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Theorem eldiophb 26836
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 26835 . . . 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 5169 . . 3  |-  dom Dioph  C_  NN0
3 elfvdm 5554 . . 3  |-  ( D  e.  (Dioph `  N
)  ->  N  e.  dom Dioph )
42, 3sseldi 3178 . 2  |-  ( D  e.  (Dioph `  N
)  ->  N  e.  NN0 )
5 fveq2 5525 . . . . . . 7  |-  ( n  =  N  ->  ( ZZ>=
`  n )  =  ( ZZ>= `  N )
)
6 eqidd 2284 . . . . . . 7  |-  ( n  =  N  ->  (mzPoly `  ( 1 ... k
) )  =  (mzPoly `  ( 1 ... k
) ) )
7 oveq2 5866 . . . . . . . . . . . 12  |-  ( n  =  N  ->  (
1 ... n )  =  ( 1 ... N
) )
87reseq2d 4955 . . . . . . . . . . 11  |-  ( n  =  N  ->  (
u  |`  ( 1 ... n ) )  =  ( u  |`  (
1 ... N ) ) )
98eqeq2d 2294 . . . . . . . . . 10  |-  ( n  =  N  ->  (
t  =  ( u  |`  ( 1 ... n
) )  <->  t  =  ( u  |`  ( 1 ... N ) ) ) )
109anbi1d 685 . . . . . . . . 9  |-  ( n  =  N  ->  (
( t  =  ( u  |`  ( 1 ... n ) )  /\  ( p `  u )  =  0 )  <->  ( t  =  ( u  |`  (
1 ... N ) )  /\  ( p `  u )  =  0 ) ) )
1110rexbidv 2564 . . . . . . . 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 2397 . . . . . . 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 5910 . . . . . 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 4906 . . . . 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 5883 . . . . . . 7  |-  ( NN0 
^m  ( 1 ... N ) )  e. 
_V
1615pwex 4193 . . . . . 6  |-  ~P ( NN0  ^m  ( 1 ... N ) )  e. 
_V
17 eqid 2283 . . . . . . . 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 5954 . . . . . . 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 6792 . . . . . . . . . . . . . . . . 17  |-  ( u  e.  ( NN0  ^m  ( 1 ... k
) )  ->  u : ( 1 ... k ) --> NN0 )
20 fzss2 10831 . . . . . . . . . . . . . . . . 17  |-  ( k  e.  ( ZZ>= `  N
)  ->  ( 1 ... N )  C_  ( 1 ... k
) )
21 fssres 5408 . . . . . . . . . . . . . . . . 17  |-  ( ( u : ( 1 ... k ) --> NN0 
/\  ( 1 ... N )  C_  (
1 ... k ) )  ->  ( u  |`  ( 1 ... N
) ) : ( 1 ... N ) --> NN0 )
2219, 20, 21syl2anr 464 . . . . . . . . . . . . . . . 16  |-  ( ( k  e.  ( ZZ>= `  N )  /\  u  e.  ( NN0  ^m  (
1 ... k ) ) )  ->  ( u  |`  ( 1 ... N
) ) : ( 1 ... N ) --> NN0 )
23 nn0ex 9971 . . . . . . . . . . . . . . . . 17  |-  NN0  e.  _V
24 ovex 5883 . . . . . . . . . . . . . . . . 17  |-  ( 1 ... N )  e. 
_V
2523, 24elmap 6796 . . . . . . . . . . . . . . . 16  |-  ( ( u  |`  ( 1 ... N ) )  e.  ( NN0  ^m  ( 1 ... N
) )  <->  ( u  |`  ( 1 ... N
) ) : ( 1 ... N ) --> NN0 )
2622, 25sylibr 203 . . . . . . . . . . . . . . 15  |-  ( ( k  e.  ( ZZ>= `  N )  /\  u  e.  ( NN0  ^m  (
1 ... k ) ) )  ->  ( u  |`  ( 1 ... N
) )  e.  ( NN0  ^m  ( 1 ... N ) ) )
27 eleq1 2343 . . . . . . . . . . . . . . . 16  |-  ( t  =  ( u  |`  ( 1 ... N
) )  ->  (
t  e.  ( NN0 
^m  ( 1 ... N ) )  <->  ( u  |`  ( 1 ... N
) )  e.  ( NN0  ^m  ( 1 ... N ) ) ) )
2827adantr 451 . . . . . . . . . . . . . . 15  |-  ( ( t  =  ( u  |`  ( 1 ... N
) )  /\  (
p `  u )  =  0 )  -> 
( t  e.  ( NN0  ^m  ( 1 ... N ) )  <-> 
( u  |`  (
1 ... N ) )  e.  ( NN0  ^m  ( 1 ... N
) ) ) )
2926, 28syl5ibrcom 213 . . . . . . . . . . . . . 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 2667 . . . . . . . . . . . . 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 3247 . . . . . . . . . . . 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 4175 . . . . . . . . . . . 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 203 . . . . . . . . . . 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 2343 . . . . . . . . . . 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 213 . . . . . . . . . 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 2670 . . . . . . . . 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 2661 . . . . . . . 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 3248 . . . . . . 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 3208 . . . . . 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 4159 . . . . 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 5602 . . . 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 2350 . . 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 5883 . . . . . 6  |-  ( NN0 
^m  ( 1 ... k ) )  e. 
_V
4443abrexex 5763 . . . . 5  |-  { t  |  E. u  e.  ( NN0  ^m  (
1 ... k ) ) t  =  ( u  |`  ( 1 ... N
) ) }  e.  _V
45 simpl 443 . . . . . . 7  |-  ( ( t  =  ( u  |`  ( 1 ... N
) )  /\  (
p `  u )  =  0 )  -> 
t  =  ( u  |`  ( 1 ... N
) ) )
4645reximi 2650 . . . . . 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 3245 . . . . 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 4159 . . . 4  |-  { t  |  E. u  e.  ( NN0  ^m  (
1 ... k ) ) ( t  =  ( u  |`  ( 1 ... N ) )  /\  ( p `  u )  =  0 ) }  e.  _V
4917, 48elrnmpt2 5957 . . 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 252 . 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 623 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 176    /\ wa 358    = wceq 1623    e. wcel 1684   {cab 2269   E.wrex 2544    C_ wss 3152   ~Pcpw 3625   dom cdm 4689   ran crn 4690    |` cres 4691   -->wf 5251   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860    ^m cmap 6772   0cc0 8737   1c1 8738   NN0cn0 9965   ZZ>=cuz 10230   ...cfz 10782  mzPolycmzp 26800  Diophcdioph 26834
This theorem is referenced by:  eldioph  26837  eldioph2b  26842  eldiophelnn0  26843
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-i2m1 8805  ax-1ne0 8806  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-recs 6388  df-rdg 6423  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-neg 9040  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-dioph 26835
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