Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  eldioph Unicode version

Theorem eldioph 26428
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 956 . 2  |-  ( ( N  e.  NN0  /\  K  e.  ( ZZ>= `  N )  /\  P  e.  (mzPoly `  ( 1 ... K ) ) )  ->  N  e.  NN0 )
2 simp2 957 . . 3  |-  ( ( N  e.  NN0  /\  K  e.  ( ZZ>= `  N )  /\  P  e.  (mzPoly `  ( 1 ... K ) ) )  ->  K  e.  (
ZZ>= `  N ) )
3 simp3 958 . . . 4  |-  ( ( N  e.  NN0  /\  K  e.  ( ZZ>= `  N )  /\  P  e.  (mzPoly `  ( 1 ... K ) ) )  ->  P  e.  (mzPoly `  ( 1 ... K
) ) )
4 eqidd 2367 . . . 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 5631 . . . . . . . . . 10  |-  ( p  =  P  ->  (
p `  u )  =  ( P `  u ) )
65eqeq1d 2374 . . . . . . . . 9  |-  ( p  =  P  ->  (
( p `  u
)  =  0  <->  ( P `  u )  =  0 ) )
76anbi2d 684 . . . . . . . 8  |-  ( p  =  P  ->  (
( t  =  ( u  |`  ( 1 ... N ) )  /\  ( p `  u )  =  0 )  <->  ( t  =  ( u  |`  (
1 ... N ) )  /\  ( P `  u )  =  0 ) ) )
87rexbidv 2649 . . . . . . 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 2480 . . . . . 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 2377 . . . . 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 2969 . . . 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 642 . . 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 5989 . . . . . 6  |-  ( k  =  K  ->  (
1 ... k )  =  ( 1 ... K
) )
1413fveq2d 5636 . . . . 5  |-  ( k  =  K  ->  (mzPoly `  ( 1 ... k
) )  =  (mzPoly `  ( 1 ... K
) ) )
1513oveq2d 5997 . . . . . . . 8  |-  ( k  =  K  ->  ( NN0  ^m  ( 1 ... k ) )  =  ( NN0  ^m  (
1 ... K ) ) )
1615rexeqdv 2828 . . . . . . 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 2480 . . . . . 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 2377 . . . . 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 2834 . . . 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 2969 . . 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 642 . 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 26427 . 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 645 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 358    /\ w3a 935    = wceq 1647    e. wcel 1715   {cab 2352   E.wrex 2629    |` cres 4794   ` cfv 5358  (class class class)co 5981    ^m cmap 6915   0cc0 8884   1c1 8885   NN0cn0 10114   ZZ>=cuz 10381   ...cfz 10935  mzPolycmzp 26391  Diophcdioph 26425
This theorem is referenced by:  eldioph2  26432  eq0rabdioph  26447
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-cnex 8940  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-i2m1 8952  ax-1ne0 8953  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-recs 6530  df-rdg 6565  df-er 6802  df-map 6917  df-en 7007  df-dom 7008  df-sdom 7009  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-le 9020  df-neg 9187  df-nn 9894  df-n0 10115  df-z 10176  df-uz 10382  df-fz 10936  df-dioph 26426
  Copyright terms: Public domain W3C validator