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Theorem 2rexfrabdioph 29043
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 2sbcrex 29032 . . . . 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 )
21a1i 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 ) )
32rabbiia 2959 . . 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 }
4 rexfrabdioph.1 . . . . . 6  |-  M  =  ( N  +  1 )
5 peano2nn0 10616 . . . . . 6  |-  ( N  e.  NN0  ->  ( N  +  1 )  e. 
NN0 )
64, 5syl5eqel 2525 . . . . 5  |-  ( N  e.  NN0  ->  M  e. 
NN0 )
76adantr 462 . . . 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 )
8 sbcrot3 29038 . . . . . . . . 9  |-  ( [. ( t `  L
)  /  w ]. [. ( a  |`  (
1 ... N ) )  /  u ]. [. (
a `  M )  /  v ]. ph  <->  [. ( a  |`  ( 1 ... N
) )  /  u ]. [. ( a `  M )  /  v ]. [. ( t `  L )  /  w ]. ph )
98sbcbii 3243 . . . . . . . 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 )
10 reseq1 5100 . . . . . . . . . 10  |-  ( a  =  ( t  |`  ( 1 ... M
) )  ->  (
a  |`  ( 1 ... N ) )  =  ( ( t  |`  ( 1 ... M
) )  |`  (
1 ... N ) ) )
1110sbccomieg 29040 . . . . . . . . 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 )
12 fzssp1 11497 . . . . . . . . . . . 12  |-  ( 1 ... N )  C_  ( 1 ... ( N  +  1 ) )
134oveq2i 6101 . . . . . . . . . . . 12  |-  ( 1 ... M )  =  ( 1 ... ( N  +  1 ) )
1412, 13sseqtr4i 3386 . . . . . . . . . . 11  |-  ( 1 ... N )  C_  ( 1 ... M
)
15 resabs1 5136 . . . . . . . . . . 11  |-  ( ( 1 ... N ) 
C_  ( 1 ... M )  ->  (
( t  |`  (
1 ... M ) )  |`  ( 1 ... N
) )  =  ( t  |`  ( 1 ... N ) ) )
16 dfsbcq 3185 . . . . . . . . . . 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 ) )
1714, 15, 16mp2b 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 )
18 vex 2973 . . . . . . . . . . . . . 14  |-  t  e. 
_V
1918resex 5147 . . . . . . . . . . . . 13  |-  ( t  |`  ( 1 ... M
) )  e.  _V
20 fveq1 5687 . . . . . . . . . . . . . 14  |-  ( a  =  ( t  |`  ( 1 ... M
) )  ->  (
a `  M )  =  ( ( t  |`  ( 1 ... M
) ) `  M
) )
2120sbcco3g 3692 . . . . . . . . . . . . 13  |-  ( ( t  |`  ( 1 ... M ) )  e.  _V  ->  ( [. ( t  |`  (
1 ... M ) )  /  a ]. [. (
a `  M )  /  v ]. [. (
t `  L )  /  w ]. ph  <->  [. ( ( t  |`  ( 1 ... M ) ) `
 M )  / 
v ]. [. ( t `
 L )  /  w ]. ph ) )
2219, 21ax-mp 5 . . . . . . . . . . . 12  |-  ( [. ( t  |`  (
1 ... M ) )  /  a ]. [. (
a `  M )  /  v ]. [. (
t `  L )  /  w ]. ph  <->  [. ( ( t  |`  ( 1 ... M ) ) `
 M )  / 
v ]. [. ( t `
 L )  /  w ]. ph )
23 nn0p1nn 10615 . . . . . . . . . . . . . . 15  |-  ( N  e.  NN0  ->  ( N  +  1 )  e.  NN )
244, 23syl5eqel 2525 . . . . . . . . . . . . . 14  |-  ( N  e.  NN0  ->  M  e.  NN )
25 elfz1end 11475 . . . . . . . . . . . . . 14  |-  ( M  e.  NN  <->  M  e.  ( 1 ... M
) )
2624, 25sylib 196 . . . . . . . . . . . . 13  |-  ( N  e.  NN0  ->  M  e.  ( 1 ... M
) )
27 fvres 5701 . . . . . . . . . . . . 13  |-  ( M  e.  ( 1 ... M )  ->  (
( t  |`  (
1 ... M ) ) `
 M )  =  ( t `  M
) )
28 dfsbcq 3185 . . . . . . . . . . . . 13  |-  ( ( ( t  |`  (
1 ... M ) ) `
 M )  =  ( t `  M
)  ->  ( [. ( ( t  |`  ( 1 ... M
) ) `  M
)  /  v ]. [. ( t `  L
)  /  w ]. ph  <->  [. ( t `  M
)  /  v ]. [. ( t `  L
)  /  w ]. ph ) )
2926, 27, 283syl 20 . . . . . . . . . . . 12  |-  ( N  e.  NN0  ->  ( [. ( ( t  |`  ( 1 ... M
) ) `  M
)  /  v ]. [. ( t `  L
)  /  w ]. ph  <->  [. ( t `  M
)  /  v ]. [. ( t `  L
)  /  w ]. ph ) )
3022, 29syl5bb 257 . . . . . . . . . . 11  |-  ( N  e.  NN0  ->  ( [. ( t  |`  (
1 ... M ) )  /  a ]. [. (
a `  M )  /  v ]. [. (
t `  L )  /  w ]. ph  <->  [. ( t `
 M )  / 
v ]. [. ( t `
 L )  /  w ]. ph ) )
3130sbcbidv 3242 . . . . . . . . . 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 ) )
3217, 31syl5bb 257 . . . . . . . . 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 ) )
3311, 32syl5bb 257 . . . . . . . 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 ) )
349, 33syl5rbb 258 . . . . . . 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 ) )
3534rabbidv 2962 . . . . . 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 } )
3635eleq1d 2507 . . . . 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 ) ) )
3736biimpa 481 . . . 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 ) )
38 rexfrabdioph.2 . . . . 5  |-  L  =  ( M  +  1 )
3938rexfrabdioph 29042 . . . 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 ) )
407, 37, 39syl2anc 656 . . 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 ) )
413, 40syl5eqel 2525 . 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 ) )
424rexfrabdioph 29042 . 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 )
)
4341, 42syldan 467 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1364    e. wcel 1761   E.wrex 2714   {crab 2717   _Vcvv 2970   [.wsbc 3183    C_ wss 3325    |` cres 4838   ` cfv 5415  (class class class)co 6090    ^m cmap 7210   1c1 9279    + caddc 9281   NNcn 10318   NN0cn0 10575   ...cfz 11433  Diophcdioph 29002
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-map 7212  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-card 8105  df-cda 8333  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-n0 10576  df-z 10643  df-uz 10858  df-fz 11434  df-hash 12100  df-mzpcl 28968  df-mzp 28969  df-dioph 29003
This theorem is referenced by:  3rexfrabdioph  29044  4rexfrabdioph  29045  6rexfrabdioph  29046
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