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Theorem eldioph 30619
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 996 . 2  |-  ( ( N  e.  NN0  /\  K  e.  ( ZZ>= `  N )  /\  P  e.  (mzPoly `  ( 1 ... K ) ) )  ->  N  e.  NN0 )
2 simp2 997 . . 3  |-  ( ( N  e.  NN0  /\  K  e.  ( ZZ>= `  N )  /\  P  e.  (mzPoly `  ( 1 ... K ) ) )  ->  K  e.  (
ZZ>= `  N ) )
3 simp3 998 . . . 4  |-  ( ( N  e.  NN0  /\  K  e.  ( ZZ>= `  N )  /\  P  e.  (mzPoly `  ( 1 ... K ) ) )  ->  P  e.  (mzPoly `  ( 1 ... K
) ) )
4 eqidd 2468 . . . 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 5871 . . . . . . . . . 10  |-  ( p  =  P  ->  (
p `  u )  =  ( P `  u ) )
65eqeq1d 2469 . . . . . . . . 9  |-  ( p  =  P  ->  (
( p `  u
)  =  0  <->  ( P `  u )  =  0 ) )
76anbi2d 703 . . . . . . . 8  |-  ( p  =  P  ->  (
( t  =  ( u  |`  ( 1 ... N ) )  /\  ( p `  u )  =  0 )  <->  ( t  =  ( u  |`  (
1 ... N ) )  /\  ( P `  u )  =  0 ) ) )
87rexbidv 2978 . . . . . . 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 2603 . . . . . 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 2481 . . . . 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 3219 . . . 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 661 . . 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 6303 . . . . . 6  |-  ( k  =  K  ->  (
1 ... k )  =  ( 1 ... K
) )
1413fveq2d 5876 . . . . 5  |-  ( k  =  K  ->  (mzPoly `  ( 1 ... k
) )  =  (mzPoly `  ( 1 ... K
) ) )
1513oveq2d 6311 . . . . . . . 8  |-  ( k  =  K  ->  ( NN0  ^m  ( 1 ... k ) )  =  ( NN0  ^m  (
1 ... K ) ) )
1615rexeqdv 3070 . . . . . . 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 2603 . . . . . 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 2481 . . . . 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 3078 . . . 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 3219 . . 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 661 . 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 30618 . 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 664 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   {cab 2452   E.wrex 2818    |` cres 5007   ` cfv 5594  (class class class)co 6295    ^m cmap 7432   0cc0 9504   1c1 9505   NN0cn0 10807   ZZ>=cuz 11094   ...cfz 11684  mzPolycmzp 30582  Diophcdioph 30616
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 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-i2m1 9572  ax-1ne0 9573  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  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 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-er 7323  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-neg 9820  df-nn 10549  df-n0 10808  df-z 10877  df-uz 11095  df-fz 11685  df-dioph 30617
This theorem is referenced by:  eldioph2  30623  eq0rabdioph  30638
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