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Theorem 2rexfrabdioph 26468
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 2876 . . . . . . 7  |-  a  e. 
_V
21resex 5098 . . . . . 6  |-  ( a  |`  ( 1 ... N
) )  e.  _V
3 fvex 5646 . . . . . 6  |-  ( a `
 M )  e. 
_V
42, 32sbcrex 26455 . . . . 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 2863 . . 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 10153 . . . . . 6  |-  ( N  e.  NN0  ->  ( N  +  1 )  e. 
NN0 )
97, 8syl5eqel 2450 . . . . 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 2876 . . . . . . . . . 10  |-  t  e. 
_V
1211resex 5098 . . . . . . . . 9  |-  ( t  |`  ( 1 ... M
) )  e.  _V
13 fvex 5646 . . . . . . . . . 10  |-  ( t `
 L )  e. 
_V
1413, 2, 3sbcrot3OLD 26463 . . . . . . . . 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 26458 . . . . . . . 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 5098 . . . . . . . . . 10  |-  ( ( t  |`  ( 1 ... M ) )  |`  ( 1 ... N
) )  e.  _V
17 reseq1 5052 . . . . . . . . . . 11  |-  ( a  =  ( t  |`  ( 1 ... M
) )  ->  (
a  |`  ( 1 ... N ) )  =  ( ( t  |`  ( 1 ... M
) )  |`  (
1 ... N ) ) )
1817sbccomiegOLD 26465 . . . . . . . . . 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 10987 . . . . . . . . . . . 12  |-  ( 1 ... N )  C_  ( 1 ... ( N  +  1 ) )
217oveq2i 5992 . . . . . . . . . . . 12  |-  ( 1 ... M )  =  ( 1 ... ( N  +  1 ) )
2220, 21sseqtr4i 3297 . . . . . . . . . . 11  |-  ( 1 ... N )  C_  ( 1 ... M
)
23 resabs1 5087 . . . . . . . . . . 11  |-  ( ( 1 ... N ) 
C_  ( 1 ... M )  ->  (
( t  |`  (
1 ... M ) )  |`  ( 1 ... N
) )  =  ( t  |`  ( 1 ... N ) ) )
24 dfsbcq 3079 . . . . . . . . . . 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 1551 . . . . . . . . . . . . 13  |-  A. a
( a `  M
)  e.  _V
27 fveq1 5631 . . . . . . . . . . . . . 14  |-  ( a  =  ( t  |`  ( 1 ... M
) )  ->  (
a `  M )  =  ( ( t  |`  ( 1 ... M
) ) `  M
) )
2827sbcco3gOLD 3223 . . . . . . . . . . . . 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 10152 . . . . . . . . . . . . . . 15  |-  ( N  e.  NN0  ->  ( N  +  1 )  e.  NN )
317, 30syl5eqel 2450 . . . . . . . . . . . . . 14  |-  ( N  e.  NN0  ->  M  e.  NN )
32 elfz1end 10973 . . . . . . . . . . . . . 14  |-  ( M  e.  NN  <->  M  e.  ( 1 ... M
) )
3331, 32sylib 188 . . . . . . . . . . . . 13  |-  ( N  e.  NN0  ->  M  e.  ( 1 ... M
) )
34 fvres 5649 . . . . . . . . . . . . 13  |-  ( M  e.  ( 1 ... M )  ->  (
( t  |`  (
1 ... M ) ) `
 M )  =  ( t `  M
) )
35 dfsbcq 3079 . . . . . . . . . . . . 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 3131 . . . . . . . . . 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 2865 . . . . . 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 2432 . . . . 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 26467 . . . 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 2450 . 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 26467 . 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 1545    = wceq 1647    e. wcel 1715   E.wrex 2629   {crab 2632   _Vcvv 2873   [.wsbc 3077    C_ wss 3238    |` cres 4794   ` cfv 5358  (class class class)co 5981    ^m cmap 6915   1c1 8885    + caddc 8887   NNcn 9893   NN0cn0 10114   ...cfz 10935  Diophcdioph 26425
This theorem is referenced by:  3rexfrabdioph  26469  4rexfrabdioph  26470  6rexfrabdioph  26471
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-inf2 7489  ax-cnex 8940  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960  ax-pre-mulgt0 8961
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-int 3965  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-of 6205  df-1st 6249  df-2nd 6250  df-riota 6446  df-recs 6530  df-rdg 6565  df-1o 6621  df-oadd 6625  df-er 6802  df-map 6917  df-en 7007  df-dom 7008  df-sdom 7009  df-fin 7010  df-card 7719  df-cda 7941  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-le 9020  df-sub 9186  df-neg 9187  df-nn 9894  df-n0 10115  df-z 10176  df-uz 10382  df-fz 10936  df-hash 11506  df-mzpcl 26392  df-mzp 26393  df-dioph 26426
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