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Theorem eldiophss 29118
Description: Diophantine sets are sets of tuples of nonnegative integers. (Contributed by Stefan O'Rear, 10-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.)
Assertion
Ref Expression
eldiophss  |-  ( A  e.  (Dioph `  B
)  ->  A  C_  ( NN0  ^m  ( 1 ... B ) ) )

Proof of Theorem eldiophss
Dummy variables  a 
b  c  d are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eldioph3b 29108 . 2  |-  ( A  e.  (Dioph `  B
)  <->  ( B  e. 
NN0  /\  E. a  e.  (mzPoly `  NN ) A  =  { b  |  E. c  e.  ( NN0  ^m  NN ) ( b  =  ( c  |`  ( 1 ... B ) )  /\  ( a `  c )  =  0 ) } ) )
2 simpr 461 . . . . . 6  |-  ( ( ( B  e.  NN0  /\  a  e.  (mzPoly `  NN ) )  /\  A  =  { b  |  E. c  e.  ( NN0  ^m  NN ) ( b  =  ( c  |`  ( 1 ... B
) )  /\  (
a `  c )  =  0 ) } )  ->  A  =  { b  |  E. c  e.  ( NN0  ^m  NN ) ( b  =  ( c  |`  ( 1 ... B
) )  /\  (
a `  c )  =  0 ) } )
3 vex 2980 . . . . . . . . . 10  |-  d  e. 
_V
4 eqeq1 2449 . . . . . . . . . . . 12  |-  ( b  =  d  ->  (
b  =  ( c  |`  ( 1 ... B
) )  <->  d  =  ( c  |`  (
1 ... B ) ) ) )
54anbi1d 704 . . . . . . . . . . 11  |-  ( b  =  d  ->  (
( b  =  ( c  |`  ( 1 ... B ) )  /\  ( a `  c )  =  0 )  <->  ( d  =  ( c  |`  (
1 ... B ) )  /\  ( a `  c )  =  0 ) ) )
65rexbidv 2741 . . . . . . . . . 10  |-  ( b  =  d  ->  ( E. c  e.  ( NN0  ^m  NN ) ( b  =  ( c  |`  ( 1 ... B
) )  /\  (
a `  c )  =  0 )  <->  E. c  e.  ( NN0  ^m  NN ) ( d  =  ( c  |`  (
1 ... B ) )  /\  ( a `  c )  =  0 ) ) )
73, 6elab 3111 . . . . . . . . 9  |-  ( d  e.  { b  |  E. c  e.  ( NN0  ^m  NN ) ( b  =  ( c  |`  ( 1 ... B ) )  /\  ( a `  c )  =  0 ) }  <->  E. c  e.  ( NN0  ^m  NN ) ( d  =  ( c  |`  (
1 ... B ) )  /\  ( a `  c )  =  0 ) )
8 simpr 461 . . . . . . . . . . . . 13  |-  ( ( ( ( B  e. 
NN0  /\  a  e.  (mzPoly `  NN ) )  /\  c  e.  ( NN0  ^m  NN ) )  /\  d  =  ( c  |`  (
1 ... B ) ) )  ->  d  =  ( c  |`  (
1 ... B ) ) )
9 elfznn 11483 . . . . . . . . . . . . . . . 16  |-  ( a  e.  ( 1 ... B )  ->  a  e.  NN )
109ssriv 3365 . . . . . . . . . . . . . . 15  |-  ( 1 ... B )  C_  NN
11 elmapssres 7242 . . . . . . . . . . . . . . 15  |-  ( ( c  e.  ( NN0 
^m  NN )  /\  ( 1 ... B
)  C_  NN )  ->  ( c  |`  (
1 ... B ) )  e.  ( NN0  ^m  ( 1 ... B
) ) )
1210, 11mpan2 671 . . . . . . . . . . . . . 14  |-  ( c  e.  ( NN0  ^m  NN )  ->  ( c  |`  ( 1 ... B
) )  e.  ( NN0  ^m  ( 1 ... B ) ) )
1312ad2antlr 726 . . . . . . . . . . . . 13  |-  ( ( ( ( B  e. 
NN0  /\  a  e.  (mzPoly `  NN ) )  /\  c  e.  ( NN0  ^m  NN ) )  /\  d  =  ( c  |`  (
1 ... B ) ) )  ->  ( c  |`  ( 1 ... B
) )  e.  ( NN0  ^m  ( 1 ... B ) ) )
148, 13eqeltrd 2517 . . . . . . . . . . . 12  |-  ( ( ( ( B  e. 
NN0  /\  a  e.  (mzPoly `  NN ) )  /\  c  e.  ( NN0  ^m  NN ) )  /\  d  =  ( c  |`  (
1 ... B ) ) )  ->  d  e.  ( NN0  ^m  ( 1 ... B ) ) )
1514ex 434 . . . . . . . . . . 11  |-  ( ( ( B  e.  NN0  /\  a  e.  (mzPoly `  NN ) )  /\  c  e.  ( NN0  ^m  NN ) )  ->  (
d  =  ( c  |`  ( 1 ... B
) )  ->  d  e.  ( NN0  ^m  (
1 ... B ) ) ) )
1615adantrd 468 . . . . . . . . . 10  |-  ( ( ( B  e.  NN0  /\  a  e.  (mzPoly `  NN ) )  /\  c  e.  ( NN0  ^m  NN ) )  ->  (
( d  =  ( c  |`  ( 1 ... B ) )  /\  ( a `  c )  =  0 )  ->  d  e.  ( NN0  ^m  ( 1 ... B ) ) ) )
1716rexlimdva 2846 . . . . . . . . 9  |-  ( ( B  e.  NN0  /\  a  e.  (mzPoly `  NN ) )  ->  ( E. c  e.  ( NN0  ^m  NN ) ( d  =  ( c  |`  ( 1 ... B
) )  /\  (
a `  c )  =  0 )  -> 
d  e.  ( NN0 
^m  ( 1 ... B ) ) ) )
187, 17syl5bi 217 . . . . . . . 8  |-  ( ( B  e.  NN0  /\  a  e.  (mzPoly `  NN ) )  ->  (
d  e.  { b  |  E. c  e.  ( NN0  ^m  NN ) ( b  =  ( c  |`  (
1 ... B ) )  /\  ( a `  c )  =  0 ) }  ->  d  e.  ( NN0  ^m  (
1 ... B ) ) ) )
1918ssrdv 3367 . . . . . . 7  |-  ( ( B  e.  NN0  /\  a  e.  (mzPoly `  NN ) )  ->  { b  |  E. c  e.  ( NN0  ^m  NN ) ( b  =  ( c  |`  (
1 ... B ) )  /\  ( a `  c )  =  0 ) }  C_  ( NN0  ^m  ( 1 ... B ) ) )
2019adantr 465 . . . . . 6  |-  ( ( ( B  e.  NN0  /\  a  e.  (mzPoly `  NN ) )  /\  A  =  { b  |  E. c  e.  ( NN0  ^m  NN ) ( b  =  ( c  |`  ( 1 ... B
) )  /\  (
a `  c )  =  0 ) } )  ->  { b  |  E. c  e.  ( NN0  ^m  NN ) ( b  =  ( c  |`  ( 1 ... B ) )  /\  ( a `  c )  =  0 ) }  C_  ( NN0  ^m  ( 1 ... B ) ) )
212, 20eqsstrd 3395 . . . . 5  |-  ( ( ( B  e.  NN0  /\  a  e.  (mzPoly `  NN ) )  /\  A  =  { b  |  E. c  e.  ( NN0  ^m  NN ) ( b  =  ( c  |`  ( 1 ... B
) )  /\  (
a `  c )  =  0 ) } )  ->  A  C_  ( NN0  ^m  ( 1 ... B ) ) )
2221ex 434 . . . 4  |-  ( ( B  e.  NN0  /\  a  e.  (mzPoly `  NN ) )  ->  ( A  =  { b  |  E. c  e.  ( NN0  ^m  NN ) ( b  =  ( c  |`  ( 1 ... B ) )  /\  ( a `  c )  =  0 ) }  ->  A  C_  ( NN0  ^m  (
1 ... B ) ) ) )
2322rexlimdva 2846 . . 3  |-  ( B  e.  NN0  ->  ( E. a  e.  (mzPoly `  NN ) A  =  {
b  |  E. c  e.  ( NN0  ^m  NN ) ( b  =  ( c  |`  (
1 ... B ) )  /\  ( a `  c )  =  0 ) }  ->  A  C_  ( NN0  ^m  (
1 ... B ) ) ) )
2423imp 429 . 2  |-  ( ( B  e.  NN0  /\  E. a  e.  (mzPoly `  NN ) A  =  {
b  |  E. c  e.  ( NN0  ^m  NN ) ( b  =  ( c  |`  (
1 ... B ) )  /\  ( a `  c )  =  0 ) } )  ->  A  C_  ( NN0  ^m  ( 1 ... B
) ) )
251, 24sylbi 195 1  |-  ( A  e.  (Dioph `  B
)  ->  A  C_  ( NN0  ^m  ( 1 ... B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   {cab 2429   E.wrex 2721    C_ wss 3333    |` cres 4847   ` cfv 5423  (class class class)co 6096    ^m cmap 7219   0cc0 9287   1c1 9288   NNcn 10327   NN0cn0 10584   ...cfz 11442  mzPolycmzp 29063  Diophcdioph 29098
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 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-inf2 7852  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
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 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-of 6325  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-map 7221  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-card 8114  df-cda 8342  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-n0 10585  df-z 10652  df-uz 10867  df-fz 11443  df-hash 12109  df-mzpcl 29064  df-mzp 29065  df-dioph 29099
This theorem is referenced by: (None)
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