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Theorem elnn0rabdioph 28986
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 2753 . . . . . 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 2951 . . . 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 2569 . . . 4  |-  F/_ t
( NN0  ^m  (
1 ... N ) )
6 nfcv 2569 . . . 4  |-  F/_ a
( NN0  ^m  (
1 ... N ) )
7 nfv 1672 . . . 4  |-  F/ a E. b  e.  NN0  b  =  A
8 nfcv 2569 . . . . 5  |-  F/_ t NN0
9 nfcsb1v 3292 . . . . . 6  |-  F/_ t [_ a  /  t ]_ A
109nfeq2 2580 . . . . 5  |-  F/ t  b  =  [_ a  /  t ]_ A
118, 10nfrex 2761 . . . 4  |-  F/ t E. b  e.  NN0  b  =  [_ a  / 
t ]_ A
12 csbeq1a 3285 . . . . . 6  |-  ( t  =  a  ->  A  =  [_ a  /  t ]_ A )
1312eqeq2d 2444 . . . . 5  |-  ( t  =  a  ->  (
b  =  A  <->  b  =  [_ a  /  t ]_ A ) )
1413rexbidv 2726 . . . 4  |-  ( t  =  a  ->  ( E. b  e.  NN0  b  =  A  <->  E. b  e.  NN0  b  =  [_ a  /  t ]_ A
) )
155, 6, 7, 11, 14cbvrab 2960 . . 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 2481 . 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 10608 . . . . 5  |-  ( N  e.  NN0  ->  ( N  +  1 )  e. 
NN0 )
1817adantr 462 . . . 4  |-  ( ( N  e.  NN0  /\  ( t  e.  ( ZZ  ^m  ( 1 ... N ) ) 
|->  A )  e.  (mzPoly `  ( 1 ... N
) ) )  -> 
( N  +  1 )  e.  NN0 )
19 ovex 6105 . . . . 5  |-  ( 1 ... ( N  + 
1 ) )  e. 
_V
20 nn0p1nn 10607 . . . . . . 7  |-  ( N  e.  NN0  ->  ( N  +  1 )  e.  NN )
21 elfz1end 11466 . . . . . . 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 462 . . . . 5  |-  ( ( N  e.  NN0  /\  ( t  e.  ( ZZ  ^m  ( 1 ... N ) ) 
|->  A )  e.  (mzPoly `  ( 1 ... N
) ) )  -> 
( N  +  1 )  e.  ( 1 ... ( N  + 
1 ) ) )
24 mzpproj 28918 . . . . 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 656 . . . 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 2433 . . . . 5  |-  ( N  +  1 )  =  ( N  +  1 )
2726rabdiophlem2 28985 . . . 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 28961 . . . 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 1211 . . 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 2439 . . . 4  |-  ( b  =  ( c `  ( N  +  1
) )  ->  (
b  =  [_ a  /  t ]_ A  <->  ( c `  ( N  +  1 ) )  =  [_ a  / 
t ]_ A ) )
31 csbeq1 3279 . . . . 5  |-  ( a  =  ( c  |`  ( 1 ... N
) )  ->  [_ a  /  t ]_ A  =  [_ ( c  |`  ( 1 ... N
) )  /  t ]_ A )
3231eqeq2d 2444 . . . 4  |-  ( a  =  ( c  |`  ( 1 ... N
) )  ->  (
( c `  ( N  +  1 ) )  =  [_ a  /  t ]_ A  <->  ( c `  ( N  +  1 ) )  =  [_ ( c  |`  ( 1 ... N
) )  /  t ]_ A ) )
3326, 30, 32rexrabdioph 28977 . . 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 467 . 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 2507 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 1362    e. wcel 1755   E.wrex 2706   {crab 2709   _Vcvv 2962   [_csb 3276    e. cmpt 4338    |` cres 4829   ` cfv 5406  (class class class)co 6080    ^m cmap 7202   1c1 9271    + caddc 9273   NNcn 10310   NN0cn0 10567   ZZcz 10634   ...cfz 11424  mzPolycmzp 28903  Diophcdioph 28938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-inf2 7835  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-of 6309  df-om 6466  df-1st 6566  df-2nd 6567  df-recs 6818  df-rdg 6852  df-1o 6908  df-oadd 6912  df-er 7089  df-map 7204  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-card 8097  df-cda 8325  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-nn 10311  df-n0 10568  df-z 10635  df-uz 10850  df-fz 11425  df-hash 12088  df-mzpcl 28904  df-mzp 28905  df-dioph 28939
This theorem is referenced by:  lerabdioph  28988
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