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Theorem elqaalem3 22843
Description: Lemma for elqaa 22844. (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 9590 . . . . . . . 8  |-  CC  e.  _V
32a1i 11 . . . . . . 7  |-  ( ph  ->  CC  e.  _V )
4 elqaa.6 . . . . . . . . 9  |-  R  =  (  seq 0 (  x.  ,  N ) `
 (deg `  F
) )
5 fvex 5882 . . . . . . . . 9  |-  (  seq 0 (  x.  ,  N ) `  (deg `  F ) )  e. 
_V
64, 5eqeltri 2541 . . . . . . . 8  |-  R  e. 
_V
76a1i 11 . . . . . . 7  |-  ( (
ph  /\  z  e.  CC )  ->  R  e. 
_V )
8 fvex 5882 . . . . . . . 8  |-  ( F `
 z )  e. 
_V
98a1i 11 . . . . . . 7  |-  ( (
ph  /\  z  e.  CC )  ->  ( F `
 z )  e. 
_V )
10 fconstmpt 5052 . . . . . . . 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 3483 . . . . . . . . 9  |-  ( ph  ->  F  e.  (Poly `  QQ ) )
14 plyf 22721 . . . . . . . . 9  |-  ( F  e.  (Poly `  QQ )  ->  F : CC --> CC )
1513, 14syl 16 . . . . . . . 8  |-  ( ph  ->  F : CC --> CC )
1615feqmptd 5926 . . . . . . 7  |-  ( ph  ->  F  =  ( z  e.  CC  |->  ( F `
 z ) ) )
173, 7, 9, 11, 16offval2 6555 . . . . . 6  |-  ( ph  ->  ( ( CC  X.  { R } )  oF  x.  F )  =  ( z  e.  CC  |->  ( R  x.  ( F `  z ) ) ) )
18 fzfid 12086 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  CC )  ->  ( 0 ... (deg `  F
) )  e.  Fin )
19 nn0uz 11140 . . . . . . . . . . . . . 14  |-  NN0  =  ( ZZ>= `  0 )
20 0zd 10897 . . . . . . . . . . . . . 14  |-  ( ph  ->  0  e.  ZZ )
21 ssrab2 3581 . . . . . . . . . . . . . . 15  |-  { n  e.  NN  |  ( ( B `  m )  x.  n )  e.  ZZ }  C_  NN
22 fveq2 5872 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( k  =  m  ->  ( B `  k )  =  ( B `  m ) )
2322oveq1d 6311 . . . . . . . . . . . . . . . . . . . . 21  |-  ( k  =  m  ->  (
( B `  k
)  x.  n )  =  ( ( B `
 m )  x.  n ) )
2423eleq1d 2526 . . . . . . . . . . . . . . . . . . . 20  |-  ( k  =  m  ->  (
( ( B `  k )  x.  n
)  e.  ZZ  <->  ( ( B `  m )  x.  n )  e.  ZZ ) )
2524rabbidv 3101 . . . . . . . . . . . . . . . . . . 19  |-  ( k  =  m  ->  { n  e.  NN  |  ( ( B `  k )  x.  n )  e.  ZZ }  =  {
n  e.  NN  | 
( ( B `  m )  x.  n
)  e.  ZZ }
)
2625supeq1d 7923 . . . . . . . . . . . . . . . . . 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 9683 . . . . . . . . . . . . . . . . . . 19  |-  `'  <  Or  RR
2928supex 7940 . . . . . . . . . . . . . . . . . 18  |-  sup ( { n  e.  NN  |  ( ( B `
 m )  x.  n )  e.  ZZ } ,  RR ,  `'  <  )  e.  _V
3026, 27, 29fvmpt 5956 . . . . . . . . . . . . . . . . 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 11141 . . . . . . . . . . . . . . . . . 18  |-  NN  =  ( ZZ>= `  1 )
3321, 32sseqtri 3531 . . . . . . . . . . . . . . . . 17  |-  { n  e.  NN  |  ( ( B `  m )  x.  n )  e.  ZZ }  C_  ( ZZ>=
`  1 )
34 0z 10896 . . . . . . . . . . . . . . . . . . . . . 22  |-  0  e.  ZZ
35 zq 11213 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( 0  e.  ZZ  ->  0  e.  QQ )
3634, 35ax-mp 5 . . . . . . . . . . . . . . . . . . . . 21  |-  0  e.  QQ
37 elqaa.4 . . . . . . . . . . . . . . . . . . . . . 22  |-  B  =  (coeff `  F )
3837coef2 22754 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( F  e.  (Poly `  QQ )  /\  0  e.  QQ )  ->  B : NN0 --> QQ )
3913, 36, 38sylancl 662 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  B : NN0 --> QQ )
4039ffvelrnda 6032 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  m  e.  NN0 )  ->  ( B `  m )  e.  QQ )
41 qmulz 11210 . . . . . . . . . . . . . . . . . . 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 3814 . . . . . . . . . . . . . . . . . 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 11190 . . . . . . . . . . . . . . . . 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 2545 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  m  e.  NN0 )  ->  ( N `  m )  e.  {
n  e.  NN  | 
( ( B `  m )  x.  n
)  e.  ZZ }
)
4821, 47sseldi 3497 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  m  e.  NN0 )  ->  ( N `  m )  e.  NN )
49 nnmulcl 10579 . . . . . . . . . . . . . . 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 12131 . . . . . . . . . . . . 13  |-  ( ph  ->  seq 0 (  x.  ,  N ) : NN0 --> NN )
52 dgrcl 22756 . . . . . . . . . . . . . 14  |-  ( F  e.  (Poly `  QQ )  ->  (deg `  F
)  e.  NN0 )
5313, 52syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  (deg `  F )  e.  NN0 )
5451, 53ffvelrnd 6033 . . . . . . . . . . . 12  |-  ( ph  ->  (  seq 0 (  x.  ,  N ) `
 (deg `  F
) )  e.  NN )
554, 54syl5eqel 2549 . . . . . . . . . . 11  |-  ( ph  ->  R  e.  NN )
5655nncnd 10572 . . . . . . . . . 10  |-  ( ph  ->  R  e.  CC )
5756adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  CC )  ->  R  e.  CC )
58 elfznn0 11797 . . . . . . . . . 10  |-  ( m  e.  ( 0 ... (deg `  F )
)  ->  m  e.  NN0 )
5937coef3 22755 . . . . . . . . . . . . . 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 6032 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  CC )  /\  m  e.  NN0 )  ->  ( B `  m )  e.  CC )
63 expcl 12187 . . . . . . . . . . . 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 9633 . . . . . . . . . 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 13611 . . . . . . . 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 2457 . . . . . . . . . . 11  |-  (deg `  F )  =  (deg
`  F )
6937, 68coeid2 22762 . . . . . . . . . 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 6312 . . . . . . . 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 9636 . . . . . . . . . 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 13537 . . . . . . . 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 2508 . . . . . . 7  |-  ( (
ph  /\  z  e.  CC )  ->  ( R  x.  ( F `  z ) )  = 
sum_ m  e.  (
0 ... (deg `  F
) ) ( ( R  x.  ( B `
 m ) )  x.  ( z ^
m ) ) )
7776mpteq2dva 4543 . . . . . 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 2498 . . . . 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 10893 . . . . . . 7  |-  ZZ  C_  CC
8079a1i 11 . . . . . 6  |-  ( ph  ->  ZZ  C_  CC )
8156adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  m  e.  NN0 )  ->  R  e.  CC )
8248nncnd 10572 . . . . . . . . . . 11  |-  ( (
ph  /\  m  e.  NN0 )  ->  ( N `  m )  e.  CC )
8348nnne0d 10601 . . . . . . . . . . 11  |-  ( (
ph  /\  m  e.  NN0 )  ->  ( N `  m )  =/=  0
)
8481, 82, 83divcan2d 10343 . . . . . . . . . 10  |-  ( (
ph  /\  m  e.  NN0 )  ->  ( ( N `  m )  x.  ( R  /  ( N `  m )
) )  =  R )
8584oveq2d 6312 . . . . . . . . 9  |-  ( (
ph  /\  m  e.  NN0 )  ->  ( ( B `  m )  x.  ( ( N `  m )  x.  ( R  /  ( N `  m ) ) ) )  =  ( ( B `  m )  x.  R ) )
8660ffvelrnda 6032 . . . . . . . . . 10  |-  ( (
ph  /\  m  e.  NN0 )  ->  ( B `  m )  e.  CC )
8781, 82, 83divcld 10341 . . . . . . . . . 10  |-  ( (
ph  /\  m  e.  NN0 )  ->  ( R  /  ( N `  m ) )  e.  CC )
8886, 82, 87mulassd 9636 . . . . . . . . 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 9634 . . . . . . . . 9  |-  ( (
ph  /\  m  e.  NN0 )  ->  ( R  x.  ( B `  m
) )  =  ( ( B `  m
)  x.  R ) )
9085, 88, 893eqtr4rd 2509 . . . . . . . 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 6304 . . . . . . . . . . . . 13  |-  ( n  =  ( N `  m )  ->  (
( B `  m
)  x.  n )  =  ( ( B `
 m )  x.  ( N `  m
) ) )
9392eleq1d 2526 . . . . . . . . . . . 12  |-  ( n  =  ( N `  m )  ->  (
( ( B `  m )  x.  n
)  e.  ZZ  <->  ( ( B `  m )  x.  ( N `  m
) )  e.  ZZ ) )
9493elrab 3257 . . . . . . . . . . 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 2457 . . . . . . . . . 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 22842 . . . . . . . . 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 10563 . . . . . . . . . . 11  |-  ( R  e.  NN  ->  R  e.  RR )
104 nnrp 11254 . . . . . . . . . . 11  |-  ( ( N `  m )  e.  NN  ->  ( N `  m )  e.  RR+ )
105 mod0 12006 . . . . . . . . . . 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 10996 . . . . . . 7  |-  ( (
ph  /\  m  e.  ( 0 ... (deg `  F ) ) )  ->  ( ( ( B `  m )  x.  ( N `  m ) )  x.  ( R  /  ( N `  m )
) )  e.  ZZ )
11091, 109eqeltrd 2545 . . . . . 6  |-  ( (
ph  /\  m  e.  ( 0 ... (deg `  F ) ) )  ->  ( R  x.  ( B `  m ) )  e.  ZZ )
11180, 53, 110elplyd 22725 . . . . 5  |-  ( ph  ->  ( z  e.  CC  |->  sum_
m  e.  ( 0 ... (deg `  F
) ) ( ( R  x.  ( B `
 m ) )  x.  ( z ^
m ) ) )  e.  (Poly `  ZZ ) )
11278, 111eqeltrd 2545 . . . 4  |-  ( ph  ->  ( ( CC  X.  { R } )  oF  x.  F )  e.  (Poly `  ZZ ) )
113 eldifsn 4157 . . . . . . 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 6303 . . . . . . 7  |-  ( ( ( CC  X.  { R } )  oF  x.  F )  =  0p  ->  (
( ( CC  X.  { R } )  oF  x.  F )  oF  /  ( CC  X.  { R }
) )  =  ( 0p  oF  /  ( CC  X.  { R } ) ) )
11715ffvelrnda 6032 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  CC )  ->  ( F `
 z )  e.  CC )
11855nnne0d 10601 . . . . . . . . . . . 12  |-  ( ph  ->  R  =/=  0 )
119118adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  CC )  ->  R  =/=  0 )
120117, 57, 119divcan3d 10346 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  CC )  ->  ( ( R  x.  ( F `
 z ) )  /  R )  =  ( F `  z
) )
121120mpteq2dva 4543 . . . . . . . . 9  |-  ( ph  ->  ( z  e.  CC  |->  ( ( R  x.  ( F `  z ) )  /  R ) )  =  ( z  e.  CC  |->  ( F `
 z ) ) )
122 ovex 6324 . . . . . . . . . . 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 6555 . . . . . . . . 9  |-  ( ph  ->  ( ( ( CC 
X.  { R }
)  oF  x.  F )  oF  /  ( CC  X.  { R } ) )  =  ( z  e.  CC  |->  ( ( R  x.  ( F `  z ) )  /  R ) ) )
125121, 124, 163eqtr4d 2508 . . . . . . . 8  |-  ( ph  ->  ( ( ( CC 
X.  { R }
)  oF  x.  F )  oF  /  ( CC  X.  { R } ) )  =  F )
12656, 118div0d 10340 . . . . . . . . . 10  |-  ( ph  ->  ( 0  /  R
)  =  0 )
127126mpteq2dv 4544 . . . . . . . . 9  |-  ( ph  ->  ( z  e.  CC  |->  ( 0  /  R
) )  =  ( z  e.  CC  |->  0 ) )
128 0cnd 9606 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  CC )  ->  0  e.  CC )
129 df-0p 22203 . . . . . . . . . . . 12  |-  0p  =  ( CC  X.  { 0 } )
130 fconstmpt 5052 . . . . . . . . . . . 12  |-  ( CC 
X.  { 0 } )  =  ( z  e.  CC  |->  0 )
131129, 130eqtri 2486 . . . . . . . . . . 11  |-  0p  =  ( z  e.  CC  |->  0 )
132131a1i 11 . . . . . . . . . 10  |-  ( ph  ->  0p  =  ( z  e.  CC  |->  0 ) )
1333, 128, 7, 132, 11offval2 6555 . . . . . . . . 9  |-  ( ph  ->  ( 0p  oF  /  ( CC 
X.  { R }
) )  =  ( z  e.  CC  |->  ( 0  /  R ) ) )
134127, 133, 1323eqtr4d 2508 . . . . . . . 8  |-  ( ph  ->  ( 0p  oF  /  ( CC 
X.  { R }
) )  =  0p )
135125, 134eqeq12d 2479 . . . . . . 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 2681 . . . . 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 4157 . . . 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 5777 . . . . . . 7  |-  ( CC 
X.  { R }
) : CC --> { R }
142 ffn 5737 . . . . . . 7  |-  ( ( CC  X.  { R } ) : CC --> { R }  ->  ( CC  X.  { R }
)  Fn  CC )
143141, 142mp1i 12 . . . . . 6  |-  ( ph  ->  ( CC  X.  { R } )  Fn  CC )
144 ffn 5737 . . . . . . 7  |-  ( F : CC --> CC  ->  F  Fn  CC )
14515, 144syl 16 . . . . . 6  |-  ( ph  ->  F  Fn  CC )
146 inidm 3703 . . . . . 6  |-  ( CC 
i^i  CC )  =  CC
1476fvconst2 6128 . . . . . . 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 6548 . . . . 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 9796 . . . 4  |-  ( ph  ->  ( R  x.  0 )  =  0 )
153151, 152eqtrd 2498 . . 3  |-  ( ph  ->  ( ( ( CC 
X.  { R }
)  oF  x.  F ) `  A
)  =  0 )
154 fveq1 5871 . . . . 5  |-  ( f  =  ( ( CC 
X.  { R }
)  oF  x.  F )  ->  (
f `  A )  =  ( ( ( CC  X.  { R } )  oF  x.  F ) `  A ) )
155154eqeq1d 2459 . . . 4  |-  ( f  =  ( ( CC 
X.  { R }
)  oF  x.  F )  ->  (
( f `  A
)  =  0  <->  (
( ( CC  X.  { R } )  oF  x.  F ) `
 A )  =  0 ) )
156155rspcev 3210 . . 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 22838 . 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 1395    e. wcel 1819    =/= wne 2652   E.wrex 2808   {crab 2811   _Vcvv 3109    \ cdif 3468    C_ wss 3471   (/)c0 3793   {csn 4032    |-> cmpt 4515    X. cxp 5006   `'ccnv 5007    Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6296    |-> cmpt2 6298    oFcof 6537   supcsup 7918   CCcc 9507   RRcr 9508   0cc0 9509   1c1 9510    x. cmul 9514    < clt 9645    / cdiv 10227   NNcn 10556   NN0cn0 10816   ZZcz 10885   ZZ>=cuz 11106   QQcq 11207   RR+crp 11245   ...cfz 11697    mod cmo 11999    seqcseq 12110   ^cexp 12169   sum_csu 13520   0pc0p 22202  Polycply 22707  coeffccoe 22709  degcdgr 22710   AAcaa 22836
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587  ax-addf 9588
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-pm 7441  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-sup 7919  df-oi 7953  df-card 8337  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-q 11208  df-rp 11246  df-fz 11698  df-fzo 11822  df-fl 11932  df-mod 12000  df-seq 12111  df-exp 12170  df-hash 12409  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081  df-clim 13323  df-rlim 13324  df-sum 13521  df-0p 22203  df-ply 22711  df-coe 22713  df-dgr 22714  df-aa 22837
This theorem is referenced by:  elqaa  22844
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