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Theorem aannenlem2 23297
Description: Lemma for aannen 23299. (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
aannenlem2  |-  AA  =  U. ran  H
Distinct variable group:    a, b, c, d, e
Allowed substitution hints:    H( e, a, b, c, d)

Proof of Theorem aannenlem2
Dummy variables  f 
g  h  i are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp3 1011 . . . . . . . . . 10  |-  ( ( h  e.  ( (Poly `  ZZ )  \  {
0p } )  /\  ( h `  g )  =  0  /\  g  e.  CC )  ->  g  e.  CC )
2 eldifi 3523 . . . . . . . . . . . . . 14  |-  ( h  e.  ( (Poly `  ZZ )  \  { 0p } )  ->  h  e.  (Poly `  ZZ ) )
32adantr 471 . . . . . . . . . . . . 13  |-  ( ( h  e.  ( (Poly `  ZZ )  \  {
0p } )  /\  g  e.  CC )  ->  h  e.  (Poly `  ZZ ) )
433adant2 1028 . . . . . . . . . . . 12  |-  ( ( h  e.  ( (Poly `  ZZ )  \  {
0p } )  /\  ( h `  g )  =  0  /\  g  e.  CC )  ->  h  e.  (Poly `  ZZ ) )
5 eldifsni 4067 . . . . . . . . . . . . . . 15  |-  ( h  e.  ( (Poly `  ZZ )  \  { 0p } )  ->  h  =/=  0p )
65adantr 471 . . . . . . . . . . . . . 14  |-  ( ( h  e.  ( (Poly `  ZZ )  \  {
0p } )  /\  g  e.  CC )  ->  h  =/=  0p )
7 0nn0 10874 . . . . . . . . . . . . . . . . . 18  |-  0  e.  NN0
8 dgrcl 23199 . . . . . . . . . . . . . . . . . . 19  |-  ( h  e.  (Poly `  ZZ )  ->  (deg `  h
)  e.  NN0 )
93, 8syl 17 . . . . . . . . . . . . . . . . . 18  |-  ( ( h  e.  ( (Poly `  ZZ )  \  {
0p } )  /\  g  e.  CC )  ->  (deg `  h
)  e.  NN0 )
10 prssi 4097 . . . . . . . . . . . . . . . . . 18  |-  ( ( 0  e.  NN0  /\  (deg `  h )  e. 
NN0 )  ->  { 0 ,  (deg `  h
) }  C_  NN0 )
117, 9, 10sylancr 674 . . . . . . . . . . . . . . . . 17  |-  ( ( h  e.  ( (Poly `  ZZ )  \  {
0p } )  /\  g  e.  CC )  ->  { 0 ,  (deg `  h ) }  C_  NN0 )
12 ssrab2 3482 . . . . . . . . . . . . . . . . . 18  |-  { g  e.  NN0  |  E. i  e.  ( 0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } 
C_  NN0
1312a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( ( h  e.  ( (Poly `  ZZ )  \  {
0p } )  /\  g  e.  CC )  ->  { g  e. 
NN0  |  E. i  e.  ( 0 ... (deg `  h ) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } 
C_  NN0 )
1411, 13unssd 3578 . . . . . . . . . . . . . . . 16  |-  ( ( h  e.  ( (Poly `  ZZ )  \  {
0p } )  /\  g  e.  CC )  ->  ( { 0 ,  (deg `  h
) }  u.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } )  C_  NN0 )
15 nn0ssre 10863 . . . . . . . . . . . . . . . . 17  |-  NN0  C_  RR
16 ressxr 9671 . . . . . . . . . . . . . . . . 17  |-  RR  C_  RR*
1715, 16sstri 3409 . . . . . . . . . . . . . . . 16  |-  NN0  C_  RR*
1814, 17syl6ss 3412 . . . . . . . . . . . . . . 15  |-  ( ( h  e.  ( (Poly `  ZZ )  \  {
0p } )  /\  g  e.  CC )  ->  ( { 0 ,  (deg `  h
) }  u.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } )  C_  RR* )
19 fvex 5858 . . . . . . . . . . . . . . . . 17  |-  (deg `  h )  e.  _V
2019prid2 4050 . . . . . . . . . . . . . . . 16  |-  (deg `  h )  e.  {
0 ,  (deg `  h ) }
21 elun1 3569 . . . . . . . . . . . . . . . 16  |-  ( (deg
`  h )  e. 
{ 0 ,  (deg
`  h ) }  ->  (deg `  h
)  e.  ( { 0 ,  (deg `  h ) }  u.  { g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) )
2220, 21ax-mp 5 . . . . . . . . . . . . . . 15  |-  (deg `  h )  e.  ( { 0 ,  (deg
`  h ) }  u.  { g  e. 
NN0  |  E. i  e.  ( 0 ... (deg `  h ) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } )
23 supxrub 11600 . . . . . . . . . . . . . . 15  |-  ( ( ( { 0 ,  (deg `  h ) }  u.  { g  e.  NN0  |  E. i  e.  ( 0 ... (deg `  h ) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } )  C_  RR*  /\  (deg `  h )  e.  ( { 0 ,  (deg
`  h ) }  u.  { g  e. 
NN0  |  E. i  e.  ( 0 ... (deg `  h ) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) )  ->  (deg `  h )  <_  sup ( ( { 0 ,  (deg `  h
) }  u.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  ) )
2418, 22, 23sylancl 673 . . . . . . . . . . . . . 14  |-  ( ( h  e.  ( (Poly `  ZZ )  \  {
0p } )  /\  g  e.  CC )  ->  (deg `  h
)  <_  sup (
( { 0 ,  (deg `  h ) }  u.  { g  e.  NN0  |  E. i  e.  ( 0 ... (deg `  h ) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  ) )
2518adantr 471 . . . . . . . . . . . . . . . 16  |-  ( ( ( h  e.  ( (Poly `  ZZ )  \  { 0p }
)  /\  g  e.  CC )  /\  e  e.  NN0 )  ->  ( { 0 ,  (deg
`  h ) }  u.  { g  e. 
NN0  |  E. i  e.  ( 0 ... (deg `  h ) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } )  C_  RR* )
26 fveq2 5848 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( (coeff `  h ) `  e )  =  0  ->  ( abs `  (
(coeff `  h ) `  e ) )  =  ( abs `  0
) )
27 abs0 13359 . . . . . . . . . . . . . . . . . . . 20  |-  ( abs `  0 )  =  0
2826, 27syl6eq 2502 . . . . . . . . . . . . . . . . . . 19  |-  ( ( (coeff `  h ) `  e )  =  0  ->  ( abs `  (
(coeff `  h ) `  e ) )  =  0 )
29 c0ex 9624 . . . . . . . . . . . . . . . . . . . . 21  |-  0  e.  _V
3029prid1 4049 . . . . . . . . . . . . . . . . . . . 20  |-  0  e.  { 0 ,  (deg
`  h ) }
31 elun1 3569 . . . . . . . . . . . . . . . . . . . 20  |-  ( 0  e.  { 0 ,  (deg `  h ) }  ->  0  e.  ( { 0 ,  (deg
`  h ) }  u.  { g  e. 
NN0  |  E. i  e.  ( 0 ... (deg `  h ) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) )
3230, 31ax-mp 5 . . . . . . . . . . . . . . . . . . 19  |-  0  e.  ( { 0 ,  (deg `  h ) }  u.  { g  e.  NN0  |  E. i  e.  ( 0 ... (deg `  h ) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } )
3328, 32syl6eqel 2538 . . . . . . . . . . . . . . . . . 18  |-  ( ( (coeff `  h ) `  e )  =  0  ->  ( abs `  (
(coeff `  h ) `  e ) )  e.  ( { 0 ,  (deg `  h ) }  u.  { g  e.  NN0  |  E. i  e.  ( 0 ... (deg `  h ) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) )
3433adantl 472 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( h  e.  ( (Poly `  ZZ )  \  { 0p } )  /\  g  e.  CC )  /\  e  e.  NN0 )  /\  (
(coeff `  h ) `  e )  =  0 )  ->  ( abs `  ( (coeff `  h
) `  e )
)  e.  ( { 0 ,  (deg `  h ) }  u.  { g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) )
35 0z 10938 . . . . . . . . . . . . . . . . . . . . . . 23  |-  0  e.  ZZ
36 eqid 2452 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  (coeff `  h )  =  (coeff `  h )
3736coef2 23197 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( h  e.  (Poly `  ZZ )  /\  0  e.  ZZ )  ->  (coeff `  h ) : NN0 --> ZZ )
383, 35, 37sylancl 673 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( h  e.  ( (Poly `  ZZ )  \  {
0p } )  /\  g  e.  CC )  ->  (coeff `  h
) : NN0 --> ZZ )
3938ffvelrnda 6006 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( h  e.  ( (Poly `  ZZ )  \  { 0p }
)  /\  g  e.  CC )  /\  e  e.  NN0 )  ->  (
(coeff `  h ) `  e )  e.  ZZ )
40 nn0abscl 13386 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( (coeff `  h ) `  e )  e.  ZZ  ->  ( abs `  (
(coeff `  h ) `  e ) )  e. 
NN0 )
4139, 40syl 17 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( h  e.  ( (Poly `  ZZ )  \  { 0p }
)  /\  g  e.  CC )  /\  e  e.  NN0 )  ->  ( abs `  ( (coeff `  h ) `  e
) )  e.  NN0 )
4241adantr 471 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( h  e.  ( (Poly `  ZZ )  \  { 0p } )  /\  g  e.  CC )  /\  e  e.  NN0 )  /\  (
(coeff `  h ) `  e )  =/=  0
)  ->  ( abs `  ( (coeff `  h
) `  e )
)  e.  NN0 )
43 simplr 767 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( h  e.  ( (Poly `  ZZ )  \  { 0p } )  /\  g  e.  CC )  /\  e  e.  NN0 )  /\  (
(coeff `  h ) `  e )  =/=  0
)  ->  e  e.  NN0 )
449ad2antrr 737 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( h  e.  ( (Poly `  ZZ )  \  { 0p } )  /\  g  e.  CC )  /\  e  e.  NN0 )  /\  (
(coeff `  h ) `  e )  =/=  0
)  ->  (deg `  h
)  e.  NN0 )
453ad2antrr 737 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( h  e.  ( (Poly `  ZZ )  \  { 0p } )  /\  g  e.  CC )  /\  e  e.  NN0 )  /\  (
(coeff `  h ) `  e )  =/=  0
)  ->  h  e.  (Poly `  ZZ ) )
46 simpr 467 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( h  e.  ( (Poly `  ZZ )  \  { 0p } )  /\  g  e.  CC )  /\  e  e.  NN0 )  /\  (
(coeff `  h ) `  e )  =/=  0
)  ->  ( (coeff `  h ) `  e
)  =/=  0 )
47 eqid 2452 . . . . . . . . . . . . . . . . . . . . . . 23  |-  (deg `  h )  =  (deg
`  h )
4836, 47dgrub 23200 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( h  e.  (Poly `  ZZ )  /\  e  e.  NN0  /\  ( (coeff `  h ) `  e
)  =/=  0 )  ->  e  <_  (deg `  h ) )
4945, 43, 46, 48syl3anc 1271 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( h  e.  ( (Poly `  ZZ )  \  { 0p } )  /\  g  e.  CC )  /\  e  e.  NN0 )  /\  (
(coeff `  h ) `  e )  =/=  0
)  ->  e  <_  (deg
`  h ) )
50 elfz2nn0 11876 . . . . . . . . . . . . . . . . . . . . 21  |-  ( e  e.  ( 0 ... (deg `  h )
)  <->  ( e  e. 
NN0  /\  (deg `  h
)  e.  NN0  /\  e  <_  (deg `  h
) ) )
5143, 44, 49, 50syl3anbrc 1193 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( h  e.  ( (Poly `  ZZ )  \  { 0p } )  /\  g  e.  CC )  /\  e  e.  NN0 )  /\  (
(coeff `  h ) `  e )  =/=  0
)  ->  e  e.  ( 0 ... (deg `  h ) ) )
52 eqid 2452 . . . . . . . . . . . . . . . . . . . 20  |-  ( abs `  ( (coeff `  h
) `  e )
)  =  ( abs `  ( (coeff `  h
) `  e )
)
53 fveq2 5848 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( i  =  e  ->  (
(coeff `  h ) `  i )  =  ( (coeff `  h ) `  e ) )
5453fveq2d 5852 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( i  =  e  ->  ( abs `  ( (coeff `  h ) `  i
) )  =  ( abs `  ( (coeff `  h ) `  e
) ) )
5554eqeq2d 2462 . . . . . . . . . . . . . . . . . . . . 21  |-  ( i  =  e  ->  (
( abs `  (
(coeff `  h ) `  e ) )  =  ( abs `  (
(coeff `  h ) `  i ) )  <->  ( abs `  ( (coeff `  h
) `  e )
)  =  ( abs `  ( (coeff `  h
) `  e )
) ) )
5655rspcev 3118 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( e  e.  ( 0 ... (deg `  h
) )  /\  ( abs `  ( (coeff `  h ) `  e
) )  =  ( abs `  ( (coeff `  h ) `  e
) ) )  ->  E. i  e.  (
0 ... (deg `  h
) ) ( abs `  ( (coeff `  h
) `  e )
)  =  ( abs `  ( (coeff `  h
) `  i )
) )
5751, 52, 56sylancl 673 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( h  e.  ( (Poly `  ZZ )  \  { 0p } )  /\  g  e.  CC )  /\  e  e.  NN0 )  /\  (
(coeff `  h ) `  e )  =/=  0
)  ->  E. i  e.  ( 0 ... (deg `  h ) ) ( abs `  ( (coeff `  h ) `  e
) )  =  ( abs `  ( (coeff `  h ) `  i
) ) )
58 eqeq1 2456 . . . . . . . . . . . . . . . . . . . . 21  |-  ( g  =  ( abs `  (
(coeff `  h ) `  e ) )  -> 
( g  =  ( abs `  ( (coeff `  h ) `  i
) )  <->  ( abs `  ( (coeff `  h
) `  e )
)  =  ( abs `  ( (coeff `  h
) `  i )
) ) )
5958rexbidv 2873 . . . . . . . . . . . . . . . . . . . 20  |-  ( g  =  ( abs `  (
(coeff `  h ) `  e ) )  -> 
( E. i  e.  ( 0 ... (deg `  h ) ) g  =  ( abs `  (
(coeff `  h ) `  i ) )  <->  E. i  e.  ( 0 ... (deg `  h ) ) ( abs `  ( (coeff `  h ) `  e
) )  =  ( abs `  ( (coeff `  h ) `  i
) ) ) )
6059elrab 3164 . . . . . . . . . . . . . . . . . . 19  |-  ( ( abs `  ( (coeff `  h ) `  e
) )  e.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) }  <-> 
( ( abs `  (
(coeff `  h ) `  e ) )  e. 
NN0  /\  E. i  e.  ( 0 ... (deg `  h ) ) ( abs `  ( (coeff `  h ) `  e
) )  =  ( abs `  ( (coeff `  h ) `  i
) ) ) )
6142, 57, 60sylanbrc 675 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( h  e.  ( (Poly `  ZZ )  \  { 0p } )  /\  g  e.  CC )  /\  e  e.  NN0 )  /\  (
(coeff `  h ) `  e )  =/=  0
)  ->  ( abs `  ( (coeff `  h
) `  e )
)  e.  { g  e.  NN0  |  E. i  e.  ( 0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } )
62 elun2 3570 . . . . . . . . . . . . . . . . . 18  |-  ( ( abs `  ( (coeff `  h ) `  e
) )  e.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) }  ->  ( abs `  (
(coeff `  h ) `  e ) )  e.  ( { 0 ,  (deg `  h ) }  u.  { g  e.  NN0  |  E. i  e.  ( 0 ... (deg `  h ) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) )
6361, 62syl 17 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( h  e.  ( (Poly `  ZZ )  \  { 0p } )  /\  g  e.  CC )  /\  e  e.  NN0 )  /\  (
(coeff `  h ) `  e )  =/=  0
)  ->  ( abs `  ( (coeff `  h
) `  e )
)  e.  ( { 0 ,  (deg `  h ) }  u.  { g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) )
6434, 63pm2.61dane 2711 . . . . . . . . . . . . . . . 16  |-  ( ( ( h  e.  ( (Poly `  ZZ )  \  { 0p }
)  /\  g  e.  CC )  /\  e  e.  NN0 )  ->  ( abs `  ( (coeff `  h ) `  e
) )  e.  ( { 0 ,  (deg
`  h ) }  u.  { g  e. 
NN0  |  E. i  e.  ( 0 ... (deg `  h ) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) )
65 supxrub 11600 . . . . . . . . . . . . . . . 16  |-  ( ( ( { 0 ,  (deg `  h ) }  u.  { g  e.  NN0  |  E. i  e.  ( 0 ... (deg `  h ) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } )  C_  RR*  /\  ( abs `  ( (coeff `  h ) `  e
) )  e.  ( { 0 ,  (deg
`  h ) }  u.  { g  e. 
NN0  |  E. i  e.  ( 0 ... (deg `  h ) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) )  ->  ( abs `  ( (coeff `  h ) `  e
) )  <_  sup ( ( { 0 ,  (deg `  h
) }  u.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  ) )
6625, 64, 65syl2anc 671 . . . . . . . . . . . . . . 15  |-  ( ( ( h  e.  ( (Poly `  ZZ )  \  { 0p }
)  /\  g  e.  CC )  /\  e  e.  NN0 )  ->  ( abs `  ( (coeff `  h ) `  e
) )  <_  sup ( ( { 0 ,  (deg `  h
) }  u.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  ) )
6766ralrimiva 2790 . . . . . . . . . . . . . 14  |-  ( ( h  e.  ( (Poly `  ZZ )  \  {
0p } )  /\  g  e.  CC )  ->  A. e  e.  NN0  ( abs `  ( (coeff `  h ) `  e
) )  <_  sup ( ( { 0 ,  (deg `  h
) }  u.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  ) )
686, 24, 673jca 1189 . . . . . . . . . . . . 13  |-  ( ( h  e.  ( (Poly `  ZZ )  \  {
0p } )  /\  g  e.  CC )  ->  ( h  =/=  0p  /\  (deg `  h )  <_  sup ( ( { 0 ,  (deg `  h
) }  u.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  )  /\  A. e  e. 
NN0  ( abs `  (
(coeff `  h ) `  e ) )  <_  sup ( ( { 0 ,  (deg `  h
) }  u.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  ) ) )
69683adant2 1028 . . . . . . . . . . . 12  |-  ( ( h  e.  ( (Poly `  ZZ )  \  {
0p } )  /\  ( h `  g )  =  0  /\  g  e.  CC )  ->  ( h  =/=  0p  /\  (deg `  h )  <_  sup ( ( { 0 ,  (deg `  h
) }  u.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  )  /\  A. e  e. 
NN0  ( abs `  (
(coeff `  h ) `  e ) )  <_  sup ( ( { 0 ,  (deg `  h
) }  u.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  ) ) )
70 neeq1 2686 . . . . . . . . . . . . . 14  |-  ( d  =  h  ->  (
d  =/=  0p  <-> 
h  =/=  0p ) )
71 fveq2 5848 . . . . . . . . . . . . . . 15  |-  ( d  =  h  ->  (deg `  d )  =  (deg
`  h ) )
7271breq1d 4384 . . . . . . . . . . . . . 14  |-  ( d  =  h  ->  (
(deg `  d )  <_  sup ( ( { 0 ,  (deg `  h ) }  u.  { g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  )  <-> 
(deg `  h )  <_  sup ( ( { 0 ,  (deg `  h ) }  u.  { g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  ) ) )
73 fveq2 5848 . . . . . . . . . . . . . . . . . 18  |-  ( d  =  h  ->  (coeff `  d )  =  (coeff `  h ) )
7473fveq1d 5850 . . . . . . . . . . . . . . . . 17  |-  ( d  =  h  ->  (
(coeff `  d ) `  e )  =  ( (coeff `  h ) `  e ) )
7574fveq2d 5852 . . . . . . . . . . . . . . . 16  |-  ( d  =  h  ->  ( abs `  ( (coeff `  d ) `  e
) )  =  ( abs `  ( (coeff `  h ) `  e
) ) )
7675breq1d 4384 . . . . . . . . . . . . . . 15  |-  ( d  =  h  ->  (
( abs `  (
(coeff `  d ) `  e ) )  <_  sup ( ( { 0 ,  (deg `  h
) }  u.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  )  <-> 
( abs `  (
(coeff `  h ) `  e ) )  <_  sup ( ( { 0 ,  (deg `  h
) }  u.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  ) ) )
7776ralbidv 2810 . . . . . . . . . . . . . 14  |-  ( d  =  h  ->  ( A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  sup ( ( { 0 ,  (deg `  h
) }  u.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  )  <->  A. e  e.  NN0  ( abs `  ( (coeff `  h ) `  e
) )  <_  sup ( ( { 0 ,  (deg `  h
) }  u.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  ) ) )
7870, 72, 773anbi123d 1343 . . . . . . . . . . . . 13  |-  ( d  =  h  ->  (
( d  =/=  0p  /\  (deg `  d
)  <_  sup (
( { 0 ,  (deg `  h ) }  u.  { g  e.  NN0  |  E. i  e.  ( 0 ... (deg `  h ) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  )  /\  A. e  e. 
NN0  ( abs `  (
(coeff `  d ) `  e ) )  <_  sup ( ( { 0 ,  (deg `  h
) }  u.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  ) )  <->  ( h  =/=  0p  /\  (deg `  h )  <_  sup ( ( { 0 ,  (deg `  h
) }  u.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  )  /\  A. e  e. 
NN0  ( abs `  (
(coeff `  h ) `  e ) )  <_  sup ( ( { 0 ,  (deg `  h
) }  u.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  ) ) ) )
7978elrab 3164 . . . . . . . . . . . 12  |-  ( h  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0p  /\  (deg `  d )  <_  sup ( ( { 0 ,  (deg `  h
) }  u.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  )  /\  A. e  e. 
NN0  ( abs `  (
(coeff `  d ) `  e ) )  <_  sup ( ( { 0 ,  (deg `  h
) }  u.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  ) ) }  <->  ( h  e.  (Poly `  ZZ )  /\  ( h  =/=  0p  /\  (deg `  h
)  <_  sup (
( { 0 ,  (deg `  h ) }  u.  { g  e.  NN0  |  E. i  e.  ( 0 ... (deg `  h ) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  )  /\  A. e  e. 
NN0  ( abs `  (
(coeff `  h ) `  e ) )  <_  sup ( ( { 0 ,  (deg `  h
) }  u.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  ) ) ) )
804, 69, 79sylanbrc 675 . . . . . . . . . . 11  |-  ( ( h  e.  ( (Poly `  ZZ )  \  {
0p } )  /\  ( h `  g )  =  0  /\  g  e.  CC )  ->  h  e.  {
d  e.  (Poly `  ZZ )  |  (
d  =/=  0p  /\  (deg `  d
)  <_  sup (
( { 0 ,  (deg `  h ) }  u.  { g  e.  NN0  |  E. i  e.  ( 0 ... (deg `  h ) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  )  /\  A. e  e. 
NN0  ( abs `  (
(coeff `  d ) `  e ) )  <_  sup ( ( { 0 ,  (deg `  h
) }  u.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  ) ) } )
81 simp2 1010 . . . . . . . . . . 11  |-  ( ( h  e.  ( (Poly `  ZZ )  \  {
0p } )  /\  ( h `  g )  =  0  /\  g  e.  CC )  ->  ( h `  g )  =  0 )
82 fveq1 5847 . . . . . . . . . . . . 13  |-  ( c  =  h  ->  (
c `  g )  =  ( h `  g ) )
8382eqeq1d 2454 . . . . . . . . . . . 12  |-  ( c  =  h  ->  (
( c `  g
)  =  0  <->  (
h `  g )  =  0 ) )
8483rspcev 3118 . . . . . . . . . . 11  |-  ( ( h  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0p  /\  (deg `  d )  <_  sup ( ( { 0 ,  (deg `  h
) }  u.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  )  /\  A. e  e. 
NN0  ( abs `  (
(coeff `  d ) `  e ) )  <_  sup ( ( { 0 ,  (deg `  h
) }  u.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  ) ) }  /\  (
h `  g )  =  0 )  ->  E. c  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0p  /\  (deg `  d )  <_  sup ( ( { 0 ,  (deg `  h
) }  u.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  )  /\  A. e  e. 
NN0  ( abs `  (
(coeff `  d ) `  e ) )  <_  sup ( ( { 0 ,  (deg `  h
) }  u.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  ) ) }  ( c `
 g )  =  0 )
8580, 81, 84syl2anc 671 . . . . . . . . . 10  |-  ( ( h  e.  ( (Poly `  ZZ )  \  {
0p } )  /\  ( h `  g )  =  0  /\  g  e.  CC )  ->  E. c  e.  {
d  e.  (Poly `  ZZ )  |  (
d  =/=  0p  /\  (deg `  d
)  <_  sup (
( { 0 ,  (deg `  h ) }  u.  { g  e.  NN0  |  E. i  e.  ( 0 ... (deg `  h ) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  )  /\  A. e  e. 
NN0  ( abs `  (
(coeff `  d ) `  e ) )  <_  sup ( ( { 0 ,  (deg `  h
) }  u.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  ) ) }  ( c `
 g )  =  0 )
86 fveq2 5848 . . . . . . . . . . . . 13  |-  ( b  =  g  ->  (
c `  b )  =  ( c `  g ) )
8786eqeq1d 2454 . . . . . . . . . . . 12  |-  ( b  =  g  ->  (
( c `  b
)  =  0  <->  (
c `  g )  =  0 ) )
8887rexbidv 2873 . . . . . . . . . . 11  |-  ( b  =  g  ->  ( E. c  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0p  /\  (deg `  d )  <_  sup ( ( { 0 ,  (deg `  h
) }  u.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  )  /\  A. e  e. 
NN0  ( abs `  (
(coeff `  d ) `  e ) )  <_  sup ( ( { 0 ,  (deg `  h
) }  u.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  ) ) }  ( c `
 b )  =  0  <->  E. c  e.  {
d  e.  (Poly `  ZZ )  |  (
d  =/=  0p  /\  (deg `  d
)  <_  sup (
( { 0 ,  (deg `  h ) }  u.  { g  e.  NN0  |  E. i  e.  ( 0 ... (deg `  h ) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  )  /\  A. e  e. 
NN0  ( abs `  (
(coeff `  d ) `  e ) )  <_  sup ( ( { 0 ,  (deg `  h
) }  u.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  ) ) }  ( c `
 g )  =  0 ) )
8988elrab 3164 . . . . . . . . . 10  |-  ( g  e.  { b  e.  CC  |  E. c  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0p  /\  (deg `  d
)  <_  sup (
( { 0 ,  (deg `  h ) }  u.  { g  e.  NN0  |  E. i  e.  ( 0 ... (deg `  h ) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  )  /\  A. e  e. 
NN0  ( abs `  (
(coeff `  d ) `  e ) )  <_  sup ( ( { 0 ,  (deg `  h
) }  u.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  ) ) }  ( c `
 b )  =  0 }  <->  ( g  e.  CC  /\  E. c  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0p  /\  (deg `  d
)  <_  sup (
( { 0 ,  (deg `  h ) }  u.  { g  e.  NN0  |  E. i  e.  ( 0 ... (deg `  h ) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  )  /\  A. e  e. 
NN0  ( abs `  (
(coeff `  d ) `  e ) )  <_  sup ( ( { 0 ,  (deg `  h
) }  u.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  ) ) }  ( c `
 g )  =  0 ) )
901, 85, 89sylanbrc 675 . . . . . . . . 9  |-  ( ( h  e.  ( (Poly `  ZZ )  \  {
0p } )  /\  ( h `  g )  =  0  /\  g  e.  CC )  ->  g  e.  {
b  e.  CC  |  E. c  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0p  /\  (deg `  d )  <_  sup ( ( { 0 ,  (deg `  h
) }  u.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  )  /\  A. e  e. 
NN0  ( abs `  (
(coeff `  d ) `  e ) )  <_  sup ( ( { 0 ,  (deg `  h
) }  u.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  ) ) }  ( c `
 b )  =  0 } )
91 prfi 7833 . . . . . . . . . . . . . . 15  |-  { 0 ,  (deg `  h
) }  e.  Fin
92 fzfi 12179 . . . . . . . . . . . . . . . . 17  |-  ( 0 ... (deg `  h
) )  e.  Fin
93 abrexfi 7861 . . . . . . . . . . . . . . . . 17  |-  ( ( 0 ... (deg `  h ) )  e. 
Fin  ->  { g  |  E. i  e.  ( 0 ... (deg `  h ) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) }  e.  Fin )
9492, 93ax-mp 5 . . . . . . . . . . . . . . . 16  |-  { g  |  E. i  e.  ( 0 ... (deg `  h ) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) }  e.  Fin
95 rabssab 3484 . . . . . . . . . . . . . . . 16  |-  { g  e.  NN0  |  E. i  e.  ( 0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } 
C_  { g  |  E. i  e.  ( 0 ... (deg `  h ) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) }
96 ssfi 7779 . . . . . . . . . . . . . . . 16  |-  ( ( { g  |  E. i  e.  ( 0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) }  e.  Fin  /\  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } 
C_  { g  |  E. i  e.  ( 0 ... (deg `  h ) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } )  ->  { g  e.  NN0  |  E. i  e.  ( 0 ... (deg `  h ) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) }  e.  Fin )
9794, 95, 96mp2an 683 . . . . . . . . . . . . . . 15  |-  { g  e.  NN0  |  E. i  e.  ( 0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) }  e.  Fin
98 unfi 7825 . . . . . . . . . . . . . . 15  |-  ( ( { 0 ,  (deg
`  h ) }  e.  Fin  /\  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) }  e.  Fin )  -> 
( { 0 ,  (deg `  h ) }  u.  { g  e.  NN0  |  E. i  e.  ( 0 ... (deg `  h ) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } )  e.  Fin )
9991, 97, 98mp2an 683 . . . . . . . . . . . . . 14  |-  ( { 0 ,  (deg `  h ) }  u.  { g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } )  e.  Fin
10099a1i 11 . . . . . . . . . . . . 13  |-  ( ( h  e.  ( (Poly `  ZZ )  \  {
0p } )  /\  g  e.  CC )  ->  ( { 0 ,  (deg `  h
) }  u.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } )  e.  Fin )
10122ne0ii 3706 . . . . . . . . . . . . . 14  |-  ( { 0 ,  (deg `  h ) }  u.  { g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } )  =/=  (/)
102101a1i 11 . . . . . . . . . . . . 13  |-  ( ( h  e.  ( (Poly `  ZZ )  \  {
0p } )  /\  g  e.  CC )  ->  ( { 0 ,  (deg `  h
) }  u.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } )  =/=  (/) )
103 xrltso 11430 . . . . . . . . . . . . . 14  |-  <  Or  RR*
104 fisupcl 7972 . . . . . . . . . . . . . 14  |-  ( (  <  Or  RR*  /\  (
( { 0 ,  (deg `  h ) }  u.  { g  e.  NN0  |  E. i  e.  ( 0 ... (deg `  h ) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } )  e.  Fin  /\  ( { 0 ,  (deg
`  h ) }  u.  { g  e. 
NN0  |  E. i  e.  ( 0 ... (deg `  h ) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } )  =/=  (/)  /\  ( { 0 ,  (deg
`  h ) }  u.  { g  e. 
NN0  |  E. i  e.  ( 0 ... (deg `  h ) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } )  C_  RR* ) )  ->  sup ( ( { 0 ,  (deg `  h ) }  u.  { g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  )  e.  ( { 0 ,  (deg `  h
) }  u.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) )
105103, 104mpan 681 . . . . . . . . . . . . 13  |-  ( ( ( { 0 ,  (deg `  h ) }  u.  { g  e.  NN0  |  E. i  e.  ( 0 ... (deg `  h ) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } )  e.  Fin  /\  ( { 0 ,  (deg
`  h ) }  u.  { g  e. 
NN0  |  E. i  e.  ( 0 ... (deg `  h ) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } )  =/=  (/)  /\  ( { 0 ,  (deg
`  h ) }  u.  { g  e. 
NN0  |  E. i  e.  ( 0 ... (deg `  h ) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } )  C_  RR* )  ->  sup ( ( { 0 ,  (deg `  h
) }  u.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  )  e.  ( { 0 ,  (deg `  h
) }  u.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) )
106100, 102, 18, 105syl3anc 1271 . . . . . . . . . . . 12  |-  ( ( h  e.  ( (Poly `  ZZ )  \  {
0p } )  /\  g  e.  CC )  ->  sup ( ( { 0 ,  (deg `  h ) }  u.  { g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  )  e.  ( { 0 ,  (deg `  h
) }  u.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) )
10714, 106sseldd 3401 . . . . . . . . . . 11  |-  ( ( h  e.  ( (Poly `  ZZ )  \  {
0p } )  /\  g  e.  CC )  ->  sup ( ( { 0 ,  (deg `  h ) }  u.  { g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  )  e.  NN0 )
1081073adant2 1028 . . . . . . . . . 10  |-  ( ( h  e.  ( (Poly `  ZZ )  \  {
0p } )  /\  ( h `  g )  =  0  /\  g  e.  CC )  ->  sup ( ( { 0 ,  (deg `  h ) }  u.  { g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  )  e.  NN0 )
109 eqidd 2453 . . . . . . . . . 10  |-  ( ( h  e.  ( (Poly `  ZZ )  \  {
0p } )  /\  ( h `  g )  =  0  /\  g  e.  CC )  ->  { b  e.  CC  |  E. c  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0p  /\  (deg `  d
)  <_  sup (
( { 0 ,  (deg `  h ) }  u.  { g  e.  NN0  |  E. i  e.  ( 0 ... (deg `  h ) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  )  /\  A. e  e. 
NN0  ( abs `  (
(coeff `  d ) `  e ) )  <_  sup ( ( { 0 ,  (deg `  h
) }  u.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  ) ) }  ( c `
 b )  =  0 }  =  {
b  e.  CC  |  E. c  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0p  /\  (deg `  d )  <_  sup ( ( { 0 ,  (deg `  h
) }  u.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  )  /\  A. e  e. 
NN0  ( abs `  (
(coeff `  d ) `  e ) )  <_  sup ( ( { 0 ,  (deg `  h
) }  u.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  ) ) }  ( c `
 b )  =  0 } )
110 breq2 4378 . . . . . . . . . . . . . . . 16  |-  ( a  =  sup ( ( { 0 ,  (deg
`  h ) }  u.  { g  e. 
NN0  |  E. i  e.  ( 0 ... (deg `  h ) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  )  ->  ( (deg `  d )  <_  a  <->  (deg
`  d )  <_  sup ( ( { 0 ,  (deg `  h
) }  u.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  ) ) )
111 breq2 4378 . . . . . . . . . . . . . . . . 17  |-  ( a  =  sup ( ( { 0 ,  (deg
`  h ) }  u.  { g  e. 
NN0  |  E. i  e.  ( 0 ... (deg `  h ) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  )  ->  ( ( abs `  ( (coeff `  d
) `  e )
)  <_  a  <->  ( abs `  ( (coeff `  d
) `  e )
)  <_  sup (
( { 0 ,  (deg `  h ) }  u.  { g  e.  NN0  |  E. i  e.  ( 0 ... (deg `  h ) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  ) ) )
112111ralbidv 2810 . . . . . . . . . . . . . . . 16  |-  ( a  =  sup ( ( { 0 ,  (deg
`  h ) }  u.  { g  e. 
NN0  |  E. i  e.  ( 0 ... (deg `  h ) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  )  ->  ( A. e  e.  NN0  ( abs `  (
(coeff `  d ) `  e ) )  <_ 
a  <->  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  sup ( ( { 0 ,  (deg `  h
) }  u.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  ) ) )
113110, 1123anbi23d 1346 . . . . . . . . . . . . . . 15  |-  ( a  =  sup ( ( { 0 ,  (deg
`  h ) }  u.  { g  e. 
NN0  |  E. i  e.  ( 0 ... (deg `  h ) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  )  ->  ( ( d  =/=  0p  /\  (deg `  d )  <_ 
a  /\  A. e  e.  NN0  ( abs `  (
(coeff `  d ) `  e ) )  <_ 
a )  <->  ( d  =/=  0p  /\  (deg `  d )  <_  sup ( ( { 0 ,  (deg `  h
) }  u.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  )  /\  A. e  e. 
NN0  ( abs `  (
(coeff `  d ) `  e ) )  <_  sup ( ( { 0 ,  (deg `  h
) }  u.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  ) ) ) )
114113rabbidv 3004 . . . . . . . . . . . . . 14  |-  ( a  =  sup ( ( { 0 ,  (deg
`  h ) }  u.  { g  e. 
NN0  |  E. i  e.  ( 0 ... (deg `  h ) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  )  ->  { 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
)  <_  sup (
( { 0 ,  (deg `  h ) }  u.  { g  e.  NN0  |  E. i  e.  ( 0 ... (deg `  h ) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  )  /\  A. e  e. 
NN0  ( abs `  (
(coeff `  d ) `  e ) )  <_  sup ( ( { 0 ,  (deg `  h
) }  u.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  ) ) } )
115114rexeqdv 2962 . . . . . . . . . . . . 13  |-  ( a  =  sup ( ( { 0 ,  (deg
`  h ) }  u.  { g  e. 
NN0  |  E. i  e.  ( 0 ... (deg `  h ) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  )  ->  ( 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
)  <_  sup (
( { 0 ,  (deg `  h ) }  u.  { g  e.  NN0  |  E. i  e.  ( 0 ... (deg `  h ) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  )  /\  A. e  e. 
NN0  ( abs `  (
(coeff `  d ) `  e ) )  <_  sup ( ( { 0 ,  (deg `  h
) }  u.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  ) ) }  ( c `
 b )  =  0 ) )
116115rabbidv 3004 . . . . . . . . . . . 12  |-  ( a  =  sup ( ( { 0 ,  (deg
`  h ) }  u.  { g  e. 
NN0  |  E. i  e.  ( 0 ... (deg `  h ) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  )  ->  { 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 )  <_  sup ( ( { 0 ,  (deg `  h
) }  u.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  )  /\  A. e  e. 
NN0  ( abs `  (
(coeff `  d ) `  e ) )  <_  sup ( ( { 0 ,  (deg `  h
) }  u.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  ) ) }  ( c `
 b )  =  0 } )
117116eqeq2d 2462 . . . . . . . . . . 11  |-  ( a  =  sup ( ( { 0 ,  (deg
`  h ) }  u.  { g  e. 
NN0  |  E. i  e.  ( 0 ... (deg `  h ) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  )  ->  ( { b  e.  CC  |  E. c  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0p  /\  (deg `  d )  <_  sup ( ( { 0 ,  (deg `  h
) }  u.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  )  /\  A. e  e. 
NN0  ( abs `  (
(coeff `  d ) `  e ) )  <_  sup ( ( { 0 ,  (deg `  h
) }  u.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  ) ) }  ( 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 }  <->  { b  e.  CC  |  E. c  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0p  /\  (deg `  d
)  <_  sup (
( { 0 ,  (deg `  h ) }  u.  { g  e.  NN0  |  E. i  e.  ( 0 ... (deg `  h ) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  )  /\  A. e  e. 
NN0  ( abs `  (
(coeff `  d ) `  e ) )  <_  sup ( ( { 0 ,  (deg `  h
) }  u.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  ) ) }  ( c `
 b )  =  0 }  =  {
b  e.  CC  |  E. c  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0p  /\  (deg `  d )  <_  sup ( ( { 0 ,  (deg `  h
) }  u.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  )  /\  A. e  e. 
NN0  ( abs `  (
(coeff `  d ) `  e ) )  <_  sup ( ( { 0 ,  (deg `  h
) }  u.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  ) ) }  ( c `
 b )  =  0 } ) )
118117rspcev 3118 . . . . . . . . . 10  |-  ( ( sup ( ( { 0 ,  (deg `  h ) }  u.  { g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  )  e.  NN0  /\  { b  e.  CC  |  E. c  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0p  /\  (deg `  d )  <_  sup ( ( { 0 ,  (deg `  h
) }  u.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  )  /\  A. e  e. 
NN0  ( abs `  (
(coeff `  d ) `  e ) )  <_  sup ( ( { 0 ,  (deg `  h
) }  u.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  ) ) }  ( c `
 b )  =  0 }  =  {
b  e.  CC  |  E. c  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0p  /\  (deg `  d )  <_  sup ( ( { 0 ,  (deg `  h
) }  u.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  )  /\  A. e  e. 
NN0  ( abs `  (
(coeff `  d ) `  e ) )  <_  sup ( ( { 0 ,  (deg `  h
) }  u.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  ) ) }  ( c `
 b )  =  0 } )  ->  E. a  e.  NN0  { b  e.  CC  |  E. c  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0p  /\  (deg `  d )  <_  sup ( ( { 0 ,  (deg `  h
) }  u.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  )  /\  A. e  e. 
NN0  ( abs `  (
(coeff `  d ) `  e ) )  <_  sup ( ( { 0 ,  (deg `  h
) }  u.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  ) ) }  ( 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 } )
119108, 109, 118syl2anc 671 . . . . . . . . 9  |-  ( ( h  e.  ( (Poly `  ZZ )  \  {
0p } )  /\  ( h `  g )  =  0  /\  g  e.  CC )  ->  E. a  e.  NN0  { b  e.  CC  |  E. c  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0p  /\  (deg `  d )  <_  sup ( ( { 0 ,  (deg `  h
) }  u.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  )  /\  A. e  e. 
NN0  ( abs `  (
(coeff `  d ) `  e ) )  <_  sup ( ( { 0 ,  (deg `  h
) }  u.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  ) ) }  ( 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 } )
120 cnex 9607 . . . . . . . . . . 11  |-  CC  e.  _V
121120rabex 4527 . . . . . . . . . 10  |-  { b  e.  CC  |  E. c  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0p  /\  (deg `  d )  <_  sup ( ( { 0 ,  (deg `  h
) }  u.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  )  /\  A. e  e. 
NN0  ( abs `  (
(coeff `  d ) `  e ) )  <_  sup ( ( { 0 ,  (deg `  h
) }  u.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  ) ) }  ( c `
 b )  =  0 }  e.  _V
122 eleq2 2519 . . . . . . . . . . 11  |-  ( f  =  { b  e.  CC  |  E. c  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0p  /\  (deg `  d
)  <_  sup (
( { 0 ,  (deg `  h ) }  u.  { g  e.  NN0  |  E. i  e.  ( 0 ... (deg `  h ) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  )  /\  A. e  e. 
NN0  ( abs `  (
(coeff `  d ) `  e ) )  <_  sup ( ( { 0 ,  (deg `  h
) }  u.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  ) ) }  ( c `
 b )  =  0 }  ->  (
g  e.  f  <->  g  e.  { b  e.  CC  |  E. c  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0p  /\  (deg `  d )  <_  sup ( ( { 0 ,  (deg `  h
) }  u.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  )  /\  A. e  e. 
NN0  ( abs `  (
(coeff `  d ) `  e ) )  <_  sup ( ( { 0 ,  (deg `  h
) }  u.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  ) ) }  ( c `
 b )  =  0 } ) )
123 eqeq1 2456 . . . . . . . . . . . 12  |-  ( f  =  { b  e.  CC  |  E. c  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0p  /\  (deg `  d
)  <_  sup (
( { 0 ,  (deg `  h ) }  u.  { g  e.  NN0  |  E. i  e.  ( 0 ... (deg `  h ) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  )  /\  A. e  e. 
NN0  ( abs `  (
(coeff `  d ) `  e ) )  <_  sup ( ( { 0 ,  (deg `  h
) }  u.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  ) ) }  ( c `
 b )  =  0 }  ->  (
f  =  { 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
)  <_  sup (
( { 0 ,  (deg `  h ) }  u.  { g  e.  NN0  |  E. i  e.  ( 0 ... (deg `  h ) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  )  /\  A. e  e. 
NN0  ( abs `  (
(coeff `  d ) `  e ) )  <_  sup ( ( { 0 ,  (deg `  h
) }  u.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  ) ) }  ( 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 } ) )
124123rexbidv 2873 . . . . . . . . . . 11  |-  ( f  =  { b  e.  CC  |  E. c  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0p  /\  (deg `  d
)  <_  sup (
( { 0 ,  (deg `  h ) }  u.  { g  e.  NN0  |  E. i  e.  ( 0 ... (deg `  h ) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  )  /\  A. e  e. 
NN0  ( abs `  (
(coeff `  d ) `  e ) )  <_  sup ( ( { 0 ,  (deg `  h
) }  u.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  ) ) }  ( c `
 b )  =  0 }  ->  ( E. a  e.  NN0  f  =  { 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. a  e.  NN0  { b  e.  CC  |  E. c  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0p  /\  (deg `  d
)  <_  sup (
( { 0 ,  (deg `  h ) }  u.  { g  e.  NN0  |  E. i  e.  ( 0 ... (deg `  h ) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  )  /\  A. e  e. 
NN0  ( abs `  (
(coeff `  d ) `  e ) )  <_  sup ( ( { 0 ,  (deg `  h
) }  u.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  ) ) }  ( 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 } ) )
125122, 124anbi12d 722 . . . . . . . . . 10  |-  ( f  =  { b  e.  CC  |  E. c  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0p  /\  (deg `  d
)  <_  sup (
( { 0 ,  (deg `  h ) }  u.  { g  e.  NN0  |  E. i  e.  ( 0 ... (deg `  h ) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  )  /\  A. e  e. 
NN0  ( abs `  (
(coeff `  d ) `  e ) )  <_  sup ( ( { 0 ,  (deg `  h
) }  u.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  ) ) }  ( c `
 b )  =  0 }  ->  (
( g  e.  f  /\  E. a  e. 
NN0  f  =  {
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 } )  <->  ( g  e.  { b  e.  CC  |  E. c  e.  {
d  e.  (Poly `  ZZ )  |  (
d  =/=  0p  /\  (deg `  d
)  <_  sup (
( { 0 ,  (deg `  h ) }  u.  { g  e.  NN0  |  E. i  e.  ( 0 ... (deg `  h ) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  )  /\  A. e  e. 
NN0  ( abs `  (
(coeff `  d ) `  e ) )  <_  sup ( ( { 0 ,  (deg `  h
) }  u.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  ) ) }  ( c `
 b )  =  0 }  /\  E. a  e.  NN0  { b  e.  CC  |  E. c  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0p  /\  (deg `  d )  <_  sup ( ( { 0 ,  (deg `  h
) }  u.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  )  /\  A. e  e. 
NN0  ( abs `  (
(coeff `  d ) `  e ) )  <_  sup ( ( { 0 ,  (deg `  h
) }  u.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  ) ) }  ( 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 } ) ) )
126121, 125spcev 3109 . . . . . . . . 9  |-  ( ( g  e.  { b  e.  CC  |  E. c  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0p  /\  (deg `  d )  <_  sup ( ( { 0 ,  (deg `  h
) }  u.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  )  /\  A. e  e. 
NN0  ( abs `  (
(coeff `  d ) `  e ) )  <_  sup ( ( { 0 ,  (deg `  h
) }  u.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  ) ) }  ( c `
 b )  =  0 }  /\  E. a  e.  NN0  { b  e.  CC  |  E. c  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0p  /\  (deg `  d )  <_  sup ( ( { 0 ,  (deg `  h
) }  u.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  )  /\  A. e  e. 
NN0  ( abs `  (
(coeff `  d ) `  e ) )  <_  sup ( ( { 0 ,  (deg `  h
) }  u.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  ) ) }  ( 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 } )  ->  E. f ( g  e.  f  /\  E. a  e.  NN0  f  =  {
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 } ) )
12790, 119, 126syl2anc 671 . . . . . . . 8  |-  ( ( h  e.  ( (Poly `  ZZ )  \  {
0p } )  /\  ( h `  g )  =  0  /\  g  e.  CC )  ->  E. f ( g  e.  f  /\  E. a  e.  NN0  f  =  { 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 } ) )
1281273exp 1209 . . . . . . 7  |-  ( h  e.  ( (Poly `  ZZ )  \  { 0p } )  -> 
( ( h `  g )  =  0  ->  ( g  e.  CC  ->  E. f
( g  e.  f  /\  E. a  e. 
NN0  f  =  {
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 } ) ) ) )
129128rexlimiv 2848 . . . . . 6  |-  ( E. h  e.  ( (Poly `  ZZ )  \  {
0p } ) ( h `  g
)  =  0  -> 
( g  e.  CC  ->  E. f ( g  e.  f  /\  E. a  e.  NN0  f  =  { 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 } ) ) )
130129impcom 436 . . . . 5  |-  ( ( g  e.  CC  /\  E. h  e.  ( (Poly `  ZZ )  \  {
0p } ) ( h `  g
)  =  0 )  ->  E. f ( g  e.  f  /\  E. a  e.  NN0  f  =  { 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 } ) )
131 eleq2 2519 . . . . . . . . 9  |-  ( f  =  { 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 }  ->  (
g  e.  f  <->  g  e.  { 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 } ) )
13287rexbidv 2873 . . . . . . . . . . 11  |-  ( b  =  g  ->  ( 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 `
 g )  =  0 ) )
133132elrab 3164 . . . . . . . . . 10  |-  ( g  e.  { 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 }  <->  ( g  e.  CC  /\  E. c  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0p  /\  (deg `  d
)  <_  a  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  a
) }  ( c `
 g )  =  0 ) )
134 simp1 1009 . . . . . . . . . . . . . . 15  |-  ( ( h  =/=  0p  /\  (deg `  h
)  <_  a  /\  A. e  e.  NN0  ( abs `  ( (coeff `  h ) `  e
) )  <_  a
)  ->  h  =/=  0p )
135134anim2i 577 . . . . . . . . . . . . . 14  |-  ( ( h  e.  (Poly `  ZZ )  /\  (
h  =/=  0p  /\  (deg `  h
)  <_  a  /\  A. e  e.  NN0  ( abs `  ( (coeff `  h ) `  e
) )  <_  a
) )  ->  (
h  e.  (Poly `  ZZ )  /\  h  =/=  0p ) )
13671breq1d 4384 . . . . . . . . . . . . . . . 16  |-  ( d  =  h  ->  (
(deg `  d )  <_  a  <->  (deg `  h )  <_  a ) )
13775breq1d 4384 . . . . . . . . . . . . . . . . 17  |-  ( d  =  h  ->  (
( abs `  (
(coeff `  d ) `  e ) )  <_ 
a  <->  ( abs `  (
(coeff `  h ) `  e ) )  <_ 
a ) )
138137ralbidv 2810 . . . . . . . . . . . . . . . 16  |-  ( d  =  h  ->  ( A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  a  <->  A. e  e.  NN0  ( abs `  ( (coeff `  h ) `  e
) )  <_  a
) )
13970, 136, 1383anbi123d 1343 . . . . . . . . . . . . . . 15  |-  ( d  =  h  ->  (
( d  =/=  0p  /\  (deg `  d
)  <_  a  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  a
)  <->  ( h  =/=  0p  /\  (deg `  h )  <_  a  /\  A. e  e.  NN0  ( abs `  ( (coeff `  h ) `  e
) )  <_  a
) ) )
140139elrab 3164 . . . . . . . . . . . . . 14  |-  ( h  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0p  /\  (deg `  d )  <_  a  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  a
) }  <->  ( h  e.  (Poly `  ZZ )  /\  ( h  =/=  0p  /\  (deg `  h
)  <_  a  /\  A. e  e.  NN0  ( abs `  ( (coeff `  h ) `  e
) )  <_  a
) ) )
141 eldifsn 4066 . . . . . . . . . . . . . 14  |-  ( h  e.  ( (Poly `  ZZ )  \  { 0p } )  <->  ( h  e.  (Poly `  ZZ )  /\  h  =/=  0p ) )
142135, 140, 1413imtr4i 274 . . . . . . . . . . . . 13  |-  ( h  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0p  /\  (deg `  d )  <_  a  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  a
) }  ->  h  e.  ( (Poly `  ZZ )  \  { 0p } ) )
143142ssriv 3404 . . . . . . . . . . . 12  |-  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0p  /\  (deg `  d )  <_  a  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  a
) }  C_  (
(Poly `  ZZ )  \  { 0p }
)
144 ssrexv 3462 . . . . . . . . . . . . 13  |-  ( { d  e.  (Poly `  ZZ )  |  (
d  =/=  0p  /\  (deg `  d
)  <_  a  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  a
) }  C_  (
(Poly `  ZZ )  \  { 0p }
)  ->  ( E. c  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0p  /\  (deg `  d )  <_  a  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  a
) }  ( c `
 g )  =  0  ->  E. c  e.  ( (Poly `  ZZ )  \  { 0p } ) ( c `
 g )  =  0 ) )
14583cbvrexv 2988 . . . . . . . . . . . . 13  |-  ( E. c  e.  ( (Poly `  ZZ )  \  {
0p } ) ( c `  g
)  =  0  <->  E. h  e.  ( (Poly `  ZZ )  \  {
0p } ) ( h `  g
)  =  0 )
146144, 145syl6ib 234 . . . . . . . . . . . 12  |-  ( { d  e.  (Poly `  ZZ )  |  (
d  =/=  0p  /\  (deg `  d
)  <_  a  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  a
) }  C_  (
(Poly `  ZZ )  \  { 0p }
)  ->  ( E. c  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0p  /\  (deg `  d )  <_  a  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  a
) }  ( c `
 g )  =  0  ->  E. h  e.  ( (Poly `  ZZ )  \  { 0p } ) ( h `
 g )  =  0 ) )
147143, 146ax-mp 5 . . . . . . . . . . 11  |-  ( E. c  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0p  /\  (deg `  d )  <_  a  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  a
) }  ( c `
 g )  =  0  ->  E. h  e.  ( (Poly `  ZZ )  \  { 0p } ) ( h `
 g )  =  0 )
148147anim2i 577 . . . . . . . . . 10  |-  ( ( g  e.  CC  /\  E. c  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0p  /\  (deg `  d )  <_  a  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  a
) }  ( c `
 g )  =  0 )  ->  (
g  e.  CC  /\  E. h  e.  ( (Poly `  ZZ )  \  {
0p } ) ( h `  g
)  =  0 ) )
149133, 148sylbi 200 . . . . . . . . 9  |-  ( g  e.  { 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 }  ->  (
g  e.  CC  /\  E. h  e.  ( (Poly `  ZZ )  \  {
0p } ) ( h `  g
)  =  0 ) )
150131, 149syl6bi 236 . . . . . . . 8  |-  ( f  =  { 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 }  ->  (
g  e.  f  -> 
( g  e.  CC  /\ 
E. h  e.  ( (Poly `  ZZ )  \  { 0p }
) ( h `  g )  =  0 ) ) )
151150rexlimivw 2850 . . . . . . 7  |-  ( E. a  e.  NN0  f  =  { 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 }  ->  (
g  e.  f  -> 
( g  e.  CC  /\ 
E. h  e.  ( (Poly `  ZZ )  \  { 0p }
) ( h `  g )  =  0 ) ) )
152151impcom 436 . . . . . 6  |-  ( ( g  e.  f  /\  E. a  e.  NN0  f  =  { 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 } )  -> 
( g  e.  CC  /\ 
E. h  e.  ( (Poly `  ZZ )  \  { 0p }
) ( h `  g )  =  0 ) )
153152exlimiv 1780 . . . . 5  |-  ( E. f ( g  e.  f  /\  E. a  e.  NN0  f  =  {
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 } )  -> 
( g  e.  CC  /\ 
E. h  e.  ( (Poly `  ZZ )  \  { 0p }
) ( h `  g )  =  0 ) )
154130, 153impbii 192 . . . 4  |-  ( ( g  e.  CC  /\  E. h  e.  ( (Poly `  ZZ )  \  {
0p } ) ( h `  g
)  =  0 )  <->  E. f ( g  e.  f  /\  E. a  e.  NN0  f  =  {
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 } ) )
155 elaa 23281 . . . 4  |-  ( g  e.  AA  <->  ( g  e.  CC  /\  E. h  e.  ( (Poly `  ZZ )  \  { 0p } ) ( h `
 g )  =  0 ) )
156 eluniab 4179 . . . 4  |-  ( g  e.  U. { f  |  E. a  e. 
NN0  f  =  {
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. f
( g  e.  f  /\  E. a  e. 
NN0  f  =  {
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 } ) )
157154, 155, 1563bitr4i 285 . . 3  |-  ( g  e.  AA  <->  g  e.  U. { f  |  E. a  e.  NN0  f  =  { 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 } } )
158157eqriv 2449 . 2  |-  AA  =  U. { f  |  E. a  e.  NN0  f  =  { 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 } }
159 aannenlem.a . . . 4  |-  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 } )
160159rnmpt 5058 . . 3  |-  ran  H  =  { f  |  E. a  e.  NN0  f  =  { 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 } }
161160unieqi 4177 . 2  |-  U. ran  H  =  U. { f  |  E. a  e. 
NN0  f  =  {
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 } }
162158, 161eqtr4i 2477 1  |-  AA  =  U. ran  H
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 375    /\ w3a 986    = wceq 1448   E.wex 1667    e. wcel 1891   {cab 2438    =/= wne 2622   A.wral 2737   E.wrex 2738   {crab 2741    \ cdif 3369    u. cun 3370    C_ wss 3372   (/)c0 3699   {csn 3936   {cpr 3938   U.cuni 4168   class class class wbr 4374    |-> cmpt 4433    Or wor 4732   ran crn 4813   -->wf 5557   ` cfv 5561  (class class class)co 6276   Fincfn 7556   supcsup 7941   CCcc 9524   RRcr 9525   0cc0 9526   RR*cxr 9661    < clt 9662    <_ cle 9663   NN0cn0 10859   ZZcz 10927   ...cfz 11775   abscabs 13308   0pc0p 22639  Polycply 23150  coeffccoe 23152  degcdgr 23153   AAcaa 23279
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1673  ax-4 1686  ax-5 1762  ax-6 1809  ax-7 1855  ax-8 1893  ax-9 1900  ax-10 1919  ax-11 1924  ax-12 1937  ax-13 2092  ax-ext 2432  ax-rep 4487  ax-sep 4497  ax-nul 4506  ax-pow 4554  ax-pr 4612  ax-un 6571  ax-inf2 8133  ax-cnex 9582  ax-resscn 9583  ax-1cn 9584  ax-icn 9585  ax-addcl 9586  ax-addrcl 9587  ax-mulcl 9588  ax-mulrcl 9589  ax-mulcom 9590  ax-addass 9591  ax-mulass 9592  ax-distr 9593  ax-i2m1 9594  ax-1ne0 9595  ax-1rid 9596  ax-rnegex 9597  ax-rrecex 9598  ax-cnre 9599  ax-pre-lttri 9600  ax-pre-lttrn 9601  ax-pre-ltadd 9602  ax-pre-mulgt0 9603  ax-pre-sup 9604  ax-addf 9605
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 987  df-3an 988  df-tru 1451  df-fal 1454  df-ex 1668  df-nf 1672  df-sb 1802  df-eu 2304  df-mo 2305  df-clab 2439  df-cleq 2445  df-clel 2448  df-nfc 2582  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3015  df-sbc 3236  df-csb 3332  df-dif 3375  df-un 3377  df-in 3379  df-ss 3386  df-pss 3388  df-nul 3700  df-if 3850  df-pw 3921  df-sn 3937  df-pr 3939  df-tp 3941  df-op 3943  df-uni 4169  df-int 4205  df-iun 4250  df-br 4375  df-opab 4434  df-mpt 4435  df-tr 4470  df-eprel 4723  df-id 4727  df-po 4733  df-so 4734  df-fr 4771  df-se 4772  df-we 4773  df-xp 4818  df-rel 4819  df-cnv 4820  df-co 4821  df-dm 4822  df-rn 4823  df-res 4824  df-ima 4825  df-pred 5359  df-ord 5405  df-on 5406  df-lim 5407  df-suc 5408  df-iota 5525  df-fun 5563  df-fn 5564  df-f 5565  df-f1 5566  df-fo 5567  df-f1o 5568  df-fv 5569  df-isom 5570  df-riota 6238  df-ov 6279  df-oprab 6280  df-mpt2 6281  df-of 6519  df-om 6681  df-1st 6781  df-2nd 6782  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-1o 7169  df-oadd 7173  df-er 7350  df-map 7461  df-pm 7462  df-en 7557  df-dom 7558  df-sdom 7559  df-fin 7560  df-sup 7943  df-inf 7944  df-oi 8012  df-card 8360  df-pnf 9664  df-mnf 9665  df-xr 9666  df-ltxr 9667  df-le 9668  df-sub 9849  df-neg 9850  df-div 10259  df-nn 10599  df-2 10657  df-3 10658  df-n0 10860  df-z 10928  df-uz 11150  df-rp 11293  df-fz 11776  df-fzo 11909  df-fl 12022  df-seq 12208  df-exp 12267  df-hash 12510  df-cj 13173  df-re 13174  df-im 13175  df-sqrt 13309  df-abs 13310  df-clim 13563  df-rlim 13564  df-sum 13764  df-0p 22640  df-ply 23154  df-coe 23156  df-dgr 23157  df-aa 23280
This theorem is referenced by:  aannenlem3  23298
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