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Theorem eldiophss 26957
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 26947 . 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 447 . . . . . 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 2804 . . . . . . . . . 10  |-  d  e. 
_V
4 eqeq1 2302 . . . . . . . . . . . 12  |-  ( b  =  d  ->  (
b  =  ( c  |`  ( 1 ... B
) )  <->  d  =  ( c  |`  (
1 ... B ) ) ) )
54anbi1d 685 . . . . . . . . . . 11  |-  ( b  =  d  ->  (
( b  =  ( c  |`  ( 1 ... B ) )  /\  ( a `  c )  =  0 )  <->  ( d  =  ( c  |`  (
1 ... B ) )  /\  ( a `  c )  =  0 ) ) )
65rexbidv 2577 . . . . . . . . . 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 2927 . . . . . . . . 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 447 . . . . . . . . . . . . 13  |-  ( ( ( ( B  e. 
NN0  /\  a  e.  (mzPoly `  NN ) )  /\  c  e.  ( NN0  ^m  NN ) )  /\  d  =  ( c  |`  (
1 ... B ) ) )  ->  d  =  ( c  |`  (
1 ... B ) ) )
9 elfznn 10835 . . . . . . . . . . . . . . . 16  |-  ( a  e.  ( 1 ... B )  ->  a  e.  NN )
109ssriv 3197 . . . . . . . . . . . . . . 15  |-  ( 1 ... B )  C_  NN
11 elmapssres 26895 . . . . . . . . . . . . . . 15  |-  ( ( c  e.  ( NN0 
^m  NN )  /\  ( 1 ... B
)  C_  NN )  ->  ( c  |`  (
1 ... B ) )  e.  ( NN0  ^m  ( 1 ... B
) ) )
1210, 11mpan2 652 . . . . . . . . . . . . . 14  |-  ( c  e.  ( NN0  ^m  NN )  ->  ( c  |`  ( 1 ... B
) )  e.  ( NN0  ^m  ( 1 ... B ) ) )
1312ad2antlr 707 . . . . . . . . . . . . 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 2370 . . . . . . . . . . . 12  |-  ( ( ( ( B  e. 
NN0  /\  a  e.  (mzPoly `  NN ) )  /\  c  e.  ( NN0  ^m  NN ) )  /\  d  =  ( c  |`  (
1 ... B ) ) )  ->  d  e.  ( NN0  ^m  ( 1 ... B ) ) )
1514ex 423 . . . . . . . . . . 11  |-  ( ( ( B  e.  NN0  /\  a  e.  (mzPoly `  NN ) )  /\  c  e.  ( NN0  ^m  NN ) )  ->  (
d  =  ( c  |`  ( 1 ... B
) )  ->  d  e.  ( NN0  ^m  (
1 ... B ) ) ) )
1615adantrd 454 . . . . . . . . . 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 2680 . . . . . . . . 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 208 . . . . . . . 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 3198 . . . . . . 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 451 . . . . . 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 3225 . . . . 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 423 . . . 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 2680 . . 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 418 . 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 187 1  |-  ( A  e.  (Dioph `  B
)  ->  A  C_  ( NN0  ^m  ( 1 ... B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   {cab 2282   E.wrex 2557    C_ wss 3165    |` cres 4707   ` cfv 5271  (class class class)co 5874    ^m cmap 6788   0cc0 8753   1c1 8754   NNcn 9762   NN0cn0 9981   ...cfz 10798  mzPolycmzp 26903  Diophcdioph 26937
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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