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Theorem ig1pval 20095
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 2964 . . . . 5  |-  ( R  e.  V  ->  R  e.  _V )
3 fveq2 5728 . . . . . . . . . 10  |-  ( r  =  R  ->  (Poly1 `  r )  =  (Poly1 `  R ) )
4 ig1pval.p . . . . . . . . . 10  |-  P  =  (Poly1 `  R )
53, 4syl6eqr 2486 . . . . . . . . 9  |-  ( r  =  R  ->  (Poly1 `  r )  =  P )
65fveq2d 5732 . . . . . . . 8  |-  ( r  =  R  ->  (LIdeal `  (Poly1 `  r ) )  =  (LIdeal `  P
) )
7 ig1pval.u . . . . . . . 8  |-  U  =  (LIdeal `  P )
86, 7syl6eqr 2486 . . . . . . 7  |-  ( r  =  R  ->  (LIdeal `  (Poly1 `  r ) )  =  U )
95fveq2d 5732 . . . . . . . . . . 11  |-  ( r  =  R  ->  ( 0g `  (Poly1 `  r ) )  =  ( 0g `  P ) )
10 ig1pval.z . . . . . . . . . . 11  |-  .0.  =  ( 0g `  P )
119, 10syl6eqr 2486 . . . . . . . . . 10  |-  ( r  =  R  ->  ( 0g `  (Poly1 `  r ) )  =  .0.  )
1211sneqd 3827 . . . . . . . . 9  |-  ( r  =  R  ->  { ( 0g `  (Poly1 `  r
) ) }  =  {  .0.  } )
1312eqeq2d 2447 . . . . . . . 8  |-  ( r  =  R  ->  (
i  =  { ( 0g `  (Poly1 `  r
) ) }  <->  i  =  {  .0.  } ) )
14 fveq2 5728 . . . . . . . . . . 11  |-  ( r  =  R  ->  (Monic1p `  r )  =  (Monic1p `  R ) )
15 ig1pval.m . . . . . . . . . . 11  |-  M  =  (Monic1p `  R )
1614, 15syl6eqr 2486 . . . . . . . . . 10  |-  ( r  =  R  ->  (Monic1p `  r )  =  M )
1716ineq2d 3542 . . . . . . . . 9  |-  ( r  =  R  ->  (
i  i^i  (Monic1p `  r
) )  =  ( i  i^i  M ) )
18 fveq2 5728 . . . . . . . . . . . 12  |-  ( r  =  R  ->  ( deg1  `  r )  =  ( deg1  `  R ) )
19 ig1pval.d . . . . . . . . . . . 12  |-  D  =  ( deg1  `  R )
2018, 19syl6eqr 2486 . . . . . . . . . . 11  |-  ( r  =  R  ->  ( deg1  `  r )  =  D )
2120fveq1d 5730 . . . . . . . . . 10  |-  ( r  =  R  ->  (
( deg1  `
 r ) `  g )  =  ( D `  g ) )
2212difeq2d 3465 . . . . . . . . . . . 12  |-  ( r  =  R  ->  (
i  \  { ( 0g `  (Poly1 `  r ) ) } )  =  ( i  \  {  .0.  } ) )
2320, 22imaeq12d 5204 . . . . . . . . . . 11  |-  ( r  =  R  ->  (
( deg1  `
 r ) "
( i  \  {
( 0g `  (Poly1 `  r ) ) } ) )  =  ( D " ( i 
\  {  .0.  }
) ) )
2423supeq1d 7451 . . . . . . . . . 10  |-  ( r  =  R  ->  sup ( ( ( deg1  `  r
) " ( i 
\  { ( 0g
`  (Poly1 `  r ) ) } ) ) ,  RR ,  `'  <  )  =  sup ( ( D " ( i 
\  {  .0.  }
) ) ,  RR ,  `'  <  ) )
2521, 24eqeq12d 2450 . . . . . . . . 9  |-  ( r  =  R  ->  (
( ( deg1  `  r ) `  g )  =  sup ( ( ( deg1  `  r
) " ( i 
\  { ( 0g
`  (Poly1 `  r ) ) } ) ) ,  RR ,  `'  <  )  <-> 
( D `  g
)  =  sup (
( D " (
i  \  {  .0.  } ) ) ,  RR ,  `'  <  ) ) )
2617, 25riotaeqbidv 6552 . . . . . . . 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 3761 . . . . . . 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 4287 . . . . . 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 20057 . . . . . 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 5742 . . . . . . . 8  |-  (LIdeal `  P )  e.  _V
317, 30eqeltri 2506 . . . . . . 7  |-  U  e. 
_V
3231mptex 5966 . . . . . 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 5806 . . . . 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 2480 . . 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 5730 . 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 2442 . . . 4  |-  ( i  =  I  ->  (
i  =  {  .0.  }  <-> 
I  =  {  .0.  } ) )
38 ineq1 3535 . . . . 5  |-  ( i  =  I  ->  (
i  i^i  M )  =  ( I  i^i 
M ) )
39 difeq1 3458 . . . . . . . 8  |-  ( i  =  I  ->  (
i  \  {  .0.  } )  =  ( I 
\  {  .0.  }
) )
4039imaeq2d 5203 . . . . . . 7  |-  ( i  =  I  ->  ( D " ( i  \  {  .0.  } ) )  =  ( D "
( I  \  {  .0.  } ) ) )
4140supeq1d 7451 . . . . . 6  |-  ( i  =  I  ->  sup ( ( D "
( i  \  {  .0.  } ) ) ,  RR ,  `'  <  )  =  sup ( ( D " ( I 
\  {  .0.  }
) ) ,  RR ,  `'  <  ) )
4241eqeq2d 2447 . . . . 5  |-  ( i  =  I  ->  (
( D `  g
)  =  sup (
( D " (
i  \  {  .0.  } ) ) ,  RR ,  `'  <  )  <->  ( D `  g )  =  sup ( ( D "
( I  \  {  .0.  } ) ) ,  RR ,  `'  <  ) ) )
4338, 42riotaeqbidv 6552 . . . 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 3759 . . 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 2436 . . 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 5742 . . . . 5  |-  ( 0g
`  P )  e. 
_V
4710, 46eqeltri 2506 . . . 4  |-  .0.  e.  _V
48 riotaex 6553 . . . 4  |-  ( iota_ g  e.  ( I  i^i 
M ) ( D `
 g )  =  sup ( ( D
" ( I  \  {  .0.  } ) ) ,  RR ,  `'  <  ) )  e.  _V
4947, 48ifex 3797 . . 3  |-  if ( I  =  {  .0.  } ,  .0.  ,  (
iota_ g  e.  (
I  i^i  M )
( D `  g
)  =  sup (
( D " (
I  \  {  .0.  } ) ) ,  RR ,  `'  <  ) ) )  e.  _V
5044, 45, 49fvmpt 5806 . 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 2488 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 359    = wceq 1652    e. wcel 1725   _Vcvv 2956    \ cdif 3317    i^i cin 3319   ifcif 3739   {csn 3814    e. cmpt 4266   `'ccnv 4877   "cima 4881   ` cfv 5454   iota_crio 6542   supcsup 7445   RRcr 8989    < clt 9120   0gc0g 13723  LIdealclidl 16242  Poly1cpl1 16571   deg1 cdg1 19977  Monic1pcmn1 20048  idlGen1pcig1p 20052
This theorem is referenced by:  ig1pval2  20096  ig1pval3  20097
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-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-riota 6549  df-sup 7446  df-ig1p 20057
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