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Theorem aannenlem2 22590
Description: Lemma for aannen 22592. (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 997 . . . . . . . . . 10  |-  ( ( h  e.  ( (Poly `  ZZ )  \  {
0p } )  /\  ( h `  g )  =  0  /\  g  e.  CC )  ->  g  e.  CC )
2 eldifi 3608 . . . . . . . . . . . . . 14  |-  ( h  e.  ( (Poly `  ZZ )  \  { 0p } )  ->  h  e.  (Poly `  ZZ ) )
32adantr 465 . . . . . . . . . . . . 13  |-  ( ( h  e.  ( (Poly `  ZZ )  \  {
0p } )  /\  g  e.  CC )  ->  h  e.  (Poly `  ZZ ) )
433adant2 1014 . . . . . . . . . . . 12  |-  ( ( h  e.  ( (Poly `  ZZ )  \  {
0p } )  /\  ( h `  g )  =  0  /\  g  e.  CC )  ->  h  e.  (Poly `  ZZ ) )
5 eldifsni 4137 . . . . . . . . . . . . . . 15  |-  ( h  e.  ( (Poly `  ZZ )  \  { 0p } )  ->  h  =/=  0p )
65adantr 465 . . . . . . . . . . . . . 14  |-  ( ( h  e.  ( (Poly `  ZZ )  \  {
0p } )  /\  g  e.  CC )  ->  h  =/=  0p )
7 0nn0 10811 . . . . . . . . . . . . . . . . . 18  |-  0  e.  NN0
8 dgrcl 22496 . . . . . . . . . . . . . . . . . . 19  |-  ( h  e.  (Poly `  ZZ )  ->  (deg `  h
)  e.  NN0 )
93, 8syl 16 . . . . . . . . . . . . . . . . . 18  |-  ( ( h  e.  ( (Poly `  ZZ )  \  {
0p } )  /\  g  e.  CC )  ->  (deg `  h
)  e.  NN0 )
10 prssi 4167 . . . . . . . . . . . . . . . . . 18  |-  ( ( 0  e.  NN0  /\  (deg `  h )  e. 
NN0 )  ->  { 0 ,  (deg `  h
) }  C_  NN0 )
117, 9, 10sylancr 663 . . . . . . . . . . . . . . . . 17  |-  ( ( h  e.  ( (Poly `  ZZ )  \  {
0p } )  /\  g  e.  CC )  ->  { 0 ,  (deg `  h ) }  C_  NN0 )
12 ssrab2 3567 . . . . . . . . . . . . . . . . . 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 3662 . . . . . . . . . . . . . . . 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 10800 . . . . . . . . . . . . . . . . 17  |-  NN0  C_  RR
16 ressxr 9635 . . . . . . . . . . . . . . . . 17  |-  RR  C_  RR*
1715, 16sstri 3495 . . . . . . . . . . . . . . . 16  |-  NN0  C_  RR*
1814, 17syl6ss 3498 . . . . . . . . . . . . . . 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 5862 . . . . . . . . . . . . . . . . 17  |-  (deg `  h )  e.  _V
2019prid2 4120 . . . . . . . . . . . . . . . 16  |-  (deg `  h )  e.  {
0 ,  (deg `  h ) }
21 elun1 3653 . . . . . . . . . . . . . . . 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 11520 . . . . . . . . . . . . . . 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 662 . . . . . . . . . . . . . 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 465 . . . . . . . . . . . . . . . 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 5852 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( (coeff `  h ) `  e )  =  0  ->  ( abs `  (
(coeff `  h ) `  e ) )  =  ( abs `  0
) )
27 abs0 13092 . . . . . . . . . . . . . . . . . . . 20  |-  ( abs `  0 )  =  0
2826, 27syl6eq 2498 . . . . . . . . . . . . . . . . . . 19  |-  ( ( (coeff `  h ) `  e )  =  0  ->  ( abs `  (
(coeff `  h ) `  e ) )  =  0 )
29 c0ex 9588 . . . . . . . . . . . . . . . . . . . . 21  |-  0  e.  _V
3029prid1 4119 . . . . . . . . . . . . . . . . . . . 20  |-  0  e.  { 0 ,  (deg
`  h ) }
31 elun1 3653 . . . . . . . . . . . . . . . . . . . 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 2537 . . . . . . . . . . . . . . . . . 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 466 . . . . . . . . . . . . . . . . 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 10876 . . . . . . . . . . . . . . . . . . . . . . 23  |-  0  e.  ZZ
36 eqid 2441 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  (coeff `  h )  =  (coeff `  h )
3736coef2 22494 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( h  e.  (Poly `  ZZ )  /\  0  e.  ZZ )  ->  (coeff `  h ) : NN0 --> ZZ )
383, 35, 37sylancl 662 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( h  e.  ( (Poly `  ZZ )  \  {
0p } )  /\  g  e.  CC )  ->  (coeff `  h
) : NN0 --> ZZ )
3938ffvelrnda 6012 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( h  e.  ( (Poly `  ZZ )  \  { 0p }
)  /\  g  e.  CC )  /\  e  e.  NN0 )  ->  (
(coeff `  h ) `  e )  e.  ZZ )
40 nn0abscl 13119 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( (coeff `  h ) `  e )  e.  ZZ  ->  ( abs `  (
(coeff `  h ) `  e ) )  e. 
NN0 )
4139, 40syl 16 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( h  e.  ( (Poly `  ZZ )  \  { 0p }
)  /\  g  e.  CC )  /\  e  e.  NN0 )  ->  ( abs `  ( (coeff `  h ) `  e
) )  e.  NN0 )
4241adantr 465 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( h  e.  ( (Poly `  ZZ )  \  { 0p } )  /\  g  e.  CC )  /\  e  e.  NN0 )  /\  (
(coeff `  h ) `  e )  =/=  0
)  ->  ( abs `  ( (coeff `  h
) `  e )
)  e.  NN0 )
43 simplr 754 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( h  e.  ( (Poly `  ZZ )  \  { 0p } )  /\  g  e.  CC )  /\  e  e.  NN0 )  /\  (
(coeff `  h ) `  e )  =/=  0
)  ->  e  e.  NN0 )
449ad2antrr 725 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( h  e.  ( (Poly `  ZZ )  \  { 0p } )  /\  g  e.  CC )  /\  e  e.  NN0 )  /\  (
(coeff `  h ) `  e )  =/=  0
)  ->  (deg `  h
)  e.  NN0 )
453ad2antrr 725 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( h  e.  ( (Poly `  ZZ )  \  { 0p } )  /\  g  e.  CC )  /\  e  e.  NN0 )  /\  (
(coeff `  h ) `  e )  =/=  0
)  ->  h  e.  (Poly `  ZZ ) )
46 simpr 461 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( h  e.  ( (Poly `  ZZ )  \  { 0p } )  /\  g  e.  CC )  /\  e  e.  NN0 )  /\  (
(coeff `  h ) `  e )  =/=  0
)  ->  ( (coeff `  h ) `  e
)  =/=  0 )
47 eqid 2441 . . . . . . . . . . . . . . . . . . . . . . 23  |-  (deg `  h )  =  (deg
`  h )
4836, 47dgrub 22497 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( h  e.  (Poly `  ZZ )  /\  e  e.  NN0  /\  ( (coeff `  h ) `  e
)  =/=  0 )  ->  e  <_  (deg `  h ) )
4945, 43, 46, 48syl3anc 1227 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( h  e.  ( (Poly `  ZZ )  \  { 0p } )  /\  g  e.  CC )  /\  e  e.  NN0 )  /\  (
(coeff `  h ) `  e )  =/=  0
)  ->  e  <_  (deg
`  h ) )
50 elfz2nn0 11772 . . . . . . . . . . . . . . . . . . . . 21  |-  ( e  e.  ( 0 ... (deg `  h )
)  <->  ( e  e. 
NN0  /\  (deg `  h
)  e.  NN0  /\  e  <_  (deg `  h
) ) )
5143, 44, 49, 50syl3anbrc 1179 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( h  e.  ( (Poly `  ZZ )  \  { 0p } )  /\  g  e.  CC )  /\  e  e.  NN0 )  /\  (
(coeff `  h ) `  e )  =/=  0
)  ->  e  e.  ( 0 ... (deg `  h ) ) )
52 eqid 2441 . . . . . . . . . . . . . . . . . . . 20  |-  ( abs `  ( (coeff `  h
) `  e )
)  =  ( abs `  ( (coeff `  h
) `  e )
)
53 fveq2 5852 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( i  =  e  ->  (
(coeff `  h ) `  i )  =  ( (coeff `  h ) `  e ) )
5453fveq2d 5856 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( i  =  e  ->  ( abs `  ( (coeff `  h ) `  i
) )  =  ( abs `  ( (coeff `  h ) `  e
) ) )
5554eqeq2d 2455 . . . . . . . . . . . . . . . . . . . . 21  |-  ( i  =  e  ->  (
( abs `  (
(coeff `  h ) `  e ) )  =  ( abs `  (
(coeff `  h ) `  i ) )  <->  ( abs `  ( (coeff `  h
) `  e )
)  =  ( abs `  ( (coeff `  h
) `  e )
) ) )
5655rspcev 3194 . . . . . . . . . . . . . . . . . . . 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 662 . . . . . . . . . . . . . . . . . . 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 2445 . . . . . . . . . . . . . . . . . . . . 21  |-  ( g  =  ( abs `  (
(coeff `  h ) `  e ) )  -> 
( g  =  ( abs `  ( (coeff `  h ) `  i
) )  <->  ( abs `  ( (coeff `  h
) `  e )
)  =  ( abs `  ( (coeff `  h
) `  i )
) ) )
5958rexbidv 2952 . . . . . . . . . . . . . . . . . . . 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 3241 . . . . . . . . . . . . . . . . . . 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 664 . . . . . . . . . . . . . . . . . 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 3654 . . . . . . . . . . . . . . . . . 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 16 . . . . . . . . . . . . . . . . 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 2759 . . . . . . . . . . . . . . . 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 11520 . . . . . . . . . . . . . . . 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 661 . . . . . . . . . . . . . . 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 2855 . . . . . . . . . . . . . 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 1175 . . . . . . . . . . . . 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 1014 . . . . . . . . . . . 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 2722 . . . . . . . . . . . . . 14  |-  ( d  =  h  ->  (
d  =/=  0p  <-> 
h  =/=  0p ) )
71 fveq2 5852 . . . . . . . . . . . . . . 15  |-  ( d  =  h  ->  (deg `  d )  =  (deg
`  h ) )
7271breq1d 4443 . . . . . . . . . . . . . 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 5852 . . . . . . . . . . . . . . . . . 18  |-  ( d  =  h  ->  (coeff `  d )  =  (coeff `  h ) )
7473fveq1d 5854 . . . . . . . . . . . . . . . . 17  |-  ( d  =  h  ->  (
(coeff `  d ) `  e )  =  ( (coeff `  h ) `  e ) )
7574fveq2d 5856 . . . . . . . . . . . . . . . 16  |-  ( d  =  h  ->  ( abs `  ( (coeff `  d ) `  e
) )  =  ( abs `  ( (coeff `  h ) `  e
) ) )
7675breq1d 4443 . . . . . . . . . . . . . . 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 2880 . . . . . . . . . . . . . 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 1298 . . . . . . . . . . . . 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 3241 . . . . . . . . . . . 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 664 . . . . . . . . . . 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 996 . . . . . . . . . . 11  |-  ( ( h  e.  ( (Poly `  ZZ )  \  {
0p } )  /\  ( h `  g )  =  0  /\  g  e.  CC )  ->  ( h `  g )  =  0 )
82 fveq1 5851 . . . . . . . . . . . . 13  |-  ( c  =  h  ->  (
c `  g )  =  ( h `  g ) )
8382eqeq1d 2443 . . . . . . . . . . . 12  |-  ( c  =  h  ->  (
( c `  g
)  =  0  <->  (
h `  g )  =  0 ) )
8483rspcev 3194 . . . . . . . . . . 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 661 . . . . . . . . . 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 5852 . . . . . . . . . . . . 13  |-  ( b  =  g  ->  (
c `  b )  =  ( c `  g ) )
8786eqeq1d 2443 . . . . . . . . . . . 12  |-  ( b  =  g  ->  (
( c `  b
)  =  0  <->  (
c `  g )  =  0 ) )
8887rexbidv 2952 . . . . . . . . . . 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 3241 . . . . . . . . . 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 664 . . . . . . . . 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 7793 . . . . . . . . . . . . . . 15  |-  { 0 ,  (deg `  h
) }  e.  Fin
92 fzfi 12056 . . . . . . . . . . . . . . . . 17  |-  ( 0 ... (deg `  h
) )  e.  Fin
93 abrexfi 7818 . . . . . . . . . . . . . . . . 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 3569 . . . . . . . . . . . . . . . 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 7738 . . . . . . . . . . . . . . . 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 672 . . . . . . . . . . . . . . 15  |-  { g  e.  NN0  |  E. i  e.  ( 0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) }  e.  Fin
98 unfi 7785 . . . . . . . . . . . . . . 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 672 . . . . . . . . . . . . . 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 )
101 ne0i 3773 . . . . . . . . . . . . . . 15  |-  ( (deg
`  h )  e.  ( { 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 ) ) } )  =/=  (/) )
10222, 101ax-mp 5 . . . . . . . . . . . . . 14  |-  ( { 0 ,  (deg `  h ) }  u.  { g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } )  =/=  (/)
103102a1i 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 ) ) } )  =/=  (/) )
104 xrltso 11351 . . . . . . . . . . . . . 14  |-  <  Or  RR*
105 fisupcl 7925 . . . . . . . . . . . . . 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 ) ) } ) )
106104, 105mpan 670 . . . . . . . . . . . . 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 ) ) } ) )
107100, 103, 18, 106syl3anc 1227 . . . . . . . . . . . 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 ) ) } ) )
10814, 107sseldd 3487 . . . . . . . . . . 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 )
1091083adant2 1014 . . . . . . . . . 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 )
110 eqidd 2442 . . . . . . . . . 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 } )
111 breq2 4437 . . . . . . . . . . . . . . . 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* ,  <  ) ) )
112 breq2 4437 . . . . . . . . . . . . . . . . 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* ,  <  ) ) )
113112ralbidv 2880 . . . . . . . . . . . . . . . 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* ,  <  ) ) )
114111, 1133anbi23d 1301 . . . . . . . . . . . . . . 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* ,  <  ) ) ) )
115114rabbidv 3085 . . . . . . . . . . . . . 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* ,  <  ) ) } )
116115rexeqdv 3045 . . . . . . . . . . . . 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 ) )
117116rabbidv 3085 . . . . . . . . . . . 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 } )
118117eqeq2d 2455 . . . . . . . . . . 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 } ) )
119118rspcev 3194 . . . . . . . . . 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 } )
120109, 110, 119syl2anc 661 . . . . . . . . 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 } )
121 cnex 9571 . . . . . . . . . . 11  |-  CC  e.  _V
122121rabex 4584 . . . . . . . . . 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
123 eleq2 2514 . . . . . . . . . . 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 } ) )
124 eqeq1 2445 . . . . . . . . . . . 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 } ) )
125124rexbidv 2952 . . . . . . . . . . 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 } ) )
126123, 125anbi12d 710 . . . . . . . . . 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 } ) ) )
127122, 126spcev 3185 . . . . . . . . 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 } ) )
12890, 120, 127syl2anc 661 . . . . . . . 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 } ) )
1291283exp 1194 . . . . . . 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 } ) ) ) )
130129rexlimiv 2927 . . . . . 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 } ) ) )
131130impcom 430 . . . . 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 } ) )
132 eleq2 2514 . . . . . . . . 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 } ) )
13387rexbidv 2952 . . . . . . . . . . 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 ) )
134133elrab 3241 . . . . . . . . . 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 ) )
135 simp1 995 . . . . . . . . . . . . . . 15  |-  ( ( h  =/=  0p  /\  (deg `  h
)  <_  a  /\  A. e  e.  NN0  ( abs `  ( (coeff `  h ) `  e
) )  <_  a
)  ->  h  =/=  0p )
136135anim2i 569 . . . . . . . . . . . . . 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 ) )
13771breq1d 4443 . . . . . . . . . . . . . . . 16  |-  ( d  =  h  ->  (
(deg `  d )  <_  a  <->  (deg `  h )  <_  a ) )
13875breq1d 4443 . . . . . . . . . . . . . . . . 17  |-  ( d  =  h  ->  (
( abs `  (
(coeff `  d ) `  e ) )  <_ 
a  <->  ( abs `  (
(coeff `  h ) `  e ) )  <_ 
a ) )
139138ralbidv 2880 . . . . . . . . . . . . . . . 16  |-  ( d  =  h  ->  ( A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  a  <->  A. e  e.  NN0  ( abs `  ( (coeff `  h ) `  e
) )  <_  a
) )
14070, 137, 1393anbi123d 1298 . . . . . . . . . . . . . . 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
) ) )
141140elrab 3241 . . . . . . . . . . . . . 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
) ) )
142 eldifsn 4136 . . . . . . . . . . . . . 14  |-  ( h  e.  ( (Poly `  ZZ )  \  { 0p } )  <->  ( h  e.  (Poly `  ZZ )  /\  h  =/=  0p ) )
143136, 141, 1423imtr4i 266 . . . . . . . . . . . . 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 } ) )
144143ssriv 3490 . . . . . . . . . . . 12  |-  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0p  /\  (deg `  d )  <_  a  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  a
) }  C_  (
(Poly `  ZZ )  \  { 0p }
)
145 ssrexv 3547 . . . . . . . . . . . . 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 ) )
14683cbvrexv 3069 . . . . . . . . . . . . 13  |-  ( E. c  e.  ( (Poly `  ZZ )  \  {
0p } ) ( c `  g
)  =  0  <->  E. h  e.  ( (Poly `  ZZ )  \  {
0p } ) ( h `  g
)  =  0 )
147145, 146syl6ib 226 . . . . . . . . . . . 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 ) )
148144, 147ax-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 )
149148anim2i 569 . . . . . . . . . 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 ) )
150134, 149sylbi 195 . . . . . . . . 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 ) )
151132, 150syl6bi 228 . . . . . . . 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 ) ) )
152151rexlimivw 2930 . . . . . . 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 ) ) )
153152impcom 430 . . . . . 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 ) )
154153exlimiv 1707 . . . . 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 ) )
155131, 154impbii 188 . . . 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 } ) )
156 elaa 22577 . . . 4  |-  ( g  e.  AA  <->  ( g  e.  CC  /\  E. h  e.  ( (Poly `  ZZ )  \  { 0p } ) ( h `
 g )  =  0 ) )
157 eluniab 4241 . . . 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 } ) )
158155, 156, 1573bitr4i 277 . . 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 } } )
159158eqriv 2437 . 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 } }
160 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 } )
161160rnmpt 5234 . . 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 } }
162161unieqi 4239 . 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 } }
163159, 162eqtr4i 2473 1  |-  AA  =  U. ran  H
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 972    = wceq 1381   E.wex 1597    e. wcel 1802   {cab 2426    =/= wne 2636   A.wral 2791   E.wrex 2792   {crab 2795    \ cdif 3455    u. cun 3456    C_ wss 3458   (/)c0 3767   {csn 4010   {cpr 4012   U.cuni 4230   class class class wbr 4433    |-> cmpt 4491    Or wor 4785   ran crn 4986   -->wf 5570   ` cfv 5574  (class class class)co 6277   Fincfn 7514   supcsup 7898   CCcc 9488   RRcr 9489   0cc0 9490   RR*cxr 9625    < clt 9626    <_ cle 9627   NN0cn0 10796   ZZcz 10865   ...cfz 11676   abscabs 13041   0pc0p 21942  Polycply 22447  coeffccoe 22449  degcdgr 22450   AAcaa 22575
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-inf2 8056  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567  ax-pre-sup 9568  ax-addf 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-fal 1387  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-int 4268  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-se 4825  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-isom 5583  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-of 6521  df-om 6682  df-1st 6781  df-2nd 6782  df-recs 7040  df-rdg 7074  df-1o 7128  df-oadd 7132  df-er 7309  df-map 7420  df-pm 7421  df-en 7515  df-dom 7516  df-sdom 7517  df-fin 7518  df-sup 7899  df-oi 7933  df-card 8318  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-div 10208  df-nn 10538  df-2 10595  df-3 10596  df-n0 10797  df-z 10866  df-uz 11086  df-rp 11225  df-fz 11677  df-fzo 11799  df-fl 11903  df-seq 12082  df-exp 12141  df-hash 12380  df-cj 12906  df-re 12907  df-im 12908  df-sqrt 13042  df-abs 13043  df-clim 13285  df-rlim 13286  df-sum 13483  df-0p 21943  df-ply 22451  df-coe 22453  df-dgr 22454  df-aa 22576
This theorem is referenced by:  aannenlem3  22591
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