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Theorem eldioph3 26517
Description: Inference version of eldioph3b 26516 with quantifier expanded. (Contributed by Stefan O'Rear, 10-Oct-2014.)
Assertion
Ref Expression
eldioph3  |-  ( ( N  e.  NN0  /\  P  e.  (mzPoly `  NN ) )  ->  { t  |  E. u  e.  ( NN0  ^m  NN ) ( t  =  ( u  |`  (
1 ... N ) )  /\  ( P `  u )  =  0 ) }  e.  (Dioph `  N ) )
Distinct variable groups:    u, t, N    t, P, u

Proof of Theorem eldioph3
Dummy variables  a 
b  p are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 444 . 2  |-  ( ( N  e.  NN0  /\  P  e.  (mzPoly `  NN ) )  ->  N  e.  NN0 )
2 simpr 448 . . 3  |-  ( ( N  e.  NN0  /\  P  e.  (mzPoly `  NN ) )  ->  P  e.  (mzPoly `  NN )
)
3 eqidd 2390 . . 3  |-  ( ( N  e.  NN0  /\  P  e.  (mzPoly `  NN ) )  ->  { t  |  E. u  e.  ( NN0  ^m  NN ) ( t  =  ( u  |`  (
1 ... N ) )  /\  ( P `  u )  =  0 ) }  =  {
t  |  E. u  e.  ( NN0  ^m  NN ) ( t  =  ( u  |`  (
1 ... N ) )  /\  ( P `  u )  =  0 ) } )
4 fveq1 5669 . . . . . . . . . 10  |-  ( p  =  P  ->  (
p `  b )  =  ( P `  b ) )
54eqeq1d 2397 . . . . . . . . 9  |-  ( p  =  P  ->  (
( p `  b
)  =  0  <->  ( P `  b )  =  0 ) )
65anbi2d 685 . . . . . . . 8  |-  ( p  =  P  ->  (
( a  =  ( b  |`  ( 1 ... N ) )  /\  ( p `  b )  =  0 )  <->  ( a  =  ( b  |`  (
1 ... N ) )  /\  ( P `  b )  =  0 ) ) )
76rexbidv 2672 . . . . . . 7  |-  ( p  =  P  ->  ( E. b  e.  ( NN0  ^m  NN ) ( a  =  ( b  |`  ( 1 ... N
) )  /\  (
p `  b )  =  0 )  <->  E. b  e.  ( NN0  ^m  NN ) ( a  =  ( b  |`  (
1 ... N ) )  /\  ( P `  b )  =  0 ) ) )
87abbidv 2503 . . . . . 6  |-  ( p  =  P  ->  { a  |  E. b  e.  ( NN0  ^m  NN ) ( a  =  ( b  |`  (
1 ... N ) )  /\  ( p `  b )  =  0 ) }  =  {
a  |  E. b  e.  ( NN0  ^m  NN ) ( a  =  ( b  |`  (
1 ... N ) )  /\  ( P `  b )  =  0 ) } )
9 eqeq1 2395 . . . . . . . . . 10  |-  ( a  =  t  ->  (
a  =  ( b  |`  ( 1 ... N
) )  <->  t  =  ( b  |`  (
1 ... N ) ) ) )
109anbi1d 686 . . . . . . . . 9  |-  ( a  =  t  ->  (
( a  =  ( b  |`  ( 1 ... N ) )  /\  ( P `  b )  =  0 )  <->  ( t  =  ( b  |`  (
1 ... N ) )  /\  ( P `  b )  =  0 ) ) )
1110rexbidv 2672 . . . . . . . 8  |-  ( a  =  t  ->  ( E. b  e.  ( NN0  ^m  NN ) ( a  =  ( b  |`  ( 1 ... N
) )  /\  ( P `  b )  =  0 )  <->  E. b  e.  ( NN0  ^m  NN ) ( t  =  ( b  |`  (
1 ... N ) )  /\  ( P `  b )  =  0 ) ) )
12 reseq1 5082 . . . . . . . . . . 11  |-  ( b  =  u  ->  (
b  |`  ( 1 ... N ) )  =  ( u  |`  (
1 ... N ) ) )
1312eqeq2d 2400 . . . . . . . . . 10  |-  ( b  =  u  ->  (
t  =  ( b  |`  ( 1 ... N
) )  <->  t  =  ( u  |`  ( 1 ... N ) ) ) )
14 fveq2 5670 . . . . . . . . . . 11  |-  ( b  =  u  ->  ( P `  b )  =  ( P `  u ) )
1514eqeq1d 2397 . . . . . . . . . 10  |-  ( b  =  u  ->  (
( P `  b
)  =  0  <->  ( P `  u )  =  0 ) )
1613, 15anbi12d 692 . . . . . . . . 9  |-  ( b  =  u  ->  (
( t  =  ( b  |`  ( 1 ... N ) )  /\  ( P `  b )  =  0 )  <->  ( t  =  ( u  |`  (
1 ... N ) )  /\  ( P `  u )  =  0 ) ) )
1716cbvrexv 2878 . . . . . . . 8  |-  ( E. b  e.  ( NN0 
^m  NN ) ( t  =  ( b  |`  ( 1 ... N
) )  /\  ( P `  b )  =  0 )  <->  E. u  e.  ( NN0  ^m  NN ) ( t  =  ( u  |`  (
1 ... N ) )  /\  ( P `  u )  =  0 ) )
1811, 17syl6bb 253 . . . . . . 7  |-  ( a  =  t  ->  ( E. b  e.  ( NN0  ^m  NN ) ( a  =  ( b  |`  ( 1 ... N
) )  /\  ( P `  b )  =  0 )  <->  E. u  e.  ( NN0  ^m  NN ) ( t  =  ( u  |`  (
1 ... N ) )  /\  ( P `  u )  =  0 ) ) )
1918cbvabv 2508 . . . . . 6  |-  { a  |  E. b  e.  ( NN0  ^m  NN ) ( a  =  ( b  |`  (
1 ... N ) )  /\  ( P `  b )  =  0 ) }  =  {
t  |  E. u  e.  ( NN0  ^m  NN ) ( t  =  ( u  |`  (
1 ... N ) )  /\  ( P `  u )  =  0 ) }
208, 19syl6eq 2437 . . . . 5  |-  ( p  =  P  ->  { a  |  E. b  e.  ( NN0  ^m  NN ) ( a  =  ( b  |`  (
1 ... N ) )  /\  ( p `  b )  =  0 ) }  =  {
t  |  E. u  e.  ( NN0  ^m  NN ) ( t  =  ( u  |`  (
1 ... N ) )  /\  ( P `  u )  =  0 ) } )
2120eqeq2d 2400 . . . 4  |-  ( p  =  P  ->  ( { t  |  E. u  e.  ( NN0  ^m  NN ) ( t  =  ( u  |`  ( 1 ... N
) )  /\  ( P `  u )  =  0 ) }  =  { a  |  E. b  e.  ( NN0  ^m  NN ) ( a  =  ( b  |`  ( 1 ... N ) )  /\  ( p `  b )  =  0 ) }  <->  { t  |  E. u  e.  ( NN0  ^m  NN ) ( t  =  ( u  |`  ( 1 ... N ) )  /\  ( P `  u )  =  0 ) }  =  {
t  |  E. u  e.  ( NN0  ^m  NN ) ( t  =  ( u  |`  (
1 ... N ) )  /\  ( P `  u )  =  0 ) } ) )
2221rspcev 2997 . . 3  |-  ( ( P  e.  (mzPoly `  NN )  /\  { t  |  E. u  e.  ( NN0  ^m  NN ) ( t  =  ( u  |`  (
1 ... N ) )  /\  ( P `  u )  =  0 ) }  =  {
t  |  E. u  e.  ( NN0  ^m  NN ) ( t  =  ( u  |`  (
1 ... N ) )  /\  ( P `  u )  =  0 ) } )  ->  E. p  e.  (mzPoly `  NN ) { t  |  E. u  e.  ( NN0  ^m  NN ) ( t  =  ( u  |`  (
1 ... N ) )  /\  ( P `  u )  =  0 ) }  =  {
a  |  E. b  e.  ( NN0  ^m  NN ) ( a  =  ( b  |`  (
1 ... N ) )  /\  ( p `  b )  =  0 ) } )
232, 3, 22syl2anc 643 . 2  |-  ( ( N  e.  NN0  /\  P  e.  (mzPoly `  NN ) )  ->  E. p  e.  (mzPoly `  NN ) { t  |  E. u  e.  ( NN0  ^m  NN ) ( t  =  ( u  |`  ( 1 ... N
) )  /\  ( P `  u )  =  0 ) }  =  { a  |  E. b  e.  ( NN0  ^m  NN ) ( a  =  ( b  |`  ( 1 ... N ) )  /\  ( p `  b )  =  0 ) } )
24 eldioph3b 26516 . 2  |-  ( { t  |  E. u  e.  ( NN0  ^m  NN ) ( t  =  ( u  |`  (
1 ... N ) )  /\  ( P `  u )  =  0 ) }  e.  (Dioph `  N )  <->  ( N  e.  NN0  /\  E. p  e.  (mzPoly `  NN ) { t  |  E. u  e.  ( NN0  ^m  NN ) ( t  =  ( u  |`  ( 1 ... N
) )  /\  ( P `  u )  =  0 ) }  =  { a  |  E. b  e.  ( NN0  ^m  NN ) ( a  =  ( b  |`  ( 1 ... N ) )  /\  ( p `  b )  =  0 ) } ) )
251, 23, 24sylanbrc 646 1  |-  ( ( N  e.  NN0  /\  P  e.  (mzPoly `  NN ) )  ->  { t  |  E. u  e.  ( NN0  ^m  NN ) ( t  =  ( u  |`  (
1 ... N ) )  /\  ( P `  u )  =  0 ) }  e.  (Dioph `  N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   {cab 2375   E.wrex 2652    |` cres 4822   ` cfv 5396  (class class class)co 6022    ^m cmap 6956   0cc0 8925   1c1 8926   NNcn 9934   NN0cn0 10155   ...cfz 10977  mzPolycmzp 26472  Diophcdioph 26506
This theorem is referenced by:  diophrex  26527
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-inf2 7531  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-int 3995  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-of 6246  df-1st 6290  df-2nd 6291  df-riota 6487  df-recs 6571  df-rdg 6606  df-1o 6662  df-oadd 6666  df-er 6843  df-map 6958  df-en 7048  df-dom 7049  df-sdom 7050  df-fin 7051  df-card 7761  df-cda 7983  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-nn 9935  df-n0 10156  df-z 10217  df-uz 10423  df-fz 10978  df-hash 11548  df-mzpcl 26473  df-mzp 26474  df-dioph 26507
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