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Theorem eldiophss 26833
Description: Diophantine sets are sets of tuples of natural numbers. (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 26823 . 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 448 . . . . . 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 2959 . . . . . . . . . 10  |-  d  e. 
_V
4 eqeq1 2442 . . . . . . . . . . . 12  |-  ( b  =  d  ->  (
b  =  ( c  |`  ( 1 ... B
) )  <->  d  =  ( c  |`  (
1 ... B ) ) ) )
54anbi1d 686 . . . . . . . . . . 11  |-  ( b  =  d  ->  (
( b  =  ( c  |`  ( 1 ... B ) )  /\  ( a `  c )  =  0 )  <->  ( d  =  ( c  |`  (
1 ... B ) )  /\  ( a `  c )  =  0 ) ) )
65rexbidv 2726 . . . . . . . . . 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 3082 . . . . . . . . 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 448 . . . . . . . . . . . . 13  |-  ( ( ( ( B  e. 
NN0  /\  a  e.  (mzPoly `  NN ) )  /\  c  e.  ( NN0  ^m  NN ) )  /\  d  =  ( c  |`  (
1 ... B ) ) )  ->  d  =  ( c  |`  (
1 ... B ) ) )
9 elfznn 11080 . . . . . . . . . . . . . . . 16  |-  ( a  e.  ( 1 ... B )  ->  a  e.  NN )
109ssriv 3352 . . . . . . . . . . . . . . 15  |-  ( 1 ... B )  C_  NN
11 elmapssres 26771 . . . . . . . . . . . . . . 15  |-  ( ( c  e.  ( NN0 
^m  NN )  /\  ( 1 ... B
)  C_  NN )  ->  ( c  |`  (
1 ... B ) )  e.  ( NN0  ^m  ( 1 ... B
) ) )
1210, 11mpan2 653 . . . . . . . . . . . . . 14  |-  ( c  e.  ( NN0  ^m  NN )  ->  ( c  |`  ( 1 ... B
) )  e.  ( NN0  ^m  ( 1 ... B ) ) )
1312ad2antlr 708 . . . . . . . . . . . . 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 2510 . . . . . . . . . . . 12  |-  ( ( ( ( B  e. 
NN0  /\  a  e.  (mzPoly `  NN ) )  /\  c  e.  ( NN0  ^m  NN ) )  /\  d  =  ( c  |`  (
1 ... B ) ) )  ->  d  e.  ( NN0  ^m  ( 1 ... B ) ) )
1514ex 424 . . . . . . . . . . 11  |-  ( ( ( B  e.  NN0  /\  a  e.  (mzPoly `  NN ) )  /\  c  e.  ( NN0  ^m  NN ) )  ->  (
d  =  ( c  |`  ( 1 ... B
) )  ->  d  e.  ( NN0  ^m  (
1 ... B ) ) ) )
1615adantrd 455 . . . . . . . . . 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 2830 . . . . . . . . 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 209 . . . . . . . 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 3354 . . . . . . 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 452 . . . . . 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 3382 . . . . 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 424 . . . 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 2830 . . 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 419 . 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 188 1  |-  ( A  e.  (Dioph `  B
)  ->  A  C_  ( NN0  ^m  ( 1 ... B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   {cab 2422   E.wrex 2706    C_ wss 3320    |` cres 4880   ` cfv 5454  (class class class)co 6081    ^m cmap 7018   0cc0 8990   1c1 8991   NNcn 10000   NN0cn0 10221   ...cfz 11043  mzPolycmzp 26779  Diophcdioph 26813
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-of 6305  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-card 7826  df-cda 8048  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-n0 10222  df-z 10283  df-uz 10489  df-fz 11044  df-hash 11619  df-mzpcl 26780  df-mzp 26781  df-dioph 26814
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