MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elqaalem3 Structured version   Unicode version

Theorem elqaalem3 22451
Description: Lemma for elqaa 22452. (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 9569 . . . . . . . 8  |-  CC  e.  _V
32a1i 11 . . . . . . 7  |-  ( ph  ->  CC  e.  _V )
4 elqaa.6 . . . . . . . . 9  |-  R  =  (  seq 0 (  x.  ,  N ) `
 (deg `  F
) )
5 fvex 5874 . . . . . . . . 9  |-  (  seq 0 (  x.  ,  N ) `  (deg `  F ) )  e. 
_V
64, 5eqeltri 2551 . . . . . . . 8  |-  R  e. 
_V
76a1i 11 . . . . . . 7  |-  ( (
ph  /\  z  e.  CC )  ->  R  e. 
_V )
8 fvex 5874 . . . . . . . 8  |-  ( F `
 z )  e. 
_V
98a1i 11 . . . . . . 7  |-  ( (
ph  /\  z  e.  CC )  ->  ( F `
 z )  e. 
_V )
10 fconstmpt 5042 . . . . . . . 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 3488 . . . . . . . . 9  |-  ( ph  ->  F  e.  (Poly `  QQ ) )
14 plyf 22330 . . . . . . . . 9  |-  ( F  e.  (Poly `  QQ )  ->  F : CC --> CC )
1513, 14syl 16 . . . . . . . 8  |-  ( ph  ->  F : CC --> CC )
1615feqmptd 5918 . . . . . . 7  |-  ( ph  ->  F  =  ( z  e.  CC  |->  ( F `
 z ) ) )
173, 7, 9, 11, 16offval2 6538 . . . . . 6  |-  ( ph  ->  ( ( CC  X.  { R } )  oF  x.  F )  =  ( z  e.  CC  |->  ( R  x.  ( F `  z ) ) ) )
18 fzfid 12047 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  CC )  ->  ( 0 ... (deg `  F
) )  e.  Fin )
19 nn0uz 11112 . . . . . . . . . . . . . 14  |-  NN0  =  ( ZZ>= `  0 )
20 0zd 10872 . . . . . . . . . . . . . 14  |-  ( ph  ->  0  e.  ZZ )
21 ssrab2 3585 . . . . . . . . . . . . . . 15  |-  { n  e.  NN  |  ( ( B `  m )  x.  n )  e.  ZZ }  C_  NN
22 fveq2 5864 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( k  =  m  ->  ( B `  k )  =  ( B `  m ) )
2322oveq1d 6297 . . . . . . . . . . . . . . . . . . . . 21  |-  ( k  =  m  ->  (
( B `  k
)  x.  n )  =  ( ( B `
 m )  x.  n ) )
2423eleq1d 2536 . . . . . . . . . . . . . . . . . . . 20  |-  ( k  =  m  ->  (
( ( B `  k )  x.  n
)  e.  ZZ  <->  ( ( B `  m )  x.  n )  e.  ZZ ) )
2524rabbidv 3105 . . . . . . . . . . . . . . . . . . 19  |-  ( k  =  m  ->  { n  e.  NN  |  ( ( B `  k )  x.  n )  e.  ZZ }  =  {
n  e.  NN  | 
( ( B `  m )  x.  n
)  e.  ZZ }
)
2625supeq1d 7902 . . . . . . . . . . . . . . . . . 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 9662 . . . . . . . . . . . . . . . . . . 19  |-  `'  <  Or  RR
2928supex 7919 . . . . . . . . . . . . . . . . . 18  |-  sup ( { n  e.  NN  |  ( ( B `
 m )  x.  n )  e.  ZZ } ,  RR ,  `'  <  )  e.  _V
3026, 27, 29fvmpt 5948 . . . . . . . . . . . . . . . . 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 11113 . . . . . . . . . . . . . . . . . 18  |-  NN  =  ( ZZ>= `  1 )
3321, 32sseqtri 3536 . . . . . . . . . . . . . . . . 17  |-  { n  e.  NN  |  ( ( B `  m )  x.  n )  e.  ZZ }  C_  ( ZZ>=
`  1 )
34 0z 10871 . . . . . . . . . . . . . . . . . . . . . 22  |-  0  e.  ZZ
35 zq 11184 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( 0  e.  ZZ  ->  0  e.  QQ )
3634, 35ax-mp 5 . . . . . . . . . . . . . . . . . . . . 21  |-  0  e.  QQ
37 elqaa.4 . . . . . . . . . . . . . . . . . . . . . 22  |-  B  =  (coeff `  F )
3837coef2 22363 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( F  e.  (Poly `  QQ )  /\  0  e.  QQ )  ->  B : NN0 --> QQ )
3913, 36, 38sylancl 662 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  B : NN0 --> QQ )
4039ffvelrnda 6019 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  m  e.  NN0 )  ->  ( B `  m )  e.  QQ )
41 qmulz 11181 . . . . . . . . . . . . . . . . . . 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 3805 . . . . . . . . . . . . . . . . . 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 11161 . . . . . . . . . . . . . . . . 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 2555 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  m  e.  NN0 )  ->  ( N `  m )  e.  {
n  e.  NN  | 
( ( B `  m )  x.  n
)  e.  ZZ }
)
4821, 47sseldi 3502 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  m  e.  NN0 )  ->  ( N `  m )  e.  NN )
49 nnmulcl 10555 . . . . . . . . . . . . . . 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 12092 . . . . . . . . . . . . 13  |-  ( ph  ->  seq 0 (  x.  ,  N ) : NN0 --> NN )
52 dgrcl 22365 . . . . . . . . . . . . . 14  |-  ( F  e.  (Poly `  QQ )  ->  (deg `  F
)  e.  NN0 )
5313, 52syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  (deg `  F )  e.  NN0 )
5451, 53ffvelrnd 6020 . . . . . . . . . . . 12  |-  ( ph  ->  (  seq 0 (  x.  ,  N ) `
 (deg `  F
) )  e.  NN )
554, 54syl5eqel 2559 . . . . . . . . . . 11  |-  ( ph  ->  R  e.  NN )
5655nncnd 10548 . . . . . . . . . 10  |-  ( ph  ->  R  e.  CC )
5756adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  CC )  ->  R  e.  CC )
58 elfznn0 11766 . . . . . . . . . 10  |-  ( m  e.  ( 0 ... (deg `  F )
)  ->  m  e.  NN0 )
5937coef3 22364 . . . . . . . . . . . . . 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 6019 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  CC )  /\  m  e.  NN0 )  ->  ( B `  m )  e.  CC )
63 expcl 12148 . . . . . . . . . . . 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 9612 . . . . . . . . . 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 13558 . . . . . . . 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 2467 . . . . . . . . . . 11  |-  (deg `  F )  =  (deg
`  F )
6937, 68coeid2 22371 . . . . . . . . . 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 6298 . . . . . . . 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 9615 . . . . . . . . . 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 13484 . . . . . . . 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 2518 . . . . . . 7  |-  ( (
ph  /\  z  e.  CC )  ->  ( R  x.  ( F `  z ) )  = 
sum_ m  e.  (
0 ... (deg `  F
) ) ( ( R  x.  ( B `
 m ) )  x.  ( z ^
m ) ) )
7776mpteq2dva 4533 . . . . . 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 2508 . . . . 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 10868 . . . . . . 7  |-  ZZ  C_  CC
8079a1i 11 . . . . . 6  |-  ( ph  ->  ZZ  C_  CC )
8156adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  m  e.  NN0 )  ->  R  e.  CC )
8248nncnd 10548 . . . . . . . . . . 11  |-  ( (
ph  /\  m  e.  NN0 )  ->  ( N `  m )  e.  CC )
8348nnne0d 10576 . . . . . . . . . . 11  |-  ( (
ph  /\  m  e.  NN0 )  ->  ( N `  m )  =/=  0
)
8481, 82, 83divcan2d 10318 . . . . . . . . . 10  |-  ( (
ph  /\  m  e.  NN0 )  ->  ( ( N `  m )  x.  ( R  /  ( N `  m )
) )  =  R )
8584oveq2d 6298 . . . . . . . . 9  |-  ( (
ph  /\  m  e.  NN0 )  ->  ( ( B `  m )  x.  ( ( N `  m )  x.  ( R  /  ( N `  m ) ) ) )  =  ( ( B `  m )  x.  R ) )
8660ffvelrnda 6019 . . . . . . . . . 10  |-  ( (
ph  /\  m  e.  NN0 )  ->  ( B `  m )  e.  CC )
8781, 82, 83divcld 10316 . . . . . . . . . 10  |-  ( (
ph  /\  m  e.  NN0 )  ->  ( R  /  ( N `  m ) )  e.  CC )
8886, 82, 87mulassd 9615 . . . . . . . . 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 9613 . . . . . . . . 9  |-  ( (
ph  /\  m  e.  NN0 )  ->  ( R  x.  ( B `  m
) )  =  ( ( B `  m
)  x.  R ) )
9085, 88, 893eqtr4rd 2519 . . . . . . . 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 6290 . . . . . . . . . . . . 13  |-  ( n  =  ( N `  m )  ->  (
( B `  m
)  x.  n )  =  ( ( B `
 m )  x.  ( N `  m
) ) )
9392eleq1d 2536 . . . . . . . . . . . 12  |-  ( n  =  ( N `  m )  ->  (
( ( B `  m )  x.  n
)  e.  ZZ  <->  ( ( B `  m )  x.  ( N `  m
) )  e.  ZZ ) )
9493elrab 3261 . . . . . . . . . . 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 2467 . . . . . . . . . 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 22450 . . . . . . . . 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 10539 . . . . . . . . . . 11  |-  ( R  e.  NN  ->  R  e.  RR )
104 nnrp 11225 . . . . . . . . . . 11  |-  ( ( N `  m )  e.  NN  ->  ( N `  m )  e.  RR+ )
105 mod0 11967 . . . . . . . . . . 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 10968 . . . . . . 7  |-  ( (
ph  /\  m  e.  ( 0 ... (deg `  F ) ) )  ->  ( ( ( B `  m )  x.  ( N `  m ) )  x.  ( R  /  ( N `  m )
) )  e.  ZZ )
11091, 109eqeltrd 2555 . . . . . 6  |-  ( (
ph  /\  m  e.  ( 0 ... (deg `  F ) ) )  ->  ( R  x.  ( B `  m ) )  e.  ZZ )
11180, 53, 110elplyd 22334 . . . . 5  |-  ( ph  ->  ( z  e.  CC  |->  sum_
m  e.  ( 0 ... (deg `  F
) ) ( ( R  x.  ( B `
 m ) )  x.  ( z ^
m ) ) )  e.  (Poly `  ZZ ) )
11278, 111eqeltrd 2555 . . . 4  |-  ( ph  ->  ( ( CC  X.  { R } )  oF  x.  F )  e.  (Poly `  ZZ ) )
113 eldifsn 4152 . . . . . . 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 6289 . . . . . . 7  |-  ( ( ( CC  X.  { R } )  oF  x.  F )  =  0p  ->  (
( ( CC  X.  { R } )  oF  x.  F )  oF  /  ( CC  X.  { R }
) )  =  ( 0p  oF  /  ( CC  X.  { R } ) ) )
11715ffvelrnda 6019 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  CC )  ->  ( F `
 z )  e.  CC )
11855nnne0d 10576 . . . . . . . . . . . 12  |-  ( ph  ->  R  =/=  0 )
119118adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  CC )  ->  R  =/=  0 )
120117, 57, 119divcan3d 10321 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  CC )  ->  ( ( R  x.  ( F `
 z ) )  /  R )  =  ( F `  z
) )
121120mpteq2dva 4533 . . . . . . . . 9  |-  ( ph  ->  ( z  e.  CC  |->  ( ( R  x.  ( F `  z ) )  /  R ) )  =  ( z  e.  CC  |->  ( F `
 z ) ) )
122 ovex 6307 . . . . . . . . . . 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 6538 . . . . . . . . 9  |-  ( ph  ->  ( ( ( CC 
X.  { R }
)  oF  x.  F )  oF  /  ( CC  X.  { R } ) )  =  ( z  e.  CC  |->  ( ( R  x.  ( F `  z ) )  /  R ) ) )
125121, 124, 163eqtr4d 2518 . . . . . . . 8  |-  ( ph  ->  ( ( ( CC 
X.  { R }
)  oF  x.  F )  oF  /  ( CC  X.  { R } ) )  =  F )
12656, 118div0d 10315 . . . . . . . . . 10  |-  ( ph  ->  ( 0  /  R
)  =  0 )
127126mpteq2dv 4534 . . . . . . . . 9  |-  ( ph  ->  ( z  e.  CC  |->  ( 0  /  R
) )  =  ( z  e.  CC  |->  0 ) )
128 0cnd 9585 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  CC )  ->  0  e.  CC )
129 df-0p 21812 . . . . . . . . . . . 12  |-  0p  =  ( CC  X.  { 0 } )
130 fconstmpt 5042 . . . . . . . . . . . 12  |-  ( CC 
X.  { 0 } )  =  ( z  e.  CC  |->  0 )
131129, 130eqtri 2496 . . . . . . . . . . 11  |-  0p  =  ( z  e.  CC  |->  0 )
132131a1i 11 . . . . . . . . . 10  |-  ( ph  ->  0p  =  ( z  e.  CC  |->  0 ) )
1333, 128, 7, 132, 11offval2 6538 . . . . . . . . 9  |-  ( ph  ->  ( 0p  oF  /  ( CC 
X.  { R }
) )  =  ( z  e.  CC  |->  ( 0  /  R ) ) )
134127, 133, 1323eqtr4d 2518 . . . . . . . 8  |-  ( ph  ->  ( 0p  oF  /  ( CC 
X.  { R }
) )  =  0p )
135125, 134eqeq12d 2489 . . . . . . 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 2691 . . . . 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 4152 . . . 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 5769 . . . . . . 7  |-  ( CC 
X.  { R }
) : CC --> { R }
142 ffn 5729 . . . . . . 7  |-  ( ( CC  X.  { R } ) : CC --> { R }  ->  ( CC  X.  { R }
)  Fn  CC )
143141, 142mp1i 12 . . . . . 6  |-  ( ph  ->  ( CC  X.  { R } )  Fn  CC )
144 ffn 5729 . . . . . . 7  |-  ( F : CC --> CC  ->  F  Fn  CC )
14515, 144syl 16 . . . . . 6  |-  ( ph  ->  F  Fn  CC )
146 inidm 3707 . . . . . 6  |-  ( CC 
i^i  CC )  =  CC
1476fvconst2 6114 . . . . . . 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 6531 . . . . 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 9774 . . . 4  |-  ( ph  ->  ( R  x.  0 )  =  0 )
153151, 152eqtrd 2508 . . 3  |-  ( ph  ->  ( ( ( CC 
X.  { R }
)  oF  x.  F ) `  A
)  =  0 )
154 fveq1 5863 . . . . 5  |-  ( f  =  ( ( CC 
X.  { R }
)  oF  x.  F )  ->  (
f `  A )  =  ( ( ( CC  X.  { R } )  oF  x.  F ) `  A ) )
155154eqeq1d 2469 . . . 4  |-  ( f  =  ( ( CC 
X.  { R }
)  oF  x.  F )  ->  (
( f `  A
)  =  0  <->  (
( ( CC  X.  { R } )  oF  x.  F ) `
 A )  =  0 ) )
156155rspcev 3214 . . 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 22446 . 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 1379    e. wcel 1767    =/= wne 2662   E.wrex 2815   {crab 2818   _Vcvv 3113    \ cdif 3473    C_ wss 3476   (/)c0 3785   {csn 4027    |-> cmpt 4505    X. cxp 4997   `'ccnv 4998    Fn wfn 5581   -->wf 5582   ` cfv 5586  (class class class)co 6282    |-> cmpt2 6284    oFcof 6520   supcsup 7896   CCcc 9486   RRcr 9487   0cc0 9488   1c1 9489    x. cmul 9493    < clt 9624    / cdiv 10202   NNcn 10532   NN0cn0 10791   ZZcz 10860   ZZ>=cuz 11078   QQcq 11178   RR+crp 11216   ...cfz 11668    mod cmo 11960    seqcseq 12071   ^cexp 12130   sum_csu 13467   0pc0p 21811  Polycply 22316  coeffccoe 22318  degcdgr 22319   AAcaa 22444
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566  ax-addf 9567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-of 6522  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-map 7419  df-pm 7420  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-sup 7897  df-oi 7931  df-card 8316  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11079  df-q 11179  df-rp 11217  df-fz 11669  df-fzo 11789  df-fl 11893  df-mod 11961  df-seq 12072  df-exp 12131  df-hash 12370  df-cj 12891  df-re 12892  df-im 12893  df-sqrt 13027  df-abs 13028  df-clim 13270  df-rlim 13271  df-sum 13468  df-0p 21812  df-ply 22320  df-coe 22322  df-dgr 22323  df-aa 22445
This theorem is referenced by:  elqaa  22452
  Copyright terms: Public domain W3C validator