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Theorem mplval 16173
Description: Value of the set of multivariate polynomials. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
Hypotheses
Ref Expression
mplval.p  |-  P  =  ( I mPoly  R )
mplval.s  |-  S  =  ( I mPwSer  R )
mplval.b  |-  B  =  ( Base `  S
)
mplval.z  |-  .0.  =  ( 0g `  R )
mplval.u  |-  U  =  { f  e.  B  |  ( `' f
" ( _V  \  {  .0.  } ) )  e.  Fin }
Assertion
Ref Expression
mplval  |-  P  =  ( Ss  U )
Distinct variable groups:    B, f    f, I    R, f    .0. , f
Allowed substitution hints:    P( f)    S( f)    U( f)

Proof of Theorem mplval
Dummy variables  i 
r  s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mplval.p . 2  |-  P  =  ( I mPoly  R )
2 ovex 5883 . . . . . 6  |-  ( i mPwSer 
r )  e.  _V
32a1i 10 . . . . 5  |-  ( ( i  =  I  /\  r  =  R )  ->  ( i mPwSer  r )  e.  _V )
4 id 19 . . . . . . . 8  |-  ( s  =  ( i mPwSer  r
)  ->  s  =  ( i mPwSer  r )
)
5 oveq12 5867 . . . . . . . 8  |-  ( ( i  =  I  /\  r  =  R )  ->  ( i mPwSer  r )  =  ( I mPwSer  R
) )
64, 5sylan9eqr 2337 . . . . . . 7  |-  ( ( ( i  =  I  /\  r  =  R )  /\  s  =  ( i mPwSer  r ) )  ->  s  =  ( I mPwSer  R ) )
7 mplval.s . . . . . . 7  |-  S  =  ( I mPwSer  R )
86, 7syl6eqr 2333 . . . . . 6  |-  ( ( ( i  =  I  /\  r  =  R )  /\  s  =  ( i mPwSer  r ) )  ->  s  =  S )
98fveq2d 5529 . . . . . . . . 9  |-  ( ( ( i  =  I  /\  r  =  R )  /\  s  =  ( i mPwSer  r ) )  ->  ( Base `  s )  =  (
Base `  S )
)
10 mplval.b . . . . . . . . 9  |-  B  =  ( Base `  S
)
119, 10syl6eqr 2333 . . . . . . . 8  |-  ( ( ( i  =  I  /\  r  =  R )  /\  s  =  ( i mPwSer  r ) )  ->  ( Base `  s )  =  B )
12 simplr 731 . . . . . . . . . . . . . 14  |-  ( ( ( i  =  I  /\  r  =  R )  /\  s  =  ( i mPwSer  r ) )  ->  r  =  R )
1312fveq2d 5529 . . . . . . . . . . . . 13  |-  ( ( ( i  =  I  /\  r  =  R )  /\  s  =  ( i mPwSer  r ) )  ->  ( 0g `  r )  =  ( 0g `  R ) )
14 mplval.z . . . . . . . . . . . . 13  |-  .0.  =  ( 0g `  R )
1513, 14syl6eqr 2333 . . . . . . . . . . . 12  |-  ( ( ( i  =  I  /\  r  =  R )  /\  s  =  ( i mPwSer  r ) )  ->  ( 0g `  r )  =  .0.  )
1615sneqd 3653 . . . . . . . . . . 11  |-  ( ( ( i  =  I  /\  r  =  R )  /\  s  =  ( i mPwSer  r ) )  ->  { ( 0g `  r ) }  =  {  .0.  }
)
1716difeq2d 3294 . . . . . . . . . 10  |-  ( ( ( i  =  I  /\  r  =  R )  /\  s  =  ( i mPwSer  r ) )  ->  ( _V  \  { ( 0g `  r ) } )  =  ( _V  \  {  .0.  } ) )
1817imaeq2d 5012 . . . . . . . . 9  |-  ( ( ( i  =  I  /\  r  =  R )  /\  s  =  ( i mPwSer  r ) )  ->  ( `' f " ( _V  \  { ( 0g `  r ) } ) )  =  ( `' f " ( _V 
\  {  .0.  }
) ) )
1918eleq1d 2349 . . . . . . . 8  |-  ( ( ( i  =  I  /\  r  =  R )  /\  s  =  ( i mPwSer  r ) )  ->  ( ( `' f " ( _V  \  { ( 0g
`  r ) } ) )  e.  Fin  <->  ( `' f " ( _V  \  {  .0.  }
) )  e.  Fin ) )
2011, 19rabeqbidv 2783 . . . . . . 7  |-  ( ( ( i  =  I  /\  r  =  R )  /\  s  =  ( i mPwSer  r ) )  ->  { f  e.  ( Base `  s
)  |  ( `' f " ( _V 
\  { ( 0g
`  r ) } ) )  e.  Fin }  =  { f  e.  B  |  ( `' f " ( _V 
\  {  .0.  }
) )  e.  Fin } )
21 mplval.u . . . . . . 7  |-  U  =  { f  e.  B  |  ( `' f
" ( _V  \  {  .0.  } ) )  e.  Fin }
2220, 21syl6eqr 2333 . . . . . 6  |-  ( ( ( i  =  I  /\  r  =  R )  /\  s  =  ( i mPwSer  r ) )  ->  { f  e.  ( Base `  s
)  |  ( `' f " ( _V 
\  { ( 0g
`  r ) } ) )  e.  Fin }  =  U )
238, 22oveq12d 5876 . . . . 5  |-  ( ( ( i  =  I  /\  r  =  R )  /\  s  =  ( i mPwSer  r ) )  ->  ( ss  {
f  e.  ( Base `  s )  |  ( `' f " ( _V  \  { ( 0g
`  r ) } ) )  e.  Fin } )  =  ( Ss  U ) )
243, 23csbied 3123 . . . 4  |-  ( ( i  =  I  /\  r  =  R )  ->  [_ ( i mPwSer  r
)  /  s ]_ ( ss  { f  e.  (
Base `  s )  |  ( `' f
" ( _V  \  { ( 0g `  r ) } ) )  e.  Fin }
)  =  ( Ss  U ) )
25 df-mpl 16100 . . . 4  |- mPoly  =  ( i  e.  _V , 
r  e.  _V  |->  [_ ( i mPwSer  r )  /  s ]_ (
ss 
{ f  e.  (
Base `  s )  |  ( `' f
" ( _V  \  { ( 0g `  r ) } ) )  e.  Fin }
) )
26 ovex 5883 . . . 4  |-  ( Ss  U )  e.  _V
2724, 25, 26ovmpt2a 5978 . . 3  |-  ( ( I  e.  _V  /\  R  e.  _V )  ->  ( I mPoly  R )  =  ( Ss  U ) )
28 reldmmpl 16172 . . . . . 6  |-  Rel  dom mPoly
2928ovprc 5885 . . . . 5  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( I mPoly  R )  =  (/) )
30 ress0 13202 . . . . 5  |-  ( (/)s  U )  =  (/)
3129, 30syl6eqr 2333 . . . 4  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( I mPoly  R )  =  ( (/)s  U ) )
32 reldmpsr 16109 . . . . . . 7  |-  Rel  dom mPwSer
3332ovprc 5885 . . . . . 6  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( I mPwSer  R )  =  (/) )
347, 33syl5eq 2327 . . . . 5  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  S  =  (/) )
3534oveq1d 5873 . . . 4  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( Ss  U )  =  (
(/)s  U ) )
3631, 35eqtr4d 2318 . . 3  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( I mPoly  R )  =  ( Ss  U ) )
3727, 36pm2.61i 156 . 2  |-  ( I mPoly 
R )  =  ( Ss  U )
381, 37eqtri 2303 1  |-  P  =  ( Ss  U )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 358    = wceq 1623    e. wcel 1684   {crab 2547   _Vcvv 2788   [_csb 3081    \ cdif 3149   (/)c0 3455   {csn 3640   `'ccnv 4688   "cima 4692   ` cfv 5255  (class class class)co 5858   Fincfn 6863   Basecbs 13148   ↾s cress 13149   0gc0g 13400   mPwSer cmps 16087   mPoly cmpl 16089
This theorem is referenced by:  mplbas  16174  mplval2  16176
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-slot 13152  df-base 13153  df-ress 13155  df-psr 16098  df-mpl 16100
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