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Theorem eldiophb 30874
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 30873 . . . 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 5509 . . 3  |-  dom Dioph  C_  NN0
3 elfvdm 5898 . . 3  |-  ( D  e.  (Dioph `  N
)  ->  N  e.  dom Dioph )
42, 3sseldi 3497 . 2  |-  ( D  e.  (Dioph `  N
)  ->  N  e.  NN0 )
5 fveq2 5872 . . . . . . 7  |-  ( n  =  N  ->  ( ZZ>=
`  n )  =  ( ZZ>= `  N )
)
6 eqidd 2458 . . . . . . 7  |-  ( n  =  N  ->  (mzPoly `  ( 1 ... k
) )  =  (mzPoly `  ( 1 ... k
) ) )
7 oveq2 6304 . . . . . . . . . . . 12  |-  ( n  =  N  ->  (
1 ... n )  =  ( 1 ... N
) )
87reseq2d 5283 . . . . . . . . . . 11  |-  ( n  =  N  ->  (
u  |`  ( 1 ... n ) )  =  ( u  |`  (
1 ... N ) ) )
98eqeq2d 2471 . . . . . . . . . 10  |-  ( n  =  N  ->  (
t  =  ( u  |`  ( 1 ... n
) )  <->  t  =  ( u  |`  ( 1 ... N ) ) ) )
109anbi1d 704 . . . . . . . . 9  |-  ( n  =  N  ->  (
( t  =  ( u  |`  ( 1 ... n ) )  /\  ( p `  u )  =  0 )  <->  ( t  =  ( u  |`  (
1 ... N ) )  /\  ( p `  u )  =  0 ) ) )
1110rexbidv 2968 . . . . . . . 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 2593 . . . . . . 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 6358 . . . . . 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 5240 . . . . 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 6324 . . . . . . 7  |-  ( NN0 
^m  ( 1 ... N ) )  e. 
_V
1615pwex 4639 . . . . . 6  |-  ~P ( NN0  ^m  ( 1 ... N ) )  e. 
_V
17 eqid 2457 . . . . . . . 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 6411 . . . . . . 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 7459 . . . . . . . . . . . . . . . . 17  |-  ( u  e.  ( NN0  ^m  ( 1 ... k
) )  ->  u : ( 1 ... k ) --> NN0 )
20 fzss2 11749 . . . . . . . . . . . . . . . . 17  |-  ( k  e.  ( ZZ>= `  N
)  ->  ( 1 ... N )  C_  ( 1 ... k
) )
21 fssres 5757 . . . . . . . . . . . . . . . . 17  |-  ( ( u : ( 1 ... k ) --> NN0 
/\  ( 1 ... N )  C_  (
1 ... k ) )  ->  ( u  |`  ( 1 ... N
) ) : ( 1 ... N ) --> NN0 )
2219, 20, 21syl2anr 478 . . . . . . . . . . . . . . . 16  |-  ( ( k  e.  ( ZZ>= `  N )  /\  u  e.  ( NN0  ^m  (
1 ... k ) ) )  ->  ( u  |`  ( 1 ... N
) ) : ( 1 ... N ) --> NN0 )
23 nn0ex 10822 . . . . . . . . . . . . . . . . 17  |-  NN0  e.  _V
24 ovex 6324 . . . . . . . . . . . . . . . . 17  |-  ( 1 ... N )  e. 
_V
2523, 24elmap 7466 . . . . . . . . . . . . . . . 16  |-  ( ( u  |`  ( 1 ... N ) )  e.  ( NN0  ^m  ( 1 ... N
) )  <->  ( u  |`  ( 1 ... N
) ) : ( 1 ... N ) --> NN0 )
2622, 25sylibr 212 . . . . . . . . . . . . . . 15  |-  ( ( k  e.  ( ZZ>= `  N )  /\  u  e.  ( NN0  ^m  (
1 ... k ) ) )  ->  ( u  |`  ( 1 ... N
) )  e.  ( NN0  ^m  ( 1 ... N ) ) )
27 eleq1 2529 . . . . . . . . . . . . . . . 16  |-  ( t  =  ( u  |`  ( 1 ... N
) )  ->  (
t  e.  ( NN0 
^m  ( 1 ... N ) )  <->  ( u  |`  ( 1 ... N
) )  e.  ( NN0  ^m  ( 1 ... N ) ) ) )
2827adantr 465 . . . . . . . . . . . . . . 15  |-  ( ( t  =  ( u  |`  ( 1 ... N
) )  /\  (
p `  u )  =  0 )  -> 
( t  e.  ( NN0  ^m  ( 1 ... N ) )  <-> 
( u  |`  (
1 ... N ) )  e.  ( NN0  ^m  ( 1 ... N
) ) ) )
2926, 28syl5ibrcom 222 . . . . . . . . . . . . . 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 2949 . . . . . . . . . . . . 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 3570 . . . . . . . . . . . 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 4620 . . . . . . . . . . . 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 212 . . . . . . . . . . 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 2529 . . . . . . . . . . 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 222 . . . . . . . . . 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 2952 . . . . . . . . 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 2943 . . . . . . . 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 3571 . . . . . . 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 3529 . . . . . 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 4601 . . . . 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 5956 . . . 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 2527 . . 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 6324 . . . . . 6  |-  ( NN0 
^m  ( 1 ... k ) )  e. 
_V
4443abrexex 6773 . . . . 5  |-  { t  |  E. u  e.  ( NN0  ^m  (
1 ... k ) ) t  =  ( u  |`  ( 1 ... N
) ) }  e.  _V
45 simpl 457 . . . . . . 7  |-  ( ( t  =  ( u  |`  ( 1 ... N
) )  /\  (
p `  u )  =  0 )  -> 
t  =  ( u  |`  ( 1 ... N
) ) )
4645reximi 2925 . . . . . 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 3568 . . . . 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 4601 . . . 4  |-  { t  |  E. u  e.  ( NN0  ^m  (
1 ... k ) ) ( t  =  ( u  |`  ( 1 ... N ) )  /\  ( p `  u )  =  0 ) }  e.  _V
4917, 48elrnmpt2 6414 . . 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 261 . 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 642 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 184    /\ wa 369    = wceq 1395    e. wcel 1819   {cab 2442   E.wrex 2808    C_ wss 3471   ~Pcpw 4015   dom cdm 5008   ran crn 5009    |` cres 5010   -->wf 5590   ` cfv 5594  (class class class)co 6296    |-> cmpt2 6298    ^m cmap 7438   0cc0 9509   1c1 9510   NN0cn0 10816   ZZ>=cuz 11106   ...cfz 11697  mzPolycmzp 30838  Diophcdioph 30872
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-i2m1 9577  ax-1ne0 9578  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  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 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-neg 9827  df-nn 10557  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-dioph 30873
This theorem is referenced by:  eldioph  30875  eldioph2b  30880  eldiophelnn0  30881
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