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Theorem eldiophb 35569
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 35568 . . . 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 5350 . . 3  |-  dom Dioph  C_  NN0
3 elfvdm 5908 . . 3  |-  ( D  e.  (Dioph `  N
)  ->  N  e.  dom Dioph )
42, 3sseldi 3462 . 2  |-  ( D  e.  (Dioph `  N
)  ->  N  e.  NN0 )
5 fveq2 5882 . . . . . . 7  |-  ( n  =  N  ->  ( ZZ>=
`  n )  =  ( ZZ>= `  N )
)
6 eqidd 2423 . . . . . . 7  |-  ( n  =  N  ->  (mzPoly `  ( 1 ... k
) )  =  (mzPoly `  ( 1 ... k
) ) )
7 oveq2 6314 . . . . . . . . . . . 12  |-  ( n  =  N  ->  (
1 ... n )  =  ( 1 ... N
) )
87reseq2d 5124 . . . . . . . . . . 11  |-  ( n  =  N  ->  (
u  |`  ( 1 ... n ) )  =  ( u  |`  (
1 ... N ) ) )
98eqeq2d 2436 . . . . . . . . . 10  |-  ( n  =  N  ->  (
t  =  ( u  |`  ( 1 ... n
) )  <->  t  =  ( u  |`  ( 1 ... N ) ) ) )
109anbi1d 709 . . . . . . . . 9  |-  ( n  =  N  ->  (
( t  =  ( u  |`  ( 1 ... n ) )  /\  ( p `  u )  =  0 )  <->  ( t  =  ( u  |`  (
1 ... N ) )  /\  ( p `  u )  =  0 ) ) )
1110rexbidv 2936 . . . . . . . 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 2553 . . . . . . 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 6368 . . . . . 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 5081 . . . . 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 6334 . . . . . . 7  |-  ( NN0 
^m  ( 1 ... N ) )  e. 
_V
1615pwex 4607 . . . . . 6  |-  ~P ( NN0  ^m  ( 1 ... N ) )  e. 
_V
17 eqid 2422 . . . . . . . 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 6421 . . . . . . 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 7505 . . . . . . . . . . . . . . . . 17  |-  ( u  e.  ( NN0  ^m  ( 1 ... k
) )  ->  u : ( 1 ... k ) --> NN0 )
20 fzss2 11846 . . . . . . . . . . . . . . . . 17  |-  ( k  e.  ( ZZ>= `  N
)  ->  ( 1 ... N )  C_  ( 1 ... k
) )
21 fssres 5766 . . . . . . . . . . . . . . . . 17  |-  ( ( u : ( 1 ... k ) --> NN0 
/\  ( 1 ... N )  C_  (
1 ... k ) )  ->  ( u  |`  ( 1 ... N
) ) : ( 1 ... N ) --> NN0 )
2219, 20, 21syl2anr 480 . . . . . . . . . . . . . . . 16  |-  ( ( k  e.  ( ZZ>= `  N )  /\  u  e.  ( NN0  ^m  (
1 ... k ) ) )  ->  ( u  |`  ( 1 ... N
) ) : ( 1 ... N ) --> NN0 )
23 nn0ex 10883 . . . . . . . . . . . . . . . . 17  |-  NN0  e.  _V
24 ovex 6334 . . . . . . . . . . . . . . . . 17  |-  ( 1 ... N )  e. 
_V
2523, 24elmap 7512 . . . . . . . . . . . . . . . 16  |-  ( ( u  |`  ( 1 ... N ) )  e.  ( NN0  ^m  ( 1 ... N
) )  <->  ( u  |`  ( 1 ... N
) ) : ( 1 ... N ) --> NN0 )
2622, 25sylibr 215 . . . . . . . . . . . . . . 15  |-  ( ( k  e.  ( ZZ>= `  N )  /\  u  e.  ( NN0  ^m  (
1 ... k ) ) )  ->  ( u  |`  ( 1 ... N
) )  e.  ( NN0  ^m  ( 1 ... N ) ) )
27 eleq1 2495 . . . . . . . . . . . . . . . 16  |-  ( t  =  ( u  |`  ( 1 ... N
) )  ->  (
t  e.  ( NN0 
^m  ( 1 ... N ) )  <->  ( u  |`  ( 1 ... N
) )  e.  ( NN0  ^m  ( 1 ... N ) ) ) )
2827adantr 466 . . . . . . . . . . . . . . 15  |-  ( ( t  =  ( u  |`  ( 1 ... N
) )  /\  (
p `  u )  =  0 )  -> 
( t  e.  ( NN0  ^m  ( 1 ... N ) )  <-> 
( u  |`  (
1 ... N ) )  e.  ( NN0  ^m  ( 1 ... N
) ) ) )
2926, 28syl5ibrcom 225 . . . . . . . . . . . . . 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 2914 . . . . . . . . . . . . 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 3535 . . . . . . . . . . . 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 4588 . . . . . . . . . . . 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 215 . . . . . . . . . . 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 2495 . . . . . . . . . . 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 225 . . . . . . . . . 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 2917 . . . . . . . . 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 2908 . . . . . . . 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 3536 . . . . . . 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 3494 . . . . . 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 4569 . . . . 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 5965 . . . 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 2492 . . 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 6334 . . . . . 6  |-  ( NN0 
^m  ( 1 ... k ) )  e. 
_V
4443abrexex 6782 . . . . 5  |-  { t  |  E. u  e.  ( NN0  ^m  (
1 ... k ) ) t  =  ( u  |`  ( 1 ... N
) ) }  e.  _V
45 simpl 458 . . . . . . 7  |-  ( ( t  =  ( u  |`  ( 1 ... N
) )  /\  (
p `  u )  =  0 )  -> 
t  =  ( u  |`  ( 1 ... N
) ) )
4645reximi 2890 . . . . . 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 3533 . . . . 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 4569 . . . 4  |-  { t  |  E. u  e.  ( NN0  ^m  (
1 ... k ) ) ( t  =  ( u  |`  ( 1 ... N ) )  /\  ( p `  u )  =  0 ) }  e.  _V
4917, 48elrnmpt2 6424 . . 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 264 . 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 646 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 setvar class
Syntax hints:    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1872   {cab 2407   E.wrex 2772    C_ wss 3436   ~Pcpw 3981   dom cdm 4853   ran crn 4854    |` cres 4855   -->wf 5597   ` cfv 5601  (class class class)co 6306    |-> cmpt2 6308    ^m cmap 7484   0cc0 9547   1c1 9548   NN0cn0 10877   ZZ>=cuz 11167   ...cfz 11792  mzPolycmzp 35534  Diophcdioph 35567
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6598  ax-cnex 9603  ax-resscn 9604  ax-1cn 9605  ax-icn 9606  ax-addcl 9607  ax-addrcl 9608  ax-mulcl 9609  ax-mulrcl 9610  ax-i2m1 9615  ax-1ne0 9616  ax-rrecex 9619  ax-cnre 9620  ax-pre-lttri 9621  ax-pre-lttrn 9622
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6309  df-oprab 6310  df-mpt2 6311  df-om 6708  df-1st 6808  df-2nd 6809  df-wrecs 7040  df-recs 7102  df-rdg 7140  df-er 7375  df-map 7486  df-en 7582  df-dom 7583  df-sdom 7584  df-pnf 9685  df-mnf 9686  df-xr 9687  df-ltxr 9688  df-le 9689  df-neg 9871  df-nn 10618  df-n0 10878  df-z 10946  df-uz 11168  df-fz 11793  df-dioph 35568
This theorem is referenced by:  eldioph  35570  eldioph2b  35575  eldiophelnn0  35576
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