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Theorem ig1pval 19574
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 2809 . . . . 5  |-  ( R  e.  V  ->  R  e.  _V )
3 fveq2 5541 . . . . . . . . . 10  |-  ( r  =  R  ->  (Poly1 `  r )  =  (Poly1 `  R ) )
4 ig1pval.p . . . . . . . . . 10  |-  P  =  (Poly1 `  R )
53, 4syl6eqr 2346 . . . . . . . . 9  |-  ( r  =  R  ->  (Poly1 `  r )  =  P )
65fveq2d 5545 . . . . . . . 8  |-  ( r  =  R  ->  (LIdeal `  (Poly1 `  r ) )  =  (LIdeal `  P
) )
7 ig1pval.u . . . . . . . 8  |-  U  =  (LIdeal `  P )
86, 7syl6eqr 2346 . . . . . . 7  |-  ( r  =  R  ->  (LIdeal `  (Poly1 `  r ) )  =  U )
95fveq2d 5545 . . . . . . . . . . 11  |-  ( r  =  R  ->  ( 0g `  (Poly1 `  r ) )  =  ( 0g `  P ) )
10 ig1pval.z . . . . . . . . . . 11  |-  .0.  =  ( 0g `  P )
119, 10syl6eqr 2346 . . . . . . . . . 10  |-  ( r  =  R  ->  ( 0g `  (Poly1 `  r ) )  =  .0.  )
1211sneqd 3666 . . . . . . . . 9  |-  ( r  =  R  ->  { ( 0g `  (Poly1 `  r
) ) }  =  {  .0.  } )
1312eqeq2d 2307 . . . . . . . 8  |-  ( r  =  R  ->  (
i  =  { ( 0g `  (Poly1 `  r
) ) }  <->  i  =  {  .0.  } ) )
14 fveq2 5541 . . . . . . . . . . 11  |-  ( r  =  R  ->  (Monic1p `  r )  =  (Monic1p `  R ) )
15 ig1pval.m . . . . . . . . . . 11  |-  M  =  (Monic1p `  R )
1614, 15syl6eqr 2346 . . . . . . . . . 10  |-  ( r  =  R  ->  (Monic1p `  r )  =  M )
1716ineq2d 3383 . . . . . . . . 9  |-  ( r  =  R  ->  (
i  i^i  (Monic1p `  r
) )  =  ( i  i^i  M ) )
18 fveq2 5541 . . . . . . . . . . . 12  |-  ( r  =  R  ->  ( deg1  `  r )  =  ( deg1  `  R ) )
19 ig1pval.d . . . . . . . . . . . 12  |-  D  =  ( deg1  `  R )
2018, 19syl6eqr 2346 . . . . . . . . . . 11  |-  ( r  =  R  ->  ( deg1  `  r )  =  D )
2120fveq1d 5543 . . . . . . . . . 10  |-  ( r  =  R  ->  (
( deg1  `
 r ) `  g )  =  ( D `  g ) )
2220imaeq1d 5027 . . . . . . . . . . . 12  |-  ( r  =  R  ->  (
( deg1  `
 r ) "
( i  \  {
( 0g `  (Poly1 `  r ) ) } ) )  =  ( D " ( i 
\  { ( 0g
`  (Poly1 `  r ) ) } ) ) )
2312difeq2d 3307 . . . . . . . . . . . . 13  |-  ( r  =  R  ->  (
i  \  { ( 0g `  (Poly1 `  r ) ) } )  =  ( i  \  {  .0.  } ) )
2423imaeq2d 5028 . . . . . . . . . . . 12  |-  ( r  =  R  ->  ( D " ( i  \  { ( 0g `  (Poly1 `  r ) ) } ) )  =  ( D " ( i 
\  {  .0.  }
) ) )
2522, 24eqtrd 2328 . . . . . . . . . . 11  |-  ( r  =  R  ->  (
( deg1  `
 r ) "
( i  \  {
( 0g `  (Poly1 `  r ) ) } ) )  =  ( D " ( i 
\  {  .0.  }
) ) )
2625supeq1d 7215 . . . . . . . . . 10  |-  ( r  =  R  ->  sup ( ( ( deg1  `  r
) " ( i 
\  { ( 0g
`  (Poly1 `  r ) ) } ) ) ,  RR ,  `'  <  )  =  sup ( ( D " ( i 
\  {  .0.  }
) ) ,  RR ,  `'  <  ) )
2721, 26eqeq12d 2310 . . . . . . . . 9  |-  ( r  =  R  ->  (
( ( deg1  `  r ) `  g )  =  sup ( ( ( deg1  `  r
) " ( i 
\  { ( 0g
`  (Poly1 `  r ) ) } ) ) ,  RR ,  `'  <  )  <-> 
( D `  g
)  =  sup (
( D " (
i  \  {  .0.  } ) ) ,  RR ,  `'  <  ) ) )
2817, 27riotaeqbidv 6323 . . . . . . . 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 ,  `'  <  ) ) )
2913, 11, 28ifbieq12d 3600 . . . . . . 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 ,  `'  <  ) ) ) )
308, 29mpteq12dv 4114 . . . . . 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 ,  `'  <  ) ) ) ) )
31 df-ig1p 19536 . . . . . 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 ,  `'  <  ) ) ) ) )
32 fvex 5555 . . . . . . . 8  |-  (LIdeal `  P )  e.  _V
337, 32eqeltri 2366 . . . . . . 7  |-  U  e. 
_V
3433mptex 5762 . . . . . 6  |-  ( i  e.  U  |->  if ( i  =  {  .0.  } ,  .0.  ,  (
iota_ g  e.  (
i  i^i  M )
( D `  g
)  =  sup (
( D " (
i  \  {  .0.  } ) ) ,  RR ,  `'  <  ) ) ) )  e.  _V
3530, 31, 34fvmpt 5618 . . . . 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 ,  `'  <  ) ) ) ) )
362, 35syl 15 . . . 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 ,  `'  <  ) ) ) ) )
371, 36syl5eq 2340 . . 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 ,  `'  <  ) ) ) ) )
3837fveq1d 5543 . 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
) )
39 eqeq1 2302 . . . 4  |-  ( i  =  I  ->  (
i  =  {  .0.  }  <-> 
I  =  {  .0.  } ) )
40 ineq1 3376 . . . . 5  |-  ( i  =  I  ->  (
i  i^i  M )  =  ( I  i^i 
M ) )
41 difeq1 3300 . . . . . . . 8  |-  ( i  =  I  ->  (
i  \  {  .0.  } )  =  ( I 
\  {  .0.  }
) )
4241imaeq2d 5028 . . . . . . 7  |-  ( i  =  I  ->  ( D " ( i  \  {  .0.  } ) )  =  ( D "
( I  \  {  .0.  } ) ) )
4342supeq1d 7215 . . . . . 6  |-  ( i  =  I  ->  sup ( ( D "
( i  \  {  .0.  } ) ) ,  RR ,  `'  <  )  =  sup ( ( D " ( I 
\  {  .0.  }
) ) ,  RR ,  `'  <  ) )
4443eqeq2d 2307 . . . . 5  |-  ( i  =  I  ->  (
( D `  g
)  =  sup (
( D " (
i  \  {  .0.  } ) ) ,  RR ,  `'  <  )  <->  ( D `  g )  =  sup ( ( D "
( I  \  {  .0.  } ) ) ,  RR ,  `'  <  ) ) )
4540, 44riotaeqbidv 6323 . . . 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 ,  `'  <  ) ) )
4639, 45ifbieq2d 3598 . . 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 ,  `'  <  ) ) ) )
47 eqid 2296 . . 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 ,  `'  <  ) ) ) )
48 fvex 5555 . . . . 5  |-  ( 0g
`  P )  e. 
_V
4910, 48eqeltri 2366 . . . 4  |-  .0.  e.  _V
50 riotaex 6324 . . . 4  |-  ( iota_ g  e.  ( I  i^i 
M ) ( D `
 g )  =  sup ( ( D
" ( I  \  {  .0.  } ) ) ,  RR ,  `'  <  ) )  e.  _V
5149, 50ifex 3636 . . 3  |-  if ( I  =  {  .0.  } ,  .0.  ,  (
iota_ g  e.  (
I  i^i  M )
( D `  g
)  =  sup (
( D " (
I  \  {  .0.  } ) ) ,  RR ,  `'  <  ) ) )  e.  _V
5246, 47, 51fvmpt 5618 . 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 ,  `'  <  ) ) ) )
5338, 52sylan9eq 2348 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 358    = wceq 1632    e. wcel 1696   _Vcvv 2801    \ cdif 3162    i^i cin 3164   ifcif 3578   {csn 3653    e. cmpt 4093   `'ccnv 4704   "cima 4708   ` cfv 5271   iota_crio 6313   supcsup 7209   RRcr 8752    < clt 8883   0gc0g 13416  LIdealclidl 15939  Poly1cpl1 16268   deg1 cdg1 19456  Monic1pcmn1 19527  idlGen1pcig1p 19531
This theorem is referenced by:  ig1pval2  19575  ig1pval3  19576
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-riota 6320  df-sup 7210  df-ig1p 19536
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