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Theorem aannenlem1 23284
Description: Lemma for aannen 23287. (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 4406 . . . . . . 7  |-  ( a  =  A  ->  (
(deg `  d )  <_  a  <->  (deg `  d )  <_  A ) )
2 breq2 4406 . . . . . . . 8  |-  ( a  =  A  ->  (
( abs `  (
(coeff `  d ) `  e ) )  <_ 
a  <->  ( abs `  (
(coeff `  d ) `  e ) )  <_  A ) )
32ralbidv 2827 . . . . . . 7  |-  ( a  =  A  ->  ( A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  a  <->  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  A
) )
41, 33anbi23d 1342 . . . . . 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 3036 . . . . 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 2994 . . . 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 3036 . . 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 9620 . . . 4  |-  CC  e.  _V
109rabex 4554 . . 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 5948 . 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 4325 . . 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 12185 . . . . . . 7  |-  ( -u A ... A )  e. 
Fin
14 fzfi 12185 . . . . . . 7  |-  ( 0 ... A )  e. 
Fin
15 mapfi 7870 . . . . . . 7  |-  ( ( ( -u A ... A )  e.  Fin  /\  ( 0 ... A
)  e.  Fin )  ->  ( ( -u A ... A )  ^m  (
0 ... A ) )  e.  Fin )
1613, 14, 15mp2an 678 . . . . . 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 6318 . . . . . 6  |-  ( (
-u A ... A
)  ^m  ( 0 ... A ) )  e.  _V
19 neeq1 2686 . . . . . . . . . . 11  |-  ( d  =  a  ->  (
d  =/=  0p  <-> 
a  =/=  0p ) )
20 fveq2 5865 . . . . . . . . . . . 12  |-  ( d  =  a  ->  (deg `  d )  =  (deg
`  a ) )
2120breq1d 4412 . . . . . . . . . . 11  |-  ( d  =  a  ->  (
(deg `  d )  <_  A  <->  (deg `  a )  <_  A ) )
22 fveq2 5865 . . . . . . . . . . . . . . 15  |-  ( d  =  a  ->  (coeff `  d )  =  (coeff `  a ) )
2322fveq1d 5867 . . . . . . . . . . . . . 14  |-  ( d  =  a  ->  (
(coeff `  d ) `  e )  =  ( (coeff `  a ) `  e ) )
2423fveq2d 5869 . . . . . . . . . . . . 13  |-  ( d  =  a  ->  ( abs `  ( (coeff `  d ) `  e
) )  =  ( abs `  ( (coeff `  a ) `  e
) ) )
2524breq1d 4412 . . . . . . . . . . . 12  |-  ( d  =  a  ->  (
( abs `  (
(coeff `  d ) `  e ) )  <_  A 
<->  ( abs `  (
(coeff `  a ) `  e ) )  <_  A ) )
2625ralbidv 2827 . . . . . . . . . . 11  |-  ( d  =  a  ->  ( A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  A  <->  A. e  e.  NN0  ( abs `  ( (coeff `  a ) `  e
) )  <_  A
) )
2719, 21, 263anbi123d 1339 . . . . . . . . . 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 3196 . . . . . . . . 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 1010 . . . . . . . . . 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 573 . . . . . . . . 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 199 . . . . . . . 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 10948 . . . . . . . . . . . . . . 15  |-  0  e.  ZZ
33 eqid 2451 . . . . . . . . . . . . . . . 16  |-  (coeff `  a )  =  (coeff `  a )
3433coef2 23185 . . . . . . . . . . . . . . 15  |-  ( ( a  e.  (Poly `  ZZ )  /\  0  e.  ZZ )  ->  (coeff `  a ) : NN0 --> ZZ )
3532, 34mpan2 677 . . . . . . . . . . . . . 14  |-  ( a  e.  (Poly `  ZZ )  ->  (coeff `  a
) : NN0 --> ZZ )
3635ad2antrl 734 . . . . . . . . . . . . 13  |-  ( ( A  e.  NN0  /\  ( a  e.  (Poly `  ZZ )  /\  A. e  e.  NN0  ( abs `  ( (coeff `  a
) `  e )
)  <_  A )
)  ->  (coeff `  a
) : NN0 --> ZZ )
37 ffn 5728 . . . . . . . . . . . . 13  |-  ( (coeff `  a ) : NN0 --> ZZ 
->  (coeff `  a )  Fn  NN0 )
3836, 37syl 17 . . . . . . . . . . . 12  |-  ( ( A  e.  NN0  /\  ( a  e.  (Poly `  ZZ )  /\  A. e  e.  NN0  ( abs `  ( (coeff `  a
) `  e )
)  <_  A )
)  ->  (coeff `  a
)  Fn  NN0 )
3935adantl 468 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( A  e.  NN0  /\  a  e.  (Poly `  ZZ ) )  ->  (coeff `  a ) : NN0 --> ZZ )
4039ffvelrnda 6022 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( A  e.  NN0  /\  a  e.  (Poly `  ZZ ) )  /\  e  e.  NN0 )  ->  (
(coeff `  a ) `  e )  e.  ZZ )
4140zred 11040 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  e.  NN0  /\  a  e.  (Poly `  ZZ ) )  /\  e  e.  NN0 )  ->  (
(coeff `  a ) `  e )  e.  RR )
42 nn0re 10878 . . . . . . . . . . . . . . . . . . 19  |-  ( A  e.  NN0  ->  A  e.  RR )
4342ad2antrr 732 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  e.  NN0  /\  a  e.  (Poly `  ZZ ) )  /\  e  e.  NN0 )  ->  A  e.  RR )
4441, 43absled 13492 . . . . . . . . . . . . . . . . 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 10960 . . . . . . . . . . . . . . . . . . . 20  |-  ( A  e.  NN0  ->  A  e.  ZZ )
4645ad2antrr 732 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( A  e.  NN0  /\  a  e.  (Poly `  ZZ ) )  /\  e  e.  NN0 )  ->  A  e.  ZZ )
4746znegcld 11042 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  e.  NN0  /\  a  e.  (Poly `  ZZ ) )  /\  e  e.  NN0 )  ->  -u A  e.  ZZ )
48 elfz 11790 . . . . . . . . . . . . . . . . . 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 1268 . . . . . . . . . . . . . . . . 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 260 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  NN0  /\  a  e.  (Poly `  ZZ ) )  /\  e  e.  NN0 )  ->  (
( abs `  (
(coeff `  a ) `  e ) )  <_  A 
<->  ( (coeff `  a
) `  e )  e.  ( -u A ... A ) ) )
5150biimpd 211 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  NN0  /\  a  e.  (Poly `  ZZ ) )  /\  e  e.  NN0 )  ->  (
( abs `  (
(coeff `  a ) `  e ) )  <_  A  ->  ( (coeff `  a ) `  e
)  e.  ( -u A ... A ) ) )
5251ralimdva 2796 . . . . . . . . . . . . . 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 625 . . . . . . . . . . . . 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 6051 . . . . . . . . . . . . 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 667 . . . . . . . . . . . 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 5586 . . . . . . . . . . . 12  |-  ( (coeff `  a ) : NN0 --> (
-u A ... A
)  <->  ( (coeff `  a )  Fn  NN0  /\ 
ran  (coeff `  a )  C_  ( -u A ... A ) ) )
5738, 55, 56sylanbrc 670 . . . . . . . . . . 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 11887 . . . . . . . . . . . 12  |-  ( a  e.  ( 0 ... A )  ->  a  e.  NN0 )
5958ssriv 3436 . . . . . . . . . . 11  |-  ( 0 ... A )  C_  NN0
60 fssres 5749 . . . . . . . . . . 11  |-  ( ( (coeff `  a ) : NN0 --> ( -u A ... A )  /\  (
0 ... A )  C_  NN0 )  ->  ( (coeff `  a )  |`  (
0 ... A ) ) : ( 0 ... A ) --> ( -u A ... A ) )
6157, 59, 60sylancl 668 . . . . . . . . . 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 6318 . . . . . . . . . . 11  |-  ( -u A ... A )  e. 
_V
63 ovex 6318 . . . . . . . . . . 11  |-  ( 0 ... A )  e. 
_V
6462, 63elmap 7500 . . . . . . . . . 10  |-  ( ( (coeff `  a )  |`  ( 0 ... A
) )  e.  ( ( -u A ... A )  ^m  (
0 ... A ) )  <-> 
( (coeff `  a
)  |`  ( 0 ... A ) ) : ( 0 ... A
) --> ( -u A ... A ) )
6561, 64sylibr 216 . . . . . . . . 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 436 . . . . . . . 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 33 . . . . . . 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 1009 . . . . . . . . . 10  |-  ( ( a  =/=  0p  /\  (deg `  a
)  <_  A  /\  A. e  e.  NN0  ( abs `  ( (coeff `  a ) `  e
) )  <_  A
)  ->  (deg `  a
)  <_  A )
6968anim2i 573 . . . . . . . . 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 199 . . . . . . . 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 2686 . . . . . . . . . . 11  |-  ( d  =  b  ->  (
d  =/=  0p  <-> 
b  =/=  0p ) )
72 fveq2 5865 . . . . . . . . . . . 12  |-  ( d  =  b  ->  (deg `  d )  =  (deg
`  b ) )
7372breq1d 4412 . . . . . . . . . . 11  |-  ( d  =  b  ->  (
(deg `  d )  <_  A  <->  (deg `  b )  <_  A ) )
74 fveq2 5865 . . . . . . . . . . . . . . 15  |-  ( d  =  b  ->  (coeff `  d )  =  (coeff `  b ) )
7574fveq1d 5867 . . . . . . . . . . . . . 14  |-  ( d  =  b  ->  (
(coeff `  d ) `  e )  =  ( (coeff `  b ) `  e ) )
7675fveq2d 5869 . . . . . . . . . . . . 13  |-  ( d  =  b  ->  ( abs `  ( (coeff `  d ) `  e
) )  =  ( abs `  ( (coeff `  b ) `  e
) ) )
7776breq1d 4412 . . . . . . . . . . . 12  |-  ( d  =  b  ->  (
( abs `  (
(coeff `  d ) `  e ) )  <_  A 
<->  ( abs `  (
(coeff `  b ) `  e ) )  <_  A ) )
7877ralbidv 2827 . . . . . . . . . . 11  |-  ( d  =  b  ->  ( A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  A  <->  A. e  e.  NN0  ( abs `  ( (coeff `  b ) `  e
) )  <_  A
) )
7971, 73, 783anbi123d 1339 . . . . . . . . . 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 3196 . . . . . . . . 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 1009 . . . . . . . . . 10  |-  ( ( b  =/=  0p  /\  (deg `  b
)  <_  A  /\  A. e  e.  NN0  ( abs `  ( (coeff `  b ) `  e
) )  <_  A
)  ->  (deg `  b
)  <_  A )
8281anim2i 573 . . . . . . . . 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 199 . . . . . . . 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 768 . . . . . . . . . . . . 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 23152 . . . . . . . . . . . . 13  |-  ( a  e.  (Poly `  ZZ )  ->  a : CC --> CC )
86 ffn 5728 . . . . . . . . . . . . 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 770 . . . . . . . . . . . . 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 23152 . . . . . . . . . . . . 13  |-  ( b  e.  (Poly `  ZZ )  ->  b : CC --> CC )
90 ffn 5728 . . . . . . . . . . . . 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 771 . . . . . . . . . . . . . . . . . 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 467 . . . . . . . . . . . . . . . . 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 5867 . . . . . . . . . . . . . . . 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 5879 . . . . . . . . . . . . . . . . 17  |-  ( d  e.  ( 0 ... A )  ->  (
( (coeff `  a
)  |`  ( 0 ... A ) ) `  d )  =  ( (coeff `  a ) `  d ) )
9695adantl 468 . . . . . . . . . . . . . . . 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 5879 . . . . . . . . . . . . . . . . 17  |-  ( d  e.  ( 0 ... A )  ->  (
( (coeff `  b
)  |`  ( 0 ... A ) ) `  d )  =  ( (coeff `  b ) `  d ) )
9897adantl 468 . . . . . . . . . . . . . . . 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 2493 . . . . . . . . . . . . . . 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 6305 . . . . . . . . . . . . . 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 13769 . . . . . . . . . . . . 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 776 . . . . . . . . . . . . . 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 777 . . . . . . . . . . . . . . 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 23187 . . . . . . . . . . . . . . . . 17  |-  ( a  e.  (Poly `  ZZ )  ->  (deg `  a
)  e.  NN0 )
105 nn0z 10960 . . . . . . . . . . . . . . . . 17  |-  ( (deg
`  a )  e. 
NN0  ->  (deg `  a
)  e.  ZZ )
106102, 104, 1053syl 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 )
107 simplrl 770 . . . . . . . . . . . . . . . . 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 11038 . . . . . . . . . . . . . . . 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 11172 . . . . . . . . . . . . . . . 16  |-  ( ( (deg `  a )  e.  ZZ  /\  A  e.  ZZ )  ->  ( A  e.  ( ZZ>= `  (deg `  a ) )  <-> 
(deg `  a )  <_  A ) )
110106, 108, 109syl2anc 667 . . . . . . . . . . . . . . 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 236 . . . . . . . . . . . . . 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 463 . . . . . . . . . . . . . 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 2451 . . . . . . . . . . . . . . 15  |-  (deg `  a )  =  (deg
`  a )
11433, 113coeid3 23194 . . . . . . . . . . . . . 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 1268 . . . . . . . . . . . . 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 1073 . . . . . . . . . . . . . . 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 1208 . . . . . . . . . . . . . 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 771 . . . . . . . . . . . . . . . 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 467 . . . . . . . . . . . . . . 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 23187 . . . . . . . . . . . . . . . . 17  |-  ( b  e.  (Poly `  ZZ )  ->  (deg `  b
)  e.  NN0 )
121 nn0z 10960 . . . . . . . . . . . . . . . . 17  |-  ( (deg
`  b )  e. 
NN0  ->  (deg `  b
)  e.  ZZ )
122117, 120, 1213syl 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 )
123 eluz 11172 . . . . . . . . . . . . . . . 16  |-  ( ( (deg `  b )  e.  ZZ  /\  A  e.  ZZ )  ->  ( A  e.  ( ZZ>= `  (deg `  b ) )  <-> 
(deg `  b )  <_  A ) )
124122, 108, 123syl2anc 667 . . . . . . . . . . . . . . 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 236 . . . . . . . . . . . . . 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 2451 . . . . . . . . . . . . . . 15  |-  (coeff `  b )  =  (coeff `  b )
127 eqid 2451 . . . . . . . . . . . . . . 15  |-  (deg `  b )  =  (deg
`  b )
128126, 127coeid3 23194 . . . . . . . . . . . . . 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 1268 . . . . . . . . . . . . 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 2495 . . . . . . . . . . . 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 5979 . . . . . . . . . . 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 620 . . . . . . . . . 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 5865 . . . . . . . . . . 11  |-  ( a  =  b  ->  (coeff `  a )  =  (coeff `  b ) )
134133reseq1d 5104 . . . . . . . . . 10  |-  ( a  =  b  ->  (
(coeff `  a )  |`  ( 0 ... A
) )  =  ( (coeff `  b )  |`  ( 0 ... A
) ) )
135132, 134impbid1 207 . . . . . . . . 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 437 . . . . . . . 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 662 . . . . . . 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 7610 . . . . . 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 20 . . . . 5  |-  ( A  e.  NN0  ->  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0p  /\  (deg `  d )  <_  A  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  A
) }  ~<_  ( (
-u A ... A
)  ^m  ( 0 ... A ) ) )
140 domfi 7793 . . . . 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 667 . . . 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 2686 . . . . . . . . 9  |-  ( d  =  c  ->  (
d  =/=  0p  <-> 
c  =/=  0p ) )
143 fveq2 5865 . . . . . . . . . 10  |-  ( d  =  c  ->  (deg `  d )  =  (deg
`  c ) )
144143breq1d 4412 . . . . . . . . 9  |-  ( d  =  c  ->  (
(deg `  d )  <_  A  <->  (deg `  c )  <_  A ) )
145 fveq2 5865 . . . . . . . . . . . . 13  |-  ( d  =  c  ->  (coeff `  d )  =  (coeff `  c ) )
146145fveq1d 5867 . . . . . . . . . . . 12  |-  ( d  =  c  ->  (
(coeff `  d ) `  e )  =  ( (coeff `  c ) `  e ) )
147146fveq2d 5869 . . . . . . . . . . 11  |-  ( d  =  c  ->  ( abs `  ( (coeff `  d ) `  e
) )  =  ( abs `  ( (coeff `  c ) `  e
) ) )
148147breq1d 4412 . . . . . . . . . 10  |-  ( d  =  c  ->  (
( abs `  (
(coeff `  d ) `  e ) )  <_  A 
<->  ( abs `  (
(coeff `  c ) `  e ) )  <_  A ) )
149148ralbidv 2827 . . . . . . . . 9  |-  ( d  =  c  ->  ( A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  A  <->  A. e  e.  NN0  ( abs `  ( (coeff `  c ) `  e
) )  <_  A
) )
150142, 144, 1493anbi123d 1339 . . . . . . . 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 3196 . . . . . . 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 1008 . . . . . . . 8  |-  ( ( c  =/=  0p  /\  (deg `  c
)  <_  A  /\  A. e  e.  NN0  ( abs `  ( (coeff `  c ) `  e
) )  <_  A
)  ->  c  =/=  0p )
153152anim2i 573 . . . . . . 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 199 . . . . . 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 23152 . . . . . . . . . . . . 13  |-  ( c  e.  (Poly `  ZZ )  ->  c : CC --> CC )
156 ffn 5728 . . . . . . . . . . . . 13  |-  ( c : CC --> CC  ->  c  Fn  CC )
157155, 156syl 17 . . . . . . . . . . . 12  |-  ( c  e.  (Poly `  ZZ )  ->  c  Fn  CC )
158157adantr 467 . . . . . . . . . . 11  |-  ( ( c  e.  (Poly `  ZZ )  /\  c  =/=  0p )  -> 
c  Fn  CC )
159 fniniseg 6003 . . . . . . . . . . 11  |-  ( c  Fn  CC  ->  (
a  e.  ( `' c " { 0 } )  <->  ( a  e.  CC  /\  ( c `
 a )  =  0 ) ) )
160158, 159syl 17 . . . . . . . . . 10  |-  ( ( c  e.  (Poly `  ZZ )  /\  c  =/=  0p )  -> 
( a  e.  ( `' c " {
0 } )  <->  ( a  e.  CC  /\  ( c `
 a )  =  0 ) ) )
161 fveq2 5865 . . . . . . . . . . . 12  |-  ( b  =  a  ->  (
c `  b )  =  ( c `  a ) )
162161eqeq1d 2453 . . . . . . . . . . 11  |-  ( b  =  a  ->  (
( c `  b
)  =  0  <->  (
c `  a )  =  0 ) )
163162elrab 3196 . . . . . . . . . 10  |-  ( a  e.  { b  e.  CC  |  ( c `
 b )  =  0 }  <->  ( a  e.  CC  /\  ( c `
 a )  =  0 ) )
164160, 163syl6rbbr 268 . . . . . . . . 9  |-  ( ( c  e.  (Poly `  ZZ )  /\  c  =/=  0p )  -> 
( a  e.  {
b  e.  CC  | 
( c `  b
)  =  0 }  <-> 
a  e.  ( `' c " { 0 } ) ) )
165164eqrdv 2449 . . . . . . . 8  |-  ( ( c  e.  (Poly `  ZZ )  /\  c  =/=  0p )  ->  { b  e.  CC  |  ( c `  b )  =  0 }  =  ( `' c " { 0 } ) )
166 eqid 2451 . . . . . . . . . 10  |-  ( `' c " { 0 } )  =  ( `' c " {
0 } )
167166fta1 23261 . . . . . . . . 9  |-  ( ( c  e.  (Poly `  ZZ )  /\  c  =/=  0p )  -> 
( ( `' c
" { 0 } )  e.  Fin  /\  ( # `  ( `' c " { 0 } ) )  <_ 
(deg `  c )
) )
168167simpld 461 . . . . . . . 8  |-  ( ( c  e.  (Poly `  ZZ )  /\  c  =/=  0p )  -> 
( `' c " { 0 } )  e.  Fin )
169165, 168eqeltrd 2529 . . . . . . 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 33 . . . . 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 2800 . . . 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 7862 . . . 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 667 . . 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 2534 . 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 2529 1  |-  ( A  e.  NN0  ->  ( H `
 A )  e. 
Fin )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887    =/= wne 2622   A.wral 2737   E.wrex 2738   {crab 2741   _Vcvv 3045    C_ wss 3404   {csn 3968   U_ciun 4278   class class class wbr 4402    |-> cmpt 4461   `'ccnv 4833   ran crn 4835    |` cres 4836   "cima 4837    Fn wfn 5577   -->wf 5578   ` cfv 5582  (class class class)co 6290    ^m cmap 7472    ~<_ cdom 7567   Fincfn 7569   CCcc 9537   RRcr 9538   0cc0 9539    x. cmul 9544    <_ cle 9676   -ucneg 9861   NN0cn0 10869   ZZcz 10937   ZZ>=cuz 11159   ...cfz 11784   ^cexp 12272   #chash 12515   abscabs 13297   sum_csu 13752   0pc0p 22627  Polycply 23138  coeffccoe 23140  degcdgr 23141
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617  ax-addf 9618
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-of 6531  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-2o 7183  df-oadd 7186  df-er 7363  df-map 7474  df-pm 7475  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-sup 7956  df-inf 7957  df-oi 8025  df-card 8373  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-fz 11785  df-fzo 11916  df-fl 12028  df-seq 12214  df-exp 12273  df-hash 12516  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-clim 13552  df-rlim 13553  df-sum 13753  df-0p 22628  df-ply 23142  df-idp 23143  df-coe 23144  df-dgr 23145  df-quot 23244
This theorem is referenced by:  aannenlem3  23286
  Copyright terms: Public domain W3C validator