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Theorem ig1pval 21649
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 2986 . . . . 5  |-  ( R  e.  V  ->  R  e.  _V )
3 fveq2 5696 . . . . . . . . . 10  |-  ( r  =  R  ->  (Poly1 `  r )  =  (Poly1 `  R ) )
4 ig1pval.p . . . . . . . . . 10  |-  P  =  (Poly1 `  R )
53, 4syl6eqr 2493 . . . . . . . . 9  |-  ( r  =  R  ->  (Poly1 `  r )  =  P )
65fveq2d 5700 . . . . . . . 8  |-  ( r  =  R  ->  (LIdeal `  (Poly1 `  r ) )  =  (LIdeal `  P
) )
7 ig1pval.u . . . . . . . 8  |-  U  =  (LIdeal `  P )
86, 7syl6eqr 2493 . . . . . . 7  |-  ( r  =  R  ->  (LIdeal `  (Poly1 `  r ) )  =  U )
95fveq2d 5700 . . . . . . . . . . 11  |-  ( r  =  R  ->  ( 0g `  (Poly1 `  r ) )  =  ( 0g `  P ) )
10 ig1pval.z . . . . . . . . . . 11  |-  .0.  =  ( 0g `  P )
119, 10syl6eqr 2493 . . . . . . . . . 10  |-  ( r  =  R  ->  ( 0g `  (Poly1 `  r ) )  =  .0.  )
1211sneqd 3894 . . . . . . . . 9  |-  ( r  =  R  ->  { ( 0g `  (Poly1 `  r
) ) }  =  {  .0.  } )
1312eqeq2d 2454 . . . . . . . 8  |-  ( r  =  R  ->  (
i  =  { ( 0g `  (Poly1 `  r
) ) }  <->  i  =  {  .0.  } ) )
14 fveq2 5696 . . . . . . . . . . 11  |-  ( r  =  R  ->  (Monic1p `  r )  =  (Monic1p `  R ) )
15 ig1pval.m . . . . . . . . . . 11  |-  M  =  (Monic1p `  R )
1614, 15syl6eqr 2493 . . . . . . . . . 10  |-  ( r  =  R  ->  (Monic1p `  r )  =  M )
1716ineq2d 3557 . . . . . . . . 9  |-  ( r  =  R  ->  (
i  i^i  (Monic1p `  r
) )  =  ( i  i^i  M ) )
18 fveq2 5696 . . . . . . . . . . . 12  |-  ( r  =  R  ->  ( deg1  `  r )  =  ( deg1  `  R ) )
19 ig1pval.d . . . . . . . . . . . 12  |-  D  =  ( deg1  `  R )
2018, 19syl6eqr 2493 . . . . . . . . . . 11  |-  ( r  =  R  ->  ( deg1  `  r )  =  D )
2120fveq1d 5698 . . . . . . . . . 10  |-  ( r  =  R  ->  (
( deg1  `
 r ) `  g )  =  ( D `  g ) )
2212difeq2d 3479 . . . . . . . . . . . 12  |-  ( r  =  R  ->  (
i  \  { ( 0g `  (Poly1 `  r ) ) } )  =  ( i  \  {  .0.  } ) )
2320, 22imaeq12d 5175 . . . . . . . . . . 11  |-  ( r  =  R  ->  (
( deg1  `
 r ) "
( i  \  {
( 0g `  (Poly1 `  r ) ) } ) )  =  ( D " ( i 
\  {  .0.  }
) ) )
2423supeq1d 7701 . . . . . . . . . 10  |-  ( r  =  R  ->  sup ( ( ( deg1  `  r
) " ( i 
\  { ( 0g
`  (Poly1 `  r ) ) } ) ) ,  RR ,  `'  <  )  =  sup ( ( D " ( i 
\  {  .0.  }
) ) ,  RR ,  `'  <  ) )
2521, 24eqeq12d 2457 . . . . . . . . 9  |-  ( r  =  R  ->  (
( ( deg1  `  r ) `  g )  =  sup ( ( ( deg1  `  r
) " ( i 
\  { ( 0g
`  (Poly1 `  r ) ) } ) ) ,  RR ,  `'  <  )  <-> 
( D `  g
)  =  sup (
( D " (
i  \  {  .0.  } ) ) ,  RR ,  `'  <  ) ) )
2617, 25riotaeqbidv 6060 . . . . . . . 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 3821 . . . . . . 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 4375 . . . . . 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 21611 . . . . . 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 5706 . . . . . . . 8  |-  (LIdeal `  P )  e.  _V
317, 30eqeltri 2513 . . . . . . 7  |-  U  e. 
_V
3231mptex 5953 . . . . . 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 5779 . . . . 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 2487 . . 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 5698 . 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 2449 . . . 4  |-  ( i  =  I  ->  (
i  =  {  .0.  }  <-> 
I  =  {  .0.  } ) )
38 ineq1 3550 . . . . 5  |-  ( i  =  I  ->  (
i  i^i  M )  =  ( I  i^i 
M ) )
39 difeq1 3472 . . . . . . . 8  |-  ( i  =  I  ->  (
i  \  {  .0.  } )  =  ( I 
\  {  .0.  }
) )
4039imaeq2d 5174 . . . . . . 7  |-  ( i  =  I  ->  ( D " ( i  \  {  .0.  } ) )  =  ( D "
( I  \  {  .0.  } ) ) )
4140supeq1d 7701 . . . . . 6  |-  ( i  =  I  ->  sup ( ( D "
( i  \  {  .0.  } ) ) ,  RR ,  `'  <  )  =  sup ( ( D " ( I 
\  {  .0.  }
) ) ,  RR ,  `'  <  ) )
4241eqeq2d 2454 . . . . 5  |-  ( i  =  I  ->  (
( D `  g
)  =  sup (
( D " (
i  \  {  .0.  } ) ) ,  RR ,  `'  <  )  <->  ( D `  g )  =  sup ( ( D "
( I  \  {  .0.  } ) ) ,  RR ,  `'  <  ) ) )
4338, 42riotaeqbidv 6060 . . . 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 3819 . . 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 2443 . . 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 5706 . . . . 5  |-  ( 0g
`  P )  e. 
_V
4710, 46eqeltri 2513 . . . 4  |-  .0.  e.  _V
48 riotaex 6061 . . . 4  |-  ( iota_ g  e.  ( I  i^i 
M ) ( D `
 g )  =  sup ( ( D
" ( I  \  {  .0.  } ) ) ,  RR ,  `'  <  ) )  e.  _V
4947, 48ifex 3863 . . 3  |-  if ( I  =  {  .0.  } ,  .0.  ,  (
iota_ g  e.  (
I  i^i  M )
( D `  g
)  =  sup (
( D " (
I  \  {  .0.  } ) ) ,  RR ,  `'  <  ) ) )  e.  _V
5044, 45, 49fvmpt 5779 . 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 2495 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   _Vcvv 2977    \ cdif 3330    i^i cin 3332   ifcif 3796   {csn 3882    e. cmpt 4355   `'ccnv 4844   "cima 4848   ` cfv 5423   iota_crio 6056   supcsup 7695   RRcr 9286    < clt 9423   0gc0g 14383  LIdealclidl 17256  Poly1cpl1 17638   deg1 cdg1 21528  Monic1pcmn1 21602  idlGen1pcig1p 21606
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-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pr 4536
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-sup 7696  df-ig1p 21611
This theorem is referenced by:  ig1pval2  21650  ig1pval3  21651
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