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Theorem eldioph 26714
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 957 . 2  |-  ( ( N  e.  NN0  /\  K  e.  ( ZZ>= `  N )  /\  P  e.  (mzPoly `  ( 1 ... K ) ) )  ->  N  e.  NN0 )
2 simp2 958 . . 3  |-  ( ( N  e.  NN0  /\  K  e.  ( ZZ>= `  N )  /\  P  e.  (mzPoly `  ( 1 ... K ) ) )  ->  K  e.  (
ZZ>= `  N ) )
3 simp3 959 . . . 4  |-  ( ( N  e.  NN0  /\  K  e.  ( ZZ>= `  N )  /\  P  e.  (mzPoly `  ( 1 ... K ) ) )  ->  P  e.  (mzPoly `  ( 1 ... K
) ) )
4 eqidd 2413 . . . 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 5694 . . . . . . . . . 10  |-  ( p  =  P  ->  (
p `  u )  =  ( P `  u ) )
65eqeq1d 2420 . . . . . . . . 9  |-  ( p  =  P  ->  (
( p `  u
)  =  0  <->  ( P `  u )  =  0 ) )
76anbi2d 685 . . . . . . . 8  |-  ( p  =  P  ->  (
( t  =  ( u  |`  ( 1 ... N ) )  /\  ( p `  u )  =  0 )  <->  ( t  =  ( u  |`  (
1 ... N ) )  /\  ( P `  u )  =  0 ) ) )
87rexbidv 2695 . . . . . . 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 2526 . . . . . 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 2423 . . . . 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 3020 . . . 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 643 . . 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 6056 . . . . . 6  |-  ( k  =  K  ->  (
1 ... k )  =  ( 1 ... K
) )
1413fveq2d 5699 . . . . 5  |-  ( k  =  K  ->  (mzPoly `  ( 1 ... k
) )  =  (mzPoly `  ( 1 ... K
) ) )
1513oveq2d 6064 . . . . . . . 8  |-  ( k  =  K  ->  ( NN0  ^m  ( 1 ... k ) )  =  ( NN0  ^m  (
1 ... K ) ) )
1615rexeqdv 2879 . . . . . . 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 2526 . . . . . 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 2423 . . . . 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 2885 . . . 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 3020 . . 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 643 . 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 26713 . 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 646 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 359    /\ w3a 936    = wceq 1649    e. wcel 1721   {cab 2398   E.wrex 2675    |` cres 4847   ` cfv 5421  (class class class)co 6048    ^m cmap 6985   0cc0 8954   1c1 8955   NN0cn0 10185   ZZ>=cuz 10452   ...cfz 11007  mzPolycmzp 26677  Diophcdioph 26711
This theorem is referenced by:  eldioph2  26718  eq0rabdioph  26733
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-i2m1 9022  ax-1ne0 9023  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-recs 6600  df-rdg 6635  df-er 6872  df-map 6987  df-en 7077  df-dom 7078  df-sdom 7079  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-neg 9258  df-nn 9965  df-n0 10186  df-z 10247  df-uz 10453  df-fz 11008  df-dioph 26712
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