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Theorem aannenlem2 19725
Description: Lemma for aannen 19727. (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 957 . . . . . . . . . 10  |-  ( ( h  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  ( h `  g )  =  0  /\  g  e.  CC )  ->  g  e.  CC )
2 eldifi 3311 . . . . . . . . . . . . . 14  |-  ( h  e.  ( (Poly `  ZZ )  \  { 0 p } )  ->  h  e.  (Poly `  ZZ ) )
32adantr 451 . . . . . . . . . . . . 13  |-  ( ( h  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  g  e.  CC )  ->  h  e.  (Poly `  ZZ ) )
433adant2 974 . . . . . . . . . . . 12  |-  ( ( h  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  ( h `  g )  =  0  /\  g  e.  CC )  ->  h  e.  (Poly `  ZZ ) )
5 eldifsni 3763 . . . . . . . . . . . . . . 15  |-  ( h  e.  ( (Poly `  ZZ )  \  { 0 p } )  ->  h  =/=  0 p )
65adantr 451 . . . . . . . . . . . . . 14  |-  ( ( h  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  g  e.  CC )  ->  h  =/=  0 p )
7 0nn0 9996 . . . . . . . . . . . . . . . . . 18  |-  0  e.  NN0
8 dgrcl 19631 . . . . . . . . . . . . . . . . . . 19  |-  ( h  e.  (Poly `  ZZ )  ->  (deg `  h
)  e.  NN0 )
93, 8syl 15 . . . . . . . . . . . . . . . . . 18  |-  ( ( h  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  g  e.  CC )  ->  (deg `  h
)  e.  NN0 )
10 prssi 3787 . . . . . . . . . . . . . . . . . 18  |-  ( ( 0  e.  NN0  /\  (deg `  h )  e. 
NN0 )  ->  { 0 ,  (deg `  h
) }  C_  NN0 )
117, 9, 10sylancr 644 . . . . . . . . . . . . . . . . 17  |-  ( ( h  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  g  e.  CC )  ->  { 0 ,  (deg `  h ) }  C_  NN0 )
12 ssrab2 3271 . . . . . . . . . . . . . . . . . 18  |-  { g  e.  NN0  |  E. i  e.  ( 0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } 
C_  NN0
1312a1i 10 . . . . . . . . . . . . . . . . 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 3364 . . . . . . . . . . . . . . . 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 9985 . . . . . . . . . . . . . . . . 17  |-  NN0  C_  RR
16 ressxr 8892 . . . . . . . . . . . . . . . . 17  |-  RR  C_  RR*
1715, 16sstri 3201 . . . . . . . . . . . . . . . 16  |-  NN0  C_  RR*
1814, 17syl6ss 3204 . . . . . . . . . . . . . . 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 5555 . . . . . . . . . . . . . . . . 17  |-  (deg `  h )  e.  _V
2019prid2 3748 . . . . . . . . . . . . . . . 16  |-  (deg `  h )  e.  {
0 ,  (deg `  h ) }
21 elun1 3355 . . . . . . . . . . . . . . . 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 10659 . . . . . . . . . . . . . . 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 643 . . . . . . . . . . . . . 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 451 . . . . . . . . . . . . . . . 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 5541 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( (coeff `  h ) `  e )  =  0  ->  ( abs `  (
(coeff `  h ) `  e ) )  =  ( abs `  0
) )
27 abs0 11786 . . . . . . . . . . . . . . . . . . . 20  |-  ( abs `  0 )  =  0
2826, 27syl6eq 2344 . . . . . . . . . . . . . . . . . . 19  |-  ( ( (coeff `  h ) `  e )  =  0  ->  ( abs `  (
(coeff `  h ) `  e ) )  =  0 )
29 c0ex 8848 . . . . . . . . . . . . . . . . . . . . 21  |-  0  e.  _V
3029prid1 3747 . . . . . . . . . . . . . . . . . . . 20  |-  0  e.  { 0 ,  (deg
`  h ) }
31 elun1 3355 . . . . . . . . . . . . . . . . . . . 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 2384 . . . . . . . . . . . . . . . . . 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 452 . . . . . . . . . . . . . . . . 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 10051 . . . . . . . . . . . . . . . . . . . . . . 23  |-  0  e.  ZZ
36 eqid 2296 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  (coeff `  h )  =  (coeff `  h )
3736coef2 19629 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( h  e.  (Poly `  ZZ )  /\  0  e.  ZZ )  ->  (coeff `  h ) : NN0 --> ZZ )
383, 35, 37sylancl 643 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( h  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  g  e.  CC )  ->  (coeff `  h
) : NN0 --> ZZ )
3938ffvelrnda 5681 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( h  e.  ( (Poly `  ZZ )  \  { 0 p }
)  /\  g  e.  CC )  /\  e  e.  NN0 )  ->  (
(coeff `  h ) `  e )  e.  ZZ )
40 nn0abscl 11813 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( (coeff `  h ) `  e )  e.  ZZ  ->  ( abs `  (
(coeff `  h ) `  e ) )  e. 
NN0 )
4139, 40syl 15 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( h  e.  ( (Poly `  ZZ )  \  { 0 p }
)  /\  g  e.  CC )  /\  e  e.  NN0 )  ->  ( abs `  ( (coeff `  h ) `  e
) )  e.  NN0 )
4241adantr 451 . . . . . . . . . . . . . . . . . . 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 731 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( h  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  g  e.  CC )  /\  e  e.  NN0 )  /\  (
(coeff `  h ) `  e )  =/=  0
)  ->  e  e.  NN0 )
449ad2antrr 706 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( h  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  g  e.  CC )  /\  e  e.  NN0 )  /\  (
(coeff `  h ) `  e )  =/=  0
)  ->  (deg `  h
)  e.  NN0 )
453ad2antrr 706 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( h  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  g  e.  CC )  /\  e  e.  NN0 )  /\  (
(coeff `  h ) `  e )  =/=  0
)  ->  h  e.  (Poly `  ZZ ) )
46 simpr 447 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( h  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  g  e.  CC )  /\  e  e.  NN0 )  /\  (
(coeff `  h ) `  e )  =/=  0
)  ->  ( (coeff `  h ) `  e
)  =/=  0 )
47 eqid 2296 . . . . . . . . . . . . . . . . . . . . . . 23  |-  (deg `  h )  =  (deg
`  h )
4836, 47dgrub 19632 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( h  e.  (Poly `  ZZ )  /\  e  e.  NN0  /\  ( (coeff `  h ) `  e
)  =/=  0 )  ->  e  <_  (deg `  h ) )
4945, 43, 46, 48syl3anc 1182 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( h  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  g  e.  CC )  /\  e  e.  NN0 )  /\  (
(coeff `  h ) `  e )  =/=  0
)  ->  e  <_  (deg
`  h ) )
50 elfz2nn0 10837 . . . . . . . . . . . . . . . . . . . . 21  |-  ( e  e.  ( 0 ... (deg `  h )
)  <->  ( e  e. 
NN0  /\  (deg `  h
)  e.  NN0  /\  e  <_  (deg `  h
) ) )
5143, 44, 49, 50syl3anbrc 1136 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( h  e.  ( (Poly `  ZZ )  \  { 0 p } )  /\  g  e.  CC )  /\  e  e.  NN0 )  /\  (
(coeff `  h ) `  e )  =/=  0
)  ->  e  e.  ( 0 ... (deg `  h ) ) )
52 eqid 2296 . . . . . . . . . . . . . . . . . . . 20  |-  ( abs `  ( (coeff `  h
) `  e )
)  =  ( abs `  ( (coeff `  h
) `  e )
)
53 fveq2 5541 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( i  =  e  ->  (
(coeff `  h ) `  i )  =  ( (coeff `  h ) `  e ) )
5453fveq2d 5545 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( i  =  e  ->  ( abs `  ( (coeff `  h ) `  i
) )  =  ( abs `  ( (coeff `  h ) `  e
) ) )
5554eqeq2d 2307 . . . . . . . . . . . . . . . . . . . . 21  |-  ( i  =  e  ->  (
( abs `  (
(coeff `  h ) `  e ) )  =  ( abs `  (
(coeff `  h ) `  i ) )  <->  ( abs `  ( (coeff `  h
) `  e )
)  =  ( abs `  ( (coeff `  h
) `  e )
) ) )
5655rspcev 2897 . . . . . . . . . . . . . . . . . . . 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 643 . . . . . . . . . . . . . . . . . . 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 2302 . . . . . . . . . . . . . . . . . . . . 21  |-  ( g  =  ( abs `  (
(coeff `  h ) `  e ) )  -> 
( g  =  ( abs `  ( (coeff `  h ) `  i
) )  <->  ( abs `  ( (coeff `  h
) `  e )
)  =  ( abs `  ( (coeff `  h
) `  i )
) ) )
5958rexbidv 2577 . . . . . . . . . . . . . . . . . . . 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 2936 . . . . . . . . . . . . . . . . . . 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 645 . . . . . . . . . . . . . . . . . 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 3356 . . . . . . . . . . . . . . . . . 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 15 . . . . . . . . . . . . . . . . 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 2537 . . . . . . . . . . . . . . . 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 10659 . . . . . . . . . . . . . . . 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 642 . . . . . . . . . . . . . . 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 2639 . . . . . . . . . . . . . 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 1132 . . . . . . . . . . . . 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 974 . . . . . . . . . . . 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 2467 . . . . . . . . . . . . . 14  |-  ( d  =  h  ->  (
d  =/=  0 p  <-> 
h  =/=  0 p ) )
71 fveq2 5541 . . . . . . . . . . . . . . 15  |-  ( d  =  h  ->  (deg `  d )  =  (deg
`  h ) )
7271breq1d 4049 . . . . . . . . . . . . . 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 5541 . . . . . . . . . . . . . . . . . 18  |-  ( d  =  h  ->  (coeff `  d )  =  (coeff `  h ) )
7473fveq1d 5543 . . . . . . . . . . . . . . . . 17  |-  ( d  =  h  ->  (
(coeff `  d ) `  e )  =  ( (coeff `  h ) `  e ) )
7574fveq2d 5545 . . . . . . . . . . . . . . . 16  |-  ( d  =  h  ->  ( abs `  ( (coeff `  d ) `  e
) )  =  ( abs `  ( (coeff `  h ) `  e
) ) )
7675breq1d 4049 . . . . . . . . . . . . . . 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 2576 . . . . . . . . . . . . . 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 1252 . . . . . . . . . . . . 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 2936 . . . . . . . . . . . 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 645 . . . . . . . . . . 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 956 . . . . . . . . . . 11  |-  ( ( h  e.  ( (Poly `  ZZ )  \  {
0 p } )  /\  ( h `  g )  =  0  /\  g  e.  CC )  ->  ( h `  g )  =  0 )
82 fveq1 5540 . . . . . . . . . . . . 13  |-  ( c  =  h  ->  (
c `  g )  =  ( h `  g ) )
8382eqeq1d 2304 . . . . . . . . . . . 12  |-  ( c  =  h  ->  (
( c `  g
)  =  0  <->  (
h `  g )  =  0 ) )
8483rspcev 2897 . . . . . . . . . . 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 642 . . . . . . . . . 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 5541 . . . . . . . . . . . . 13  |-  ( b  =  g  ->  (
c `  b )  =  ( c `  g ) )
8786eqeq1d 2304 . . . . . . . . . . . 12  |-  ( b  =  g  ->  (
( c `  b
)  =  0  <->  (
c `  g )  =  0 ) )
8887rexbidv 2577 . . . . . . . . . . 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 2936 . . . . . . . . . 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 645 . . . . . . . . 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 7147 . . . . . . . . . . . . . . 15  |-  { 0 ,  (deg `  h
) }  e.  Fin
92 fzfi 11050 . . . . . . . . . . . . . . . . 17  |-  ( 0 ... (deg `  h
) )  e.  Fin
93 abrexfi 7172 . . . . . . . . . . . . . . . . 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 3272 . . . . . . . . . . . . . . . 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 7099 . . . . . . . . . . . . . . . 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 653 . . . . . . . . . . . . . . 15  |-  { g  e.  NN0  |  E. i  e.  ( 0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) }  e.  Fin
98 unfi 7140 . . . . . . . . . . . . . . 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 653 . . . . . . . . . . . . . 14  |-  ( { 0 ,  (deg `  h ) }  u.  { g  e.  NN0  |  E. i  e.  (
0 ... (deg `  h
) ) g  =  ( abs `  (
(coeff `  h ) `  i ) ) } )  e.  Fin
10099a1i 10 . . . . . . . . . . . . 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 3474 . . . . . . . . . . . . . . 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 10 . . . . . . . . . . . . 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 10491 . . . . . . . . . . . . . 14  |-  <  Or  RR*
105 fisupcl 7234 . . . . . . . . . . . . . 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 651 . . . . . . . . . . . . 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 1182 . . . . . . . . . . . 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 3194 . . . . . . . . . . 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 974 . . . . . . . . . 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 2297 . . . . . . . . . 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 4043 . . . . . . . . . . . . . . . 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 4043 . . . . . . . . . . . . . . . . 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 2576 . . . . . . . . . . . . . . . 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 1255 . . . . . . . . . . . . . . 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 2793 . . . . . . . . . . . . . 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 2756 . . . . . . . . . . . . 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 2793 . . . . . . . . . . . 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 2307 . . . . . . . . . . 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 2897 . . . . . . . . . 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 642 . . . . . . . . 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 8834 . . . . . . . . . . 11  |-  CC  e.  _V
122121rabex 4181 . . . . . . . . . 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 2357 . . . . . . . . . . 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 2302 . . . . . . . . . . . 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 2577 . . . . . . . . . . 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 691 . . . . . . . . . 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 2888 . . . . . . . . 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 642 . . . . . . . 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 1150 . . . . . . 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 2674 . . . . . 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 419 . . . . 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 2357 . . . . . . . . 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 2577 . . . . . . . . . . 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 2936 . . . . . . . . . 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 955 . . . . . . . . . . . . . . 15  |-  ( ( h  =/=  0 p  /\  (deg `  h
)  <_  a  /\  A. e  e.  NN0  ( abs `  ( (coeff `  h ) `  e
) )  <_  a
)  ->  h  =/=  0 p )
136135anim2i 552 . . . . . . . . . . . . . 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 4049 . . . . . . . . . . . . . . . 16  |-  ( d  =  h  ->  (
(deg `  d )  <_  a  <->  (deg `  h )  <_  a ) )
13875breq1d 4049 . . . . . . . . . . . . . . . . 17  |-  ( d  =  h  ->  (
( abs `  (
(coeff `  d ) `  e ) )  <_ 
a  <->  ( abs `  (
(coeff `  h ) `  e ) )  <_ 
a ) )
139138ralbidv 2576 . . . . . . . . . . . . . . . 16  |-  ( d  =  h  ->  ( A. e  e.  NN0  ( abs `  ( (coeff `  d ) `  e
) )  <_  a  <->  A. e  e.  NN0  ( abs `  ( (coeff `  h ) `  e
) )  <_  a
) )
14070, 137, 1393anbi123d 1252 . . . . . . . . . . . . . . 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 2936 . . . . . . . . . . . . . 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 3762 . . . . . . . . . . . . . 14  |-  ( h  e.  ( (Poly `  ZZ )  \  { 0 p } )  <->  ( h  e.  (Poly `  ZZ )  /\  h  =/=  0 p ) )
143136, 141, 1423imtr4i 257 . . . . . . . . . . . . 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 3197 . . . . . . . . . . . 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 3251 . . . . . . . . . . . . 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 2778 . . . . . . . . . . . . 13  |-  ( E. c  e.  ( (Poly `  ZZ )  \  {
0 p } ) ( c `  g
)  =  0  <->  E. h  e.  ( (Poly `  ZZ )  \  {
0 p } ) ( h `  g
)  =  0 )
147145, 146syl6ib 217 . . . . . . . . . . . 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 552 . . . . . . . . . 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 187 . . . . . . . . 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 219 . . . . . . . 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 2676 . . . . . . 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 419 . . . . . 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 1624 . . . . 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 180 . . . 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 19712 . . . 4  |-  ( g  e.  AA  <->  ( g  e.  CC  /\  E. h  e.  ( (Poly `  ZZ )  \  { 0 p } ) ( h `
 g )  =  0 ) )
157 eluniab 3855 . . . 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 268 . . 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 2293 . 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 4941 . . 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 3853 . 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 2319 1  |-  AA  =  U. ran  H
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934   E.wex 1531    = wceq 1632    e. wcel 1696   {cab 2282    =/= wne 2459   A.wral 2556   E.wrex 2557   {crab 2560    \ cdif 3162    u. cun 3163    C_ wss 3165   (/)c0 3468   {csn 3653   {cpr 3654   U.cuni 3843   class class class wbr 4039    e. cmpt 4093    Or wor 4329   ran crn 4706   -->wf 5267   ` cfv 5271  (class class class)co 5874   Fincfn 6879   supcsup 7209   CCcc 8751   RRcr 8752   0cc0 8753   RR*cxr 8882    < clt 8883    <_ cle 8884   NN0cn0 9981   ZZcz 10040   ...cfz 10798   abscabs 11735   0 pc0p 19040  Polycply 19582  coeffccoe 19584  degcdgr 19585   AAcaa 19710
This theorem is referenced by:  aannenlem3  19726
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-fz 10799  df-fzo 10887  df-fl 10941  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-rlim 11979  df-sum 12175  df-0p 19041  df-ply 19586  df-coe 19588  df-dgr 19589  df-aa 19711
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