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Theorem eldioph 26854
Description: Condition for a set to be Diophantine (unpacking existential quantifier) (Contributed by Stefan O'Rear, 5-Oct-2014.)
Assertion
Ref Expression
eldioph  |-  ( ( N  e.  NN0  /\  K  e.  ( ZZ>= `  N )  /\  P  e.  (mzPoly `  ( 1 ... K ) ) )  ->  { t  |  E. u  e.  ( NN0  ^m  ( 1 ... K ) ) ( t  =  ( u  |`  ( 1 ... N ) )  /\  ( P `  u )  =  0 ) }  e.  (Dioph `  N ) )
Distinct variable groups:    t, N, u    t, K, u    t, P, u

Proof of Theorem eldioph
Dummy variables  k  p are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 958 . 2  |-  ( ( N  e.  NN0  /\  K  e.  ( ZZ>= `  N )  /\  P  e.  (mzPoly `  ( 1 ... K ) ) )  ->  N  e.  NN0 )
2 simp2 959 . . 3  |-  ( ( N  e.  NN0  /\  K  e.  ( ZZ>= `  N )  /\  P  e.  (mzPoly `  ( 1 ... K ) ) )  ->  K  e.  (
ZZ>= `  N ) )
3 simp3 960 . . . 4  |-  ( ( N  e.  NN0  /\  K  e.  ( ZZ>= `  N )  /\  P  e.  (mzPoly `  ( 1 ... K ) ) )  ->  P  e.  (mzPoly `  ( 1 ... K
) ) )
4 eqidd 2443 . . . 4  |-  ( ( N  e.  NN0  /\  K  e.  ( ZZ>= `  N )  /\  P  e.  (mzPoly `  ( 1 ... K ) ) )  ->  { t  |  E. u  e.  ( NN0  ^m  ( 1 ... K ) ) ( t  =  ( u  |`  ( 1 ... N ) )  /\  ( P `  u )  =  0 ) }  =  {
t  |  E. u  e.  ( NN0  ^m  (
1 ... K ) ) ( t  =  ( u  |`  ( 1 ... N ) )  /\  ( P `  u )  =  0 ) } )
5 fveq1 5756 . . . . . . . . . 10  |-  ( p  =  P  ->  (
p `  u )  =  ( P `  u ) )
65eqeq1d 2450 . . . . . . . . 9  |-  ( p  =  P  ->  (
( p `  u
)  =  0  <->  ( P `  u )  =  0 ) )
76anbi2d 686 . . . . . . . 8  |-  ( p  =  P  ->  (
( t  =  ( u  |`  ( 1 ... N ) )  /\  ( p `  u )  =  0 )  <->  ( t  =  ( u  |`  (
1 ... N ) )  /\  ( P `  u )  =  0 ) ) )
87rexbidv 2732 . . . . . . 7  |-  ( p  =  P  ->  ( E. u  e.  ( NN0  ^m  ( 1 ... K ) ) ( t  =  ( u  |`  ( 1 ... N
) )  /\  (
p `  u )  =  0 )  <->  E. u  e.  ( NN0  ^m  (
1 ... K ) ) ( t  =  ( u  |`  ( 1 ... N ) )  /\  ( P `  u )  =  0 ) ) )
98abbidv 2556 . . . . . 6  |-  ( p  =  P  ->  { t  |  E. u  e.  ( NN0  ^m  (
1 ... K ) ) ( t  =  ( u  |`  ( 1 ... N ) )  /\  ( p `  u )  =  0 ) }  =  {
t  |  E. u  e.  ( NN0  ^m  (
1 ... K ) ) ( t  =  ( u  |`  ( 1 ... N ) )  /\  ( P `  u )  =  0 ) } )
109eqeq2d 2453 . . . . 5  |-  ( p  =  P  ->  ( { t  |  E. u  e.  ( NN0  ^m  ( 1 ... K
) ) ( t  =  ( u  |`  ( 1 ... N
) )  /\  ( P `  u )  =  0 ) }  =  { t  |  E. u  e.  ( NN0  ^m  ( 1 ... K ) ) ( t  =  ( u  |`  ( 1 ... N ) )  /\  ( p `  u )  =  0 ) }  <->  { t  |  E. u  e.  ( NN0  ^m  ( 1 ... K ) ) ( t  =  ( u  |`  ( 1 ... N ) )  /\  ( P `  u )  =  0 ) }  =  {
t  |  E. u  e.  ( NN0  ^m  (
1 ... K ) ) ( t  =  ( u  |`  ( 1 ... N ) )  /\  ( P `  u )  =  0 ) } ) )
1110rspcev 3058 . . . 4  |-  ( ( P  e.  (mzPoly `  ( 1 ... K
) )  /\  {
t  |  E. u  e.  ( NN0  ^m  (
1 ... K ) ) ( t  =  ( u  |`  ( 1 ... N ) )  /\  ( P `  u )  =  0 ) }  =  {
t  |  E. u  e.  ( NN0  ^m  (
1 ... K ) ) ( t  =  ( u  |`  ( 1 ... N ) )  /\  ( P `  u )  =  0 ) } )  ->  E. p  e.  (mzPoly `  ( 1 ... K
) ) { t  |  E. u  e.  ( NN0  ^m  (
1 ... K ) ) ( t  =  ( u  |`  ( 1 ... N ) )  /\  ( P `  u )  =  0 ) }  =  {
t  |  E. u  e.  ( NN0  ^m  (
1 ... K ) ) ( t  =  ( u  |`  ( 1 ... N ) )  /\  ( p `  u )  =  0 ) } )
123, 4, 11syl2anc 644 . . 3  |-  ( ( N  e.  NN0  /\  K  e.  ( ZZ>= `  N )  /\  P  e.  (mzPoly `  ( 1 ... K ) ) )  ->  E. p  e.  (mzPoly `  ( 1 ... K
) ) { t  |  E. u  e.  ( NN0  ^m  (
1 ... K ) ) ( t  =  ( u  |`  ( 1 ... N ) )  /\  ( P `  u )  =  0 ) }  =  {
t  |  E. u  e.  ( NN0  ^m  (
1 ... K ) ) ( t  =  ( u  |`  ( 1 ... N ) )  /\  ( p `  u )  =  0 ) } )
13 oveq2 6118 . . . . . 6  |-  ( k  =  K  ->  (
1 ... k )  =  ( 1 ... K
) )
1413fveq2d 5761 . . . . 5  |-  ( k  =  K  ->  (mzPoly `  ( 1 ... k
) )  =  (mzPoly `  ( 1 ... K
) ) )
1513oveq2d 6126 . . . . . . . 8  |-  ( k  =  K  ->  ( NN0  ^m  ( 1 ... k ) )  =  ( NN0  ^m  (
1 ... K ) ) )
1615rexeqdv 2917 . . . . . . 7  |-  ( k  =  K  ->  ( E. u  e.  ( NN0  ^m  ( 1 ... k ) ) ( t  =  ( u  |`  ( 1 ... N
) )  /\  (
p `  u )  =  0 )  <->  E. u  e.  ( NN0  ^m  (
1 ... K ) ) ( t  =  ( u  |`  ( 1 ... N ) )  /\  ( p `  u )  =  0 ) ) )
1716abbidv 2556 . . . . . 6  |-  ( k  =  K  ->  { t  |  E. u  e.  ( NN0  ^m  (
1 ... k ) ) ( t  =  ( u  |`  ( 1 ... N ) )  /\  ( p `  u )  =  0 ) }  =  {
t  |  E. u  e.  ( NN0  ^m  (
1 ... K ) ) ( t  =  ( u  |`  ( 1 ... N ) )  /\  ( p `  u )  =  0 ) } )
1817eqeq2d 2453 . . . . 5  |-  ( k  =  K  ->  ( { t  |  E. u  e.  ( NN0  ^m  ( 1 ... K
) ) ( t  =  ( u  |`  ( 1 ... N
) )  /\  ( P `  u )  =  0 ) }  =  { t  |  E. u  e.  ( NN0  ^m  ( 1 ... k ) ) ( t  =  ( u  |`  ( 1 ... N ) )  /\  ( p `  u )  =  0 ) }  <->  { t  |  E. u  e.  ( NN0  ^m  ( 1 ... K ) ) ( t  =  ( u  |`  ( 1 ... N ) )  /\  ( P `  u )  =  0 ) }  =  {
t  |  E. u  e.  ( NN0  ^m  (
1 ... K ) ) ( t  =  ( u  |`  ( 1 ... N ) )  /\  ( p `  u )  =  0 ) } ) )
1914, 18rexeqbidv 2923 . . . 4  |-  ( k  =  K  ->  ( E. p  e.  (mzPoly `  ( 1 ... k
) ) { t  |  E. u  e.  ( NN0  ^m  (
1 ... K ) ) ( t  =  ( u  |`  ( 1 ... N ) )  /\  ( P `  u )  =  0 ) }  =  {
t  |  E. u  e.  ( NN0  ^m  (
1 ... k ) ) ( t  =  ( u  |`  ( 1 ... N ) )  /\  ( p `  u )  =  0 ) }  <->  E. p  e.  (mzPoly `  ( 1 ... K ) ) { t  |  E. u  e.  ( NN0  ^m  (
1 ... K ) ) ( t  =  ( u  |`  ( 1 ... N ) )  /\  ( P `  u )  =  0 ) }  =  {
t  |  E. u  e.  ( NN0  ^m  (
1 ... K ) ) ( t  =  ( u  |`  ( 1 ... N ) )  /\  ( p `  u )  =  0 ) } ) )
2019rspcev 3058 . . 3  |-  ( ( K  e.  ( ZZ>= `  N )  /\  E. p  e.  (mzPoly `  (
1 ... K ) ) { t  |  E. u  e.  ( NN0  ^m  ( 1 ... K
) ) ( t  =  ( u  |`  ( 1 ... N
) )  /\  ( P `  u )  =  0 ) }  =  { t  |  E. u  e.  ( NN0  ^m  ( 1 ... K ) ) ( t  =  ( u  |`  ( 1 ... N ) )  /\  ( p `  u )  =  0 ) } )  ->  E. k  e.  ( ZZ>=
`  N ) E. p  e.  (mzPoly `  ( 1 ... k
) ) { t  |  E. u  e.  ( NN0  ^m  (
1 ... K ) ) ( t  =  ( u  |`  ( 1 ... N ) )  /\  ( P `  u )  =  0 ) }  =  {
t  |  E. u  e.  ( NN0  ^m  (
1 ... k ) ) ( t  =  ( u  |`  ( 1 ... N ) )  /\  ( p `  u )  =  0 ) } )
212, 12, 20syl2anc 644 . 2  |-  ( ( N  e.  NN0  /\  K  e.  ( ZZ>= `  N )  /\  P  e.  (mzPoly `  ( 1 ... K ) ) )  ->  E. k  e.  (
ZZ>= `  N ) E. p  e.  (mzPoly `  ( 1 ... k
) ) { t  |  E. u  e.  ( NN0  ^m  (
1 ... K ) ) ( t  =  ( u  |`  ( 1 ... N ) )  /\  ( P `  u )  =  0 ) }  =  {
t  |  E. u  e.  ( NN0  ^m  (
1 ... k ) ) ( t  =  ( u  |`  ( 1 ... N ) )  /\  ( p `  u )  =  0 ) } )
22 eldiophb 26853 . 2  |-  ( { t  |  E. u  e.  ( NN0  ^m  (
1 ... K ) ) ( t  =  ( u  |`  ( 1 ... N ) )  /\  ( P `  u )  =  0 ) }  e.  (Dioph `  N )  <->  ( N  e.  NN0  /\  E. k  e.  ( ZZ>= `  N ) E. p  e.  (mzPoly `  ( 1 ... k
) ) { t  |  E. u  e.  ( NN0  ^m  (
1 ... K ) ) ( t  =  ( u  |`  ( 1 ... N ) )  /\  ( P `  u )  =  0 ) }  =  {
t  |  E. u  e.  ( NN0  ^m  (
1 ... k ) ) ( t  =  ( u  |`  ( 1 ... N ) )  /\  ( p `  u )  =  0 ) } ) )
231, 21, 22sylanbrc 647 1  |-  ( ( N  e.  NN0  /\  K  e.  ( ZZ>= `  N )  /\  P  e.  (mzPoly `  ( 1 ... K ) ) )  ->  { t  |  E. u  e.  ( NN0  ^m  ( 1 ... K ) ) ( t  =  ( u  |`  ( 1 ... N ) )  /\  ( P `  u )  =  0 ) }  e.  (Dioph `  N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1727   {cab 2428   E.wrex 2712    |` cres 4909   ` cfv 5483  (class class class)co 6110    ^m cmap 7047   0cc0 9021   1c1 9022   NN0cn0 10252   ZZ>=cuz 10519   ...cfz 11074  mzPolycmzp 26817  Diophcdioph 26851
This theorem is referenced by:  eldioph2  26858  eq0rabdioph  26873
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730  ax-cnex 9077  ax-resscn 9078  ax-1cn 9079  ax-icn 9080  ax-addcl 9081  ax-addrcl 9082  ax-mulcl 9083  ax-mulrcl 9084  ax-i2m1 9089  ax-1ne0 9090  ax-rrecex 9093  ax-cnre 9094  ax-pre-lttri 9095  ax-pre-lttrn 9096
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2716  df-rex 2717  df-reu 2718  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-tp 3846  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-tr 4328  df-eprel 4523  df-id 4527  df-po 4532  df-so 4533  df-fr 4570  df-we 4572  df-ord 4613  df-on 4614  df-lim 4615  df-suc 4616  df-om 4875  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-1st 6378  df-2nd 6379  df-recs 6662  df-rdg 6697  df-er 6934  df-map 7049  df-en 7139  df-dom 7140  df-sdom 7141  df-pnf 9153  df-mnf 9154  df-xr 9155  df-ltxr 9156  df-le 9157  df-neg 9325  df-nn 10032  df-n0 10253  df-z 10314  df-uz 10520  df-fz 11075  df-dioph 26852
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