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Theorem eldioph3 30861
Description: Inference version of eldioph3b 30860 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 2458 . . 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 5871 . . . . . . . . . 10  |-  ( p  =  P  ->  (
p `  b )  =  ( P `  b ) )
54eqeq1d 2459 . . . . . . . . 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 2968 . . . . . . 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 2593 . . . . . 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 2461 . . . . . . . . . 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 2968 . . . . . . . 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 5277 . . . . . . . . . . 11  |-  ( b  =  u  ->  (
b  |`  ( 1 ... N ) )  =  ( u  |`  (
1 ... N ) ) )
1312eqeq2d 2471 . . . . . . . . . 10  |-  ( b  =  u  ->  (
t  =  ( b  |`  ( 1 ... N
) )  <->  t  =  ( u  |`  ( 1 ... N ) ) ) )
14 fveq2 5872 . . . . . . . . . . 11  |-  ( b  =  u  ->  ( P `  b )  =  ( P `  u ) )
1514eqeq1d 2459 . . . . . . . . . 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 3085 . . . . . . . 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 2600 . . . . . 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 2514 . . . . 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 2471 . . . 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 3210 . . 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 30860 . 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 1395    e. wcel 1819   {cab 2442   E.wrex 2808    |` cres 5010   ` cfv 5594  (class class class)co 6296    ^m cmap 7438   0cc0 9509   1c1 9510   NNcn 10556   NN0cn0 10816   ...cfz 11697  mzPolycmzp 30816  Diophcdioph 30850
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-hash 12408  df-mzpcl 30817  df-mzp 30818  df-dioph 30851
This theorem is referenced by:  diophrex  30871
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