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Theorem 2rexfrabdioph 26877
Description: Diophantine set builder for existential quantifier, explicit substitution, two variables. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.)
Hypotheses
Ref Expression
rexfrabdioph.1  |-  M  =  ( N  +  1 )
rexfrabdioph.2  |-  L  =  ( M  +  1 )
Assertion
Ref Expression
2rexfrabdioph  |-  ( ( N  e.  NN0  /\  { t  e.  ( NN0 
^m  ( 1 ... L ) )  | 
[. ( t  |`  ( 1 ... N
) )  /  u ]. [. ( t `  M )  /  v ]. [. ( t `  L )  /  w ]. ph }  e.  (Dioph `  L ) )  ->  { u  e.  ( NN0  ^m  ( 1 ... N ) )  |  E. v  e.  NN0  E. w  e.  NN0  ph }  e.  (Dioph `  N )
)
Distinct variable groups:    u, t,
v, w, L    t, M, u, v, w    t, N, u, v, w    ph, t
Allowed substitution hints:    ph( w, v, u)

Proof of Theorem 2rexfrabdioph
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 vex 2791 . . . . . . 7  |-  a  e. 
_V
21resex 4995 . . . . . 6  |-  ( a  |`  ( 1 ... N
) )  e.  _V
3 fvex 5539 . . . . . 6  |-  ( a `
 M )  e. 
_V
42, 32sbcrex 26864 . . . . 5  |-  ( [. ( a  |`  (
1 ... N ) )  /  u ]. [. (
a `  M )  /  v ]. E. w  e.  NN0  ph  <->  E. w  e.  NN0  [. ( a  |`  ( 1 ... N
) )  /  u ]. [. ( a `  M )  /  v ]. ph )
54a1i 10 . . . 4  |-  ( a  e.  ( NN0  ^m  ( 1 ... M
) )  ->  ( [. ( a  |`  (
1 ... N ) )  /  u ]. [. (
a `  M )  /  v ]. E. w  e.  NN0  ph  <->  E. w  e.  NN0  [. ( a  |`  ( 1 ... N
) )  /  u ]. [. ( a `  M )  /  v ]. ph ) )
65rabbiia 2778 . . 3  |-  { a  e.  ( NN0  ^m  ( 1 ... M
) )  |  [. ( a  |`  (
1 ... N ) )  /  u ]. [. (
a `  M )  /  v ]. E. w  e.  NN0  ph }  =  { a  e.  ( NN0  ^m  ( 1 ... M ) )  |  E. w  e. 
NN0  [. ( a  |`  ( 1 ... N
) )  /  u ]. [. ( a `  M )  /  v ]. ph }
7 rexfrabdioph.1 . . . . . 6  |-  M  =  ( N  +  1 )
8 peano2nn0 10004 . . . . . 6  |-  ( N  e.  NN0  ->  ( N  +  1 )  e. 
NN0 )
97, 8syl5eqel 2367 . . . . 5  |-  ( N  e.  NN0  ->  M  e. 
NN0 )
109adantr 451 . . . 4  |-  ( ( N  e.  NN0  /\  { t  e.  ( NN0 
^m  ( 1 ... L ) )  | 
[. ( t  |`  ( 1 ... N
) )  /  u ]. [. ( t `  M )  /  v ]. [. ( t `  L )  /  w ]. ph }  e.  (Dioph `  L ) )  ->  M  e.  NN0 )
11 vex 2791 . . . . . . . . . 10  |-  t  e. 
_V
1211resex 4995 . . . . . . . . 9  |-  ( t  |`  ( 1 ... M
) )  e.  _V
13 fvex 5539 . . . . . . . . . 10  |-  ( t `
 L )  e. 
_V
1413, 2, 3sbcrot3OLD 26872 . . . . . . . . 9  |-  ( [. ( t `  L
)  /  w ]. [. ( a  |`  (
1 ... N ) )  /  u ]. [. (
a `  M )  /  v ]. ph  <->  [. ( a  |`  ( 1 ... N
) )  /  u ]. [. ( a `  M )  /  v ]. [. ( t `  L )  /  w ]. ph )
1512, 14sbcbiiiOLD 26867 . . . . . . . 8  |-  ( [. ( t  |`  (
1 ... M ) )  /  a ]. [. (
t `  L )  /  w ]. [. (
a  |`  ( 1 ... N ) )  /  u ]. [. ( a `
 M )  / 
v ]. ph  <->  [. ( t  |`  ( 1 ... M
) )  /  a ]. [. ( a  |`  ( 1 ... N
) )  /  u ]. [. ( a `  M )  /  v ]. [. ( t `  L )  /  w ]. ph )
1612resex 4995 . . . . . . . . . 10  |-  ( ( t  |`  ( 1 ... M ) )  |`  ( 1 ... N
) )  e.  _V
17 reseq1 4949 . . . . . . . . . . 11  |-  ( a  =  ( t  |`  ( 1 ... M
) )  ->  (
a  |`  ( 1 ... N ) )  =  ( ( t  |`  ( 1 ... M
) )  |`  (
1 ... N ) ) )
1817sbccomiegOLD 26874 . . . . . . . . . 10  |-  ( ( ( t  |`  (
1 ... M ) )  e.  _V  /\  (
( t  |`  (
1 ... M ) )  |`  ( 1 ... N
) )  e.  _V )  ->  ( [. (
t  |`  ( 1 ... M ) )  / 
a ]. [. ( a  |`  ( 1 ... N
) )  /  u ]. [. ( a `  M )  /  v ]. [. ( t `  L )  /  w ]. ph  <->  [. ( ( t  |`  ( 1 ... M
) )  |`  (
1 ... N ) )  /  u ]. [. (
t  |`  ( 1 ... M ) )  / 
a ]. [. ( a `
 M )  / 
v ]. [. ( t `
 L )  /  w ]. ph ) )
1912, 16, 18mp2an 653 . . . . . . . . 9  |-  ( [. ( t  |`  (
1 ... M ) )  /  a ]. [. (
a  |`  ( 1 ... N ) )  /  u ]. [. ( a `
 M )  / 
v ]. [. ( t `
 L )  /  w ]. ph  <->  [. ( ( t  |`  ( 1 ... M ) )  |`  ( 1 ... N
) )  /  u ]. [. ( t  |`  ( 1 ... M
) )  /  a ]. [. ( a `  M )  /  v ]. [. ( t `  L )  /  w ]. ph )
20 fzssp1 10834 . . . . . . . . . . . 12  |-  ( 1 ... N )  C_  ( 1 ... ( N  +  1 ) )
217oveq2i 5869 . . . . . . . . . . . 12  |-  ( 1 ... M )  =  ( 1 ... ( N  +  1 ) )
2220, 21sseqtr4i 3211 . . . . . . . . . . 11  |-  ( 1 ... N )  C_  ( 1 ... M
)
23 resabs1 4984 . . . . . . . . . . 11  |-  ( ( 1 ... N ) 
C_  ( 1 ... M )  ->  (
( t  |`  (
1 ... M ) )  |`  ( 1 ... N
) )  =  ( t  |`  ( 1 ... N ) ) )
24 dfsbcq 2993 . . . . . . . . . . 11  |-  ( ( ( t  |`  (
1 ... M ) )  |`  ( 1 ... N
) )  =  ( t  |`  ( 1 ... N ) )  ->  ( [. (
( t  |`  (
1 ... M ) )  |`  ( 1 ... N
) )  /  u ]. [. ( t  |`  ( 1 ... M
) )  /  a ]. [. ( a `  M )  /  v ]. [. ( t `  L )  /  w ]. ph  <->  [. ( t  |`  ( 1 ... N
) )  /  u ]. [. ( t  |`  ( 1 ... M
) )  /  a ]. [. ( a `  M )  /  v ]. [. ( t `  L )  /  w ]. ph ) )
2522, 23, 24mp2b 9 . . . . . . . . . 10  |-  ( [. ( ( t  |`  ( 1 ... M
) )  |`  (
1 ... N ) )  /  u ]. [. (
t  |`  ( 1 ... M ) )  / 
a ]. [. ( a `
 M )  / 
v ]. [. ( t `
 L )  /  w ]. ph  <->  [. ( t  |`  ( 1 ... N
) )  /  u ]. [. ( t  |`  ( 1 ... M
) )  /  a ]. [. ( a `  M )  /  v ]. [. ( t `  L )  /  w ]. ph )
263ax-gen 1533 . . . . . . . . . . . . 13  |-  A. a
( a `  M
)  e.  _V
27 fveq1 5524 . . . . . . . . . . . . . 14  |-  ( a  =  ( t  |`  ( 1 ... M
) )  ->  (
a `  M )  =  ( ( t  |`  ( 1 ... M
) ) `  M
) )
2827sbcco3gOLD 3137 . . . . . . . . . . . . 13  |-  ( ( ( t  |`  (
1 ... M ) )  e.  _V  /\  A. a ( a `  M )  e.  _V )  ->  ( [. (
t  |`  ( 1 ... M ) )  / 
a ]. [. ( a `
 M )  / 
v ]. [. ( t `
 L )  /  w ]. ph  <->  [. ( ( t  |`  ( 1 ... M ) ) `
 M )  / 
v ]. [. ( t `
 L )  /  w ]. ph ) )
2912, 26, 28mp2an 653 . . . . . . . . . . . 12  |-  ( [. ( t  |`  (
1 ... M ) )  /  a ]. [. (
a `  M )  /  v ]. [. (
t `  L )  /  w ]. ph  <->  [. ( ( t  |`  ( 1 ... M ) ) `
 M )  / 
v ]. [. ( t `
 L )  /  w ]. ph )
30 nn0p1nn 10003 . . . . . . . . . . . . . . 15  |-  ( N  e.  NN0  ->  ( N  +  1 )  e.  NN )
317, 30syl5eqel 2367 . . . . . . . . . . . . . 14  |-  ( N  e.  NN0  ->  M  e.  NN )
32 elfz1end 10820 . . . . . . . . . . . . . 14  |-  ( M  e.  NN  <->  M  e.  ( 1 ... M
) )
3331, 32sylib 188 . . . . . . . . . . . . 13  |-  ( N  e.  NN0  ->  M  e.  ( 1 ... M
) )
34 fvres 5542 . . . . . . . . . . . . 13  |-  ( M  e.  ( 1 ... M )  ->  (
( t  |`  (
1 ... M ) ) `
 M )  =  ( t `  M
) )
35 dfsbcq 2993 . . . . . . . . . . . . 13  |-  ( ( ( t  |`  (
1 ... M ) ) `
 M )  =  ( t `  M
)  ->  ( [. ( ( t  |`  ( 1 ... M
) ) `  M
)  /  v ]. [. ( t `  L
)  /  w ]. ph  <->  [. ( t `  M
)  /  v ]. [. ( t `  L
)  /  w ]. ph ) )
3633, 34, 353syl 18 . . . . . . . . . . . 12  |-  ( N  e.  NN0  ->  ( [. ( ( t  |`  ( 1 ... M
) ) `  M
)  /  v ]. [. ( t `  L
)  /  w ]. ph  <->  [. ( t `  M
)  /  v ]. [. ( t `  L
)  /  w ]. ph ) )
3729, 36syl5bb 248 . . . . . . . . . . 11  |-  ( N  e.  NN0  ->  ( [. ( t  |`  (
1 ... M ) )  /  a ]. [. (
a `  M )  /  v ]. [. (
t `  L )  /  w ]. ph  <->  [. ( t `
 M )  / 
v ]. [. ( t `
 L )  /  w ]. ph ) )
3837sbcbidv 3045 . . . . . . . . . 10  |-  ( N  e.  NN0  ->  ( [. ( t  |`  (
1 ... N ) )  /  u ]. [. (
t  |`  ( 1 ... M ) )  / 
a ]. [. ( a `
 M )  / 
v ]. [. ( t `
 L )  /  w ]. ph  <->  [. ( t  |`  ( 1 ... N
) )  /  u ]. [. ( t `  M )  /  v ]. [. ( t `  L )  /  w ]. ph ) )
3925, 38syl5bb 248 . . . . . . . . 9  |-  ( N  e.  NN0  ->  ( [. ( ( t  |`  ( 1 ... M
) )  |`  (
1 ... N ) )  /  u ]. [. (
t  |`  ( 1 ... M ) )  / 
a ]. [. ( a `
 M )  / 
v ]. [. ( t `
 L )  /  w ]. ph  <->  [. ( t  |`  ( 1 ... N
) )  /  u ]. [. ( t `  M )  /  v ]. [. ( t `  L )  /  w ]. ph ) )
4019, 39syl5bb 248 . . . . . . . 8  |-  ( N  e.  NN0  ->  ( [. ( t  |`  (
1 ... M ) )  /  a ]. [. (
a  |`  ( 1 ... N ) )  /  u ]. [. ( a `
 M )  / 
v ]. [. ( t `
 L )  /  w ]. ph  <->  [. ( t  |`  ( 1 ... N
) )  /  u ]. [. ( t `  M )  /  v ]. [. ( t `  L )  /  w ]. ph ) )
4115, 40syl5rbb 249 . . . . . . 7  |-  ( N  e.  NN0  ->  ( [. ( t  |`  (
1 ... N ) )  /  u ]. [. (
t `  M )  /  v ]. [. (
t `  L )  /  w ]. ph  <->  [. ( t  |`  ( 1 ... M
) )  /  a ]. [. ( t `  L )  /  w ]. [. ( a  |`  ( 1 ... N
) )  /  u ]. [. ( a `  M )  /  v ]. ph ) )
4241rabbidv 2780 . . . . . 6  |-  ( N  e.  NN0  ->  { t  e.  ( NN0  ^m  ( 1 ... L
) )  |  [. ( t  |`  (
1 ... N ) )  /  u ]. [. (
t `  M )  /  v ]. [. (
t `  L )  /  w ]. ph }  =  { t  e.  ( NN0  ^m  ( 1 ... L ) )  |  [. ( t  |`  ( 1 ... M
) )  /  a ]. [. ( t `  L )  /  w ]. [. ( a  |`  ( 1 ... N
) )  /  u ]. [. ( a `  M )  /  v ]. ph } )
4342eleq1d 2349 . . . . 5  |-  ( N  e.  NN0  ->  ( { t  e.  ( NN0 
^m  ( 1 ... L ) )  | 
[. ( t  |`  ( 1 ... N
) )  /  u ]. [. ( t `  M )  /  v ]. [. ( t `  L )  /  w ]. ph }  e.  (Dioph `  L )  <->  { t  e.  ( NN0  ^m  (
1 ... L ) )  |  [. ( t  |`  ( 1 ... M
) )  /  a ]. [. ( t `  L )  /  w ]. [. ( a  |`  ( 1 ... N
) )  /  u ]. [. ( a `  M )  /  v ]. ph }  e.  (Dioph `  L ) ) )
4443biimpa 470 . . . 4  |-  ( ( N  e.  NN0  /\  { t  e.  ( NN0 
^m  ( 1 ... L ) )  | 
[. ( t  |`  ( 1 ... N
) )  /  u ]. [. ( t `  M )  /  v ]. [. ( t `  L )  /  w ]. ph }  e.  (Dioph `  L ) )  ->  { t  e.  ( NN0  ^m  ( 1 ... L ) )  |  [. ( t  |`  ( 1 ... M
) )  /  a ]. [. ( t `  L )  /  w ]. [. ( a  |`  ( 1 ... N
) )  /  u ]. [. ( a `  M )  /  v ]. ph }  e.  (Dioph `  L ) )
45 rexfrabdioph.2 . . . . 5  |-  L  =  ( M  +  1 )
4645rexfrabdioph 26876 . . . 4  |-  ( ( M  e.  NN0  /\  { t  e.  ( NN0 
^m  ( 1 ... L ) )  | 
[. ( t  |`  ( 1 ... M
) )  /  a ]. [. ( t `  L )  /  w ]. [. ( a  |`  ( 1 ... N
) )  /  u ]. [. ( a `  M )  /  v ]. ph }  e.  (Dioph `  L ) )  ->  { a  e.  ( NN0  ^m  ( 1 ... M ) )  |  E. w  e. 
NN0  [. ( a  |`  ( 1 ... N
) )  /  u ]. [. ( a `  M )  /  v ]. ph }  e.  (Dioph `  M ) )
4710, 44, 46syl2anc 642 . . 3  |-  ( ( N  e.  NN0  /\  { t  e.  ( NN0 
^m  ( 1 ... L ) )  | 
[. ( t  |`  ( 1 ... N
) )  /  u ]. [. ( t `  M )  /  v ]. [. ( t `  L )  /  w ]. ph }  e.  (Dioph `  L ) )  ->  { a  e.  ( NN0  ^m  ( 1 ... M ) )  |  E. w  e. 
NN0  [. ( a  |`  ( 1 ... N
) )  /  u ]. [. ( a `  M )  /  v ]. ph }  e.  (Dioph `  M ) )
486, 47syl5eqel 2367 . 2  |-  ( ( N  e.  NN0  /\  { t  e.  ( NN0 
^m  ( 1 ... L ) )  | 
[. ( t  |`  ( 1 ... N
) )  /  u ]. [. ( t `  M )  /  v ]. [. ( t `  L )  /  w ]. ph }  e.  (Dioph `  L ) )  ->  { a  e.  ( NN0  ^m  ( 1 ... M ) )  |  [. ( a  |`  ( 1 ... N
) )  /  u ]. [. ( a `  M )  /  v ]. E. w  e.  NN0  ph }  e.  (Dioph `  M ) )
497rexfrabdioph 26876 . 2  |-  ( ( N  e.  NN0  /\  { a  e.  ( NN0 
^m  ( 1 ... M ) )  | 
[. ( a  |`  ( 1 ... N
) )  /  u ]. [. ( a `  M )  /  v ]. E. w  e.  NN0  ph }  e.  (Dioph `  M ) )  ->  { u  e.  ( NN0  ^m  ( 1 ... N ) )  |  E. v  e.  NN0  E. w  e.  NN0  ph }  e.  (Dioph `  N )
)
5048, 49syldan 456 1  |-  ( ( N  e.  NN0  /\  { t  e.  ( NN0 
^m  ( 1 ... L ) )  | 
[. ( t  |`  ( 1 ... N
) )  /  u ]. [. ( t `  M )  /  v ]. [. ( t `  L )  /  w ]. ph }  e.  (Dioph `  L ) )  ->  { u  e.  ( NN0  ^m  ( 1 ... N ) )  |  E. v  e.  NN0  E. w  e.  NN0  ph }  e.  (Dioph `  N )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1527    = wceq 1623    e. wcel 1684   E.wrex 2544   {crab 2547   _Vcvv 2788   [.wsbc 2991    C_ wss 3152    |` cres 4691   ` cfv 5255  (class class class)co 5858    ^m cmap 6772   1c1 8738    + caddc 8740   NNcn 9746   NN0cn0 9965   ...cfz 10782  Diophcdioph 26834
This theorem is referenced by:  3rexfrabdioph  26878  4rexfrabdioph  26879  6rexfrabdioph  26880
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-inf2 7342  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-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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-rmo 2551  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-int 3863  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-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-hash 11338  df-mzpcl 26801  df-mzp 26802  df-dioph 26835
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