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Theorem eldioph3 29102
Description: Inference version of eldioph3b 29101 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 457 . 2  |-  ( ( N  e.  NN0  /\  P  e.  (mzPoly `  NN ) )  ->  N  e.  NN0 )
2 simpr 461 . . 3  |-  ( ( N  e.  NN0  /\  P  e.  (mzPoly `  NN ) )  ->  P  e.  (mzPoly `  NN )
)
3 eqidd 2443 . . 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 5689 . . . . . . . . . 10  |-  ( p  =  P  ->  (
p `  b )  =  ( P `  b ) )
54eqeq1d 2450 . . . . . . . . 9  |-  ( p  =  P  ->  (
( p `  b
)  =  0  <->  ( P `  b )  =  0 ) )
65anbi2d 703 . . . . . . . 8  |-  ( p  =  P  ->  (
( a  =  ( b  |`  ( 1 ... N ) )  /\  ( p `  b )  =  0 )  <->  ( a  =  ( b  |`  (
1 ... N ) )  /\  ( P `  b )  =  0 ) ) )
76rexbidv 2735 . . . . . . 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 2556 . . . . . 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 2448 . . . . . . . . . 10  |-  ( a  =  t  ->  (
a  =  ( b  |`  ( 1 ... N
) )  <->  t  =  ( b  |`  (
1 ... N ) ) ) )
109anbi1d 704 . . . . . . . . 9  |-  ( a  =  t  ->  (
( a  =  ( b  |`  ( 1 ... N ) )  /\  ( P `  b )  =  0 )  <->  ( t  =  ( b  |`  (
1 ... N ) )  /\  ( P `  b )  =  0 ) ) )
1110rexbidv 2735 . . . . . . . 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 5103 . . . . . . . . . . 11  |-  ( b  =  u  ->  (
b  |`  ( 1 ... N ) )  =  ( u  |`  (
1 ... N ) ) )
1312eqeq2d 2453 . . . . . . . . . 10  |-  ( b  =  u  ->  (
t  =  ( b  |`  ( 1 ... N
) )  <->  t  =  ( u  |`  ( 1 ... N ) ) ) )
14 fveq2 5690 . . . . . . . . . . 11  |-  ( b  =  u  ->  ( P `  b )  =  ( P `  u ) )
1514eqeq1d 2450 . . . . . . . . . 10  |-  ( b  =  u  ->  (
( P `  b
)  =  0  <->  ( P `  u )  =  0 ) )
1613, 15anbi12d 710 . . . . . . . . 9  |-  ( b  =  u  ->  (
( t  =  ( b  |`  ( 1 ... N ) )  /\  ( P `  b )  =  0 )  <->  ( t  =  ( u  |`  (
1 ... N ) )  /\  ( P `  u )  =  0 ) ) )
1716cbvrexv 2947 . . . . . . . 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 261 . . . . . . 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 2561 . . . . . 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 2490 . . . . 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 2453 . . . 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 3072 . . 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 661 . 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 29101 . 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 664 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   {cab 2428   E.wrex 2715    |` cres 4841   ` cfv 5417  (class class class)co 6090    ^m cmap 7213   0cc0 9281   1c1 9282   NNcn 10321   NN0cn0 10578   ...cfz 11436  mzPolycmzp 29056  Diophcdioph 29091
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 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-inf2 7846  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358
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 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-int 4128  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6831  df-rdg 6865  df-1o 6919  df-oadd 6923  df-er 7100  df-map 7215  df-en 7310  df-dom 7311  df-sdom 7312  df-fin 7313  df-card 8108  df-cda 8336  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-nn 10322  df-n0 10579  df-z 10646  df-uz 10861  df-fz 11437  df-hash 12103  df-mzpcl 29057  df-mzp 29058  df-dioph 29092
This theorem is referenced by:  diophrex  29112
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