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Theorem eldiophb 26815
Description: Initial expression of Diophantine property of a set. (Contributed by Stefan O'Rear, 5-Oct-2014.) (Revised by Mario Carneiro, 22-Sep-2015.)
Assertion
Ref Expression
eldiophb  |-  ( D  e.  (Dioph `  N
)  <->  ( N  e. 
NN0  /\  E. k  e.  ( ZZ>= `  N ) E. p  e.  (mzPoly `  ( 1 ... k
) ) D  =  { t  |  E. u  e.  ( NN0  ^m  ( 1 ... k
) ) ( t  =  ( u  |`  ( 1 ... N
) )  /\  (
p `  u )  =  0 ) } ) )
Distinct variable groups:    D, k, p    k, N, p, t, u
Allowed substitution hints:    D( u, t)

Proof of Theorem eldiophb
Dummy variables  n  d are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-dioph 26814 . . . 4  |- Dioph  =  ( n  e.  NN0  |->  ran  (
k  e.  ( ZZ>= `  n ) ,  p  e.  (mzPoly `  ( 1 ... k ) )  |->  { t  |  E. u  e.  ( NN0  ^m  (
1 ... k ) ) ( t  =  ( u  |`  ( 1 ... n ) )  /\  ( p `  u )  =  0 ) } ) )
21dmmptss 5366 . . 3  |-  dom Dioph  C_  NN0
3 elfvdm 5757 . . 3  |-  ( D  e.  (Dioph `  N
)  ->  N  e.  dom Dioph )
42, 3sseldi 3346 . 2  |-  ( D  e.  (Dioph `  N
)  ->  N  e.  NN0 )
5 fveq2 5728 . . . . . . 7  |-  ( n  =  N  ->  ( ZZ>=
`  n )  =  ( ZZ>= `  N )
)
6 eqidd 2437 . . . . . . 7  |-  ( n  =  N  ->  (mzPoly `  ( 1 ... k
) )  =  (mzPoly `  ( 1 ... k
) ) )
7 oveq2 6089 . . . . . . . . . . . 12  |-  ( n  =  N  ->  (
1 ... n )  =  ( 1 ... N
) )
87reseq2d 5146 . . . . . . . . . . 11  |-  ( n  =  N  ->  (
u  |`  ( 1 ... n ) )  =  ( u  |`  (
1 ... N ) ) )
98eqeq2d 2447 . . . . . . . . . 10  |-  ( n  =  N  ->  (
t  =  ( u  |`  ( 1 ... n
) )  <->  t  =  ( u  |`  ( 1 ... N ) ) ) )
109anbi1d 686 . . . . . . . . 9  |-  ( n  =  N  ->  (
( t  =  ( u  |`  ( 1 ... n ) )  /\  ( p `  u )  =  0 )  <->  ( t  =  ( u  |`  (
1 ... N ) )  /\  ( p `  u )  =  0 ) ) )
1110rexbidv 2726 . . . . . . . 8  |-  ( n  =  N  ->  ( 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 ) ) )
1211abbidv 2550 . . . . . . 7  |-  ( n  =  N  ->  { 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 ) } )
135, 6, 12mpt2eq123dv 6136 . . . . . 6  |-  ( n  =  N  ->  (
k  e.  ( ZZ>= `  n ) ,  p  e.  (mzPoly `  ( 1 ... k ) )  |->  { t  |  E. u  e.  ( NN0  ^m  (
1 ... k ) ) ( t  =  ( u  |`  ( 1 ... n ) )  /\  ( p `  u )  =  0 ) } )  =  ( k  e.  (
ZZ>= `  N ) ,  p  e.  (mzPoly `  ( 1 ... k
) )  |->  { t  |  E. u  e.  ( NN0  ^m  (
1 ... k ) ) ( t  =  ( u  |`  ( 1 ... N ) )  /\  ( p `  u )  =  0 ) } ) )
1413rneqd 5097 . . . . 5  |-  ( n  =  N  ->  ran  ( k  e.  (
ZZ>= `  n ) ,  p  e.  (mzPoly `  ( 1 ... k
) )  |->  { t  |  E. u  e.  ( NN0  ^m  (
1 ... k ) ) ( t  =  ( u  |`  ( 1 ... n ) )  /\  ( p `  u )  =  0 ) } )  =  ran  ( k  e.  ( ZZ>= `  N ) ,  p  e.  (mzPoly `  ( 1 ... k
) )  |->  { t  |  E. u  e.  ( NN0  ^m  (
1 ... k ) ) ( t  =  ( u  |`  ( 1 ... N ) )  /\  ( p `  u )  =  0 ) } ) )
15 ovex 6106 . . . . . . 7  |-  ( NN0 
^m  ( 1 ... N ) )  e. 
_V
1615pwex 4382 . . . . . 6  |-  ~P ( NN0  ^m  ( 1 ... N ) )  e. 
_V
17 eqid 2436 . . . . . . . 8  |-  ( k  e.  ( ZZ>= `  N
) ,  p  e.  (mzPoly `  ( 1 ... k ) )  |->  { t  |  E. u  e.  ( NN0  ^m  (
1 ... k ) ) ( t  =  ( u  |`  ( 1 ... N ) )  /\  ( p `  u )  =  0 ) } )  =  ( k  e.  (
ZZ>= `  N ) ,  p  e.  (mzPoly `  ( 1 ... k
) )  |->  { t  |  E. u  e.  ( NN0  ^m  (
1 ... k ) ) ( t  =  ( u  |`  ( 1 ... N ) )  /\  ( p `  u )  =  0 ) } )
1817rnmpt2 6180 . . . . . . 7  |-  ran  (
k  e.  ( ZZ>= `  N ) ,  p  e.  (mzPoly `  ( 1 ... k ) )  |->  { t  |  E. u  e.  ( NN0  ^m  (
1 ... k ) ) ( t  =  ( u  |`  ( 1 ... N ) )  /\  ( p `  u )  =  0 ) } )  =  { d  |  E. k  e.  ( ZZ>= `  N ) E. p  e.  (mzPoly `  ( 1 ... k ) ) d  =  { t  |  E. u  e.  ( NN0  ^m  ( 1 ... k ) ) ( t  =  ( u  |`  ( 1 ... N ) )  /\  ( p `  u )  =  0 ) } }
19 elmapi 7038 . . . . . . . . . . . . . . . . 17  |-  ( u  e.  ( NN0  ^m  ( 1 ... k
) )  ->  u : ( 1 ... k ) --> NN0 )
20 fzss2 11092 . . . . . . . . . . . . . . . . 17  |-  ( k  e.  ( ZZ>= `  N
)  ->  ( 1 ... N )  C_  ( 1 ... k
) )
21 fssres 5610 . . . . . . . . . . . . . . . . 17  |-  ( ( u : ( 1 ... k ) --> NN0 
/\  ( 1 ... N )  C_  (
1 ... k ) )  ->  ( u  |`  ( 1 ... N
) ) : ( 1 ... N ) --> NN0 )
2219, 20, 21syl2anr 465 . . . . . . . . . . . . . . . 16  |-  ( ( k  e.  ( ZZ>= `  N )  /\  u  e.  ( NN0  ^m  (
1 ... k ) ) )  ->  ( u  |`  ( 1 ... N
) ) : ( 1 ... N ) --> NN0 )
23 nn0ex 10227 . . . . . . . . . . . . . . . . 17  |-  NN0  e.  _V
24 ovex 6106 . . . . . . . . . . . . . . . . 17  |-  ( 1 ... N )  e. 
_V
2523, 24elmap 7042 . . . . . . . . . . . . . . . 16  |-  ( ( u  |`  ( 1 ... N ) )  e.  ( NN0  ^m  ( 1 ... N
) )  <->  ( u  |`  ( 1 ... N
) ) : ( 1 ... N ) --> NN0 )
2622, 25sylibr 204 . . . . . . . . . . . . . . 15  |-  ( ( k  e.  ( ZZ>= `  N )  /\  u  e.  ( NN0  ^m  (
1 ... k ) ) )  ->  ( u  |`  ( 1 ... N
) )  e.  ( NN0  ^m  ( 1 ... N ) ) )
27 eleq1 2496 . . . . . . . . . . . . . . . 16  |-  ( t  =  ( u  |`  ( 1 ... N
) )  ->  (
t  e.  ( NN0 
^m  ( 1 ... N ) )  <->  ( u  |`  ( 1 ... N
) )  e.  ( NN0  ^m  ( 1 ... N ) ) ) )
2827adantr 452 . . . . . . . . . . . . . . 15  |-  ( ( t  =  ( u  |`  ( 1 ... N
) )  /\  (
p `  u )  =  0 )  -> 
( t  e.  ( NN0  ^m  ( 1 ... N ) )  <-> 
( u  |`  (
1 ... N ) )  e.  ( NN0  ^m  ( 1 ... N
) ) ) )
2926, 28syl5ibrcom 214 . . . . . . . . . . . . . 14  |-  ( ( k  e.  ( ZZ>= `  N )  /\  u  e.  ( NN0  ^m  (
1 ... k ) ) )  ->  ( (
t  =  ( u  |`  ( 1 ... N
) )  /\  (
p `  u )  =  0 )  -> 
t  e.  ( NN0 
^m  ( 1 ... N ) ) ) )
3029rexlimdva 2830 . . . . . . . . . . . . 13  |-  ( k  e.  ( ZZ>= `  N
)  ->  ( E. u  e.  ( NN0  ^m  ( 1 ... k
) ) ( t  =  ( u  |`  ( 1 ... N
) )  /\  (
p `  u )  =  0 )  -> 
t  e.  ( NN0 
^m  ( 1 ... N ) ) ) )
3130abssdv 3417 . . . . . . . . . . . 12  |-  ( k  e.  ( ZZ>= `  N
)  ->  { t  |  E. u  e.  ( NN0  ^m  ( 1 ... k ) ) ( t  =  ( u  |`  ( 1 ... N ) )  /\  ( p `  u )  =  0 ) }  C_  ( NN0  ^m  ( 1 ... N ) ) )
3215elpw2 4364 . . . . . . . . . . . 12  |-  ( { t  |  E. u  e.  ( NN0  ^m  (
1 ... k ) ) ( t  =  ( u  |`  ( 1 ... N ) )  /\  ( p `  u )  =  0 ) }  e.  ~P ( NN0  ^m  ( 1 ... N ) )  <->  { t  |  E. u  e.  ( NN0  ^m  ( 1 ... k
) ) ( t  =  ( u  |`  ( 1 ... N
) )  /\  (
p `  u )  =  0 ) } 
C_  ( NN0  ^m  ( 1 ... N
) ) )
3331, 32sylibr 204 . . . . . . . . . . 11  |-  ( k  e.  ( ZZ>= `  N
)  ->  { t  |  E. u  e.  ( NN0  ^m  ( 1 ... k ) ) ( t  =  ( u  |`  ( 1 ... N ) )  /\  ( p `  u )  =  0 ) }  e.  ~P ( NN0  ^m  ( 1 ... N ) ) )
34 eleq1 2496 . . . . . . . . . . 11  |-  ( d  =  { t  |  E. u  e.  ( NN0  ^m  ( 1 ... k ) ) ( t  =  ( u  |`  ( 1 ... N ) )  /\  ( p `  u )  =  0 ) }  ->  (
d  e.  ~P ( NN0  ^m  ( 1 ... N ) )  <->  { t  |  E. u  e.  ( NN0  ^m  ( 1 ... k ) ) ( t  =  ( u  |`  ( 1 ... N ) )  /\  ( p `  u )  =  0 ) }  e.  ~P ( NN0  ^m  ( 1 ... N ) ) ) )
3533, 34syl5ibrcom 214 . . . . . . . . . 10  |-  ( k  e.  ( ZZ>= `  N
)  ->  ( d  =  { t  |  E. u  e.  ( NN0  ^m  ( 1 ... k
) ) ( t  =  ( u  |`  ( 1 ... N
) )  /\  (
p `  u )  =  0 ) }  ->  d  e.  ~P ( NN0  ^m  ( 1 ... N ) ) ) )
3635rexlimdvw 2833 . . . . . . . . 9  |-  ( k  e.  ( ZZ>= `  N
)  ->  ( E. p  e.  (mzPoly `  (
1 ... k ) ) d  =  { t  |  E. u  e.  ( NN0  ^m  (
1 ... k ) ) ( t  =  ( u  |`  ( 1 ... N ) )  /\  ( p `  u )  =  0 ) }  ->  d  e.  ~P ( NN0  ^m  ( 1 ... N
) ) ) )
3736rexlimiv 2824 . . . . . . . 8  |-  ( E. k  e.  ( ZZ>= `  N ) E. p  e.  (mzPoly `  ( 1 ... k ) ) d  =  { t  |  E. u  e.  ( NN0  ^m  ( 1 ... k ) ) ( t  =  ( u  |`  ( 1 ... N ) )  /\  ( p `  u )  =  0 ) }  ->  d  e.  ~P ( NN0  ^m  ( 1 ... N
) ) )
3837abssi 3418 . . . . . . 7  |-  { d  |  E. k  e.  ( ZZ>= `  N ) E. p  e.  (mzPoly `  ( 1 ... k
) ) d  =  { t  |  E. u  e.  ( NN0  ^m  ( 1 ... k
) ) ( t  =  ( u  |`  ( 1 ... N
) )  /\  (
p `  u )  =  0 ) } }  C_  ~P ( NN0  ^m  ( 1 ... N ) )
3918, 38eqsstri 3378 . . . . . 6  |-  ran  (
k  e.  ( ZZ>= `  N ) ,  p  e.  (mzPoly `  ( 1 ... k ) )  |->  { t  |  E. u  e.  ( NN0  ^m  (
1 ... k ) ) ( t  =  ( u  |`  ( 1 ... N ) )  /\  ( p `  u )  =  0 ) } )  C_  ~P ( NN0  ^m  (
1 ... N ) )
4016, 39ssexi 4348 . . . . 5  |-  ran  (
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. 
_V
4114, 1, 40fvmpt 5806 . . . 4  |-  ( N  e.  NN0  ->  (Dioph `  N )  =  ran  ( k  e.  (
ZZ>= `  N ) ,  p  e.  (mzPoly `  ( 1 ... k
) )  |->  { t  |  E. u  e.  ( NN0  ^m  (
1 ... k ) ) ( t  =  ( u  |`  ( 1 ... N ) )  /\  ( p `  u )  =  0 ) } ) )
4241eleq2d 2503 . . 3  |-  ( N  e.  NN0  ->  ( D  e.  (Dioph `  N
)  <->  D  e.  ran  ( k  e.  (
ZZ>= `  N ) ,  p  e.  (mzPoly `  ( 1 ... k
) )  |->  { t  |  E. u  e.  ( NN0  ^m  (
1 ... k ) ) ( t  =  ( u  |`  ( 1 ... N ) )  /\  ( p `  u )  =  0 ) } ) ) )
43 ovex 6106 . . . . . 6  |-  ( NN0 
^m  ( 1 ... k ) )  e. 
_V
4443abrexex 5983 . . . . 5  |-  { t  |  E. u  e.  ( NN0  ^m  (
1 ... k ) ) t  =  ( u  |`  ( 1 ... N
) ) }  e.  _V
45 simpl 444 . . . . . . 7  |-  ( ( t  =  ( u  |`  ( 1 ... N
) )  /\  (
p `  u )  =  0 )  -> 
t  =  ( u  |`  ( 1 ... N
) ) )
4645reximi 2813 . . . . . 6  |-  ( 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
) ) )
4746ss2abi 3415 . . . . 5  |-  { t  |  E. u  e.  ( NN0  ^m  (
1 ... k ) ) ( t  =  ( u  |`  ( 1 ... N ) )  /\  ( p `  u )  =  0 ) }  C_  { t  |  E. u  e.  ( NN0  ^m  (
1 ... k ) ) t  =  ( u  |`  ( 1 ... N
) ) }
4844, 47ssexi 4348 . . . 4  |-  { t  |  E. u  e.  ( NN0  ^m  (
1 ... k ) ) ( t  =  ( u  |`  ( 1 ... N ) )  /\  ( p `  u )  =  0 ) }  e.  _V
4917, 48elrnmpt2 6183 . . 3  |-  ( D  e.  ran  ( 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. k  e.  ( ZZ>= `  N ) E. p  e.  (mzPoly `  ( 1 ... k
) ) D  =  { t  |  E. u  e.  ( NN0  ^m  ( 1 ... k
) ) ( t  =  ( u  |`  ( 1 ... N
) )  /\  (
p `  u )  =  0 ) } )
5042, 49syl6bb 253 . 2  |-  ( N  e.  NN0  ->  ( D  e.  (Dioph `  N
)  <->  E. k  e.  (
ZZ>= `  N ) E. p  e.  (mzPoly `  ( 1 ... k
) ) D  =  { t  |  E. u  e.  ( NN0  ^m  ( 1 ... k
) ) ( t  =  ( u  |`  ( 1 ... N
) )  /\  (
p `  u )  =  0 ) } ) )
514, 50biadan2 624 1  |-  ( D  e.  (Dioph `  N
)  <->  ( N  e. 
NN0  /\  E. k  e.  ( ZZ>= `  N ) E. p  e.  (mzPoly `  ( 1 ... k
) ) D  =  { t  |  E. u  e.  ( NN0  ^m  ( 1 ... k
) ) ( t  =  ( u  |`  ( 1 ... N
) )  /\  (
p `  u )  =  0 ) } ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   {cab 2422   E.wrex 2706    C_ wss 3320   ~Pcpw 3799   dom cdm 4878   ran crn 4879    |` cres 4880   -->wf 5450   ` cfv 5454  (class class class)co 6081    e. cmpt2 6083    ^m cmap 7018   0cc0 8990   1c1 8991   NN0cn0 10221   ZZ>=cuz 10488   ...cfz 11043  mzPolycmzp 26779  Diophcdioph 26813
This theorem is referenced by:  eldioph  26816  eldioph2b  26821  eldiophelnn0  26822
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-i2m1 9058  ax-1ne0 9059  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-recs 6633  df-rdg 6668  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-neg 9294  df-nn 10001  df-n0 10222  df-z 10283  df-uz 10489  df-fz 11044  df-dioph 26814
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