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Theorem aannenlem2 23283
Description: Lemma for aannen 23285. (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 1007 . . . . . . . . . 10  |-  ( ( h  e.  ( (Poly `  ZZ )  \  {
0p } )  /\  ( h `  g )  =  0  /\  g  e.  CC )  ->  g  e.  CC )
2 eldifi 3587 . . . . . . . . . . . . . 14  |-  ( h  e.  ( (Poly `  ZZ )  \  { 0p } )  ->  h  e.  (Poly `  ZZ ) )
32adantr 466 . . . . . . . . . . . . 13  |-  ( ( h  e.  ( (Poly `  ZZ )  \  {
0p } )  /\  g  e.  CC )  ->  h  e.  (Poly `  ZZ ) )
433adant2 1024 . . . . . . . . . . . 12  |-  ( ( h  e.  ( (Poly `  ZZ )  \  {
0p } )  /\  ( h `  g )  =  0  /\  g  e.  CC )  ->  h  e.  (Poly `  ZZ ) )
5 eldifsni 4126 . . . . . . . . . . . . . . 15  |-  ( h  e.  ( (Poly `  ZZ )  \  { 0p } )  ->  h  =/=  0p )
65adantr 466 . . . . . . . . . . . . . 14  |-  ( ( h  e.  ( (Poly `  ZZ )  \  {
0p } )  /\  g  e.  CC )  ->  h  =/=  0p )
7 0nn0 10891 . . . . . . . . . . . . . . . . . 18  |-  0  e.  NN0
8 dgrcl 23185 . . . . . . . . . . . . . . . . . . 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 4156 . . . . . . . . . . . . . . . . . 18  |-  ( ( 0  e.  NN0  /\  (deg `  h )  e. 
NN0 )  ->  { 0 ,  (deg `  h
) }  C_  NN0 )
117, 9, 10sylancr 667 . . . . . . . . . . . . . . . . 17  |-  ( ( h  e.  ( (Poly `  ZZ )  \  {
0p } )  /\  g  e.  CC )  ->  { 0 ,  (deg `  h ) }  C_  NN0 )
12 ssrab2 3546 . . . . . . . . . . . . . . . . . 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 3642 . . . . . . . . . . . . . . . 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 10880 . . . . . . . . . . . . . . . . 17  |-  NN0  C_  RR
16 ressxr 9691 . . . . . . . . . . . . . . . . 17  |-  RR  C_  RR*
1715, 16sstri 3473 . . . . . . . . . . . . . . . 16  |-  NN0  C_  RR*
1814, 17syl6ss 3476 . . . . . . . . . . . . . . 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 5891 . . . . . . . . . . . . . . . . 17  |-  (deg `  h )  e.  _V
2019prid2 4109 . . . . . . . . . . . . . . . 16  |-  (deg `  h )  e.  {
0 ,  (deg `  h ) }
21 elun1 3633 . . . . . . . . . . . . . . . 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 11617 . . . . . . . . . . . . . . 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 666 . . . . . . . . . . . . . 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 466 . . . . . . . . . . . . . . . 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 5881 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( (coeff `  h ) `  e )  =  0  ->  ( abs `  (
(coeff `  h ) `  e ) )  =  ( abs `  0
) )
27 abs0 13348 . . . . . . . . . . . . . . . . . . . 20  |-  ( abs `  0 )  =  0
2826, 27syl6eq 2479 . . . . . . . . . . . . . . . . . . 19  |-  ( ( (coeff `  h ) `  e )  =  0  ->  ( abs `  (
(coeff `  h ) `  e ) )  =  0 )
29 c0ex 9644 . . . . . . . . . . . . . . . . . . . . 21  |-  0  e.  _V
3029prid1 4108 . . . . . . . . . . . . . . . . . . . 20  |-  0  e.  { 0 ,  (deg
`  h ) }
31 elun1 3633 . . . . . . . . . . . . . . . . . . . 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 2515 . . . . . . . . . . . . . . . . . 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 467 . . . . . . . . . . . . . . . . 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 10955 . . . . . . . . . . . . . . . . . . . . . . 23  |-  0  e.  ZZ
36 eqid 2422 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  (coeff `  h )  =  (coeff `  h )
3736coef2 23183 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( h  e.  (Poly `  ZZ )  /\  0  e.  ZZ )  ->  (coeff `  h ) : NN0 --> ZZ )
383, 35, 37sylancl 666 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( h  e.  ( (Poly `  ZZ )  \  {
0p } )  /\  g  e.  CC )  ->  (coeff `  h
) : NN0 --> ZZ )
3938ffvelrnda 6037 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( h  e.  ( (Poly `  ZZ )  \  { 0p }
)  /\  g  e.  CC )  /\  e  e.  NN0 )  ->  (
(coeff `  h ) `  e )  e.  ZZ )
40 nn0abscl 13375 . . . . . . . . . . . . . . . . . . . . 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 466 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( h  e.  ( (Poly `  ZZ )  \  { 0p } )  /\  g  e.  CC )  /\  e  e.  NN0 )  /\  (
(coeff `  h ) `  e )  =/=  0
)  ->  ( abs `  ( (coeff `  h
) `  e )
)  e.  NN0 )
43 simplr 760 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( h  e.  ( (Poly `  ZZ )  \  { 0p } )  /\  g  e.  CC )  /\  e  e.  NN0 )  /\  (
(coeff `  h ) `  e )  =/=  0
)  ->  e  e.  NN0 )
449ad2antrr 730 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( h  e.  ( (Poly `  ZZ )  \  { 0p } )  /\  g  e.  CC )  /\  e  e.  NN0 )  /\  (
(coeff `  h ) `  e )  =/=  0
)  ->  (deg `  h
)  e.  NN0 )
453ad2antrr 730 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( h  e.  ( (Poly `  ZZ )  \  { 0p } )  /\  g  e.  CC )  /\  e  e.  NN0 )  /\  (
(coeff `  h ) `  e )  =/=  0
)  ->  h  e.  (Poly `  ZZ ) )
46 simpr 462 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( h  e.  ( (Poly `  ZZ )  \  { 0p } )  /\  g  e.  CC )  /\  e  e.  NN0 )  /\  (
(coeff `  h ) `  e )  =/=  0
)  ->  ( (coeff `  h ) `  e
)  =/=  0 )
47 eqid 2422 . . . . . . . . . . . . . . . . . . . . . . 23  |-  (deg `  h )  =  (deg
`  h )
4836, 47dgrub 23186 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( h  e.  (Poly `  ZZ )  /\  e  e.  NN0  /\  ( (coeff `  h ) `  e
)  =/=  0 )  ->  e  <_  (deg `  h ) )
4945, 43, 46, 48syl3anc 1264 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( h  e.  ( (Poly `  ZZ )  \  { 0p } )  /\  g  e.  CC )  /\  e  e.  NN0 )  /\  (
(coeff `  h ) `  e )  =/=  0
)  ->  e  <_  (deg
`  h ) )
50 elfz2nn0 11892 . . . . . . . . . . . . . . . . . . . . 21  |-  ( e  e.  ( 0 ... (deg `  h )
)  <->  ( e  e. 
NN0  /\  (deg `  h
)  e.  NN0  /\  e  <_  (deg `  h
) ) )
5143, 44, 49, 50syl3anbrc 1189 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( h  e.  ( (Poly `  ZZ )  \  { 0p } )  /\  g  e.  CC )  /\  e  e.  NN0 )  /\  (
(coeff `  h ) `  e )  =/=  0
)  ->  e  e.  ( 0 ... (deg `  h ) ) )
52 eqid 2422 . . . . . . . . . . . . . . . . . . . 20  |-  ( abs `  ( (coeff `  h
) `  e )
)  =  ( abs `  ( (coeff `  h
) `  e )
)
53 fveq2 5881 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( i  =  e  ->  (
(coeff `  h ) `  i )  =  ( (coeff `  h ) `  e ) )
5453fveq2d 5885 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( i  =  e  ->  ( abs `  ( (coeff `  h ) `  i
) )  =  ( abs `  ( (coeff `  h ) `  e
) ) )
5554eqeq2d 2436 . . . . . . . . . . . . . . . . . . . . 21  |-  ( i  =  e  ->  (
( abs `  (
(coeff `  h ) `  e ) )  =  ( abs `  (
(coeff `  h ) `  i ) )  <->  ( abs `  ( (coeff `  h
) `  e )
)  =  ( abs `  ( (coeff `  h
) `  e )
) ) )
5655rspcev 3182 . . . . . . . . . . . . . . . . . . . 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 666 . . . . . . . . . . . . . . . . . . 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 2426 . . . . . . . . . . . . . . . . . . . . 21  |-  ( g  =  ( abs `  (
(coeff `  h ) `  e ) )  -> 
( g  =  ( abs `  ( (coeff `  h ) `  i
) )  <->  ( abs `  ( (coeff `  h
) `  e )
)  =  ( abs `  ( (coeff `  h
) `  i )
) ) )
5958rexbidv 2936 . . . . . . . . . . . . . . . . . . . 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 3228 . . . . . . . . . . . . . . . . . . 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 668 . . . . . . . . . . . . . . . . . 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 3634 . . . . . . . . . . . . . . . . . 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 2738 . . . . . . . . . . . . . . . 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 11617 . . . . . . . . . . . . . . . 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 665 . . . . . . . . . . . . . . 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 2836 . . . . . . . . . . . . . 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 1185 . . . . . . . . . . . . 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 1024 . . . . . . . . . . . 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 2701 . . . . . . . . . . . . . 14  |-  ( d  =  h  ->  (
d  =/=  0p  <-> 
h  =/=  0p ) )
71 fveq2 5881 . . . . . . . . . . . . . . 15  |-  ( d  =  h  ->  (deg `  d )  =  (deg
`  h ) )
7271breq1d 4433 . . . . . . . . . . . . . 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 5881 . . . . . . . . . . . . . . . . . 18  |-  ( d  =  h  ->  (coeff `  d )  =  (coeff `  h ) )
7473fveq1d 5883 . . . . . . . . . . . . . . . . 17  |-  ( d  =  h  ->  (
(coeff `  d ) `  e )  =  ( (coeff `  h ) `  e ) )
7574fveq2d 5885 . . . . . . . . . . . . . . . 16  |-  ( d  =  h  ->  ( abs `  ( (coeff `  d ) `  e
) )  =  ( abs `  ( (coeff `  h ) `  e
) ) )
7675breq1d 4433 . . . . . . . . . . . . . . 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 2861 . . . . . . . . . . . . . 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 1335 . . . . . . . . . . . . 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 3228 . . . . . . . . . . . 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 668 . . . . . . . . . . 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 1006 . . . . . . . . . . 11  |-  ( ( h  e.  ( (Poly `  ZZ )  \  {
0p } )  /\  ( h `  g )  =  0  /\  g  e.  CC )  ->  ( h `  g )  =  0 )
82 fveq1 5880 . . . . . . . . . . . . 13  |-  ( c  =  h  ->  (
c `  g )  =  ( h `  g ) )
8382eqeq1d 2424 . . . . . . . . . . . 12  |-  ( c  =  h  ->  (
( c `  g
)  =  0  <->  (
h `  g )  =  0 ) )
8483rspcev 3182 . . . . . . . . . . 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 665 . . . . . . . . . 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 5881 . . . . . . . . . . . . 13  |-  ( b  =  g  ->  (
c `  b )  =  ( c `  g ) )
8786eqeq1d 2424 . . . . . . . . . . . 12  |-  ( b  =  g  ->  (
( c `  b
)  =  0  <->  (
c `  g )  =  0 ) )
8887rexbidv 2936 . . . . . . . . . . 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 3228 . . . . . . . . . 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 668 . . . . . . . . 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 7855 . . . . . . . . . . . . . . 15  |-  { 0 ,  (deg `  h
) }  e.  Fin
92 fzfi 12191 . . . . . . . . . . . . . . . . 17  |-  ( 0 ... (deg `  h
) )  e.  Fin
93 abrexfi 7883 . . . . . . . . . . . . . . . . 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 3548 . . . . . . . . . . . . . . . 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 7801 . . . . . . . . . . . . . . . 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 676 . . . . . . . . . . . . . . 15  |-  { g  e.  NN0  |  E. i  e.  ( 0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) }  e.  Fin
98 unfi 7847 . . . . . . . . . . . . . . 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 676 . . . . . . . . . . . . . 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 3768 . . . . . . . . . . . . . 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 11447 . . . . . . . . . . . . . 14  |-  <  Or  RR*
104 fisupcl 7994 . . . . . . . . . . . . . 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 674 . . . . . . . . . . . . 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 1264 . . . . . . . . . . . 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 3465 . . . . . . . . . . 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 1024 . . . . . . . . . 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 2423 . . . . . . . . . 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 4427 . . . . . . . . . . . . . . . 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 4427 . . . . . . . . . . . . . . . . 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 2861 . . . . . . . . . . . . . . . 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 1338 . . . . . . . . . . . . . . 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 3071 . . . . . . . . . . . . . 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 3029 . . . . . . . . . . . . 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 3071 . . . . . . . . . . . 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 2436 . . . . . . . . . . 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 3182 . . . . . . . . . 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 665 . . . . . . . . 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 9627 . . . . . . . . . . 11  |-  CC  e.  _V
121120rabex 4575 . . . . . . . . . 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 2496 . . . . . . . . . . 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 2426 . . . . . . . . . . . 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 2936 . . . . . . . . . . 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 715 . . . . . . . . . 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 3173 . . . . . . . . 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 665 . . . . . . . 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 1204 . . . . . . 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 2908 . . . . . 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 431 . . . . 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 2496 . . . . . . . . 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 2936 . . . . . . . . . . 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 3228 . . . . . . . . . 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 1005 . . . . . . . . . . . . . . 15  |-  ( ( h  =/=  0p  /\  (deg `  h
)  <_  a  /\  A. e  e.  NN0  ( abs `  ( (coeff `  h ) `  e
) )  <_  a
)  ->  h  =/=  0p )
135134anim2i 571 . . . . . . . . . . . . . 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 4433 . . . . . . . . . . . . . . . 16  |-  ( d  =  h  ->  (
(deg `  d )  <_  a  <->  (deg `  h )  <_  a ) )
13775breq1d 4433 . . . . . . . . . . . . . . . . 17  |-  ( d  =  h  ->  (
( abs `  (
(coeff `  d ) `  e ) )  <_ 
a  <->  ( abs `  (
(coeff `  h ) `  e ) )  <_ 
a ) )
138137ralbidv 2861 . . . . . . . . . . . . . . . 16  |-  ( d  =  h  ->  ( A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  a  <->  A. e  e.  NN0  ( abs `  ( (coeff `  h ) `  e
) )  <_  a
) )
13970, 136, 1383anbi123d 1335 . . . . . . . . . . . . . . 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 3228 . . . . . . . . . . . . . 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 4125 . . . . . . . . . . . . . 14  |-  ( h  e.  ( (Poly `  ZZ )  \  { 0p } )  <->  ( h  e.  (Poly `  ZZ )  /\  h  =/=  0p ) )
142135, 140, 1413imtr4i 269 . . . . . . . . . . . . 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 3468 . . . . . . . . . . . 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 3526 . . . . . . . . . . . . 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 3055 . . . . . . . . . . . . 13  |-  ( E. c  e.  ( (Poly `  ZZ )  \  {
0p } ) ( c `  g
)  =  0  <->  E. h  e.  ( (Poly `  ZZ )  \  {
0p } ) ( h `  g
)  =  0 )
146144, 145syl6ib 229 . . . . . . . . . . . 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 571 . . . . . . . . . 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 198 . . . . . . . . 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 231 . . . . . . . 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 2911 . . . . . . 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 431 . . . . . 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 1770 . . . . 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 190 . . . 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 23267 . . . 4  |-  ( g  e.  AA  <->  ( g  e.  CC  /\  E. h  e.  ( (Poly `  ZZ )  \  { 0p } ) ( h `
 g )  =  0 ) )
156 eluniab 4230 . . . 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 280 . . 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 2418 . 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 5099 . . 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 4228 . 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 2454 1  |-  AA  =  U. ran  H
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437   E.wex 1657    e. wcel 1872   {cab 2407    =/= wne 2614   A.wral 2771   E.wrex 2772   {crab 2775    \ cdif 3433    u. cun 3434    C_ wss 3436   (/)c0 3761   {csn 3998   {cpr 4000   U.cuni 4219   class class class wbr 4423    |-> cmpt 4482    Or wor 4773   ran crn 4854   -->wf 5597   ` cfv 5601  (class class class)co 6305   Fincfn 7580   supcsup 7963   CCcc 9544   RRcr 9545   0cc0 9546   RR*cxr 9681    < clt 9682    <_ cle 9683   NN0cn0 10876   ZZcz 10944   ...cfz 11791   abscabs 13297   0pc0p 22625  Polycply 23136  coeffccoe 23138  degcdgr 23139   AAcaa 23265
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597  ax-inf2 8155  ax-cnex 9602  ax-resscn 9603  ax-1cn 9604  ax-icn 9605  ax-addcl 9606  ax-addrcl 9607  ax-mulcl 9608  ax-mulrcl 9609  ax-mulcom 9610  ax-addass 9611  ax-mulass 9612  ax-distr 9613  ax-i2m1 9614  ax-1ne0 9615  ax-1rid 9616  ax-rnegex 9617  ax-rrecex 9618  ax-cnre 9619  ax-pre-lttri 9620  ax-pre-lttrn 9621  ax-pre-ltadd 9622  ax-pre-mulgt0 9623  ax-pre-sup 9624  ax-addf 9625
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-int 4256  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-se 4813  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-of 6545  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7039  df-recs 7101  df-rdg 7139  df-1o 7193  df-oadd 7197  df-er 7374  df-map 7485  df-pm 7486  df-en 7581  df-dom 7582  df-sdom 7583  df-fin 7584  df-sup 7965  df-inf 7966  df-oi 8034  df-card 8381  df-pnf 9684  df-mnf 9685  df-xr 9686  df-ltxr 9687  df-le 9688  df-sub 9869  df-neg 9870  df-div 10277  df-nn 10617  df-2 10675  df-3 10676  df-n0 10877  df-z 10945  df-uz 11167  df-rp 11310  df-fz 11792  df-fzo 11923  df-fl 12034  df-seq 12220  df-exp 12279  df-hash 12522  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-clim 13551  df-rlim 13552  df-sum 13752  df-0p 22626  df-ply 23140  df-coe 23142  df-dgr 23143  df-aa 23266
This theorem is referenced by:  aannenlem3  23284
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