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Theorem ig1pval 20100
Description: Substitutions for the polynomial ideal generator function. (Contributed by Stefan O'Rear, 29-Mar-2015.)
Hypotheses
Ref Expression
ig1pval.p  |-  P  =  (Poly1 `  R )
ig1pval.g  |-  G  =  (idlGen1p `
 R )
ig1pval.z  |-  .0.  =  ( 0g `  P )
ig1pval.u  |-  U  =  (LIdeal `  P )
ig1pval.d  |-  D  =  ( deg1  `  R )
ig1pval.m  |-  M  =  (Monic1p `  R )
Assertion
Ref Expression
ig1pval  |-  ( ( R  e.  V  /\  I  e.  U )  ->  ( G `  I
)  =  if ( I  =  {  .0.  } ,  .0.  ,  (
iota_ g  e.  (
I  i^i  M )
( D `  g
)  =  sup (
( D " (
I  \  {  .0.  } ) ) ,  RR ,  `'  <  ) ) ) )
Distinct variable groups:    g, I    g, M    R, g
Allowed substitution hints:    D( g)    P( g)    U( g)    G( g)    V( g)    .0. ( g)

Proof of Theorem ig1pval
Dummy variables  i 
r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ig1pval.g . . . 4  |-  G  =  (idlGen1p `
 R )
2 elex 2966 . . . . 5  |-  ( R  e.  V  ->  R  e.  _V )
3 fveq2 5731 . . . . . . . . . 10  |-  ( r  =  R  ->  (Poly1 `  r )  =  (Poly1 `  R ) )
4 ig1pval.p . . . . . . . . . 10  |-  P  =  (Poly1 `  R )
53, 4syl6eqr 2488 . . . . . . . . 9  |-  ( r  =  R  ->  (Poly1 `  r )  =  P )
65fveq2d 5735 . . . . . . . 8  |-  ( r  =  R  ->  (LIdeal `  (Poly1 `  r ) )  =  (LIdeal `  P
) )
7 ig1pval.u . . . . . . . 8  |-  U  =  (LIdeal `  P )
86, 7syl6eqr 2488 . . . . . . 7  |-  ( r  =  R  ->  (LIdeal `  (Poly1 `  r ) )  =  U )
95fveq2d 5735 . . . . . . . . . . 11  |-  ( r  =  R  ->  ( 0g `  (Poly1 `  r ) )  =  ( 0g `  P ) )
10 ig1pval.z . . . . . . . . . . 11  |-  .0.  =  ( 0g `  P )
119, 10syl6eqr 2488 . . . . . . . . . 10  |-  ( r  =  R  ->  ( 0g `  (Poly1 `  r ) )  =  .0.  )
1211sneqd 3829 . . . . . . . . 9  |-  ( r  =  R  ->  { ( 0g `  (Poly1 `  r
) ) }  =  {  .0.  } )
1312eqeq2d 2449 . . . . . . . 8  |-  ( r  =  R  ->  (
i  =  { ( 0g `  (Poly1 `  r
) ) }  <->  i  =  {  .0.  } ) )
14 fveq2 5731 . . . . . . . . . . 11  |-  ( r  =  R  ->  (Monic1p `  r )  =  (Monic1p `  R ) )
15 ig1pval.m . . . . . . . . . . 11  |-  M  =  (Monic1p `  R )
1614, 15syl6eqr 2488 . . . . . . . . . 10  |-  ( r  =  R  ->  (Monic1p `  r )  =  M )
1716ineq2d 3544 . . . . . . . . 9  |-  ( r  =  R  ->  (
i  i^i  (Monic1p `  r
) )  =  ( i  i^i  M ) )
18 fveq2 5731 . . . . . . . . . . . 12  |-  ( r  =  R  ->  ( deg1  `  r )  =  ( deg1  `  R ) )
19 ig1pval.d . . . . . . . . . . . 12  |-  D  =  ( deg1  `  R )
2018, 19syl6eqr 2488 . . . . . . . . . . 11  |-  ( r  =  R  ->  ( deg1  `  r )  =  D )
2120fveq1d 5733 . . . . . . . . . 10  |-  ( r  =  R  ->  (
( deg1  `
 r ) `  g )  =  ( D `  g ) )
2212difeq2d 3467 . . . . . . . . . . . 12  |-  ( r  =  R  ->  (
i  \  { ( 0g `  (Poly1 `  r ) ) } )  =  ( i  \  {  .0.  } ) )
2320, 22imaeq12d 5207 . . . . . . . . . . 11  |-  ( r  =  R  ->  (
( deg1  `
 r ) "
( i  \  {
( 0g `  (Poly1 `  r ) ) } ) )  =  ( D " ( i 
\  {  .0.  }
) ) )
2423supeq1d 7454 . . . . . . . . . 10  |-  ( r  =  R  ->  sup ( ( ( deg1  `  r
) " ( i 
\  { ( 0g
`  (Poly1 `  r ) ) } ) ) ,  RR ,  `'  <  )  =  sup ( ( D " ( i 
\  {  .0.  }
) ) ,  RR ,  `'  <  ) )
2521, 24eqeq12d 2452 . . . . . . . . 9  |-  ( r  =  R  ->  (
( ( deg1  `  r ) `  g )  =  sup ( ( ( deg1  `  r
) " ( i 
\  { ( 0g
`  (Poly1 `  r ) ) } ) ) ,  RR ,  `'  <  )  <-> 
( D `  g
)  =  sup (
( D " (
i  \  {  .0.  } ) ) ,  RR ,  `'  <  ) ) )
2617, 25riotaeqbidv 6555 . . . . . . . 8  |-  ( r  =  R  ->  ( iota_ g  e.  ( i  i^i  (Monic1p `  r ) ) ( ( deg1  `  r ) `  g )  =  sup ( ( ( deg1  `  r
) " ( i 
\  { ( 0g
`  (Poly1 `  r ) ) } ) ) ,  RR ,  `'  <  ) )  =  ( iota_ g  e.  ( i  i^i 
M ) ( D `
 g )  =  sup ( ( D
" ( i  \  {  .0.  } ) ) ,  RR ,  `'  <  ) ) )
2713, 11, 26ifbieq12d 3763 . . . . . . 7  |-  ( r  =  R  ->  if ( i  =  {
( 0g `  (Poly1 `  r ) ) } ,  ( 0g `  (Poly1 `  r ) ) ,  ( iota_ g  e.  ( i  i^i  (Monic1p `  r
) ) ( ( deg1  `  r ) `  g
)  =  sup (
( ( deg1  `  r ) " ( i  \  { ( 0g `  (Poly1 `  r ) ) } ) ) ,  RR ,  `'  <  ) ) )  =  if ( i  =  {  .0.  } ,  .0.  ,  (
iota_ g  e.  (
i  i^i  M )
( D `  g
)  =  sup (
( D " (
i  \  {  .0.  } ) ) ,  RR ,  `'  <  ) ) ) )
288, 27mpteq12dv 4290 . . . . . 6  |-  ( r  =  R  ->  (
i  e.  (LIdeal `  (Poly1 `  r ) )  |->  if ( i  =  {
( 0g `  (Poly1 `  r ) ) } ,  ( 0g `  (Poly1 `  r ) ) ,  ( iota_ g  e.  ( i  i^i  (Monic1p `  r
) ) ( ( deg1  `  r ) `  g
)  =  sup (
( ( deg1  `  r ) " ( i  \  { ( 0g `  (Poly1 `  r ) ) } ) ) ,  RR ,  `'  <  ) ) ) )  =  ( i  e.  U  |->  if ( i  =  {  .0.  } ,  .0.  , 
( iota_ g  e.  ( i  i^i  M ) ( D `  g
)  =  sup (
( D " (
i  \  {  .0.  } ) ) ,  RR ,  `'  <  ) ) ) ) )
29 df-ig1p 20062 . . . . . 6  |- idlGen1p  =  ( r  e.  _V  |->  ( i  e.  (LIdeal `  (Poly1 `  r
) )  |->  if ( i  =  { ( 0g `  (Poly1 `  r
) ) } , 
( 0g `  (Poly1 `  r ) ) ,  ( iota_ g  e.  ( i  i^i  (Monic1p `  r
) ) ( ( deg1  `  r ) `  g
)  =  sup (
( ( deg1  `  r ) " ( i  \  { ( 0g `  (Poly1 `  r ) ) } ) ) ,  RR ,  `'  <  ) ) ) ) )
30 fvex 5745 . . . . . . . 8  |-  (LIdeal `  P )  e.  _V
317, 30eqeltri 2508 . . . . . . 7  |-  U  e. 
_V
3231mptex 5969 . . . . . 6  |-  ( i  e.  U  |->  if ( i  =  {  .0.  } ,  .0.  ,  (
iota_ g  e.  (
i  i^i  M )
( D `  g
)  =  sup (
( D " (
i  \  {  .0.  } ) ) ,  RR ,  `'  <  ) ) ) )  e.  _V
3328, 29, 32fvmpt 5809 . . . . 5  |-  ( R  e.  _V  ->  (idlGen1p `  R )  =  ( i  e.  U  |->  if ( i  =  {  .0.  } ,  .0.  , 
( iota_ g  e.  ( i  i^i  M ) ( D `  g
)  =  sup (
( D " (
i  \  {  .0.  } ) ) ,  RR ,  `'  <  ) ) ) ) )
342, 33syl 16 . . . 4  |-  ( R  e.  V  ->  (idlGen1p `  R )  =  ( i  e.  U  |->  if ( i  =  {  .0.  } ,  .0.  , 
( iota_ g  e.  ( i  i^i  M ) ( D `  g
)  =  sup (
( D " (
i  \  {  .0.  } ) ) ,  RR ,  `'  <  ) ) ) ) )
351, 34syl5eq 2482 . . 3  |-  ( R  e.  V  ->  G  =  ( i  e.  U  |->  if ( i  =  {  .0.  } ,  .0.  ,  ( iota_ g  e.  ( i  i^i 
M ) ( D `
 g )  =  sup ( ( D
" ( i  \  {  .0.  } ) ) ,  RR ,  `'  <  ) ) ) ) )
3635fveq1d 5733 . 2  |-  ( R  e.  V  ->  ( G `  I )  =  ( ( i  e.  U  |->  if ( i  =  {  .0.  } ,  .0.  ,  (
iota_ g  e.  (
i  i^i  M )
( D `  g
)  =  sup (
( D " (
i  \  {  .0.  } ) ) ,  RR ,  `'  <  ) ) ) ) `  I
) )
37 eqeq1 2444 . . . 4  |-  ( i  =  I  ->  (
i  =  {  .0.  }  <-> 
I  =  {  .0.  } ) )
38 ineq1 3537 . . . . 5  |-  ( i  =  I  ->  (
i  i^i  M )  =  ( I  i^i 
M ) )
39 difeq1 3460 . . . . . . . 8  |-  ( i  =  I  ->  (
i  \  {  .0.  } )  =  ( I 
\  {  .0.  }
) )
4039imaeq2d 5206 . . . . . . 7  |-  ( i  =  I  ->  ( D " ( i  \  {  .0.  } ) )  =  ( D "
( I  \  {  .0.  } ) ) )
4140supeq1d 7454 . . . . . 6  |-  ( i  =  I  ->  sup ( ( D "
( i  \  {  .0.  } ) ) ,  RR ,  `'  <  )  =  sup ( ( D " ( I 
\  {  .0.  }
) ) ,  RR ,  `'  <  ) )
4241eqeq2d 2449 . . . . 5  |-  ( i  =  I  ->  (
( D `  g
)  =  sup (
( D " (
i  \  {  .0.  } ) ) ,  RR ,  `'  <  )  <->  ( D `  g )  =  sup ( ( D "
( I  \  {  .0.  } ) ) ,  RR ,  `'  <  ) ) )
4338, 42riotaeqbidv 6555 . . . 4  |-  ( i  =  I  ->  ( iota_ g  e.  ( i  i^i  M ) ( D `  g )  =  sup ( ( D " ( i 
\  {  .0.  }
) ) ,  RR ,  `'  <  ) )  =  ( iota_ g  e.  ( I  i^i  M
) ( D `  g )  =  sup ( ( D "
( I  \  {  .0.  } ) ) ,  RR ,  `'  <  ) ) )
4437, 43ifbieq2d 3761 . . 3  |-  ( i  =  I  ->  if ( i  =  {  .0.  } ,  .0.  , 
( iota_ g  e.  ( i  i^i  M ) ( D `  g
)  =  sup (
( D " (
i  \  {  .0.  } ) ) ,  RR ,  `'  <  ) ) )  =  if ( I  =  {  .0.  } ,  .0.  ,  (
iota_ g  e.  (
I  i^i  M )
( D `  g
)  =  sup (
( D " (
I  \  {  .0.  } ) ) ,  RR ,  `'  <  ) ) ) )
45 eqid 2438 . . 3  |-  ( i  e.  U  |->  if ( i  =  {  .0.  } ,  .0.  ,  (
iota_ g  e.  (
i  i^i  M )
( D `  g
)  =  sup (
( D " (
i  \  {  .0.  } ) ) ,  RR ,  `'  <  ) ) ) )  =  ( i  e.  U  |->  if ( i  =  {  .0.  } ,  .0.  , 
( iota_ g  e.  ( i  i^i  M ) ( D `  g
)  =  sup (
( D " (
i  \  {  .0.  } ) ) ,  RR ,  `'  <  ) ) ) )
46 fvex 5745 . . . . 5  |-  ( 0g
`  P )  e. 
_V
4710, 46eqeltri 2508 . . . 4  |-  .0.  e.  _V
48 riotaex 6556 . . . 4  |-  ( iota_ g  e.  ( I  i^i 
M ) ( D `
 g )  =  sup ( ( D
" ( I  \  {  .0.  } ) ) ,  RR ,  `'  <  ) )  e.  _V
4947, 48ifex 3799 . . 3  |-  if ( I  =  {  .0.  } ,  .0.  ,  (
iota_ g  e.  (
I  i^i  M )
( D `  g
)  =  sup (
( D " (
I  \  {  .0.  } ) ) ,  RR ,  `'  <  ) ) )  e.  _V
5044, 45, 49fvmpt 5809 . 2  |-  ( I  e.  U  ->  (
( i  e.  U  |->  if ( i  =  {  .0.  } ,  .0.  ,  ( iota_ g  e.  ( i  i^i  M
) ( D `  g )  =  sup ( ( D "
( i  \  {  .0.  } ) ) ,  RR ,  `'  <  ) ) ) ) `  I )  =  if ( I  =  {  .0.  } ,  .0.  , 
( iota_ g  e.  ( I  i^i  M ) ( D `  g
)  =  sup (
( D " (
I  \  {  .0.  } ) ) ,  RR ,  `'  <  ) ) ) )
5136, 50sylan9eq 2490 1  |-  ( ( R  e.  V  /\  I  e.  U )  ->  ( G `  I
)  =  if ( I  =  {  .0.  } ,  .0.  ,  (
iota_ g  e.  (
I  i^i  M )
( D `  g
)  =  sup (
( D " (
I  \  {  .0.  } ) ) ,  RR ,  `'  <  ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   _Vcvv 2958    \ cdif 3319    i^i cin 3321   ifcif 3741   {csn 3816    e. cmpt 4269   `'ccnv 4880   "cima 4884   ` cfv 5457   iota_crio 6545   supcsup 7448   RRcr 8994    < clt 9125   0gc0g 13728  LIdealclidl 16247  Poly1cpl1 16576   deg1 cdg1 19982  Monic1pcmn1 20053  idlGen1pcig1p 20057
This theorem is referenced by:  ig1pval2  20101  ig1pval3  20102
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-riota 6552  df-sup 7449  df-ig1p 20062
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