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Theorem elqaalem3 21802
Description: Lemma for elqaa 21803. (Contributed by Mario Carneiro, 23-Jul-2014.)
Hypotheses
Ref Expression
elqaa.1  |-  ( ph  ->  A  e.  CC )
elqaa.2  |-  ( ph  ->  F  e.  ( (Poly `  QQ )  \  {
0p } ) )
elqaa.3  |-  ( ph  ->  ( F `  A
)  =  0 )
elqaa.4  |-  B  =  (coeff `  F )
elqaa.5  |-  N  =  ( k  e.  NN0  |->  sup ( { n  e.  NN  |  ( ( B `  k )  x.  n )  e.  ZZ } ,  RR ,  `'  <  ) )
elqaa.6  |-  R  =  (  seq 0 (  x.  ,  N ) `
 (deg `  F
) )
Assertion
Ref Expression
elqaalem3  |-  ( ph  ->  A  e.  AA )
Distinct variable groups:    k, n, A    B, k, n    ph, k    k, N, n    R, k
Allowed substitution hints:    ph( n)    R( n)    F( k, n)

Proof of Theorem elqaalem3
Dummy variables  f  m  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elqaa.1 . 2  |-  ( ph  ->  A  e.  CC )
2 cnex 9378 . . . . . . . 8  |-  CC  e.  _V
32a1i 11 . . . . . . 7  |-  ( ph  ->  CC  e.  _V )
4 elqaa.6 . . . . . . . . 9  |-  R  =  (  seq 0 (  x.  ,  N ) `
 (deg `  F
) )
5 fvex 5716 . . . . . . . . 9  |-  (  seq 0 (  x.  ,  N ) `  (deg `  F ) )  e. 
_V
64, 5eqeltri 2513 . . . . . . . 8  |-  R  e. 
_V
76a1i 11 . . . . . . 7  |-  ( (
ph  /\  z  e.  CC )  ->  R  e. 
_V )
8 fvex 5716 . . . . . . . 8  |-  ( F `
 z )  e. 
_V
98a1i 11 . . . . . . 7  |-  ( (
ph  /\  z  e.  CC )  ->  ( F `
 z )  e. 
_V )
10 fconstmpt 4897 . . . . . . . 8  |-  ( CC 
X.  { R }
)  =  ( z  e.  CC  |->  R )
1110a1i 11 . . . . . . 7  |-  ( ph  ->  ( CC  X.  { R } )  =  ( z  e.  CC  |->  R ) )
12 elqaa.2 . . . . . . . . . 10  |-  ( ph  ->  F  e.  ( (Poly `  QQ )  \  {
0p } ) )
1312eldifad 3355 . . . . . . . . 9  |-  ( ph  ->  F  e.  (Poly `  QQ ) )
14 plyf 21681 . . . . . . . . 9  |-  ( F  e.  (Poly `  QQ )  ->  F : CC --> CC )
1513, 14syl 16 . . . . . . . 8  |-  ( ph  ->  F : CC --> CC )
1615feqmptd 5759 . . . . . . 7  |-  ( ph  ->  F  =  ( z  e.  CC  |->  ( F `
 z ) ) )
173, 7, 9, 11, 16offval2 6351 . . . . . 6  |-  ( ph  ->  ( ( CC  X.  { R } )  oF  x.  F )  =  ( z  e.  CC  |->  ( R  x.  ( F `  z ) ) ) )
18 fzfid 11810 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  CC )  ->  ( 0 ... (deg `  F
) )  e.  Fin )
19 nn0uz 10910 . . . . . . . . . . . . . 14  |-  NN0  =  ( ZZ>= `  0 )
20 0zd 10673 . . . . . . . . . . . . . 14  |-  ( ph  ->  0  e.  ZZ )
21 ssrab2 3452 . . . . . . . . . . . . . . 15  |-  { n  e.  NN  |  ( ( B `  m )  x.  n )  e.  ZZ }  C_  NN
22 fveq2 5706 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( k  =  m  ->  ( B `  k )  =  ( B `  m ) )
2322oveq1d 6121 . . . . . . . . . . . . . . . . . . . . 21  |-  ( k  =  m  ->  (
( B `  k
)  x.  n )  =  ( ( B `
 m )  x.  n ) )
2423eleq1d 2509 . . . . . . . . . . . . . . . . . . . 20  |-  ( k  =  m  ->  (
( ( B `  k )  x.  n
)  e.  ZZ  <->  ( ( B `  m )  x.  n )  e.  ZZ ) )
2524rabbidv 2979 . . . . . . . . . . . . . . . . . . 19  |-  ( k  =  m  ->  { n  e.  NN  |  ( ( B `  k )  x.  n )  e.  ZZ }  =  {
n  e.  NN  | 
( ( B `  m )  x.  n
)  e.  ZZ }
)
2625supeq1d 7711 . . . . . . . . . . . . . . . . . 18  |-  ( k  =  m  ->  sup ( { n  e.  NN  |  ( ( B `
 k )  x.  n )  e.  ZZ } ,  RR ,  `'  <  )  =  sup ( { n  e.  NN  |  ( ( B `
 m )  x.  n )  e.  ZZ } ,  RR ,  `'  <  ) )
27 elqaa.5 . . . . . . . . . . . . . . . . . 18  |-  N  =  ( k  e.  NN0  |->  sup ( { n  e.  NN  |  ( ( B `  k )  x.  n )  e.  ZZ } ,  RR ,  `'  <  ) )
28 gtso 9471 . . . . . . . . . . . . . . . . . . 19  |-  `'  <  Or  RR
2928supex 7728 . . . . . . . . . . . . . . . . . 18  |-  sup ( { n  e.  NN  |  ( ( B `
 m )  x.  n )  e.  ZZ } ,  RR ,  `'  <  )  e.  _V
3026, 27, 29fvmpt 5789 . . . . . . . . . . . . . . . . 17  |-  ( m  e.  NN0  ->  ( N `
 m )  =  sup ( { n  e.  NN  |  ( ( B `  m )  x.  n )  e.  ZZ } ,  RR ,  `'  <  ) )
3130adantl 466 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  m  e.  NN0 )  ->  ( N `  m )  =  sup ( { n  e.  NN  |  ( ( B `
 m )  x.  n )  e.  ZZ } ,  RR ,  `'  <  ) )
32 nnuz 10911 . . . . . . . . . . . . . . . . . 18  |-  NN  =  ( ZZ>= `  1 )
3321, 32sseqtri 3403 . . . . . . . . . . . . . . . . 17  |-  { n  e.  NN  |  ( ( B `  m )  x.  n )  e.  ZZ }  C_  ( ZZ>=
`  1 )
34 0z 10672 . . . . . . . . . . . . . . . . . . . . . 22  |-  0  e.  ZZ
35 zq 10974 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( 0  e.  ZZ  ->  0  e.  QQ )
3634, 35ax-mp 5 . . . . . . . . . . . . . . . . . . . . 21  |-  0  e.  QQ
37 elqaa.4 . . . . . . . . . . . . . . . . . . . . . 22  |-  B  =  (coeff `  F )
3837coef2 21714 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( F  e.  (Poly `  QQ )  /\  0  e.  QQ )  ->  B : NN0 --> QQ )
3913, 36, 38sylancl 662 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  B : NN0 --> QQ )
4039ffvelrnda 5858 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  m  e.  NN0 )  ->  ( B `  m )  e.  QQ )
41 qmulz 10971 . . . . . . . . . . . . . . . . . . 19  |-  ( ( B `  m )  e.  QQ  ->  E. n  e.  NN  ( ( B `
 m )  x.  n )  e.  ZZ )
4240, 41syl 16 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  m  e.  NN0 )  ->  E. n  e.  NN  ( ( B `
 m )  x.  n )  e.  ZZ )
43 rabn0 3672 . . . . . . . . . . . . . . . . . 18  |-  ( { n  e.  NN  | 
( ( B `  m )  x.  n
)  e.  ZZ }  =/=  (/)  <->  E. n  e.  NN  ( ( B `  m )  x.  n
)  e.  ZZ )
4442, 43sylibr 212 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  m  e.  NN0 )  ->  { n  e.  NN  |  ( ( B `  m )  x.  n )  e.  ZZ }  =/=  (/) )
45 infmssuzcl 10953 . . . . . . . . . . . . . . . . 17  |-  ( ( { n  e.  NN  |  ( ( B `
 m )  x.  n )  e.  ZZ }  C_  ( ZZ>= `  1
)  /\  { n  e.  NN  |  ( ( B `  m )  x.  n )  e.  ZZ }  =/=  (/) )  ->  sup ( { n  e.  NN  |  ( ( B `  m )  x.  n )  e.  ZZ } ,  RR ,  `'  <  )  e. 
{ n  e.  NN  |  ( ( B `
 m )  x.  n )  e.  ZZ } )
4633, 44, 45sylancr 663 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  m  e.  NN0 )  ->  sup ( { n  e.  NN  |  ( ( B `
 m )  x.  n )  e.  ZZ } ,  RR ,  `'  <  )  e.  {
n  e.  NN  | 
( ( B `  m )  x.  n
)  e.  ZZ }
)
4731, 46eqeltrd 2517 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  m  e.  NN0 )  ->  ( N `  m )  e.  {
n  e.  NN  | 
( ( B `  m )  x.  n
)  e.  ZZ }
)
4821, 47sseldi 3369 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  m  e.  NN0 )  ->  ( N `  m )  e.  NN )
49 nnmulcl 10360 . . . . . . . . . . . . . . 15  |-  ( ( m  e.  NN  /\  k  e.  NN )  ->  ( m  x.  k
)  e.  NN )
5049adantl 466 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( m  e.  NN  /\  k  e.  NN ) )  -> 
( m  x.  k
)  e.  NN )
5119, 20, 48, 50seqf 11842 . . . . . . . . . . . . 13  |-  ( ph  ->  seq 0 (  x.  ,  N ) : NN0 --> NN )
52 dgrcl 21716 . . . . . . . . . . . . . 14  |-  ( F  e.  (Poly `  QQ )  ->  (deg `  F
)  e.  NN0 )
5313, 52syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  (deg `  F )  e.  NN0 )
5451, 53ffvelrnd 5859 . . . . . . . . . . . 12  |-  ( ph  ->  (  seq 0 (  x.  ,  N ) `
 (deg `  F
) )  e.  NN )
554, 54syl5eqel 2527 . . . . . . . . . . 11  |-  ( ph  ->  R  e.  NN )
5655nncnd 10353 . . . . . . . . . 10  |-  ( ph  ->  R  e.  CC )
5756adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  CC )  ->  R  e.  CC )
58 elfznn0 11496 . . . . . . . . . 10  |-  ( m  e.  ( 0 ... (deg `  F )
)  ->  m  e.  NN0 )
5937coef3 21715 . . . . . . . . . . . . . 14  |-  ( F  e.  (Poly `  QQ )  ->  B : NN0 --> CC )
6013, 59syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  B : NN0 --> CC )
6160adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  z  e.  CC )  ->  B : NN0
--> CC )
6261ffvelrnda 5858 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  CC )  /\  m  e.  NN0 )  ->  ( B `  m )  e.  CC )
63 expcl 11898 . . . . . . . . . . . 12  |-  ( ( z  e.  CC  /\  m  e.  NN0 )  -> 
( z ^ m
)  e.  CC )
6463adantll 713 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  CC )  /\  m  e.  NN0 )  ->  (
z ^ m )  e.  CC )
6562, 64mulcld 9421 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  CC )  /\  m  e.  NN0 )  ->  (
( B `  m
)  x.  ( z ^ m ) )  e.  CC )
6658, 65sylan2 474 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  CC )  /\  m  e.  ( 0 ... (deg `  F ) ) )  ->  ( ( B `
 m )  x.  ( z ^ m
) )  e.  CC )
6718, 57, 66fsummulc2 13266 . . . . . . . 8  |-  ( (
ph  /\  z  e.  CC )  ->  ( R  x.  sum_ m  e.  ( 0 ... (deg `  F ) ) ( ( B `  m
)  x.  ( z ^ m ) ) )  =  sum_ m  e.  ( 0 ... (deg `  F ) ) ( R  x.  ( ( B `  m )  x.  ( z ^
m ) ) ) )
68 eqid 2443 . . . . . . . . . . 11  |-  (deg `  F )  =  (deg
`  F )
6937, 68coeid2 21722 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  QQ )  /\  z  e.  CC )  ->  ( F `  z )  =  sum_ m  e.  ( 0 ... (deg `  F ) ) ( ( B `  m
)  x.  ( z ^ m ) ) )
7013, 69sylan 471 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  CC )  ->  ( F `
 z )  = 
sum_ m  e.  (
0 ... (deg `  F
) ) ( ( B `  m )  x.  ( z ^
m ) ) )
7170oveq2d 6122 . . . . . . . 8  |-  ( (
ph  /\  z  e.  CC )  ->  ( R  x.  ( F `  z ) )  =  ( R  x.  sum_ m  e.  ( 0 ... (deg `  F )
) ( ( B `
 m )  x.  ( z ^ m
) ) ) )
7257adantr 465 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  CC )  /\  m  e.  NN0 )  ->  R  e.  CC )
7372, 62, 64mulassd 9424 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  CC )  /\  m  e.  NN0 )  ->  (
( R  x.  ( B `  m )
)  x.  ( z ^ m ) )  =  ( R  x.  ( ( B `  m )  x.  (
z ^ m ) ) ) )
7458, 73sylan2 474 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  CC )  /\  m  e.  ( 0 ... (deg `  F ) ) )  ->  ( ( R  x.  ( B `  m ) )  x.  ( z ^ m
) )  =  ( R  x.  ( ( B `  m )  x.  ( z ^
m ) ) ) )
7574sumeq2dv 13195 . . . . . . . 8  |-  ( (
ph  /\  z  e.  CC )  ->  sum_ m  e.  ( 0 ... (deg `  F ) ) ( ( R  x.  ( B `  m )
)  x.  ( z ^ m ) )  =  sum_ m  e.  ( 0 ... (deg `  F ) ) ( R  x.  ( ( B `  m )  x.  ( z ^
m ) ) ) )
7667, 71, 753eqtr4d 2485 . . . . . . 7  |-  ( (
ph  /\  z  e.  CC )  ->  ( R  x.  ( F `  z ) )  = 
sum_ m  e.  (
0 ... (deg `  F
) ) ( ( R  x.  ( B `
 m ) )  x.  ( z ^
m ) ) )
7776mpteq2dva 4393 . . . . . 6  |-  ( ph  ->  ( z  e.  CC  |->  ( R  x.  ( F `  z )
) )  =  ( z  e.  CC  |->  sum_
m  e.  ( 0 ... (deg `  F
) ) ( ( R  x.  ( B `
 m ) )  x.  ( z ^
m ) ) ) )
7817, 77eqtrd 2475 . . . . 5  |-  ( ph  ->  ( ( CC  X.  { R } )  oF  x.  F )  =  ( z  e.  CC  |->  sum_ m  e.  ( 0 ... (deg `  F ) ) ( ( R  x.  ( B `  m )
)  x.  ( z ^ m ) ) ) )
79 zsscn 10669 . . . . . . 7  |-  ZZ  C_  CC
8079a1i 11 . . . . . 6  |-  ( ph  ->  ZZ  C_  CC )
8156adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  m  e.  NN0 )  ->  R  e.  CC )
8248nncnd 10353 . . . . . . . . . . 11  |-  ( (
ph  /\  m  e.  NN0 )  ->  ( N `  m )  e.  CC )
8348nnne0d 10381 . . . . . . . . . . 11  |-  ( (
ph  /\  m  e.  NN0 )  ->  ( N `  m )  =/=  0
)
8481, 82, 83divcan2d 10124 . . . . . . . . . 10  |-  ( (
ph  /\  m  e.  NN0 )  ->  ( ( N `  m )  x.  ( R  /  ( N `  m )
) )  =  R )
8584oveq2d 6122 . . . . . . . . 9  |-  ( (
ph  /\  m  e.  NN0 )  ->  ( ( B `  m )  x.  ( ( N `  m )  x.  ( R  /  ( N `  m ) ) ) )  =  ( ( B `  m )  x.  R ) )
8660ffvelrnda 5858 . . . . . . . . . 10  |-  ( (
ph  /\  m  e.  NN0 )  ->  ( B `  m )  e.  CC )
8781, 82, 83divcld 10122 . . . . . . . . . 10  |-  ( (
ph  /\  m  e.  NN0 )  ->  ( R  /  ( N `  m ) )  e.  CC )
8886, 82, 87mulassd 9424 . . . . . . . . 9  |-  ( (
ph  /\  m  e.  NN0 )  ->  ( (
( B `  m
)  x.  ( N `
 m ) )  x.  ( R  / 
( N `  m
) ) )  =  ( ( B `  m )  x.  (
( N `  m
)  x.  ( R  /  ( N `  m ) ) ) ) )
8981, 86mulcomd 9422 . . . . . . . . 9  |-  ( (
ph  /\  m  e.  NN0 )  ->  ( R  x.  ( B `  m
) )  =  ( ( B `  m
)  x.  R ) )
9085, 88, 893eqtr4rd 2486 . . . . . . . 8  |-  ( (
ph  /\  m  e.  NN0 )  ->  ( R  x.  ( B `  m
) )  =  ( ( ( B `  m )  x.  ( N `  m )
)  x.  ( R  /  ( N `  m ) ) ) )
9158, 90sylan2 474 . . . . . . 7  |-  ( (
ph  /\  m  e.  ( 0 ... (deg `  F ) ) )  ->  ( R  x.  ( B `  m ) )  =  ( ( ( B `  m
)  x.  ( N `
 m ) )  x.  ( R  / 
( N `  m
) ) ) )
92 oveq2 6114 . . . . . . . . . . . . 13  |-  ( n  =  ( N `  m )  ->  (
( B `  m
)  x.  n )  =  ( ( B `
 m )  x.  ( N `  m
) ) )
9392eleq1d 2509 . . . . . . . . . . . 12  |-  ( n  =  ( N `  m )  ->  (
( ( B `  m )  x.  n
)  e.  ZZ  <->  ( ( B `  m )  x.  ( N `  m
) )  e.  ZZ ) )
9493elrab 3132 . . . . . . . . . . 11  |-  ( ( N `  m )  e.  { n  e.  NN  |  ( ( B `  m )  x.  n )  e.  ZZ }  <->  ( ( N `  m )  e.  NN  /\  ( ( B `  m )  x.  ( N `  m ) )  e.  ZZ ) )
9594simprbi 464 . . . . . . . . . 10  |-  ( ( N `  m )  e.  { n  e.  NN  |  ( ( B `  m )  x.  n )  e.  ZZ }  ->  (
( B `  m
)  x.  ( N `
 m ) )  e.  ZZ )
9647, 95syl 16 . . . . . . . . 9  |-  ( (
ph  /\  m  e.  NN0 )  ->  ( ( B `  m )  x.  ( N `  m
) )  e.  ZZ )
9758, 96sylan2 474 . . . . . . . 8  |-  ( (
ph  /\  m  e.  ( 0 ... (deg `  F ) ) )  ->  ( ( B `
 m )  x.  ( N `  m
) )  e.  ZZ )
98 elqaa.3 . . . . . . . . . 10  |-  ( ph  ->  ( F `  A
)  =  0 )
99 eqid 2443 . . . . . . . . . 10  |-  ( x  e.  _V ,  y  e.  _V  |->  ( ( x  x.  y )  mod  ( N `  m ) ) )  =  ( x  e. 
_V ,  y  e. 
_V  |->  ( ( x  x.  y )  mod  ( N `  m
) ) )
1001, 12, 98, 37, 27, 4, 99elqaalem2 21801 . . . . . . . . 9  |-  ( (
ph  /\  m  e.  ( 0 ... (deg `  F ) ) )  ->  ( R  mod  ( N `  m ) )  =  0 )
10155adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  m  e.  ( 0 ... (deg `  F ) ) )  ->  R  e.  NN )
10258, 48sylan2 474 . . . . . . . . . 10  |-  ( (
ph  /\  m  e.  ( 0 ... (deg `  F ) ) )  ->  ( N `  m )  e.  NN )
103 nnre 10344 . . . . . . . . . . 11  |-  ( R  e.  NN  ->  R  e.  RR )
104 nnrp 11015 . . . . . . . . . . 11  |-  ( ( N `  m )  e.  NN  ->  ( N `  m )  e.  RR+ )
105 mod0 11730 . . . . . . . . . . 11  |-  ( ( R  e.  RR  /\  ( N `  m )  e.  RR+ )  ->  (
( R  mod  ( N `  m )
)  =  0  <->  ( R  /  ( N `  m ) )  e.  ZZ ) )
106103, 104, 105syl2an 477 . . . . . . . . . 10  |-  ( ( R  e.  NN  /\  ( N `  m )  e.  NN )  -> 
( ( R  mod  ( N `  m ) )  =  0  <->  ( R  /  ( N `  m ) )  e.  ZZ ) )
107101, 102, 106syl2anc 661 . . . . . . . . 9  |-  ( (
ph  /\  m  e.  ( 0 ... (deg `  F ) ) )  ->  ( ( R  mod  ( N `  m ) )  =  0  <->  ( R  / 
( N `  m
) )  e.  ZZ ) )
108100, 107mpbid 210 . . . . . . . 8  |-  ( (
ph  /\  m  e.  ( 0 ... (deg `  F ) ) )  ->  ( R  / 
( N `  m
) )  e.  ZZ )
10997, 108zmulcld 10768 . . . . . . 7  |-  ( (
ph  /\  m  e.  ( 0 ... (deg `  F ) ) )  ->  ( ( ( B `  m )  x.  ( N `  m ) )  x.  ( R  /  ( N `  m )
) )  e.  ZZ )
11091, 109eqeltrd 2517 . . . . . 6  |-  ( (
ph  /\  m  e.  ( 0 ... (deg `  F ) ) )  ->  ( R  x.  ( B `  m ) )  e.  ZZ )
11180, 53, 110elplyd 21685 . . . . 5  |-  ( ph  ->  ( z  e.  CC  |->  sum_
m  e.  ( 0 ... (deg `  F
) ) ( ( R  x.  ( B `
 m ) )  x.  ( z ^
m ) ) )  e.  (Poly `  ZZ ) )
11278, 111eqeltrd 2517 . . . 4  |-  ( ph  ->  ( ( CC  X.  { R } )  oF  x.  F )  e.  (Poly `  ZZ ) )
113 eldifsn 4015 . . . . . . 7  |-  ( F  e.  ( (Poly `  QQ )  \  { 0p } )  <->  ( F  e.  (Poly `  QQ )  /\  F  =/=  0p ) )
11412, 113sylib 196 . . . . . 6  |-  ( ph  ->  ( F  e.  (Poly `  QQ )  /\  F  =/=  0p ) )
115114simprd 463 . . . . 5  |-  ( ph  ->  F  =/=  0p )
116 oveq1 6113 . . . . . . 7  |-  ( ( ( CC  X.  { R } )  oF  x.  F )  =  0p  ->  (
( ( CC  X.  { R } )  oF  x.  F )  oF  /  ( CC  X.  { R }
) )  =  ( 0p  oF  /  ( CC  X.  { R } ) ) )
11715ffvelrnda 5858 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  CC )  ->  ( F `
 z )  e.  CC )
11855nnne0d 10381 . . . . . . . . . . . 12  |-  ( ph  ->  R  =/=  0 )
119118adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  CC )  ->  R  =/=  0 )
120117, 57, 119divcan3d 10127 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  CC )  ->  ( ( R  x.  ( F `
 z ) )  /  R )  =  ( F `  z
) )
121120mpteq2dva 4393 . . . . . . . . 9  |-  ( ph  ->  ( z  e.  CC  |->  ( ( R  x.  ( F `  z ) )  /  R ) )  =  ( z  e.  CC  |->  ( F `
 z ) ) )
122 ovex 6131 . . . . . . . . . . 11  |-  ( R  x.  ( F `  z ) )  e. 
_V
123122a1i 11 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  CC )  ->  ( R  x.  ( F `  z ) )  e. 
_V )
1243, 123, 7, 17, 11offval2 6351 . . . . . . . . 9  |-  ( ph  ->  ( ( ( CC 
X.  { R }
)  oF  x.  F )  oF  /  ( CC  X.  { R } ) )  =  ( z  e.  CC  |->  ( ( R  x.  ( F `  z ) )  /  R ) ) )
125121, 124, 163eqtr4d 2485 . . . . . . . 8  |-  ( ph  ->  ( ( ( CC 
X.  { R }
)  oF  x.  F )  oF  /  ( CC  X.  { R } ) )  =  F )
12656, 118div0d 10121 . . . . . . . . . 10  |-  ( ph  ->  ( 0  /  R
)  =  0 )
127126mpteq2dv 4394 . . . . . . . . 9  |-  ( ph  ->  ( z  e.  CC  |->  ( 0  /  R
) )  =  ( z  e.  CC  |->  0 ) )
128 0cnd 9394 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  CC )  ->  0  e.  CC )
129 df-0p 21163 . . . . . . . . . . . 12  |-  0p  =  ( CC  X.  { 0 } )
130 fconstmpt 4897 . . . . . . . . . . . 12  |-  ( CC 
X.  { 0 } )  =  ( z  e.  CC  |->  0 )
131129, 130eqtri 2463 . . . . . . . . . . 11  |-  0p  =  ( z  e.  CC  |->  0 )
132131a1i 11 . . . . . . . . . 10  |-  ( ph  ->  0p  =  ( z  e.  CC  |->  0 ) )
1333, 128, 7, 132, 11offval2 6351 . . . . . . . . 9  |-  ( ph  ->  ( 0p  oF  /  ( CC 
X.  { R }
) )  =  ( z  e.  CC  |->  ( 0  /  R ) ) )
134127, 133, 1323eqtr4d 2485 . . . . . . . 8  |-  ( ph  ->  ( 0p  oF  /  ( CC 
X.  { R }
) )  =  0p )
135125, 134eqeq12d 2457 . . . . . . 7  |-  ( ph  ->  ( ( ( ( CC  X.  { R } )  oF  x.  F )  oF  /  ( CC 
X.  { R }
) )  =  ( 0p  oF  /  ( CC  X.  { R } ) )  <-> 
F  =  0p ) )
136116, 135syl5ib 219 . . . . . 6  |-  ( ph  ->  ( ( ( CC 
X.  { R }
)  oF  x.  F )  =  0p  ->  F  = 
0p ) )
137136necon3d 2661 . . . . 5  |-  ( ph  ->  ( F  =/=  0p  ->  ( ( CC 
X.  { R }
)  oF  x.  F )  =/=  0p ) )
138115, 137mpd 15 . . . 4  |-  ( ph  ->  ( ( CC  X.  { R } )  oF  x.  F )  =/=  0p )
139 eldifsn 4015 . . . 4  |-  ( ( ( CC  X.  { R } )  oF  x.  F )  e.  ( (Poly `  ZZ )  \  { 0p } )  <->  ( (
( CC  X.  { R } )  oF  x.  F )  e.  (Poly `  ZZ )  /\  ( ( CC  X.  { R } )  oF  x.  F )  =/=  0p ) )
140112, 138, 139sylanbrc 664 . . 3  |-  ( ph  ->  ( ( CC  X.  { R } )  oF  x.  F )  e.  ( (Poly `  ZZ )  \  { 0p } ) )
1416fconst 5611 . . . . . . 7  |-  ( CC 
X.  { R }
) : CC --> { R }
142 ffn 5574 . . . . . . 7  |-  ( ( CC  X.  { R } ) : CC --> { R }  ->  ( CC  X.  { R }
)  Fn  CC )
143141, 142mp1i 12 . . . . . 6  |-  ( ph  ->  ( CC  X.  { R } )  Fn  CC )
144 ffn 5574 . . . . . . 7  |-  ( F : CC --> CC  ->  F  Fn  CC )
14515, 144syl 16 . . . . . 6  |-  ( ph  ->  F  Fn  CC )
146 inidm 3574 . . . . . 6  |-  ( CC 
i^i  CC )  =  CC
1476fvconst2 5948 . . . . . . 7  |-  ( A  e.  CC  ->  (
( CC  X.  { R } ) `  A
)  =  R )
148147adantl 466 . . . . . 6  |-  ( (
ph  /\  A  e.  CC )  ->  ( ( CC  X.  { R } ) `  A
)  =  R )
14998adantr 465 . . . . . 6  |-  ( (
ph  /\  A  e.  CC )  ->  ( F `
 A )  =  0 )
150143, 145, 3, 3, 146, 148, 149ofval 6344 . . . . 5  |-  ( (
ph  /\  A  e.  CC )  ->  ( ( ( CC  X.  { R } )  oF  x.  F ) `  A )  =  ( R  x.  0 ) )
1511, 150mpdan 668 . . . 4  |-  ( ph  ->  ( ( ( CC 
X.  { R }
)  oF  x.  F ) `  A
)  =  ( R  x.  0 ) )
15256mul01d 9583 . . . 4  |-  ( ph  ->  ( R  x.  0 )  =  0 )
153151, 152eqtrd 2475 . . 3  |-  ( ph  ->  ( ( ( CC 
X.  { R }
)  oF  x.  F ) `  A
)  =  0 )
154 fveq1 5705 . . . . 5  |-  ( f  =  ( ( CC 
X.  { R }
)  oF  x.  F )  ->  (
f `  A )  =  ( ( ( CC  X.  { R } )  oF  x.  F ) `  A ) )
155154eqeq1d 2451 . . . 4  |-  ( f  =  ( ( CC 
X.  { R }
)  oF  x.  F )  ->  (
( f `  A
)  =  0  <->  (
( ( CC  X.  { R } )  oF  x.  F ) `
 A )  =  0 ) )
156155rspcev 3088 . . 3  |-  ( ( ( ( CC  X.  { R } )  oF  x.  F )  e.  ( (Poly `  ZZ )  \  { 0p } )  /\  ( ( ( CC 
X.  { R }
)  oF  x.  F ) `  A
)  =  0 )  ->  E. f  e.  ( (Poly `  ZZ )  \  { 0p }
) ( f `  A )  =  0 )
157140, 153, 156syl2anc 661 . 2  |-  ( ph  ->  E. f  e.  ( (Poly `  ZZ )  \  { 0p }
) ( f `  A )  =  0 )
158 elaa 21797 . 2  |-  ( A  e.  AA  <->  ( A  e.  CC  /\  E. f  e.  ( (Poly `  ZZ )  \  { 0p } ) ( f `
 A )  =  0 ) )
1591, 157, 158sylanbrc 664 1  |-  ( ph  ->  A  e.  AA )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2620   E.wrex 2731   {crab 2734   _Vcvv 2987    \ cdif 3340    C_ wss 3343   (/)c0 3652   {csn 3892    e. cmpt 4365    X. cxp 4853   `'ccnv 4854    Fn wfn 5428   -->wf 5429   ` cfv 5433  (class class class)co 6106    e. cmpt2 6108    oFcof 6333   supcsup 7705   CCcc 9295   RRcr 9296   0cc0 9297   1c1 9298    x. cmul 9302    < clt 9433    / cdiv 10008   NNcn 10337   NN0cn0 10594   ZZcz 10661   ZZ>=cuz 10876   QQcq 10968   RR+crp 11006   ...cfz 11452    mod cmo 11723    seqcseq 11821   ^cexp 11880   sum_csu 13178   0pc0p 21162  Polycply 21667  coeffccoe 21669  degcdgr 21670   AAcaa 21795
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4418  ax-sep 4428  ax-nul 4436  ax-pow 4485  ax-pr 4546  ax-un 6387  ax-inf2 7862  ax-cnex 9353  ax-resscn 9354  ax-1cn 9355  ax-icn 9356  ax-addcl 9357  ax-addrcl 9358  ax-mulcl 9359  ax-mulrcl 9360  ax-mulcom 9361  ax-addass 9362  ax-mulass 9363  ax-distr 9364  ax-i2m1 9365  ax-1ne0 9366  ax-1rid 9367  ax-rnegex 9368  ax-rrecex 9369  ax-cnre 9370  ax-pre-lttri 9371  ax-pre-lttrn 9372  ax-pre-ltadd 9373  ax-pre-mulgt0 9374  ax-pre-sup 9375  ax-addf 9376
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2735  df-rex 2736  df-reu 2737  df-rmo 2738  df-rab 2739  df-v 2989  df-sbc 3202  df-csb 3304  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-pss 3359  df-nul 3653  df-if 3807  df-pw 3877  df-sn 3893  df-pr 3895  df-tp 3897  df-op 3899  df-uni 4107  df-int 4144  df-iun 4188  df-br 4308  df-opab 4366  df-mpt 4367  df-tr 4401  df-eprel 4647  df-id 4651  df-po 4656  df-so 4657  df-fr 4694  df-se 4695  df-we 4696  df-ord 4737  df-on 4738  df-lim 4739  df-suc 4740  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-rn 4866  df-res 4867  df-ima 4868  df-iota 5396  df-fun 5435  df-fn 5436  df-f 5437  df-f1 5438  df-fo 5439  df-f1o 5440  df-fv 5441  df-isom 5442  df-riota 6067  df-ov 6109  df-oprab 6110  df-mpt2 6111  df-of 6335  df-om 6492  df-1st 6592  df-2nd 6593  df-recs 6847  df-rdg 6881  df-1o 6935  df-oadd 6939  df-er 7116  df-map 7231  df-pm 7232  df-en 7326  df-dom 7327  df-sdom 7328  df-fin 7329  df-sup 7706  df-oi 7739  df-card 8124  df-pnf 9435  df-mnf 9436  df-xr 9437  df-ltxr 9438  df-le 9439  df-sub 9612  df-neg 9613  df-div 10009  df-nn 10338  df-2 10395  df-3 10396  df-n0 10595  df-z 10662  df-uz 10877  df-q 10969  df-rp 11007  df-fz 11453  df-fzo 11564  df-fl 11657  df-mod 11724  df-seq 11822  df-exp 11881  df-hash 12119  df-cj 12603  df-re 12604  df-im 12605  df-sqr 12739  df-abs 12740  df-clim 12981  df-rlim 12982  df-sum 13179  df-0p 21163  df-ply 21671  df-coe 21673  df-dgr 21674  df-aa 21796
This theorem is referenced by:  elqaa  21803
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