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Theorem elqaalem3OLD 23356
Description: Lemma for elqaa 23357. (Contributed by Mario Carneiro, 23-Jul-2014.) Obsolete version of elqaalem1 23351 as of 3-Oct-2020. (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
elqaaOLD.1  |-  ( ph  ->  A  e.  CC )
elqaaOLD.2  |-  ( ph  ->  F  e.  ( (Poly `  QQ )  \  {
0p } ) )
elqaaOLD.3  |-  ( ph  ->  ( F `  A
)  =  0 )
elqaaOLD.4  |-  B  =  (coeff `  F )
elqaaOLD.5  |-  N  =  ( k  e.  NN0  |->  sup ( { n  e.  NN  |  ( ( B `  k )  x.  n )  e.  ZZ } ,  RR ,  `'  <  ) )
elqaaOLD.6  |-  R  =  (  seq 0 (  x.  ,  N ) `
 (deg `  F
) )
Assertion
Ref Expression
elqaalem3OLD  |-  ( 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 elqaalem3OLD
Dummy variables  f  m  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elqaaOLD.1 . 2  |-  ( ph  ->  A  e.  CC )
2 cnex 9638 . . . . . . . 8  |-  CC  e.  _V
32a1i 11 . . . . . . 7  |-  ( ph  ->  CC  e.  _V )
4 elqaaOLD.6 . . . . . . . . 9  |-  R  =  (  seq 0 (  x.  ,  N ) `
 (deg `  F
) )
5 fvex 5889 . . . . . . . . 9  |-  (  seq 0 (  x.  ,  N ) `  (deg `  F ) )  e. 
_V
64, 5eqeltri 2545 . . . . . . . 8  |-  R  e. 
_V
76a1i 11 . . . . . . 7  |-  ( (
ph  /\  z  e.  CC )  ->  R  e. 
_V )
8 fvex 5889 . . . . . . . 8  |-  ( F `
 z )  e. 
_V
98a1i 11 . . . . . . 7  |-  ( (
ph  /\  z  e.  CC )  ->  ( F `
 z )  e. 
_V )
10 fconstmpt 4883 . . . . . . . 8  |-  ( CC 
X.  { R }
)  =  ( z  e.  CC  |->  R )
1110a1i 11 . . . . . . 7  |-  ( ph  ->  ( CC  X.  { R } )  =  ( z  e.  CC  |->  R ) )
12 elqaaOLD.2 . . . . . . . . . 10  |-  ( ph  ->  F  e.  ( (Poly `  QQ )  \  {
0p } ) )
1312eldifad 3402 . . . . . . . . 9  |-  ( ph  ->  F  e.  (Poly `  QQ ) )
14 plyf 23231 . . . . . . . . 9  |-  ( F  e.  (Poly `  QQ )  ->  F : CC --> CC )
1513, 14syl 17 . . . . . . . 8  |-  ( ph  ->  F : CC --> CC )
1615feqmptd 5932 . . . . . . 7  |-  ( ph  ->  F  =  ( z  e.  CC  |->  ( F `
 z ) ) )
173, 7, 9, 11, 16offval2 6567 . . . . . 6  |-  ( ph  ->  ( ( CC  X.  { R } )  oF  x.  F )  =  ( z  e.  CC  |->  ( R  x.  ( F `  z ) ) ) )
18 fzfid 12224 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  CC )  ->  ( 0 ... (deg `  F
) )  e.  Fin )
19 nn0uz 11217 . . . . . . . . . . . . . 14  |-  NN0  =  ( ZZ>= `  0 )
20 0zd 10973 . . . . . . . . . . . . . 14  |-  ( ph  ->  0  e.  ZZ )
21 ssrab2 3500 . . . . . . . . . . . . . . 15  |-  { n  e.  NN  |  ( ( B `  m )  x.  n )  e.  ZZ }  C_  NN
22 fveq2 5879 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( k  =  m  ->  ( B `  k )  =  ( B `  m ) )
2322oveq1d 6323 . . . . . . . . . . . . . . . . . . . . 21  |-  ( k  =  m  ->  (
( B `  k
)  x.  n )  =  ( ( B `
 m )  x.  n ) )
2423eleq1d 2533 . . . . . . . . . . . . . . . . . . . 20  |-  ( k  =  m  ->  (
( ( B `  k )  x.  n
)  e.  ZZ  <->  ( ( B `  m )  x.  n )  e.  ZZ ) )
2524rabbidv 3022 . . . . . . . . . . . . . . . . . . 19  |-  ( k  =  m  ->  { n  e.  NN  |  ( ( B `  k )  x.  n )  e.  ZZ }  =  {
n  e.  NN  | 
( ( B `  m )  x.  n
)  e.  ZZ }
)
2625supeq1d 7978 . . . . . . . . . . . . . . . . . 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 elqaaOLD.5 . . . . . . . . . . . . . . . . . 18  |-  N  =  ( k  e.  NN0  |->  sup ( { n  e.  NN  |  ( ( B `  k )  x.  n )  e.  ZZ } ,  RR ,  `'  <  ) )
28 gtso 9733 . . . . . . . . . . . . . . . . . . 19  |-  `'  <  Or  RR
2928supex 7995 . . . . . . . . . . . . . . . . . 18  |-  sup ( { n  e.  NN  |  ( ( B `
 m )  x.  n )  e.  ZZ } ,  RR ,  `'  <  )  e.  _V
3026, 27, 29fvmpt 5963 . . . . . . . . . . . . . . . . 17  |-  ( m  e.  NN0  ->  ( N `
 m )  =  sup ( { n  e.  NN  |  ( ( B `  m )  x.  n )  e.  ZZ } ,  RR ,  `'  <  ) )
3130adantl 473 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  m  e.  NN0 )  ->  ( N `  m )  =  sup ( { n  e.  NN  |  ( ( B `
 m )  x.  n )  e.  ZZ } ,  RR ,  `'  <  ) )
32 nnuz 11218 . . . . . . . . . . . . . . . . . 18  |-  NN  =  ( ZZ>= `  1 )
3321, 32sseqtri 3450 . . . . . . . . . . . . . . . . 17  |-  { n  e.  NN  |  ( ( B `  m )  x.  n )  e.  ZZ }  C_  ( ZZ>=
`  1 )
34 0z 10972 . . . . . . . . . . . . . . . . . . . . . 22  |-  0  e.  ZZ
35 zq 11293 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( 0  e.  ZZ  ->  0  e.  QQ )
3634, 35ax-mp 5 . . . . . . . . . . . . . . . . . . . . 21  |-  0  e.  QQ
37 elqaaOLD.4 . . . . . . . . . . . . . . . . . . . . . 22  |-  B  =  (coeff `  F )
3837coef2 23264 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( F  e.  (Poly `  QQ )  /\  0  e.  QQ )  ->  B : NN0 --> QQ )
3913, 36, 38sylancl 675 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  B : NN0 --> QQ )
4039ffvelrnda 6037 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  m  e.  NN0 )  ->  ( B `  m )  e.  QQ )
41 qmulz 11290 . . . . . . . . . . . . . . . . . . 19  |-  ( ( B `  m )  e.  QQ  ->  E. n  e.  NN  ( ( B `
 m )  x.  n )  e.  ZZ )
4240, 41syl 17 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  m  e.  NN0 )  ->  E. n  e.  NN  ( ( B `
 m )  x.  n )  e.  ZZ )
43 rabn0 3755 . . . . . . . . . . . . . . . . . 18  |-  ( { n  e.  NN  | 
( ( B `  m )  x.  n
)  e.  ZZ }  =/=  (/)  <->  E. n  e.  NN  ( ( B `  m )  x.  n
)  e.  ZZ )
4442, 43sylibr 217 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  m  e.  NN0 )  ->  { n  e.  NN  |  ( ( B `  m )  x.  n )  e.  ZZ }  =/=  (/) )
45 infmssuzclOLD 11270 . . . . . . . . . . . . . . . . 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 676 . . . . . . . . . . . . . . . 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 2549 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  m  e.  NN0 )  ->  ( N `  m )  e.  {
n  e.  NN  | 
( ( B `  m )  x.  n
)  e.  ZZ }
)
4821, 47sseldi 3416 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  m  e.  NN0 )  ->  ( N `  m )  e.  NN )
49 nnmulcl 10654 . . . . . . . . . . . . . . 15  |-  ( ( m  e.  NN  /\  k  e.  NN )  ->  ( m  x.  k
)  e.  NN )
5049adantl 473 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( m  e.  NN  /\  k  e.  NN ) )  -> 
( m  x.  k
)  e.  NN )
5119, 20, 48, 50seqf 12272 . . . . . . . . . . . . 13  |-  ( ph  ->  seq 0 (  x.  ,  N ) : NN0 --> NN )
52 dgrcl 23266 . . . . . . . . . . . . . 14  |-  ( F  e.  (Poly `  QQ )  ->  (deg `  F
)  e.  NN0 )
5313, 52syl 17 . . . . . . . . . . . . 13  |-  ( ph  ->  (deg `  F )  e.  NN0 )
5451, 53ffvelrnd 6038 . . . . . . . . . . . 12  |-  ( ph  ->  (  seq 0 (  x.  ,  N ) `
 (deg `  F
) )  e.  NN )
554, 54syl5eqel 2553 . . . . . . . . . . 11  |-  ( ph  ->  R  e.  NN )
5655nncnd 10647 . . . . . . . . . 10  |-  ( ph  ->  R  e.  CC )
5756adantr 472 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  CC )  ->  R  e.  CC )
58 elfznn0 11913 . . . . . . . . . 10  |-  ( m  e.  ( 0 ... (deg `  F )
)  ->  m  e.  NN0 )
5937coef3 23265 . . . . . . . . . . . . . 14  |-  ( F  e.  (Poly `  QQ )  ->  B : NN0 --> CC )
6013, 59syl 17 . . . . . . . . . . . . 13  |-  ( ph  ->  B : NN0 --> CC )
6160adantr 472 . . . . . . . . . . . 12  |-  ( (
ph  /\  z  e.  CC )  ->  B : NN0
--> CC )
6261ffvelrnda 6037 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  CC )  /\  m  e.  NN0 )  ->  ( B `  m )  e.  CC )
63 expcl 12328 . . . . . . . . . . . 12  |-  ( ( z  e.  CC  /\  m  e.  NN0 )  -> 
( z ^ m
)  e.  CC )
6463adantll 728 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  CC )  /\  m  e.  NN0 )  ->  (
z ^ m )  e.  CC )
6562, 64mulcld 9681 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  CC )  /\  m  e.  NN0 )  ->  (
( B `  m
)  x.  ( z ^ m ) )  e.  CC )
6658, 65sylan2 482 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  CC )  /\  m  e.  ( 0 ... (deg `  F ) ) )  ->  ( ( B `
 m )  x.  ( z ^ m
) )  e.  CC )
6718, 57, 66fsummulc2 13922 . . . . . . . 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 2471 . . . . . . . . . . 11  |-  (deg `  F )  =  (deg
`  F )
6937, 68coeid2 23272 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  QQ )  /\  z  e.  CC )  ->  ( F `  z )  =  sum_ m  e.  ( 0 ... (deg `  F ) ) ( ( B `  m
)  x.  ( z ^ m ) ) )
7013, 69sylan 479 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  CC )  ->  ( F `
 z )  = 
sum_ m  e.  (
0 ... (deg `  F
) ) ( ( B `  m )  x.  ( z ^
m ) ) )
7170oveq2d 6324 . . . . . . . 8  |-  ( (
ph  /\  z  e.  CC )  ->  ( R  x.  ( F `  z ) )  =  ( R  x.  sum_ m  e.  ( 0 ... (deg `  F )
) ( ( B `
 m )  x.  ( z ^ m
) ) ) )
7257adantr 472 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  CC )  /\  m  e.  NN0 )  ->  R  e.  CC )
7372, 62, 64mulassd 9684 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  CC )  /\  m  e.  NN0 )  ->  (
( R  x.  ( B `  m )
)  x.  ( z ^ m ) )  =  ( R  x.  ( ( B `  m )  x.  (
z ^ m ) ) ) )
7458, 73sylan2 482 . . . . . . . . 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 13846 . . . . . . . 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 2515 . . . . . . 7  |-  ( (
ph  /\  z  e.  CC )  ->  ( R  x.  ( F `  z ) )  = 
sum_ m  e.  (
0 ... (deg `  F
) ) ( ( R  x.  ( B `
 m ) )  x.  ( z ^
m ) ) )
7776mpteq2dva 4482 . . . . . 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 2505 . . . . 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 10969 . . . . . . 7  |-  ZZ  C_  CC
8079a1i 11 . . . . . 6  |-  ( ph  ->  ZZ  C_  CC )
8156adantr 472 . . . . . . . . . . 11  |-  ( (
ph  /\  m  e.  NN0 )  ->  R  e.  CC )
8248nncnd 10647 . . . . . . . . . . 11  |-  ( (
ph  /\  m  e.  NN0 )  ->  ( N `  m )  e.  CC )
8348nnne0d 10676 . . . . . . . . . . 11  |-  ( (
ph  /\  m  e.  NN0 )  ->  ( N `  m )  =/=  0
)
8481, 82, 83divcan2d 10407 . . . . . . . . . 10  |-  ( (
ph  /\  m  e.  NN0 )  ->  ( ( N `  m )  x.  ( R  /  ( N `  m )
) )  =  R )
8584oveq2d 6324 . . . . . . . . 9  |-  ( (
ph  /\  m  e.  NN0 )  ->  ( ( B `  m )  x.  ( ( N `  m )  x.  ( R  /  ( N `  m ) ) ) )  =  ( ( B `  m )  x.  R ) )
8660ffvelrnda 6037 . . . . . . . . . 10  |-  ( (
ph  /\  m  e.  NN0 )  ->  ( B `  m )  e.  CC )
8781, 82, 83divcld 10405 . . . . . . . . . 10  |-  ( (
ph  /\  m  e.  NN0 )  ->  ( R  /  ( N `  m ) )  e.  CC )
8886, 82, 87mulassd 9684 . . . . . . . . 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 9682 . . . . . . . . 9  |-  ( (
ph  /\  m  e.  NN0 )  ->  ( R  x.  ( B `  m
) )  =  ( ( B `  m
)  x.  R ) )
9085, 88, 893eqtr4rd 2516 . . . . . . . 8  |-  ( (
ph  /\  m  e.  NN0 )  ->  ( R  x.  ( B `  m
) )  =  ( ( ( B `  m )  x.  ( N `  m )
)  x.  ( R  /  ( N `  m ) ) ) )
9158, 90sylan2 482 . . . . . . 7  |-  ( (
ph  /\  m  e.  ( 0 ... (deg `  F ) ) )  ->  ( R  x.  ( B `  m ) )  =  ( ( ( B `  m
)  x.  ( N `
 m ) )  x.  ( R  / 
( N `  m
) ) ) )
92 oveq2 6316 . . . . . . . . . . . . 13  |-  ( n  =  ( N `  m )  ->  (
( B `  m
)  x.  n )  =  ( ( B `
 m )  x.  ( N `  m
) ) )
9392eleq1d 2533 . . . . . . . . . . . 12  |-  ( n  =  ( N `  m )  ->  (
( ( B `  m )  x.  n
)  e.  ZZ  <->  ( ( B `  m )  x.  ( N `  m
) )  e.  ZZ ) )
9493elrab 3184 . . . . . . . . . . 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 471 . . . . . . . . . 10  |-  ( ( N `  m )  e.  { n  e.  NN  |  ( ( B `  m )  x.  n )  e.  ZZ }  ->  (
( B `  m
)  x.  ( N `
 m ) )  e.  ZZ )
9647, 95syl 17 . . . . . . . . 9  |-  ( (
ph  /\  m  e.  NN0 )  ->  ( ( B `  m )  x.  ( N `  m
) )  e.  ZZ )
9758, 96sylan2 482 . . . . . . . 8  |-  ( (
ph  /\  m  e.  ( 0 ... (deg `  F ) ) )  ->  ( ( B `
 m )  x.  ( N `  m
) )  e.  ZZ )
98 elqaaOLD.3 . . . . . . . . . 10  |-  ( ph  ->  ( F `  A
)  =  0 )
99 eqid 2471 . . . . . . . . . 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, 99elqaalem2OLD 23355 . . . . . . . . 9  |-  ( (
ph  /\  m  e.  ( 0 ... (deg `  F ) ) )  ->  ( R  mod  ( N `  m ) )  =  0 )
10155adantr 472 . . . . . . . . . 10  |-  ( (
ph  /\  m  e.  ( 0 ... (deg `  F ) ) )  ->  R  e.  NN )
10258, 48sylan2 482 . . . . . . . . . 10  |-  ( (
ph  /\  m  e.  ( 0 ... (deg `  F ) ) )  ->  ( N `  m )  e.  NN )
103 nnre 10638 . . . . . . . . . . 11  |-  ( R  e.  NN  ->  R  e.  RR )
104 nnrp 11334 . . . . . . . . . . 11  |-  ( ( N `  m )  e.  NN  ->  ( N `  m )  e.  RR+ )
105 mod0 12136 . . . . . . . . . . 11  |-  ( ( R  e.  RR  /\  ( N `  m )  e.  RR+ )  ->  (
( R  mod  ( N `  m )
)  =  0  <->  ( R  /  ( N `  m ) )  e.  ZZ ) )
106103, 104, 105syl2an 485 . . . . . . . . . 10  |-  ( ( R  e.  NN  /\  ( N `  m )  e.  NN )  -> 
( ( R  mod  ( N `  m ) )  =  0  <->  ( R  /  ( N `  m ) )  e.  ZZ ) )
107101, 102, 106syl2anc 673 . . . . . . . . 9  |-  ( (
ph  /\  m  e.  ( 0 ... (deg `  F ) ) )  ->  ( ( R  mod  ( N `  m ) )  =  0  <->  ( R  / 
( N `  m
) )  e.  ZZ ) )
108100, 107mpbid 215 . . . . . . . 8  |-  ( (
ph  /\  m  e.  ( 0 ... (deg `  F ) ) )  ->  ( R  / 
( N `  m
) )  e.  ZZ )
10997, 108zmulcld 11069 . . . . . . 7  |-  ( (
ph  /\  m  e.  ( 0 ... (deg `  F ) ) )  ->  ( ( ( B `  m )  x.  ( N `  m ) )  x.  ( R  /  ( N `  m )
) )  e.  ZZ )
11091, 109eqeltrd 2549 . . . . . 6  |-  ( (
ph  /\  m  e.  ( 0 ... (deg `  F ) ) )  ->  ( R  x.  ( B `  m ) )  e.  ZZ )
11180, 53, 110elplyd 23235 . . . . 5  |-  ( ph  ->  ( z  e.  CC  |->  sum_
m  e.  ( 0 ... (deg `  F
) ) ( ( R  x.  ( B `
 m ) )  x.  ( z ^
m ) ) )  e.  (Poly `  ZZ ) )
11278, 111eqeltrd 2549 . . . 4  |-  ( ph  ->  ( ( CC  X.  { R } )  oF  x.  F )  e.  (Poly `  ZZ ) )
113 eldifsn 4088 . . . . . . 7  |-  ( F  e.  ( (Poly `  QQ )  \  { 0p } )  <->  ( F  e.  (Poly `  QQ )  /\  F  =/=  0p ) )
11412, 113sylib 201 . . . . . 6  |-  ( ph  ->  ( F  e.  (Poly `  QQ )  /\  F  =/=  0p ) )
115114simprd 470 . . . . 5  |-  ( ph  ->  F  =/=  0p )
116 oveq1 6315 . . . . . . 7  |-  ( ( ( CC  X.  { R } )  oF  x.  F )  =  0p  ->  (
( ( CC  X.  { R } )  oF  x.  F )  oF  /  ( CC  X.  { R }
) )  =  ( 0p  oF  /  ( CC  X.  { R } ) ) )
11715ffvelrnda 6037 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  CC )  ->  ( F `
 z )  e.  CC )
11855nnne0d 10676 . . . . . . . . . . . 12  |-  ( ph  ->  R  =/=  0 )
119118adantr 472 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  CC )  ->  R  =/=  0 )
120117, 57, 119divcan3d 10410 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  CC )  ->  ( ( R  x.  ( F `
 z ) )  /  R )  =  ( F `  z
) )
121120mpteq2dva 4482 . . . . . . . . 9  |-  ( ph  ->  ( z  e.  CC  |->  ( ( R  x.  ( F `  z ) )  /  R ) )  =  ( z  e.  CC  |->  ( F `
 z ) ) )
122 ovex 6336 . . . . . . . . . . 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 6567 . . . . . . . . 9  |-  ( ph  ->  ( ( ( CC 
X.  { R }
)  oF  x.  F )  oF  /  ( CC  X.  { R } ) )  =  ( z  e.  CC  |->  ( ( R  x.  ( F `  z ) )  /  R ) ) )
125121, 124, 163eqtr4d 2515 . . . . . . . 8  |-  ( ph  ->  ( ( ( CC 
X.  { R }
)  oF  x.  F )  oF  /  ( CC  X.  { R } ) )  =  F )
12656, 118div0d 10404 . . . . . . . . . 10  |-  ( ph  ->  ( 0  /  R
)  =  0 )
127126mpteq2dv 4483 . . . . . . . . 9  |-  ( ph  ->  ( z  e.  CC  |->  ( 0  /  R
) )  =  ( z  e.  CC  |->  0 ) )
128 0cnd 9654 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  CC )  ->  0  e.  CC )
129 df-0p 22707 . . . . . . . . . . . 12  |-  0p  =  ( CC  X.  { 0 } )
130 fconstmpt 4883 . . . . . . . . . . . 12  |-  ( CC 
X.  { 0 } )  =  ( z  e.  CC  |->  0 )
131129, 130eqtri 2493 . . . . . . . . . . 11  |-  0p  =  ( z  e.  CC  |->  0 )
132131a1i 11 . . . . . . . . . 10  |-  ( ph  ->  0p  =  ( z  e.  CC  |->  0 ) )
1333, 128, 7, 132, 11offval2 6567 . . . . . . . . 9  |-  ( ph  ->  ( 0p  oF  /  ( CC 
X.  { R }
) )  =  ( z  e.  CC  |->  ( 0  /  R ) ) )
134127, 133, 1323eqtr4d 2515 . . . . . . . 8  |-  ( ph  ->  ( 0p  oF  /  ( CC 
X.  { R }
) )  =  0p )
135125, 134eqeq12d 2486 . . . . . . 7  |-  ( ph  ->  ( ( ( ( CC  X.  { R } )  oF  x.  F )  oF  /  ( CC 
X.  { R }
) )  =  ( 0p  oF  /  ( CC  X.  { R } ) )  <-> 
F  =  0p ) )
136116, 135syl5ib 227 . . . . . 6  |-  ( ph  ->  ( ( ( CC 
X.  { R }
)  oF  x.  F )  =  0p  ->  F  = 
0p ) )
137136necon3d 2664 . . . . 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 4088 . . . 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 677 . . 3  |-  ( ph  ->  ( ( CC  X.  { R } )  oF  x.  F )  e.  ( (Poly `  ZZ )  \  { 0p } ) )
1416fconst 5782 . . . . . . 7  |-  ( CC 
X.  { R }
) : CC --> { R }
142 ffn 5739 . . . . . . 7  |-  ( ( CC  X.  { R } ) : CC --> { R }  ->  ( CC  X.  { R }
)  Fn  CC )
143141, 142mp1i 13 . . . . . 6  |-  ( ph  ->  ( CC  X.  { R } )  Fn  CC )
144 ffn 5739 . . . . . . 7  |-  ( F : CC --> CC  ->  F  Fn  CC )
14515, 144syl 17 . . . . . 6  |-  ( ph  ->  F  Fn  CC )
146 inidm 3632 . . . . . 6  |-  ( CC 
i^i  CC )  =  CC
1476fvconst2 6136 . . . . . . 7  |-  ( A  e.  CC  ->  (
( CC  X.  { R } ) `  A
)  =  R )
148147adantl 473 . . . . . 6  |-  ( (
ph  /\  A  e.  CC )  ->  ( ( CC  X.  { R } ) `  A
)  =  R )
14998adantr 472 . . . . . 6  |-  ( (
ph  /\  A  e.  CC )  ->  ( F `
 A )  =  0 )
150143, 145, 3, 3, 146, 148, 149ofval 6559 . . . . 5  |-  ( (
ph  /\  A  e.  CC )  ->  ( ( ( CC  X.  { R } )  oF  x.  F ) `  A )  =  ( R  x.  0 ) )
1511, 150mpdan 681 . . . 4  |-  ( ph  ->  ( ( ( CC 
X.  { R }
)  oF  x.  F ) `  A
)  =  ( R  x.  0 ) )
15256mul01d 9850 . . . 4  |-  ( ph  ->  ( R  x.  0 )  =  0 )
153151, 152eqtrd 2505 . . 3  |-  ( ph  ->  ( ( ( CC 
X.  { R }
)  oF  x.  F ) `  A
)  =  0 )
154 fveq1 5878 . . . . 5  |-  ( f  =  ( ( CC 
X.  { R }
)  oF  x.  F )  ->  (
f `  A )  =  ( ( ( CC  X.  { R } )  oF  x.  F ) `  A ) )
155154eqeq1d 2473 . . . 4  |-  ( f  =  ( ( CC 
X.  { R }
)  oF  x.  F )  ->  (
( f `  A
)  =  0  <->  (
( ( CC  X.  { R } )  oF  x.  F ) `
 A )  =  0 ) )
156155rspcev 3136 . . 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 673 . 2  |-  ( ph  ->  E. f  e.  ( (Poly `  ZZ )  \  { 0p }
) ( f `  A )  =  0 )
158 elaa 23348 . 2  |-  ( A  e.  AA  <->  ( A  e.  CC  /\  E. f  e.  ( (Poly `  ZZ )  \  { 0p } ) ( f `
 A )  =  0 ) )
1591, 157, 158sylanbrc 677 1  |-  ( ph  ->  A  e.  AA )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904    =/= wne 2641   E.wrex 2757   {crab 2760   _Vcvv 3031    \ cdif 3387    C_ wss 3390   (/)c0 3722   {csn 3959    |-> cmpt 4454    X. cxp 4837   `'ccnv 4838    Fn wfn 5584   -->wf 5585   ` cfv 5589  (class class class)co 6308    |-> cmpt2 6310    oFcof 6548   supcsup 7972   CCcc 9555   RRcr 9556   0cc0 9557   1c1 9558    x. cmul 9562    < clt 9693    / cdiv 10291   NNcn 10631   NN0cn0 10893   ZZcz 10961   ZZ>=cuz 11182   QQcq 11287   RR+crp 11325   ...cfz 11810    mod cmo 12129    seqcseq 12251   ^cexp 12310   sum_csu 13829   0pc0p 22706  Polycply 23217  coeffccoe 23219  degcdgr 23220   AAcaa 23346
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-addf 9636
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-q 11288  df-rp 11326  df-fz 11811  df-fzo 11943  df-fl 12061  df-mod 12130  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-clim 13629  df-rlim 13630  df-sum 13830  df-0p 22707  df-ply 23221  df-coe 23223  df-dgr 23224  df-aa 23347
This theorem is referenced by: (None)
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