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Theorem aannenlem2 20246
Description: Lemma for aannen 20248. (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  =/=  0 p  /\  (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 959 . . . . . . . . . 10  |-  ( ( h  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  ( h `  g )  =  0  /\  g  e.  CC )  ->  g  e.  CC )
2 eldifi 3469 . . . . . . . . . . . . . 14  |-  ( h  e.  ( (Poly `  ZZ )  \  { 0 p } )  ->  h  e.  (Poly `  ZZ ) )
32adantr 452 . . . . . . . . . . . . 13  |-  ( ( h  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  g  e.  CC )  ->  h  e.  (Poly `  ZZ ) )
433adant2 976 . . . . . . . . . . . 12  |-  ( ( h  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  ( h `  g )  =  0  /\  g  e.  CC )  ->  h  e.  (Poly `  ZZ ) )
5 eldifsni 3928 . . . . . . . . . . . . . . 15  |-  ( h  e.  ( (Poly `  ZZ )  \  { 0 p } )  ->  h  =/=  0 p )
65adantr 452 . . . . . . . . . . . . . 14  |-  ( ( h  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  g  e.  CC )  ->  h  =/=  0 p )
7 0nn0 10236 . . . . . . . . . . . . . . . . . 18  |-  0  e.  NN0
8 dgrcl 20152 . . . . . . . . . . . . . . . . . . 19  |-  ( h  e.  (Poly `  ZZ )  ->  (deg `  h
)  e.  NN0 )
93, 8syl 16 . . . . . . . . . . . . . . . . . 18  |-  ( ( h  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  g  e.  CC )  ->  (deg `  h
)  e.  NN0 )
10 prssi 3954 . . . . . . . . . . . . . . . . . 18  |-  ( ( 0  e.  NN0  /\  (deg `  h )  e. 
NN0 )  ->  { 0 ,  (deg `  h
) }  C_  NN0 )
117, 9, 10sylancr 645 . . . . . . . . . . . . . . . . 17  |-  ( ( h  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  g  e.  CC )  ->  { 0 ,  (deg `  h ) }  C_  NN0 )
12 ssrab2 3428 . . . . . . . . . . . . . . . . . 18  |-  { g  e.  NN0  |  E. i  e.  ( 0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } 
C_  NN0
1312a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( ( h  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  g  e.  CC )  ->  { g  e. 
NN0  |  E. i  e.  ( 0 ... (deg `  h ) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } 
C_  NN0 )
1411, 13unssd 3523 . . . . . . . . . . . . . . . 16  |-  ( ( h  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  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 10225 . . . . . . . . . . . . . . . . 17  |-  NN0  C_  RR
16 ressxr 9129 . . . . . . . . . . . . . . . . 17  |-  RR  C_  RR*
1715, 16sstri 3357 . . . . . . . . . . . . . . . 16  |-  NN0  C_  RR*
1814, 17syl6ss 3360 . . . . . . . . . . . . . . 15  |-  ( ( h  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  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 5742 . . . . . . . . . . . . . . . . 17  |-  (deg `  h )  e.  _V
2019prid2 3913 . . . . . . . . . . . . . . . 16  |-  (deg `  h )  e.  {
0 ,  (deg `  h ) }
21 elun1 3514 . . . . . . . . . . . . . . . 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 8 . . . . . . . . . . . . . . 15  |-  (deg `  h )  e.  ( { 0 ,  (deg
`  h ) }  u.  { g  e. 
NN0  |  E. i  e.  ( 0 ... (deg `  h ) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } )
23 supxrub 10903 . . . . . . . . . . . . . . 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 644 . . . . . . . . . . . . . 14  |-  ( ( h  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  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 452 . . . . . . . . . . . . . . . 16  |-  ( ( ( h  e.  ( (Poly `  ZZ )  \  { 0 p }
)  /\  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 5728 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( (coeff `  h ) `  e )  =  0  ->  ( abs `  (
(coeff `  h ) `  e ) )  =  ( abs `  0
) )
27 abs0 12090 . . . . . . . . . . . . . . . . . . . 20  |-  ( abs `  0 )  =  0
2826, 27syl6eq 2484 . . . . . . . . . . . . . . . . . . 19  |-  ( ( (coeff `  h ) `  e )  =  0  ->  ( abs `  (
(coeff `  h ) `  e ) )  =  0 )
29 c0ex 9085 . . . . . . . . . . . . . . . . . . . . 21  |-  0  e.  _V
3029prid1 3912 . . . . . . . . . . . . . . . . . . . 20  |-  0  e.  { 0 ,  (deg
`  h ) }
31 elun1 3514 . . . . . . . . . . . . . . . . . . . 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 8 . . . . . . . . . . . . . . . . . . 19  |-  0  e.  ( { 0 ,  (deg `  h ) }  u.  { g  e.  NN0  |  E. i  e.  ( 0 ... (deg `  h ) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } )
3328, 32syl6eqel 2524 . . . . . . . . . . . . . . . . . 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 453 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( h  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  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 10293 . . . . . . . . . . . . . . . . . . . . . . 23  |-  0  e.  ZZ
36 eqid 2436 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  (coeff `  h )  =  (coeff `  h )
3736coef2 20150 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( h  e.  (Poly `  ZZ )  /\  0  e.  ZZ )  ->  (coeff `  h ) : NN0 --> ZZ )
383, 35, 37sylancl 644 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( h  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  g  e.  CC )  ->  (coeff `  h
) : NN0 --> ZZ )
3938ffvelrnda 5870 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( h  e.  ( (Poly `  ZZ )  \  { 0 p }
)  /\  g  e.  CC )  /\  e  e.  NN0 )  ->  (
(coeff `  h ) `  e )  e.  ZZ )
40 nn0abscl 12117 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( (coeff `  h ) `  e )  e.  ZZ  ->  ( abs `  (
(coeff `  h ) `  e ) )  e. 
NN0 )
4139, 40syl 16 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( h  e.  ( (Poly `  ZZ )  \  { 0 p }
)  /\  g  e.  CC )  /\  e  e.  NN0 )  ->  ( abs `  ( (coeff `  h ) `  e
) )  e.  NN0 )
4241adantr 452 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( h  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  g  e.  CC )  /\  e  e.  NN0 )  /\  (
(coeff `  h ) `  e )  =/=  0
)  ->  ( abs `  ( (coeff `  h
) `  e )
)  e.  NN0 )
43 simplr 732 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( h  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  g  e.  CC )  /\  e  e.  NN0 )  /\  (
(coeff `  h ) `  e )  =/=  0
)  ->  e  e.  NN0 )
449ad2antrr 707 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( h  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  g  e.  CC )  /\  e  e.  NN0 )  /\  (
(coeff `  h ) `  e )  =/=  0
)  ->  (deg `  h
)  e.  NN0 )
453ad2antrr 707 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( h  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  g  e.  CC )  /\  e  e.  NN0 )  /\  (
(coeff `  h ) `  e )  =/=  0
)  ->  h  e.  (Poly `  ZZ ) )
46 simpr 448 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( h  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  g  e.  CC )  /\  e  e.  NN0 )  /\  (
(coeff `  h ) `  e )  =/=  0
)  ->  ( (coeff `  h ) `  e
)  =/=  0 )
47 eqid 2436 . . . . . . . . . . . . . . . . . . . . . . 23  |-  (deg `  h )  =  (deg
`  h )
4836, 47dgrub 20153 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( h  e.  (Poly `  ZZ )  /\  e  e.  NN0  /\  ( (coeff `  h ) `  e
)  =/=  0 )  ->  e  <_  (deg `  h ) )
4945, 43, 46, 48syl3anc 1184 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( h  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  g  e.  CC )  /\  e  e.  NN0 )  /\  (
(coeff `  h ) `  e )  =/=  0
)  ->  e  <_  (deg
`  h ) )
50 elfz2nn0 11082 . . . . . . . . . . . . . . . . . . . . 21  |-  ( e  e.  ( 0 ... (deg `  h )
)  <->  ( e  e. 
NN0  /\  (deg `  h
)  e.  NN0  /\  e  <_  (deg `  h
) ) )
5143, 44, 49, 50syl3anbrc 1138 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( h  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  g  e.  CC )  /\  e  e.  NN0 )  /\  (
(coeff `  h ) `  e )  =/=  0
)  ->  e  e.  ( 0 ... (deg `  h ) ) )
52 eqid 2436 . . . . . . . . . . . . . . . . . . . 20  |-  ( abs `  ( (coeff `  h
) `  e )
)  =  ( abs `  ( (coeff `  h
) `  e )
)
53 fveq2 5728 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( i  =  e  ->  (
(coeff `  h ) `  i )  =  ( (coeff `  h ) `  e ) )
5453fveq2d 5732 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( i  =  e  ->  ( abs `  ( (coeff `  h ) `  i
) )  =  ( abs `  ( (coeff `  h ) `  e
) ) )
5554eqeq2d 2447 . . . . . . . . . . . . . . . . . . . . 21  |-  ( i  =  e  ->  (
( abs `  (
(coeff `  h ) `  e ) )  =  ( abs `  (
(coeff `  h ) `  i ) )  <->  ( abs `  ( (coeff `  h
) `  e )
)  =  ( abs `  ( (coeff `  h
) `  e )
) ) )
5655rspcev 3052 . . . . . . . . . . . . . . . . . . . 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 644 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( h  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  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 2442 . . . . . . . . . . . . . . . . . . . . 21  |-  ( g  =  ( abs `  (
(coeff `  h ) `  e ) )  -> 
( g  =  ( abs `  ( (coeff `  h ) `  i
) )  <->  ( abs `  ( (coeff `  h
) `  e )
)  =  ( abs `  ( (coeff `  h
) `  i )
) ) )
5958rexbidv 2726 . . . . . . . . . . . . . . . . . . . 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 3092 . . . . . . . . . . . . . . . . . . 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 646 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( h  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  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 3515 . . . . . . . . . . . . . . . . . 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 )  \  { 0 p } )  /\  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 2682 . . . . . . . . . . . . . . . 16  |-  ( ( ( h  e.  ( (Poly `  ZZ )  \  { 0 p }
)  /\  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 10903 . . . . . . . . . . . . . . . 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 643 . . . . . . . . . . . . . . 15  |-  ( ( ( h  e.  ( (Poly `  ZZ )  \  { 0 p }
)  /\  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 2789 . . . . . . . . . . . . . 14  |-  ( ( h  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  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 1134 . . . . . . . . . . . . 13  |-  ( ( h  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  g  e.  CC )  ->  ( h  =/=  0 p  /\  (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 976 . . . . . . . . . . . 12  |-  ( ( h  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  ( h `  g )  =  0  /\  g  e.  CC )  ->  ( h  =/=  0 p  /\  (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 2609 . . . . . . . . . . . . . 14  |-  ( d  =  h  ->  (
d  =/=  0 p  <-> 
h  =/=  0 p ) )
71 fveq2 5728 . . . . . . . . . . . . . . 15  |-  ( d  =  h  ->  (deg `  d )  =  (deg
`  h ) )
7271breq1d 4222 . . . . . . . . . . . . . 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 5728 . . . . . . . . . . . . . . . . . 18  |-  ( d  =  h  ->  (coeff `  d )  =  (coeff `  h ) )
7473fveq1d 5730 . . . . . . . . . . . . . . . . 17  |-  ( d  =  h  ->  (
(coeff `  d ) `  e )  =  ( (coeff `  h ) `  e ) )
7574fveq2d 5732 . . . . . . . . . . . . . . . 16  |-  ( d  =  h  ->  ( abs `  ( (coeff `  d ) `  e
) )  =  ( abs `  ( (coeff `  h ) `  e
) ) )
7675breq1d 4222 . . . . . . . . . . . . . . 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 2725 . . . . . . . . . . . . . 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 1254 . . . . . . . . . . . . 13  |-  ( d  =  h  ->  (
( d  =/=  0 p  /\  (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  =/=  0 p  /\  (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 3092 . . . . . . . . . . . 12  |-  ( h  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0 p  /\  (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  =/=  0 p  /\  (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 646 . . . . . . . . . . 11  |-  ( ( h  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  ( h `  g )  =  0  /\  g  e.  CC )  ->  h  e.  {
d  e.  (Poly `  ZZ )  |  (
d  =/=  0 p  /\  (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 958 . . . . . . . . . . 11  |-  ( ( h  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  ( h `  g )  =  0  /\  g  e.  CC )  ->  ( h `  g )  =  0 )
82 fveq1 5727 . . . . . . . . . . . . 13  |-  ( c  =  h  ->  (
c `  g )  =  ( h `  g ) )
8382eqeq1d 2444 . . . . . . . . . . . 12  |-  ( c  =  h  ->  (
( c `  g
)  =  0  <->  (
h `  g )  =  0 ) )
8483rspcev 3052 . . . . . . . . . . 11  |-  ( ( h  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0 p  /\  (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  =/=  0 p  /\  (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 643 . . . . . . . . . 10  |-  ( ( h  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  ( h `  g )  =  0  /\  g  e.  CC )  ->  E. c  e.  {
d  e.  (Poly `  ZZ )  |  (
d  =/=  0 p  /\  (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 5728 . . . . . . . . . . . . 13  |-  ( b  =  g  ->  (
c `  b )  =  ( c `  g ) )
8786eqeq1d 2444 . . . . . . . . . . . 12  |-  ( b  =  g  ->  (
( c `  b
)  =  0  <->  (
c `  g )  =  0 ) )
8887rexbidv 2726 . . . . . . . . . . 11  |-  ( b  =  g  ->  ( E. c  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0 p  /\  (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  =/=  0 p  /\  (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 3092 . . . . . . . . . 10  |-  ( g  e.  { b  e.  CC  |  E. c  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0 p  /\  (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  =/=  0 p  /\  (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 646 . . . . . . . . 9  |-  ( ( h  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  ( h `  g )  =  0  /\  g  e.  CC )  ->  g  e.  {
b  e.  CC  |  E. c  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0 p  /\  (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 7381 . . . . . . . . . . . . . . 15  |-  { 0 ,  (deg `  h
) }  e.  Fin
92 fzfi 11311 . . . . . . . . . . . . . . . . 17  |-  ( 0 ... (deg `  h
) )  e.  Fin
93 abrexfi 7407 . . . . . . . . . . . . . . . . 17  |-  ( ( 0 ... (deg `  h ) )  e. 
Fin  ->  { g  |  E. i  e.  ( 0 ... (deg `  h ) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) }  e.  Fin )
9492, 93ax-mp 8 . . . . . . . . . . . . . . . 16  |-  { g  |  E. i  e.  ( 0 ... (deg `  h ) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) }  e.  Fin
95 rabssab 3430 . . . . . . . . . . . . . . . 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 7329 . . . . . . . . . . . . . . . 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 654 . . . . . . . . . . . . . . 15  |-  { g  e.  NN0  |  E. i  e.  ( 0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) }  e.  Fin
98 unfi 7374 . . . . . . . . . . . . . . 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 654 . . . . . . . . . . . . . 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 )  \  {
0 p } )  /\  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 3634 . . . . . . . . . . . . . . 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 8 . . . . . . . . . . . . . 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 )  \  {
0 p } )  /\  g  e.  CC )  ->  ( { 0 ,  (deg `  h
) }  u.  {
g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } )  =/=  (/) )
104 xrltso 10734 . . . . . . . . . . . . . 14  |-  <  Or  RR*
105 fisupcl 7472 . . . . . . . . . . . . . 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 652 . . . . . . . . . . . . 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 1184 . . . . . . . . . . . 12  |-  ( ( h  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  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 3349 . . . . . . . . . . 11  |-  ( ( h  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  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 976 . . . . . . . . . 10  |-  ( ( h  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  ( 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 2437 . . . . . . . . . 10  |-  ( ( h  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  ( h `  g )  =  0  /\  g  e.  CC )  ->  { b  e.  CC  |  E. c  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0 p  /\  (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  =/=  0 p  /\  (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 4216 . . . . . . . . . . . . . . . 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 4216 . . . . . . . . . . . . . . . . 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 2725 . . . . . . . . . . . . . . . 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 1257 . . . . . . . . . . . . . . 15  |-  ( a  =  sup ( ( { 0 ,  (deg
`  h ) }  u.  { g  e. 
NN0  |  E. i  e.  ( 0 ... (deg `  h ) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } ) ,  RR* ,  <  )  ->  ( ( d  =/=  0 p  /\  (deg `  d )  <_ 
a  /\  A. e  e.  NN0  ( abs `  (
(coeff `  d ) `  e ) )  <_ 
a )  <->  ( d  =/=  0 p  /\  (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 2948 . . . . . . . . . . . . . 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  =/=  0 p  /\  (deg `  d )  <_  a  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  a
) }  =  {
d  e.  (Poly `  ZZ )  |  (
d  =/=  0 p  /\  (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 2911 . . . . . . . . . . . . 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  =/=  0 p  /\  (deg `  d
)  <_  a  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  a
) }  ( c `
 b )  =  0  <->  E. c  e.  {
d  e.  (Poly `  ZZ )  |  (
d  =/=  0 p  /\  (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 2948 . . . . . . . . . . . 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  =/=  0 p  /\  (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  =/=  0 p  /\  (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 2447 . . . . . . . . . . 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  =/=  0 p  /\  (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  =/=  0 p  /\  (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  =/=  0 p  /\  (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  =/=  0 p  /\  (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 3052 . . . . . . . . . 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  =/=  0 p  /\  (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  =/=  0 p  /\  (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  =/=  0 p  /\  (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  =/=  0 p  /\  (deg `  d )  <_  a  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  a
) }  ( c `
 b )  =  0 } )
120109, 110, 119syl2anc 643 . . . . . . . . 9  |-  ( ( h  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  ( h `  g )  =  0  /\  g  e.  CC )  ->  E. a  e.  NN0  { b  e.  CC  |  E. c  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0 p  /\  (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  =/=  0 p  /\  (deg `  d )  <_  a  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  a
) }  ( c `
 b )  =  0 } )
121 cnex 9071 . . . . . . . . . . 11  |-  CC  e.  _V
122121rabex 4354 . . . . . . . . . 10  |-  { b  e.  CC  |  E. c  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0 p  /\  (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 2497 . . . . . . . . . . 11  |-  ( f  =  { b  e.  CC  |  E. c  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0 p  /\  (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  =/=  0 p  /\  (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 2442 . . . . . . . . . . . 12  |-  ( f  =  { b  e.  CC  |  E. c  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0 p  /\  (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  =/=  0 p  /\  (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  =/=  0 p  /\  (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  =/=  0 p  /\  (deg `  d )  <_  a  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  a
) }  ( c `
 b )  =  0 } ) )
125124rexbidv 2726 . . . . . . . . . . 11  |-  ( f  =  { b  e.  CC  |  E. c  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0 p  /\  (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  =/=  0 p  /\  (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  =/=  0 p  /\  (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  =/=  0 p  /\  (deg `  d )  <_  a  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  a
) }  ( c `
 b )  =  0 } ) )
126123, 125anbi12d 692 . . . . . . . . . 10  |-  ( f  =  { b  e.  CC  |  E. c  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0 p  /\  (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  =/=  0 p  /\  (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  =/=  0 p  /\  (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  =/=  0 p  /\  (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  =/=  0 p  /\  (deg `  d )  <_  a  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  a
) }  ( c `
 b )  =  0 } ) ) )
127122, 126spcev 3043 . . . . . . . . 9  |-  ( ( g  e.  { b  e.  CC  |  E. c  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0 p  /\  (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  =/=  0 p  /\  (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  =/=  0 p  /\  (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  =/=  0 p  /\  (deg `  d )  <_  a  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  a
) }  ( c `
 b )  =  0 } ) )
12890, 120, 127syl2anc 643 . . . . . . . 8  |-  ( ( h  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  ( 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  =/=  0 p  /\  (deg `  d
)  <_  a  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  a
) }  ( c `
 b )  =  0 } ) )
1291283exp 1152 . . . . . . 7  |-  ( h  e.  ( (Poly `  ZZ )  \  { 0 p } )  -> 
( ( 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  =/=  0 p  /\  (deg `  d )  <_  a  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  a
) }  ( c `
 b )  =  0 } ) ) ) )
130129rexlimiv 2824 . . . . . 6  |-  ( E. h  e.  ( (Poly `  ZZ )  \  {
0 p } ) ( 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  =/=  0 p  /\  (deg `  d
)  <_  a  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  a
) }  ( c `
 b )  =  0 } ) ) )
131130impcom 420 . . . . 5  |-  ( ( g  e.  CC  /\  E. h  e.  ( (Poly `  ZZ )  \  {
0 p } ) ( h `  g
)  =  0 )  ->  E. f ( g  e.  f  /\  E. a  e.  NN0  f  =  { b  e.  CC  |  E. c  e.  {
d  e.  (Poly `  ZZ )  |  (
d  =/=  0 p  /\  (deg `  d
)  <_  a  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  a
) }  ( c `
 b )  =  0 } ) )
132 eleq2 2497 . . . . . . . . 9  |-  ( f  =  { b  e.  CC  |  E. c  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0 p  /\  (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  =/=  0 p  /\  (deg `  d )  <_  a  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  a
) }  ( c `
 b )  =  0 } ) )
13387rexbidv 2726 . . . . . . . . . . 11  |-  ( b  =  g  ->  ( E. c  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0 p  /\  (deg `  d )  <_  a  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  a
) }  ( c `
 b )  =  0  <->  E. c  e.  {
d  e.  (Poly `  ZZ )  |  (
d  =/=  0 p  /\  (deg `  d
)  <_  a  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  a
) }  ( c `
 g )  =  0 ) )
134133elrab 3092 . . . . . . . . . 10  |-  ( g  e.  { b  e.  CC  |  E. c  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0 p  /\  (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  =/=  0 p  /\  (deg `  d
)  <_  a  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  a
) }  ( c `
 g )  =  0 ) )
135 simp1 957 . . . . . . . . . . . . . . 15  |-  ( ( h  =/=  0 p  /\  (deg `  h
)  <_  a  /\  A. e  e.  NN0  ( abs `  ( (coeff `  h ) `  e
) )  <_  a
)  ->  h  =/=  0 p )
136135anim2i 553 . . . . . . . . . . . . . 14  |-  ( ( h  e.  (Poly `  ZZ )  /\  (
h  =/=  0 p  /\  (deg `  h
)  <_  a  /\  A. e  e.  NN0  ( abs `  ( (coeff `  h ) `  e
) )  <_  a
) )  ->  (
h  e.  (Poly `  ZZ )  /\  h  =/=  0 p ) )
13771breq1d 4222 . . . . . . . . . . . . . . . 16  |-  ( d  =  h  ->  (
(deg `  d )  <_  a  <->  (deg `  h )  <_  a ) )
13875breq1d 4222 . . . . . . . . . . . . . . . . 17  |-  ( d  =  h  ->  (
( abs `  (
(coeff `  d ) `  e ) )  <_ 
a  <->  ( abs `  (
(coeff `  h ) `  e ) )  <_ 
a ) )
139138ralbidv 2725 . . . . . . . . . . . . . . . 16  |-  ( d  =  h  ->  ( A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  a  <->  A. e  e.  NN0  ( abs `  ( (coeff `  h ) `  e
) )  <_  a
) )
14070, 137, 1393anbi123d 1254 . . . . . . . . . . . . . . 15  |-  ( d  =  h  ->  (
( d  =/=  0 p  /\  (deg `  d
)  <_  a  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  a
)  <->  ( h  =/=  0 p  /\  (deg `  h )  <_  a  /\  A. e  e.  NN0  ( abs `  ( (coeff `  h ) `  e
) )  <_  a
) ) )
141140elrab 3092 . . . . . . . . . . . . . 14  |-  ( h  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0 p  /\  (deg `  d )  <_  a  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  a
) }  <->  ( h  e.  (Poly `  ZZ )  /\  ( h  =/=  0 p  /\  (deg `  h
)  <_  a  /\  A. e  e.  NN0  ( abs `  ( (coeff `  h ) `  e
) )  <_  a
) ) )
142 eldifsn 3927 . . . . . . . . . . . . . 14  |-  ( h  e.  ( (Poly `  ZZ )  \  { 0 p } )  <->  ( h  e.  (Poly `  ZZ )  /\  h  =/=  0 p ) )
143136, 141, 1423imtr4i 258 . . . . . . . . . . . . 13  |-  ( h  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0 p  /\  (deg `  d )  <_  a  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  a
) }  ->  h  e.  ( (Poly `  ZZ )  \  { 0 p } ) )
144143ssriv 3352 . . . . . . . . . . . 12  |-  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0 p  /\  (deg `  d )  <_  a  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  a
) }  C_  (
(Poly `  ZZ )  \  { 0 p }
)
145 ssrexv 3408 . . . . . . . . . . . . 13  |-  ( { d  e.  (Poly `  ZZ )  |  (
d  =/=  0 p  /\  (deg `  d
)  <_  a  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  a
) }  C_  (
(Poly `  ZZ )  \  { 0 p }
)  ->  ( E. c  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0 p  /\  (deg `  d )  <_  a  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  a
) }  ( c `
 g )  =  0  ->  E. c  e.  ( (Poly `  ZZ )  \  { 0 p } ) ( c `
 g )  =  0 ) )
14683cbvrexv 2933 . . . . . . . . . . . . 13  |-  ( E. c  e.  ( (Poly `  ZZ )  \  {
0 p } ) ( c `  g
)  =  0  <->  E. h  e.  ( (Poly `  ZZ )  \  {
0 p } ) ( h `  g
)  =  0 )
147145, 146syl6ib 218 . . . . . . . . . . . 12  |-  ( { d  e.  (Poly `  ZZ )  |  (
d  =/=  0 p  /\  (deg `  d
)  <_  a  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  a
) }  C_  (
(Poly `  ZZ )  \  { 0 p }
)  ->  ( E. c  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0 p  /\  (deg `  d )  <_  a  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  a
) }  ( c `
 g )  =  0  ->  E. h  e.  ( (Poly `  ZZ )  \  { 0 p } ) ( h `
 g )  =  0 ) )
148144, 147ax-mp 8 . . . . . . . . . . 11  |-  ( E. c  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0 p  /\  (deg `  d )  <_  a  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  a
) }  ( c `
 g )  =  0  ->  E. h  e.  ( (Poly `  ZZ )  \  { 0 p } ) ( h `
 g )  =  0 )
149148anim2i 553 . . . . . . . . . 10  |-  ( ( g  e.  CC  /\  E. c  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0 p  /\  (deg `  d )  <_  a  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  a
) }  ( c `
 g )  =  0 )  ->  (
g  e.  CC  /\  E. h  e.  ( (Poly `  ZZ )  \  {
0 p } ) ( h `  g
)  =  0 ) )
150134, 149sylbi 188 . . . . . . . . 9  |-  ( g  e.  { b  e.  CC  |  E. c  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0 p  /\  (deg `  d
)  <_  a  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  a
) }  ( c `
 b )  =  0 }  ->  (
g  e.  CC  /\  E. h  e.  ( (Poly `  ZZ )  \  {
0 p } ) ( h `  g
)  =  0 ) )
151132, 150syl6bi 220 . . . . . . . 8  |-  ( f  =  { b  e.  CC  |  E. c  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0 p  /\  (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 )  \  { 0 p }
) ( h `  g )  =  0 ) ) )
152151rexlimivw 2826 . . . . . . 7  |-  ( E. a  e.  NN0  f  =  { b  e.  CC  |  E. c  e.  {
d  e.  (Poly `  ZZ )  |  (
d  =/=  0 p  /\  (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 )  \  { 0 p }
) ( h `  g )  =  0 ) ) )
153152impcom 420 . . . . . 6  |-  ( ( g  e.  f  /\  E. a  e.  NN0  f  =  { b  e.  CC  |  E. c  e.  {
d  e.  (Poly `  ZZ )  |  (
d  =/=  0 p  /\  (deg `  d
)  <_  a  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  a
) }  ( c `
 b )  =  0 } )  -> 
( g  e.  CC  /\ 
E. h  e.  ( (Poly `  ZZ )  \  { 0 p }
) ( h `  g )  =  0 ) )
154153exlimiv 1644 . . . . 5  |-  ( E. f ( g  e.  f  /\  E. a  e.  NN0  f  =  {
b  e.  CC  |  E. c  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0 p  /\  (deg `  d )  <_  a  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  a
) }  ( c `
 b )  =  0 } )  -> 
( g  e.  CC  /\ 
E. h  e.  ( (Poly `  ZZ )  \  { 0 p }
) ( h `  g )  =  0 ) )
155131, 154impbii 181 . . . 4  |-  ( ( g  e.  CC  /\  E. h  e.  ( (Poly `  ZZ )  \  {
0 p } ) ( h `  g
)  =  0 )  <->  E. f ( g  e.  f  /\  E. a  e.  NN0  f  =  {
b  e.  CC  |  E. c  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0 p  /\  (deg `  d )  <_  a  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  a
) }  ( c `
 b )  =  0 } ) )
156 elaa 20233 . . . 4  |-  ( g  e.  AA  <->  ( g  e.  CC  /\  E. h  e.  ( (Poly `  ZZ )  \  { 0 p } ) ( h `
 g )  =  0 ) )
157 eluniab 4027 . . . 4  |-  ( g  e.  U. { f  |  E. a  e. 
NN0  f  =  {
b  e.  CC  |  E. c  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0 p  /\  (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  =/=  0 p  /\  (deg `  d )  <_  a  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  a
) }  ( c `
 b )  =  0 } ) )
158155, 156, 1573bitr4i 269 . . 3  |-  ( g  e.  AA  <->  g  e.  U. { f  |  E. a  e.  NN0  f  =  { b  e.  CC  |  E. c  e.  {
d  e.  (Poly `  ZZ )  |  (
d  =/=  0 p  /\  (deg `  d
)  <_  a  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  a
) }  ( c `
 b )  =  0 } } )
159158eqriv 2433 . 2  |-  AA  =  U. { f  |  E. a  e.  NN0  f  =  { b  e.  CC  |  E. c  e.  {
d  e.  (Poly `  ZZ )  |  (
d  =/=  0 p  /\  (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  =/=  0 p  /\  (deg `  d
)  <_  a  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  a
) }  ( c `
 b )  =  0 } )
161160rnmpt 5116 . . 3  |-  ran  H  =  { f  |  E. a  e.  NN0  f  =  { b  e.  CC  |  E. c  e.  {
d  e.  (Poly `  ZZ )  |  (
d  =/=  0 p  /\  (deg `  d
)  <_  a  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  a
) }  ( c `
 b )  =  0 } }
162161unieqi 4025 . 2  |-  U. ran  H  =  U. { f  |  E. a  e. 
NN0  f  =  {
b  e.  CC  |  E. c  e.  { d  e.  (Poly `  ZZ )  |  ( d  =/=  0 p  /\  (deg `  d )  <_  a  /\  A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  a
) }  ( c `
 b )  =  0 } }
163159, 162eqtr4i 2459 1  |-  AA  =  U. ran  H
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936   E.wex 1550    = wceq 1652    e. wcel 1725   {cab 2422    =/= wne 2599   A.wral 2705   E.wrex 2706   {crab 2709    \ cdif 3317    u. cun 3318    C_ wss 3320   (/)c0 3628   {csn 3814   {cpr 3815   U.cuni 4015   class class class wbr 4212    e. cmpt 4266    Or wor 4502   ran crn 4879   -->wf 5450   ` cfv 5454  (class class class)co 6081   Fincfn 7109   supcsup 7445   CCcc 8988   RRcr 8989   0cc0 8990   RR*cxr 9119    < clt 9120    <_ cle 9121   NN0cn0 10221   ZZcz 10282   ...cfz 11043   abscabs 12039   0 pc0p 19561  Polycply 20103  coeffccoe 20105  degcdgr 20106   AAcaa 20231
This theorem is referenced by:  aannenlem3  20247
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068  ax-addf 9069
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-of 6305  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-map 7020  df-pm 7021  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-oi 7479  df-card 7826  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-rp 10613  df-fz 11044  df-fzo 11136  df-fl 11202  df-seq 11324  df-exp 11383  df-hash 11619  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-clim 12282  df-rlim 12283  df-sum 12480  df-0p 19562  df-ply 20107  df-coe 20109  df-dgr 20110  df-aa 20232
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