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Theorem elnn0rabdioph 35111
Description: Diophantine set builder for nonnegativity constraints. The first builder which uses a witness variable internally; an expression is nonnegative if there is a nonnegative integer equal to it. (Contributed by Stefan O'Rear, 11-Oct-2014.)
Assertion
Ref Expression
elnn0rabdioph  |-  ( ( N  e.  NN0  /\  ( t  e.  ( ZZ  ^m  ( 1 ... N ) ) 
|->  A )  e.  (mzPoly `  ( 1 ... N
) ) )  ->  { t  e.  ( NN0  ^m  ( 1 ... N ) )  |  A  e.  NN0 }  e.  (Dioph `  N
) )
Distinct variable group:    t, N
Allowed substitution hint:    A( t)

Proof of Theorem elnn0rabdioph
Dummy variables  a 
b  c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 risset 2934 . . . . . 6  |-  ( A  e.  NN0  <->  E. b  e.  NN0  b  =  A )
21a1i 11 . . . . 5  |-  ( t  e.  ( NN0  ^m  ( 1 ... N
) )  ->  ( A  e.  NN0  <->  E. b  e.  NN0  b  =  A ) )
32rabbiia 3050 . . . 4  |-  { t  e.  ( NN0  ^m  ( 1 ... N
) )  |  A  e.  NN0 }  =  {
t  e.  ( NN0 
^m  ( 1 ... N ) )  |  E. b  e.  NN0  b  =  A }
43a1i 11 . . 3  |-  ( ( N  e.  NN0  /\  ( t  e.  ( ZZ  ^m  ( 1 ... N ) ) 
|->  A )  e.  (mzPoly `  ( 1 ... N
) ) )  ->  { t  e.  ( NN0  ^m  ( 1 ... N ) )  |  A  e.  NN0 }  =  { t  e.  ( NN0  ^m  (
1 ... N ) )  |  E. b  e. 
NN0  b  =  A } )
5 nfcv 2566 . . . 4  |-  F/_ t
( NN0  ^m  (
1 ... N ) )
6 nfcv 2566 . . . 4  |-  F/_ a
( NN0  ^m  (
1 ... N ) )
7 nfv 1730 . . . 4  |-  F/ a E. b  e.  NN0  b  =  A
8 nfcv 2566 . . . . 5  |-  F/_ t NN0
9 nfcsb1v 3391 . . . . . 6  |-  F/_ t [_ a  /  t ]_ A
109nfeq2 2583 . . . . 5  |-  F/ t  b  =  [_ a  /  t ]_ A
118, 10nfrex 2869 . . . 4  |-  F/ t E. b  e.  NN0  b  =  [_ a  / 
t ]_ A
12 csbeq1a 3384 . . . . . 6  |-  ( t  =  a  ->  A  =  [_ a  /  t ]_ A )
1312eqeq2d 2418 . . . . 5  |-  ( t  =  a  ->  (
b  =  A  <->  b  =  [_ a  /  t ]_ A ) )
1413rexbidv 2920 . . . 4  |-  ( t  =  a  ->  ( E. b  e.  NN0  b  =  A  <->  E. b  e.  NN0  b  =  [_ a  /  t ]_ A
) )
155, 6, 7, 11, 14cbvrab 3059 . . 3  |-  { t  e.  ( NN0  ^m  ( 1 ... N
) )  |  E. b  e.  NN0  b  =  A }  =  {
a  e.  ( NN0 
^m  ( 1 ... N ) )  |  E. b  e.  NN0  b  =  [_ a  / 
t ]_ A }
164, 15syl6eq 2461 . 2  |-  ( ( N  e.  NN0  /\  ( t  e.  ( ZZ  ^m  ( 1 ... N ) ) 
|->  A )  e.  (mzPoly `  ( 1 ... N
) ) )  ->  { t  e.  ( NN0  ^m  ( 1 ... N ) )  |  A  e.  NN0 }  =  { a  e.  ( NN0  ^m  (
1 ... N ) )  |  E. b  e. 
NN0  b  =  [_ a  /  t ]_ A } )
17 peano2nn0 10879 . . . . 5  |-  ( N  e.  NN0  ->  ( N  +  1 )  e. 
NN0 )
1817adantr 465 . . . 4  |-  ( ( N  e.  NN0  /\  ( t  e.  ( ZZ  ^m  ( 1 ... N ) ) 
|->  A )  e.  (mzPoly `  ( 1 ... N
) ) )  -> 
( N  +  1 )  e.  NN0 )
19 ovex 6308 . . . . 5  |-  ( 1 ... ( N  + 
1 ) )  e. 
_V
20 nn0p1nn 10878 . . . . . . 7  |-  ( N  e.  NN0  ->  ( N  +  1 )  e.  NN )
21 elfz1end 11771 . . . . . . 7  |-  ( ( N  +  1 )  e.  NN  <->  ( N  +  1 )  e.  ( 1 ... ( N  +  1 ) ) )
2220, 21sylib 198 . . . . . 6  |-  ( N  e.  NN0  ->  ( N  +  1 )  e.  ( 1 ... ( N  +  1 ) ) )
2322adantr 465 . . . . 5  |-  ( ( N  e.  NN0  /\  ( t  e.  ( ZZ  ^m  ( 1 ... N ) ) 
|->  A )  e.  (mzPoly `  ( 1 ... N
) ) )  -> 
( N  +  1 )  e.  ( 1 ... ( N  + 
1 ) ) )
24 mzpproj 35044 . . . . 5  |-  ( ( ( 1 ... ( N  +  1 ) )  e.  _V  /\  ( N  +  1
)  e.  ( 1 ... ( N  + 
1 ) ) )  ->  ( c  e.  ( ZZ  ^m  (
1 ... ( N  + 
1 ) ) ) 
|->  ( c `  ( N  +  1 ) ) )  e.  (mzPoly `  ( 1 ... ( N  +  1 ) ) ) )
2519, 23, 24sylancr 663 . . . 4  |-  ( ( N  e.  NN0  /\  ( t  e.  ( ZZ  ^m  ( 1 ... N ) ) 
|->  A )  e.  (mzPoly `  ( 1 ... N
) ) )  -> 
( c  e.  ( ZZ  ^m  ( 1 ... ( N  + 
1 ) ) ) 
|->  ( c `  ( N  +  1 ) ) )  e.  (mzPoly `  ( 1 ... ( N  +  1 ) ) ) )
26 eqid 2404 . . . . 5  |-  ( N  +  1 )  =  ( N  +  1 )
2726rabdiophlem2 35110 . . . 4  |-  ( ( N  e.  NN0  /\  ( t  e.  ( ZZ  ^m  ( 1 ... N ) ) 
|->  A )  e.  (mzPoly `  ( 1 ... N
) ) )  -> 
( c  e.  ( ZZ  ^m  ( 1 ... ( N  + 
1 ) ) ) 
|->  [_ ( c  |`  ( 1 ... N
) )  /  t ]_ A )  e.  (mzPoly `  ( 1 ... ( N  +  1 ) ) ) )
28 eqrabdioph 35085 . . . 4  |-  ( ( ( N  +  1 )  e.  NN0  /\  ( c  e.  ( ZZ  ^m  ( 1 ... ( N  + 
1 ) ) ) 
|->  ( c `  ( N  +  1 ) ) )  e.  (mzPoly `  ( 1 ... ( N  +  1 ) ) )  /\  (
c  e.  ( ZZ 
^m  ( 1 ... ( N  +  1 ) ) )  |->  [_ ( c  |`  (
1 ... N ) )  /  t ]_ A
)  e.  (mzPoly `  ( 1 ... ( N  +  1 ) ) ) )  ->  { c  e.  ( NN0  ^m  ( 1 ... ( N  + 
1 ) ) )  |  ( c `  ( N  +  1
) )  =  [_ ( c  |`  (
1 ... N ) )  /  t ]_ A }  e.  (Dioph `  ( N  +  1 ) ) )
2918, 25, 27, 28syl3anc 1232 . . 3  |-  ( ( N  e.  NN0  /\  ( t  e.  ( ZZ  ^m  ( 1 ... N ) ) 
|->  A )  e.  (mzPoly `  ( 1 ... N
) ) )  ->  { c  e.  ( NN0  ^m  ( 1 ... ( N  + 
1 ) ) )  |  ( c `  ( N  +  1
) )  =  [_ ( c  |`  (
1 ... N ) )  /  t ]_ A }  e.  (Dioph `  ( N  +  1 ) ) )
30 eqeq1 2408 . . . 4  |-  ( b  =  ( c `  ( N  +  1
) )  ->  (
b  =  [_ a  /  t ]_ A  <->  ( c `  ( N  +  1 ) )  =  [_ a  / 
t ]_ A ) )
31 csbeq1 3378 . . . . 5  |-  ( a  =  ( c  |`  ( 1 ... N
) )  ->  [_ a  /  t ]_ A  =  [_ ( c  |`  ( 1 ... N
) )  /  t ]_ A )
3231eqeq2d 2418 . . . 4  |-  ( a  =  ( c  |`  ( 1 ... N
) )  ->  (
( c `  ( N  +  1 ) )  =  [_ a  /  t ]_ A  <->  ( c `  ( N  +  1 ) )  =  [_ ( c  |`  ( 1 ... N
) )  /  t ]_ A ) )
3326, 30, 32rexrabdioph 35102 . . 3  |-  ( ( N  e.  NN0  /\  { c  e.  ( NN0 
^m  ( 1 ... ( N  +  1 ) ) )  |  ( c `  ( N  +  1 ) )  =  [_ (
c  |`  ( 1 ... N ) )  / 
t ]_ A }  e.  (Dioph `  ( N  + 
1 ) ) )  ->  { a  e.  ( NN0  ^m  (
1 ... N ) )  |  E. b  e. 
NN0  b  =  [_ a  /  t ]_ A }  e.  (Dioph `  N
) )
3429, 33syldan 470 . 2  |-  ( ( N  e.  NN0  /\  ( t  e.  ( ZZ  ^m  ( 1 ... N ) ) 
|->  A )  e.  (mzPoly `  ( 1 ... N
) ) )  ->  { a  e.  ( NN0  ^m  ( 1 ... N ) )  |  E. b  e. 
NN0  b  =  [_ a  /  t ]_ A }  e.  (Dioph `  N
) )
3516, 34eqeltrd 2492 1  |-  ( ( N  e.  NN0  /\  ( t  e.  ( ZZ  ^m  ( 1 ... N ) ) 
|->  A )  e.  (mzPoly `  ( 1 ... N
) ) )  ->  { t  e.  ( NN0  ^m  ( 1 ... N ) )  |  A  e.  NN0 }  e.  (Dioph `  N
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 186    /\ wa 369    = wceq 1407    e. wcel 1844   E.wrex 2757   {crab 2760   _Vcvv 3061   [_csb 3375    |-> cmpt 4455    |` cres 4827   ` cfv 5571  (class class class)co 6280    ^m cmap 7459   1c1 9525    + caddc 9527   NNcn 10578   NN0cn0 10838   ZZcz 10907   ...cfz 11728  mzPolycmzp 35029  Diophcdioph 35062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-inf2 8093  ax-cnex 9580  ax-resscn 9581  ax-1cn 9582  ax-icn 9583  ax-addcl 9584  ax-addrcl 9585  ax-mulcl 9586  ax-mulrcl 9587  ax-mulcom 9588  ax-addass 9589  ax-mulass 9590  ax-distr 9591  ax-i2m1 9592  ax-1ne0 9593  ax-1rid 9594  ax-rnegex 9595  ax-rrecex 9596  ax-cnre 9597  ax-pre-lttri 9598  ax-pre-lttrn 9599  ax-pre-ltadd 9600  ax-pre-mulgt0 9601
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-nel 2603  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-int 4230  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-pred 5369  df-ord 5415  df-on 5416  df-lim 5417  df-suc 5418  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-riota 6242  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-of 6523  df-om 6686  df-1st 6786  df-2nd 6787  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-1o 7169  df-oadd 7173  df-er 7350  df-map 7461  df-en 7557  df-dom 7558  df-sdom 7559  df-fin 7560  df-card 8354  df-cda 8582  df-pnf 9662  df-mnf 9663  df-xr 9664  df-ltxr 9665  df-le 9666  df-sub 9845  df-neg 9846  df-nn 10579  df-n0 10839  df-z 10908  df-uz 11130  df-fz 11729  df-hash 12455  df-mzpcl 35030  df-mzp 35031  df-dioph 35063
This theorem is referenced by:  lerabdioph  35113
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