.  ZZ )
4746znegcld 10745 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  e.  NN0  /\  a  e.  (Poly `  ZZ ) )  /\  e  e.  NN0 )  ->  -u A  e.  ZZ )
48 elfz 11439 . . . . . . . . . . . . . . . . . 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 1213 . . . . . . . . . . . . . . . . 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 256 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  NN0  /\  a  e.  (Poly `  ZZ ) )  /\  e  e.  NN0 )  ->  (
( abs `  (
(coeff `  a ) `  e ) )  <_  A 
<->  ( (coeff `  a
) `  e )  e.  ( -u A ... A ) ) )
5150biimpd 207 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  NN0  /\  a  e.  (Poly `  ZZ ) )  /\  e  e.  NN0 )  ->  (
( abs `  (
(coeff `  a ) `  e ) )  <_  A  ->  ( (coeff `  a ) `  e
)  e.  ( -u A ... A ) ) )
5251ralimdva 2792 . . . . . . . . . . . . . 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 616 . . . . . . . . . . . . 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 5868 . . . . . . . . . . . . 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 656 . . . . . . . . . . . 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 5419 . . . . . . . . . . . 12  |-  ( (coeff `  a ) : NN0 --> (
-u A ... A
)  <->  ( (coeff `  a )  Fn  NN0  /\ 
ran  (coeff `  a )  C_  ( -u A ... A ) ) )
5738, 55, 56sylanbrc 659 . . . . . . . . . . 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 11477 . . . . . . . . . . . 12  |-  ( a  e.  ( 0 ... A )  ->  a  e.  NN0 )
5958ssriv 3357 . . . . . . . . . . 11  |-  ( 0 ... A )  C_  NN0
60 fssres 5575 . . . . . . . . . . 11  |-  ( ( (coeff `  a ) : NN0 --> ( -u A ... A )  /\  (
0 ... A )  C_  NN0 )  ->  ( (coeff `  a )  |`  (
0 ... A ) ) : ( 0 ... A ) --> ( -u A ... A ) )
6157, 59, 60sylancl 657 . . . . . . . . . 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 6115 . . . . . . . . . . 11  |-  ( -u A ... A )  e. 
_V
63 ovex 6115 . . . . . . . . . . 11  |-  ( 0 ... A )  e. 
_V
6462, 63elmap 7237 . . . . . . . . . 10  |-  ( ( (coeff `  a )  |`  ( 0 ... A
) )  e.  ( ( -u A ... A )  ^m  (
0 ... A ) )  <-> 
( (coeff `  a
)  |`  ( 0 ... A ) ) : ( 0 ... A
) --> ( -u A ... A ) )
6561, 64sylibr 212 . . . . . . . . 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 434 . . . . . . . 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 32 . . . . . . 7  |-  ( A  e.  NN0  ->  ( a  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0p  /\  (deg `  d )  <_  A  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  A
) }  ->  (
(coeff `  a )  |`  ( 0 ... A
) )  e.  ( ( -u A ... A )  ^m  (
0 ... A ) ) ) )
68 simp2 984 . . . . . . . . . 10  |-  ( ( a  =/=  0p  /\  (deg `  a
)  <_  A  /\  A. e  e.  NN0  ( abs `  ( (coeff `  a ) `  e
) )  <_  A
)  ->  (deg `  a
)  <_  A )
6968anim2i 566 . . . . . . . . 9  |-  ( ( a  e.  (Poly `  ZZ )  /\  (
a  =/=  0p  /\  (deg `  a
)  <_  A  /\  A. e  e.  NN0  ( abs `  ( (coeff `  a ) `  e
) )  <_  A
) )  ->  (
a  e.  (Poly `  ZZ )  /\  (deg `  a )  <_  A
) )
7028, 69sylbi 195 . . . . . . . 8  |-  ( a  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0p  /\  (deg `  d )  <_  A  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  A
) }  ->  (
a  e.  (Poly `  ZZ )  /\  (deg `  a )  <_  A
) )
71 neeq1 2614 . . . . . . . . . . 11  |-  ( d  =  b  ->  (
d  =/=  0p  <-> 
b  =/=  0p ) )
72 fveq2 5688 . . . . . . . . . . . 12  |-  ( d  =  b  ->  (deg `  d )  =  (deg
`  b ) )
7372breq1d 4299 . . . . . . . . . . 11  |-  ( d  =  b  ->  (
(deg `  d )  <_  A  <->  (deg `  b )  <_  A ) )
74 fveq2 5688 . . . . . . . . . . . . . . 15  |-  ( d  =  b  ->  (coeff `  d )  =  (coeff `  b ) )
7574fveq1d 5690 . . . . . . . . . . . . . 14  |-  ( d  =  b  ->  (
(coeff `  d ) `  e )  =  ( (coeff `  b ) `  e ) )
7675fveq2d 5692 . . . . . . . . . . . . 13  |-  ( d  =  b  ->  ( abs `  ( (coeff `  d ) `  e
) )  =  ( abs `  ( (coeff `  b ) `  e
) ) )
7776breq1d 4299 . . . . . . . . . . . 12  |-  ( d  =  b  ->  (
( abs `  (
(coeff `  d ) `  e ) )  <_  A 
<->  ( abs `  (
(coeff `  b ) `  e ) )  <_  A ) )
7877ralbidv 2733 . . . . . . . . . . 11  |-  ( d  =  b  ->  ( A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  A  <->  A. e  e.  NN0  ( abs `  ( (coeff `  b ) `  e
) )  <_  A
) )
7971, 73, 783anbi123d 1284 . . . . . . . . . 10  |-  ( d  =  b  ->  (
( d  =/=  0p  /\  (deg `  d
)  <_  A  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  A
)  <->  ( b  =/=  0p  /\  (deg `  b )  <_  A  /\  A. e  e.  NN0  ( abs `  ( (coeff `  b ) `  e
) )  <_  A
) ) )
8079elrab 3114 . . . . . . . . 9  |-  ( b  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0p  /\  (deg `  d )  <_  A  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  A
) }  <->  ( b  e.  (Poly `  ZZ )  /\  ( b  =/=  0p  /\  (deg `  b
)  <_  A  /\  A. e  e.  NN0  ( abs `  ( (coeff `  b ) `  e
) )  <_  A
) ) )
81 simp2 984 . . . . . . . . . 10  |-  ( ( b  =/=  0p  /\  (deg `  b
)  <_  A  /\  A. e  e.  NN0  ( abs `  ( (coeff `  b ) `  e
) )  <_  A
)  ->  (deg `  b
)  <_  A )
8281anim2i 566 . . . . . . . . 9  |-  ( ( b  e.  (Poly `  ZZ )  /\  (
b  =/=  0p  /\  (deg `  b
)  <_  A  /\  A. e  e.  NN0  ( abs `  ( (coeff `  b ) `  e
) )  <_  A
) )  ->  (
b  e.  (Poly `  ZZ )  /\  (deg `  b )  <_  A
) )
8380, 82sylbi 195 . . . . . . . 8  |-  ( b  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0p  /\  (deg `  d )  <_  A  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  A
) }  ->  (
b  e.  (Poly `  ZZ )  /\  (deg `  b )  <_  A
) )
84 simplll 752 . . . . . . . . . . . . 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 21625 . . . . . . . . . . . . 13  |-  ( a  e.  (Poly `  ZZ )  ->  a : CC --> CC )
86 ffn 5556 . . . . . . . . . . . . 13  |-  ( a : CC --> CC  ->  a  Fn  CC )
8784, 85, 863syl 20 . . . . . . . . . . . 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 754 . . . . . . . . . . . . 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 21625 . . . . . . . . . . . . 13  |-  ( b  e.  (Poly `  ZZ )  ->  b : CC --> CC )
90 ffn 5556 . . . . . . . . . . . . 13  |-  ( b : CC --> CC  ->  b  Fn  CC )
9188, 89, 903syl 20 . . . . . . . . . . . 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 755 . . . . . . . . . . . . . . . . . 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 462 . . . . . . . . . . . . . . . . 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 5690 . . . . . . . . . . . . . . . 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 5701 . . . . . . . . . . . . . . . . 17  |-  ( d  e.  ( 0 ... A )  ->  (
( (coeff `  a
)  |`  ( 0 ... A ) ) `  d )  =  ( (coeff `  a ) `  d ) )
9695adantl 463 . . . . . . . . . . . . . . . 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 5701 . . . . . . . . . . . . . . . . 17  |-  ( d  e.  ( 0 ... A )  ->  (
( (coeff `  b
)  |`  ( 0 ... A ) ) `  d )  =  ( (coeff `  b ) `  d ) )
9897adantl 463 . . . . . . . . . . . . . . . 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 2481 . . . . . . . . . . . . . . 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 6105 . . . . . . . . . . . . . 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 13176 . . . . . . . . . . . . 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 760 . . . . . . . . . . . . . 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 761 . . . . . . . . . . . . . . 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 21660 . . . . . . . . . . . . . . . . 17  |-  ( a  e.  (Poly `  ZZ )  ->  (deg `  a
)  e.  NN0 )
105 nn0z 10665 . . . . . . . . . . . . . . . . 17  |-  ( (deg
`  a )  e. 
NN0  ->  (deg `  a
)  e.  ZZ )
106102, 104, 1053syl 20 . . . . . . . . . . . . . . . 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 754 . . . . . . . . . . . . . . . . 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 10741 . . . . . . . . . . . . . . . 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 10870 . . . . . . . . . . . . . . . 16  |-  ( ( (deg `  a )  e.  ZZ  /\  A  e.  ZZ )  ->  ( A  e.  ( ZZ>= `  (deg `  a ) )  <-> 
(deg `  a )  <_  A ) )
110106, 108, 109syl2anc 656 . . . . . . . . . . . . . . 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 232 . . . . . . . . . . . . . 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 458 . . . . . . . . . . . . . 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 2441 . . . . . . . . . . . . . . 15  |-  (deg `  a )  =  (deg
`  a )
11433, 113coeid3 21667 . . . . . . . . . . . . . 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 1213 . . . . . . . . . . . . 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 1048 . . . . . . . . . . . . . . 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 1182 . . . . . . . . . . . . . 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 755 . . . . . . . . . . . . . . . 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 462 . . . . . . . . . . . . . . 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 21660 . . . . . . . . . . . . . . . . 17  |-  ( b  e.  (Poly `  ZZ )  ->  (deg `  b
)  e.  NN0 )
121 nn0z 10665 . . . . . . . . . . . . . . . . 17  |-  ( (deg
`  b )  e. 
NN0  ->  (deg `  b
)  e.  ZZ )
122117, 120, 1213syl 20 . . . . . . . . . . . . . . . 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 10870 . . . . . . . . . . . . . . . 16  |-  ( ( (deg `  b )  e.  ZZ  /\  A  e.  ZZ )  ->  ( A  e.  ( ZZ>= `  (deg `  b ) )  <-> 
(deg `  b )  <_  A ) )
124122, 108, 123syl2anc 656 . . . . . . . . . . . . . . 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 232 . . . . . . . . . . . . . 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 2441 . . . . . . . . . . . . . . 15  |-  (coeff `  b )  =  (coeff `  b )
127 eqid 2441 . . . . . . . . . . . . . . 15  |-  (deg `  b )  =  (deg
`  b )
128126, 127coeid3 21667 . . . . . . . . . . . . . 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 1213 . . . . . . . . . . . . 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 2483 . . . . . . . . . . . 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 5797 . . . . . . . . . . 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 612 . . . . . . . . . 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 5688 . . . . . . . . . . 11  |-  ( a  =  b  ->  (coeff `  a )  =  (coeff `  b ) )
134133reseq1d 5105 . . . . . . . . . 10  |-  ( a  =  b  ->  (
(coeff `  a )  |`  ( 0 ... A
) )  =  ( (coeff `  b )  |`  ( 0 ... A
) ) )
135132, 134impbid1 203 . . . . . . . . 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 435 . . . . . . . 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 651 . . . . . . 7  |-  ( A  e.  NN0  ->  ( ( a  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0p  /\  (deg `  d )  <_  A  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  A
) }  /\  b  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0p  /\  (deg `  d
)  <_  A  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  A
) } )  -> 
( ( (coeff `  a )  |`  (
0 ... A ) )  =  ( (coeff `  b )  |`  (
0 ... A ) )  <-> 
a  =  b ) ) )
13867, 137dom2d 7346 . . . . . 6  |-  ( A  e.  NN0  ->  ( ( ( -u A ... A )  ^m  (
0 ... A ) )  e.  _V  ->  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0p  /\  (deg `  d )  <_  A  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  A
) }  ~<_  ( (
-u A ... A
)  ^m  ( 0 ... A ) ) ) )
13918, 138mpi 17 . . . . 5  |-  ( A  e.  NN0  ->  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0p  /\  (deg `  d )  <_  A  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  A
) }  ~<_  ( (
-u A ... A
)  ^m  ( 0 ... A ) ) )
140 domfi 7530 . . . . 5  |-  ( ( ( ( -u A ... A )  ^m  (
0 ... A ) )  e.  Fin  /\  {
d  e.  (Poly `  ZZ )  |  (
d  =/=  0p  /\  (deg `  d
)  <_  A  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  A
) }  ~<_  ( (
-u A ... A
)  ^m  ( 0 ... A ) ) )  ->  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0p  /\  (deg `  d )  <_  A  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  A
) }  e.  Fin )
14117, 139, 140syl2anc 656 . . . 4  |-  ( A  e.  NN0  ->  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0p  /\  (deg `  d )  <_  A  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  A
) }  e.  Fin )
142 neeq1 2614 . . . . . . . . 9  |-  ( d  =  c  ->  (
d  =/=  0p  <-> 
c  =/=  0p ) )
143 fveq2 5688 . . . . . . . . . 10  |-  ( d  =  c  ->  (deg `  d )  =  (deg
`  c ) )
144143breq1d 4299 . . . . . . . . 9  |-  ( d  =  c  ->  (
(deg `  d )  <_  A  <->  (deg `  c )  <_  A ) )
145 fveq2 5688 . . . . . . . . . . . . 13  |-  ( d  =  c  ->  (coeff `  d )  =  (coeff `  c ) )
146145fveq1d 5690 . . . . . . . . . . . 12  |-  ( d  =  c  ->  (
(coeff `  d ) `  e )  =  ( (coeff `  c ) `  e ) )
147146fveq2d 5692 . . . . . . . . . . 11  |-  ( d  =  c  ->  ( abs `  ( (coeff `  d ) `  e
) )  =  ( abs `  ( (coeff `  c ) `  e
) ) )
148147breq1d 4299 . . . . . . . . . 10  |-  ( d  =  c  ->  (
( abs `  (
(coeff `  d ) `  e ) )  <_  A 
<->  ( abs `  (
(coeff `  c ) `  e ) )  <_  A ) )
149148ralbidv 2733 . . . . . . . . 9  |-  ( d  =  c  ->  ( A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  A  <->  A. e  e.  NN0  ( abs `  ( (coeff `  c ) `  e
) )  <_  A
) )
150142, 144, 1493anbi123d 1284 . . . . . . . 8  |-  ( d  =  c  ->  (
( d  =/=  0p  /\  (deg `  d
)  <_  A  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  A
)  <->  ( c  =/=  0p  /\  (deg `  c )  <_  A  /\  A. e  e.  NN0  ( abs `  ( (coeff `  c ) `  e
) )  <_  A
) ) )
151150elrab 3114 . . . . . . 7  |-  ( c  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0p  /\  (deg `  d )  <_  A  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  A
) }  <->  ( c  e.  (Poly `  ZZ )  /\  ( c  =/=  0p  /\  (deg `  c
)  <_  A  /\  A. e  e.  NN0  ( abs `  ( (coeff `  c ) `  e
) )  <_  A
) ) )
152 simp1 983 . . . . . . . 8  |-  ( ( c  =/=  0p  /\  (deg `  c
)  <_  A  /\  A. e  e.  NN0  ( abs `  ( (coeff `  c ) `  e
) )  <_  A
)  ->  c  =/=  0p )
153152anim2i 566 . . . . . . 7  |-  ( ( c  e.  (Poly `  ZZ )  /\  (
c  =/=  0p  /\  (deg `  c
)  <_  A  /\  A. e  e.  NN0  ( abs `  ( (coeff `  c ) `  e
) )  <_  A
) )  ->  (
c  e.  (Poly `  ZZ )  /\  c  =/=  0p ) )
154151, 153sylbi 195 . . . . . 6  |-  ( c  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0p  /\  (deg `  d )  <_  A  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  A
) }  ->  (
c  e.  (Poly `  ZZ )  /\  c  =/=  0p ) )
155 plyf 21625 . . . . . . . . . . . . 13  |-  ( c  e.  (Poly `  ZZ )  ->  c : CC --> CC )
156 ffn 5556 . . . . . . . . . . . . 13  |-  ( c : CC --> CC  ->  c  Fn  CC )
157155, 156syl 16 . . . . . . . . . . . 12  |-  ( c  e.  (Poly `  ZZ )  ->  c  Fn  CC )
158157adantr 462 . . . . . . . . . . 11  |-  ( ( c  e.  (Poly `  ZZ )  /\  c  =/=  0p )  -> 
c  Fn  CC )
159 fniniseg 5821 . . . . . . . . . . 11  |-  ( c  Fn  CC  ->  (
a  e.  ( `' c " { 0 } )  <->  ( a  e.  CC  /\  ( c `
 a )  =  0 ) ) )
160158, 159syl 16 . . . . . . . . . 10  |-  ( ( c  e.  (Poly `  ZZ )  /\  c  =/=  0p )  -> 
( a  e.  ( `' c " {
0 } )  <->  ( a  e.  CC  /\  ( c `
 a )  =  0 ) ) )
161 fveq2 5688 . . . . . . . . . . . 12  |-  ( b  =  a  ->  (
c `  b )  =  ( c `  a ) )
162161eqeq1d 2449 . . . . . . . . . . 11  |-  ( b  =  a  ->  (
( c `  b
)  =  0  <->  (
c `  a )  =  0 ) )
163162elrab 3114 . . . . . . . . . 10  |-  ( a  e.  { b  e.  CC  |  ( c `
 b )  =  0 }  <->  ( a  e.  CC  /\  ( c `
 a )  =  0 ) )
164160, 163syl6rbbr 264 . . . . . . . . 9  |-  ( ( c  e.  (Poly `  ZZ )  /\  c  =/=  0p )  -> 
( a  e.  {
b  e.  CC  | 
( c `  b
)  =  0 }  <-> 
a  e.  ( `' c " { 0 } ) ) )
165164eqrdv 2439 . . . . . . . 8  |-  ( ( c  e.  (Poly `  ZZ )  /\  c  =/=  0p )  ->  { b  e.  CC  |  ( c `  b )  =  0 }  =  ( `' c " { 0 } ) )
166 eqid 2441 . . . . . . . . . 10  |-  ( `' c " { 0 } )  =  ( `' c " {
0 } )
167166fta1 21733 . . . . . . . . 9  |-  ( ( c  e.  (Poly `  ZZ )  /\  c  =/=  0p )  -> 
( ( `' c
" { 0 } )  e.  Fin  /\  ( # `  ( `' c " { 0 } ) )  <_ 
(deg `  c )
) )
168167simpld 456 . . . . . . . 8  |-  ( ( c  e.  (Poly `  ZZ )  /\  c  =/=  0p )  -> 
( `' c " { 0 } )  e.  Fin )
169165, 168eqeltrd 2515 . . . . . . 7  |-  ( ( c  e.  (Poly `  ZZ )  /\  c  =/=  0p )  ->  { b  e.  CC  |  ( c `  b )  =  0 }  e.  Fin )
170169a1i 11 . . . . . 6  |-  ( A  e.  NN0  ->  ( ( c  e.  (Poly `  ZZ )  /\  c  =/=  0p )  ->  { b  e.  CC  |  ( c `  b )  =  0 }  e.  Fin )
)
171154, 170syl5 32 . . . . 5  |-  ( A  e.  NN0  ->  ( c  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0p  /\  (deg `  d )  <_  A  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  A
) }  ->  { b  e.  CC  |  ( c `  b )  =  0 }  e.  Fin ) )
172171ralrimiv 2796 . . . 4  |-  ( A  e.  NN0  ->  A. c  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0p  /\  (deg `  d
)  <_  A  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  A
) }  { b  e.  CC  |  ( c `  b )  =  0 }  e.  Fin )
173 iunfi 7595 . . . 4  |-  ( ( { d  e.  (Poly `  ZZ )  |  ( d  =/=  0p  /\  (deg `  d
)  <_  A  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  A
) }  e.  Fin  /\ 
A. c  e.  {
d  e.  (Poly `  ZZ )  |  (
d  =/=  0p  /\  (deg `  d
)  <_  A  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  A
) }  { b  e.  CC  |  ( c `  b )  =  0 }  e.  Fin )  ->  U_ c  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0p  /\  (deg `  d
)  <_  A  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  A
) }  { b  e.  CC  |  ( c `  b )  =  0 }  e.  Fin )
174141, 172, 173syl2anc 656 . . 3  |-  ( A  e.  NN0  ->  U_ c  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0p  /\  (deg `  d
)  <_  A  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  A
) }  { b  e.  CC  |  ( c `  b )  =  0 }  e.  Fin )
17512, 174syl5eqelr 2526 . 2  |-  ( A  e.  NN0  ->  { b  e.  CC  |  E. c  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0p  /\  (deg `  d )  <_  A  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  A
) }  ( c `
 b )  =  0 }  e.  Fin )
17611, 175eqeltrd 2515 1  |-  ( A  e.  NN0  ->  ( H `
 A )  e. 
Fin )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761    =/= wne 2604   A.wral 2713   E.wrex 2714   {crab 2717   _Vcvv 2970    C_ wss 3325   {csn 3874   U_ciun 4168   class class class wbr 4289    e. cmpt 4347   `'ccnv 4835   ran crn 4837    |` cres 4838   "cima 4839    Fn wfn 5410   -->wf 5411   ` cfv 5415  (class class class)co 6090    ^m cmap 7210    ~<_ cdom 7304   Fincfn 7306   CCcc 9276   RRcr 9277   0cc0 9278    x. cmul 9283    <_ cle 9415   -ucneg 9592   NN0cn0 10575   ZZcz 10642   ZZ>=cuz 10857   ...cfz 11433   ^cexp 11861   #chash 12099   abscabs 12719   sum_csu 13159   0pc0p 21106  Polycply 21611  coeffccoe 21613  degcdgr 21614
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356  ax-addf 9357
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-er 7097  df-map 7212  df-pm 7213  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-sup 7687  df-oi 7720  df-card 8105  df-cda 8333  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-n0 10576  df-z 10643  df-uz 10858  df-rp 10988  df-fz 11434  df-fzo 11545  df-fl 11638  df-seq 11803  df-exp 11862  df-hash 12100  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-clim 12962  df-rlim 12963  df-sum 13160  df-0p 21107  df-ply 21615  df-idp 21616  df-coe 21617  df-dgr 21618  df-quot 21716
This theorem is referenced by:  aannenlem3  21755
  Copyright terms: Public domain W3C validator