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Theorem elnn0rabdioph 29144
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 2766 . . . . . 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 2964 . . . 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 2582 . . . 4  |-  F/_ t
( NN0  ^m  (
1 ... N ) )
6 nfcv 2582 . . . 4  |-  F/_ a
( NN0  ^m  (
1 ... N ) )
7 nfv 1673 . . . 4  |-  F/ a E. b  e.  NN0  b  =  A
8 nfcv 2582 . . . . 5  |-  F/_ t NN0
9 nfcsb1v 3307 . . . . . 6  |-  F/_ t [_ a  /  t ]_ A
109nfeq2 2593 . . . . 5  |-  F/ t  b  =  [_ a  /  t ]_ A
118, 10nfrex 2774 . . . 4  |-  F/ t E. b  e.  NN0  b  =  [_ a  / 
t ]_ A
12 csbeq1a 3300 . . . . . 6  |-  ( t  =  a  ->  A  =  [_ a  /  t ]_ A )
1312eqeq2d 2454 . . . . 5  |-  ( t  =  a  ->  (
b  =  A  <->  b  =  [_ a  /  t ]_ A ) )
1413rexbidv 2739 . . . 4  |-  ( t  =  a  ->  ( E. b  e.  NN0  b  =  A  <->  E. b  e.  NN0  b  =  [_ a  /  t ]_ A
) )
155, 6, 7, 11, 14cbvrab 2973 . . 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 2491 . 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 10623 . . . . 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 6119 . . . . 5  |-  ( 1 ... ( N  + 
1 ) )  e. 
_V
20 nn0p1nn 10622 . . . . . . 7  |-  ( N  e.  NN0  ->  ( N  +  1 )  e.  NN )
21 elfz1end 11482 . . . . . . 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 29076 . . . . 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 2443 . . . . 5  |-  ( N  +  1 )  =  ( N  +  1 )
2726rabdiophlem2 29143 . . . 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 29119 . . . 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 1218 . . 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 2449 . . . 4  |-  ( b  =  ( c `  ( N  +  1
) )  ->  (
b  =  [_ a  /  t ]_ A  <->  ( c `  ( N  +  1 ) )  =  [_ a  / 
t ]_ A ) )
31 csbeq1 3294 . . . . 5  |-  ( a  =  ( c  |`  ( 1 ... N
) )  ->  [_ a  /  t ]_ A  =  [_ ( c  |`  ( 1 ... N
) )  /  t ]_ A )
3231eqeq2d 2454 . . . 4  |-  ( a  =  ( c  |`  ( 1 ... N
) )  ->  (
( c `  ( N  +  1 ) )  =  [_ a  /  t ]_ A  <->  ( c `  ( N  +  1 ) )  =  [_ ( c  |`  ( 1 ... N
) )  /  t ]_ A ) )
3326, 30, 32rexrabdioph 29135 . . 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 2517 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 1369    e. wcel 1756   E.wrex 2719   {crab 2722   _Vcvv 2975   [_csb 3291    e. cmpt 4353    |` cres 4845   ` cfv 5421  (class class class)co 6094    ^m cmap 7217   1c1 9286    + caddc 9288   NNcn 10325   NN0cn0 10582   ZZcz 10649   ...cfz 11440  mzPolycmzp 29061  Diophcdioph 29096
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4406  ax-sep 4416  ax-nul 4424  ax-pow 4473  ax-pr 4534  ax-un 6375  ax-inf2 7850  ax-cnex 9341  ax-resscn 9342  ax-1cn 9343  ax-icn 9344  ax-addcl 9345  ax-addrcl 9346  ax-mulcl 9347  ax-mulrcl 9348  ax-mulcom 9349  ax-addass 9350  ax-mulass 9351  ax-distr 9352  ax-i2m1 9353  ax-1ne0 9354  ax-1rid 9355  ax-rnegex 9356  ax-rrecex 9357  ax-cnre 9358  ax-pre-lttri 9359  ax-pre-lttrn 9360  ax-pre-ltadd 9361  ax-pre-mulgt0 9362
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2571  df-ne 2611  df-nel 2612  df-ral 2723  df-rex 2724  df-reu 2725  df-rmo 2726  df-rab 2727  df-v 2977  df-sbc 3190  df-csb 3292  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-pss 3347  df-nul 3641  df-if 3795  df-pw 3865  df-sn 3881  df-pr 3883  df-tp 3885  df-op 3887  df-uni 4095  df-int 4132  df-iun 4176  df-br 4296  df-opab 4354  df-mpt 4355  df-tr 4389  df-eprel 4635  df-id 4639  df-po 4644  df-so 4645  df-fr 4682  df-we 4684  df-ord 4725  df-on 4726  df-lim 4727  df-suc 4728  df-xp 4849  df-rel 4850  df-cnv 4851  df-co 4852  df-dm 4853  df-rn 4854  df-res 4855  df-ima 4856  df-iota 5384  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-riota 6055  df-ov 6097  df-oprab 6098  df-mpt2 6099  df-of 6323  df-om 6480  df-1st 6580  df-2nd 6581  df-recs 6835  df-rdg 6869  df-1o 6923  df-oadd 6927  df-er 7104  df-map 7219  df-en 7314  df-dom 7315  df-sdom 7316  df-fin 7317  df-card 8112  df-cda 8340  df-pnf 9423  df-mnf 9424  df-xr 9425  df-ltxr 9426  df-le 9427  df-sub 9600  df-neg 9601  df-nn 10326  df-n0 10583  df-z 10650  df-uz 10865  df-fz 11441  df-hash 12107  df-mzpcl 29062  df-mzp 29063  df-dioph 29097
This theorem is referenced by:  lerabdioph  29146
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