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Theorem elnn0rabdioph 30340
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 2987 . . . . . 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 3102 . . . 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 2629 . . . 4  |-  F/_ t
( NN0  ^m  (
1 ... N ) )
6 nfcv 2629 . . . 4  |-  F/_ a
( NN0  ^m  (
1 ... N ) )
7 nfv 1683 . . . 4  |-  F/ a E. b  e.  NN0  b  =  A
8 nfcv 2629 . . . . 5  |-  F/_ t NN0
9 nfcsb1v 3451 . . . . . 6  |-  F/_ t [_ a  /  t ]_ A
109nfeq2 2646 . . . . 5  |-  F/ t  b  =  [_ a  /  t ]_ A
118, 10nfrex 2927 . . . 4  |-  F/ t E. b  e.  NN0  b  =  [_ a  / 
t ]_ A
12 csbeq1a 3444 . . . . . 6  |-  ( t  =  a  ->  A  =  [_ a  /  t ]_ A )
1312eqeq2d 2481 . . . . 5  |-  ( t  =  a  ->  (
b  =  A  <->  b  =  [_ a  /  t ]_ A ) )
1413rexbidv 2973 . . . 4  |-  ( t  =  a  ->  ( E. b  e.  NN0  b  =  A  <->  E. b  e.  NN0  b  =  [_ a  /  t ]_ A
) )
155, 6, 7, 11, 14cbvrab 3111 . . 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 2524 . 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 10832 . . . . 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 6307 . . . . 5  |-  ( 1 ... ( N  + 
1 ) )  e. 
_V
20 nn0p1nn 10831 . . . . . . 7  |-  ( N  e.  NN0  ->  ( N  +  1 )  e.  NN )
21 elfz1end 11711 . . . . . . 7  |-  ( ( N  +  1 )  e.  NN  <->  ( N  +  1 )  e.  ( 1 ... ( N  +  1 ) ) )
2220, 21sylib 196 . . . . . 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 30273 . . . . 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 2467 . . . . 5  |-  ( N  +  1 )  =  ( N  +  1 )
2726rabdiophlem2 30339 . . . 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 30315 . . . 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 1228 . . 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 2471 . . . 4  |-  ( b  =  ( c `  ( N  +  1
) )  ->  (
b  =  [_ a  /  t ]_ A  <->  ( c `  ( N  +  1 ) )  =  [_ a  / 
t ]_ A ) )
31 csbeq1 3438 . . . . 5  |-  ( a  =  ( c  |`  ( 1 ... N
) )  ->  [_ a  /  t ]_ A  =  [_ ( c  |`  ( 1 ... N
) )  /  t ]_ A )
3231eqeq2d 2481 . . . 4  |-  ( a  =  ( c  |`  ( 1 ... N
) )  ->  (
( c `  ( N  +  1 ) )  =  [_ a  /  t ]_ A  <->  ( c `  ( N  +  1 ) )  =  [_ ( c  |`  ( 1 ... N
) )  /  t ]_ A ) )
3326, 30, 32rexrabdioph 30331 . . 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 2555 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 184    /\ wa 369    = wceq 1379    e. wcel 1767   E.wrex 2815   {crab 2818   _Vcvv 3113   [_csb 3435    |-> cmpt 4505    |` cres 5001   ` cfv 5586  (class class class)co 6282    ^m cmap 7417   1c1 9489    + caddc 9491   NNcn 10532   NN0cn0 10791   ZZcz 10860   ...cfz 11668  mzPolycmzp 30258  Diophcdioph 30292
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-of 6522  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-card 8316  df-cda 8544  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-n0 10792  df-z 10861  df-uz 11079  df-fz 11669  df-hash 12370  df-mzpcl 30259  df-mzp 30260  df-dioph 30293
This theorem is referenced by:  lerabdioph  30342
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