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Theorem elnn0rabdioph 26987
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 2603 . . . . . 6  |-  ( A  e.  NN0  <->  E. b  e.  NN0  b  =  A )
21a1i 10 . . . . 5  |-  ( t  e.  ( NN0  ^m  ( 1 ... N
) )  ->  ( A  e.  NN0  <->  E. b  e.  NN0  b  =  A ) )
32rabbiia 2791 . . . 4  |-  { t  e.  ( NN0  ^m  ( 1 ... N
) )  |  A  e.  NN0 }  =  {
t  e.  ( NN0 
^m  ( 1 ... N ) )  |  E. b  e.  NN0  b  =  A }
43a1i 10 . . 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 2432 . . . 4  |-  F/_ t
( NN0  ^m  (
1 ... N ) )
6 nfcv 2432 . . . 4  |-  F/_ a
( NN0  ^m  (
1 ... N ) )
7 nfv 1609 . . . 4  |-  F/ a E. b  e.  NN0  b  =  A
8 nfcv 2432 . . . . 5  |-  F/_ t NN0
9 nfcsb1v 3126 . . . . . 6  |-  F/_ t [_ a  /  t ]_ A
109nfeq2 2443 . . . . 5  |-  F/ t  b  =  [_ a  /  t ]_ A
118, 10nfrex 2611 . . . 4  |-  F/ t E. b  e.  NN0  b  =  [_ a  / 
t ]_ A
12 csbeq1a 3102 . . . . . 6  |-  ( t  =  a  ->  A  =  [_ a  /  t ]_ A )
1312eqeq2d 2307 . . . . 5  |-  ( t  =  a  ->  (
b  =  A  <->  b  =  [_ a  /  t ]_ A ) )
1413rexbidv 2577 . . . 4  |-  ( t  =  a  ->  ( E. b  e.  NN0  b  =  A  <->  E. b  e.  NN0  b  =  [_ a  /  t ]_ A
) )
155, 6, 7, 11, 14cbvrab 2799 . . 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 2344 . 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 10020 . . . . 5  |-  ( N  e.  NN0  ->  ( N  +  1 )  e. 
NN0 )
1817adantr 451 . . . 4  |-  ( ( N  e.  NN0  /\  ( t  e.  ( ZZ  ^m  ( 1 ... N ) ) 
|->  A )  e.  (mzPoly `  ( 1 ... N
) ) )  -> 
( N  +  1 )  e.  NN0 )
19 ovex 5899 . . . . 5  |-  ( 1 ... ( N  + 
1 ) )  e. 
_V
20 nn0p1nn 10019 . . . . . . 7  |-  ( N  e.  NN0  ->  ( N  +  1 )  e.  NN )
21 elfz1end 10836 . . . . . . 7  |-  ( ( N  +  1 )  e.  NN  <->  ( N  +  1 )  e.  ( 1 ... ( N  +  1 ) ) )
2220, 21sylib 188 . . . . . 6  |-  ( N  e.  NN0  ->  ( N  +  1 )  e.  ( 1 ... ( N  +  1 ) ) )
2322adantr 451 . . . . 5  |-  ( ( N  e.  NN0  /\  ( t  e.  ( ZZ  ^m  ( 1 ... N ) ) 
|->  A )  e.  (mzPoly `  ( 1 ... N
) ) )  -> 
( N  +  1 )  e.  ( 1 ... ( N  + 
1 ) ) )
24 mzpproj 26918 . . . . 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 644 . . . 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 2296 . . . . 5  |-  ( N  +  1 )  =  ( N  +  1 )
2726rabdiophlem2 26986 . . . 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 26960 . . . 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 1182 . . 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 2302 . . . 4  |-  ( b  =  ( c `  ( N  +  1
) )  ->  (
b  =  [_ a  /  t ]_ A  <->  ( c `  ( N  +  1 ) )  =  [_ a  / 
t ]_ A ) )
31 csbeq1 3097 . . . . 5  |-  ( a  =  ( c  |`  ( 1 ... N
) )  ->  [_ a  /  t ]_ A  =  [_ ( c  |`  ( 1 ... N
) )  /  t ]_ A )
3231eqeq2d 2307 . . . 4  |-  ( a  =  ( c  |`  ( 1 ... N
) )  ->  (
( c `  ( N  +  1 ) )  =  [_ a  /  t ]_ A  <->  ( c `  ( N  +  1 ) )  =  [_ ( c  |`  ( 1 ... N
) )  /  t ]_ A ) )
3326, 30, 32rexrabdioph 26978 . . 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 456 . 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 2370 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 set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   E.wrex 2557   {crab 2560   _Vcvv 2801   [_csb 3094    e. cmpt 4093    |` cres 4707   ` cfv 5271  (class class class)co 5874    ^m cmap 6788   1c1 8754    + caddc 8756   NNcn 9762   NN0cn0 9981   ZZcz 10040   ...cfz 10798  mzPolycmzp 26903  Diophcdioph 26937
This theorem is referenced by:  lerabdioph  26989
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-hash 11354  df-mzpcl 26904  df-mzp 26905  df-dioph 26938
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