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Theorem 2rexfrabdioph 26754
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 2927 . . . . . . 7  |-  a  e. 
_V
21resex 5153 . . . . . 6  |-  ( a  |`  ( 1 ... N
) )  e.  _V
3 fvex 5709 . . . . . 6  |-  ( a `
 M )  e. 
_V
42, 32sbcrex 26741 . . . . 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 11 . . . 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 2914 . . 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 10224 . . . . . 6  |-  ( N  e.  NN0  ->  ( N  +  1 )  e. 
NN0 )
97, 8syl5eqel 2496 . . . . 5  |-  ( N  e.  NN0  ->  M  e. 
NN0 )
109adantr 452 . . . 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 sbcrot3 26745 . . . . . . . . 9  |-  ( [. ( t `  L
)  /  w ]. [. ( a  |`  (
1 ... N ) )  /  u ]. [. (
a `  M )  /  v ]. ph  <->  [. ( a  |`  ( 1 ... N
) )  /  u ]. [. ( a `  M )  /  v ]. [. ( t `  L )  /  w ]. ph )
1211sbcbii 3184 . . . . . . . 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 )
13 reseq1 5107 . . . . . . . . . 10  |-  ( a  =  ( t  |`  ( 1 ... M
) )  ->  (
a  |`  ( 1 ... N ) )  =  ( ( t  |`  ( 1 ... M
) )  |`  (
1 ... N ) ) )
1413sbccomieg 26747 . . . . . . . . 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 )
15 fzssp1 11059 . . . . . . . . . . . 12  |-  ( 1 ... N )  C_  ( 1 ... ( N  +  1 ) )
167oveq2i 6059 . . . . . . . . . . . 12  |-  ( 1 ... M )  =  ( 1 ... ( N  +  1 ) )
1715, 16sseqtr4i 3349 . . . . . . . . . . 11  |-  ( 1 ... N )  C_  ( 1 ... M
)
18 resabs1 5142 . . . . . . . . . . 11  |-  ( ( 1 ... N ) 
C_  ( 1 ... M )  ->  (
( t  |`  (
1 ... M ) )  |`  ( 1 ... N
) )  =  ( t  |`  ( 1 ... N ) ) )
19 dfsbcq 3131 . . . . . . . . . . 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 ) )
2017, 18, 19mp2b 10 . . . . . . . . . 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 )
21 vex 2927 . . . . . . . . . . . . . 14  |-  t  e. 
_V
2221resex 5153 . . . . . . . . . . . . 13  |-  ( t  |`  ( 1 ... M
) )  e.  _V
233ax-gen 1552 . . . . . . . . . . . . 13  |-  A. a
( a `  M
)  e.  _V
24 fveq1 5694 . . . . . . . . . . . . . 14  |-  ( a  =  ( t  |`  ( 1 ... M
) )  ->  (
a `  M )  =  ( ( t  |`  ( 1 ... M
) ) `  M
) )
2524sbcco3gOLD 3274 . . . . . . . . . . . . 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 ) )
2622, 23, 25mp2an 654 . . . . . . . . . . . 12  |-  ( [. ( t  |`  (
1 ... M ) )  /  a ]. [. (
a `  M )  /  v ]. [. (
t `  L )  /  w ]. ph  <->  [. ( ( t  |`  ( 1 ... M ) ) `
 M )  / 
v ]. [. ( t `
 L )  /  w ]. ph )
27 nn0p1nn 10223 . . . . . . . . . . . . . . 15  |-  ( N  e.  NN0  ->  ( N  +  1 )  e.  NN )
287, 27syl5eqel 2496 . . . . . . . . . . . . . 14  |-  ( N  e.  NN0  ->  M  e.  NN )
29 elfz1end 11045 . . . . . . . . . . . . . 14  |-  ( M  e.  NN  <->  M  e.  ( 1 ... M
) )
3028, 29sylib 189 . . . . . . . . . . . . 13  |-  ( N  e.  NN0  ->  M  e.  ( 1 ... M
) )
31 fvres 5712 . . . . . . . . . . . . 13  |-  ( M  e.  ( 1 ... M )  ->  (
( t  |`  (
1 ... M ) ) `
 M )  =  ( t `  M
) )
32 dfsbcq 3131 . . . . . . . . . . . . 13  |-  ( ( ( t  |`  (
1 ... M ) ) `
 M )  =  ( t `  M
)  ->  ( [. ( ( t  |`  ( 1 ... M
) ) `  M
)  /  v ]. [. ( t `  L
)  /  w ]. ph  <->  [. ( t `  M
)  /  v ]. [. ( t `  L
)  /  w ]. ph ) )
3330, 31, 323syl 19 . . . . . . . . . . . 12  |-  ( N  e.  NN0  ->  ( [. ( ( t  |`  ( 1 ... M
) ) `  M
)  /  v ]. [. ( t `  L
)  /  w ]. ph  <->  [. ( t `  M
)  /  v ]. [. ( t `  L
)  /  w ]. ph ) )
3426, 33syl5bb 249 . . . . . . . . . . 11  |-  ( N  e.  NN0  ->  ( [. ( t  |`  (
1 ... M ) )  /  a ]. [. (
a `  M )  /  v ]. [. (
t `  L )  /  w ]. ph  <->  [. ( t `
 M )  / 
v ]. [. ( t `
 L )  /  w ]. ph ) )
3534sbcbidv 3183 . . . . . . . . . 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 ) )
3620, 35syl5bb 249 . . . . . . . . 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 ) )
3714, 36syl5bb 249 . . . . . . . 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 ) )
3812, 37syl5rbb 250 . . . . . . 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 ) )
3938rabbidv 2916 . . . . . 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 } )
4039eleq1d 2478 . . . . 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 ) ) )
4140biimpa 471 . . . 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 ) )
42 rexfrabdioph.2 . . . . 5  |-  L  =  ( M  +  1 )
4342rexfrabdioph 26753 . . . 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 ) )
4410, 41, 43syl2anc 643 . . 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 ) )
456, 44syl5eqel 2496 . 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 ) )
467rexfrabdioph 26753 . 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 )
)
4745, 46syldan 457 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 177    /\ wa 359   A.wal 1546    = wceq 1649    e. wcel 1721   E.wrex 2675   {crab 2678   _Vcvv 2924   [.wsbc 3129    C_ wss 3288    |` cres 4847   ` cfv 5421  (class class class)co 6048    ^m cmap 6985   1c1 8955    + caddc 8957   NNcn 9964   NN0cn0 10185   ...cfz 11007  Diophcdioph 26711
This theorem is referenced by:  3rexfrabdioph  26755  4rexfrabdioph  26756  6rexfrabdioph  26757
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-inf2 7560  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-of 6272  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-1o 6691  df-oadd 6695  df-er 6872  df-map 6987  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-card 7790  df-cda 8012  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-nn 9965  df-n0 10186  df-z 10247  df-uz 10453  df-fz 11008  df-hash 11582  df-mzpcl 26678  df-mzp 26679  df-dioph 26712
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