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Theorem eldioph3 26778
Description: Inference version of eldioph3b 26777 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 2436 . . 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 5719 . . . . . . . . . 10  |-  ( p  =  P  ->  (
p `  b )  =  ( P `  b ) )
54eqeq1d 2443 . . . . . . . . 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 2718 . . . . . . 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 2549 . . . . . 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 2441 . . . . . . . . . 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 2718 . . . . . . . 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 5132 . . . . . . . . . . 11  |-  ( b  =  u  ->  (
b  |`  ( 1 ... N ) )  =  ( u  |`  (
1 ... N ) ) )
1312eqeq2d 2446 . . . . . . . . . 10  |-  ( b  =  u  ->  (
t  =  ( b  |`  ( 1 ... N
) )  <->  t  =  ( u  |`  ( 1 ... N ) ) ) )
14 fveq2 5720 . . . . . . . . . . 11  |-  ( b  =  u  ->  ( P `  b )  =  ( P `  u ) )
1514eqeq1d 2443 . . . . . . . . . 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 2925 . . . . . . . 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 2554 . . . . . 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 2483 . . . . 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 2446 . . . 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 3044 . . 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 26777 . 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 1652    e. wcel 1725   {cab 2421   E.wrex 2698    |` cres 4872   ` cfv 5446  (class class class)co 6073    ^m cmap 7010   0cc0 8980   1c1 8981   NNcn 9990   NN0cn0 10211   ...cfz 11033  mzPolycmzp 26733  Diophcdioph 26767
This theorem is referenced by:  diophrex  26788
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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7586  ax-cnex 9036  ax-resscn 9037  ax-1cn 9038  ax-icn 9039  ax-addcl 9040  ax-addrcl 9041  ax-mulcl 9042  ax-mulrcl 9043  ax-mulcom 9044  ax-addass 9045  ax-mulass 9046  ax-distr 9047  ax-i2m1 9048  ax-1ne0 9049  ax-1rid 9050  ax-rnegex 9051  ax-rrecex 9052  ax-cnre 9053  ax-pre-lttri 9054  ax-pre-lttrn 9055  ax-pre-ltadd 9056  ax-pre-mulgt0 9057
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-card 7816  df-cda 8038  df-pnf 9112  df-mnf 9113  df-xr 9114  df-ltxr 9115  df-le 9116  df-sub 9283  df-neg 9284  df-nn 9991  df-n0 10212  df-z 10273  df-uz 10479  df-fz 11034  df-hash 11609  df-mzpcl 26734  df-mzp 26735  df-dioph 26768
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