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Theorem mplval 16493
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 6107 . . . . . 6  |-  ( i mPwSer 
r )  e.  _V
32a1i 11 . . . . 5  |-  ( ( i  =  I  /\  r  =  R )  ->  ( i mPwSer  r )  e.  _V )
4 id 21 . . . . . . . 8  |-  ( s  =  ( i mPwSer  r
)  ->  s  =  ( i mPwSer  r )
)
5 oveq12 6091 . . . . . . . 8  |-  ( ( i  =  I  /\  r  =  R )  ->  ( i mPwSer  r )  =  ( I mPwSer  R
) )
64, 5sylan9eqr 2491 . . . . . . 7  |-  ( ( ( i  =  I  /\  r  =  R )  /\  s  =  ( i mPwSer  r ) )  ->  s  =  ( I mPwSer  R ) )
7 mplval.s . . . . . . 7  |-  S  =  ( I mPwSer  R )
86, 7syl6eqr 2487 . . . . . 6  |-  ( ( ( i  =  I  /\  r  =  R )  /\  s  =  ( i mPwSer  r ) )  ->  s  =  S )
98fveq2d 5733 . . . . . . . . 9  |-  ( ( ( i  =  I  /\  r  =  R )  /\  s  =  ( i mPwSer  r ) )  ->  ( Base `  s )  =  (
Base `  S )
)
10 mplval.b . . . . . . . . 9  |-  B  =  ( Base `  S
)
119, 10syl6eqr 2487 . . . . . . . 8  |-  ( ( ( i  =  I  /\  r  =  R )  /\  s  =  ( i mPwSer  r ) )  ->  ( Base `  s )  =  B )
12 simplr 733 . . . . . . . . . . . . . 14  |-  ( ( ( i  =  I  /\  r  =  R )  /\  s  =  ( i mPwSer  r ) )  ->  r  =  R )
1312fveq2d 5733 . . . . . . . . . . . . 13  |-  ( ( ( i  =  I  /\  r  =  R )  /\  s  =  ( i mPwSer  r ) )  ->  ( 0g `  r )  =  ( 0g `  R ) )
14 mplval.z . . . . . . . . . . . . 13  |-  .0.  =  ( 0g `  R )
1513, 14syl6eqr 2487 . . . . . . . . . . . 12  |-  ( ( ( i  =  I  /\  r  =  R )  /\  s  =  ( i mPwSer  r ) )  ->  ( 0g `  r )  =  .0.  )
1615sneqd 3828 . . . . . . . . . . 11  |-  ( ( ( i  =  I  /\  r  =  R )  /\  s  =  ( i mPwSer  r ) )  ->  { ( 0g `  r ) }  =  {  .0.  }
)
1716difeq2d 3466 . . . . . . . . . 10  |-  ( ( ( i  =  I  /\  r  =  R )  /\  s  =  ( i mPwSer  r ) )  ->  ( _V  \  { ( 0g `  r ) } )  =  ( _V  \  {  .0.  } ) )
1817imaeq2d 5204 . . . . . . . . 9  |-  ( ( ( i  =  I  /\  r  =  R )  /\  s  =  ( i mPwSer  r ) )  ->  ( `' f " ( _V  \  { ( 0g `  r ) } ) )  =  ( `' f " ( _V 
\  {  .0.  }
) ) )
1918eleq1d 2503 . . . . . . . 8  |-  ( ( ( i  =  I  /\  r  =  R )  /\  s  =  ( i mPwSer  r ) )  ->  ( ( `' f " ( _V  \  { ( 0g
`  r ) } ) )  e.  Fin  <->  ( `' f " ( _V  \  {  .0.  }
) )  e.  Fin ) )
2011, 19rabeqbidv 2952 . . . . . . 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 2487 . . . . . 6  |-  ( ( ( i  =  I  /\  r  =  R )  /\  s  =  ( i mPwSer  r ) )  ->  { f  e.  ( Base `  s
)  |  ( `' f " ( _V 
\  { ( 0g
`  r ) } ) )  e.  Fin }  =  U )
238, 22oveq12d 6100 . . . . 5  |-  ( ( ( i  =  I  /\  r  =  R )  /\  s  =  ( i mPwSer  r ) )  ->  ( ss  {
f  e.  ( Base `  s )  |  ( `' f " ( _V  \  { ( 0g
`  r ) } ) )  e.  Fin } )  =  ( Ss  U ) )
243, 23csbied 3294 . . . 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 16420 . . . 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 6107 . . . 4  |-  ( Ss  U )  e.  _V
2724, 25, 26ovmpt2a 6205 . . 3  |-  ( ( I  e.  _V  /\  R  e.  _V )  ->  ( I mPoly  R )  =  ( Ss  U ) )
28 reldmmpl 16492 . . . . . 6  |-  Rel  dom mPoly
2928ovprc 6109 . . . . 5  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( I mPoly  R )  =  (/) )
30 ress0 13524 . . . . 5  |-  ( (/)s  U )  =  (/)
3129, 30syl6eqr 2487 . . . 4  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( I mPoly  R )  =  ( (/)s  U ) )
32 reldmpsr 16429 . . . . . . 7  |-  Rel  dom mPwSer
3332ovprc 6109 . . . . . 6  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( I mPwSer  R )  =  (/) )
347, 33syl5eq 2481 . . . . 5  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  S  =  (/) )
3534oveq1d 6097 . . . 4  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( Ss  U )  =  (
(/)s  U ) )
3631, 35eqtr4d 2472 . . 3  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( I mPoly  R )  =  ( Ss  U ) )
3727, 36pm2.61i 159 . 2  |-  ( I mPoly 
R )  =  ( Ss  U )
381, 37eqtri 2457 1  |-  P  =  ( Ss  U )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 360    = wceq 1653    e. wcel 1726   {crab 2710   _Vcvv 2957   [_csb 3252    \ cdif 3318   (/)c0 3629   {csn 3815   `'ccnv 4878   "cima 4882   ` cfv 5455  (class class class)co 6082   Fincfn 7110   Basecbs 13470   ↾s cress 13471   0gc0g 13724   mPwSer cmps 16407   mPoly cmpl 16409
This theorem is referenced by:  mplbas  16494  mplval2  16496
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404
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 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-slot 13474  df-base 13475  df-ress 13477  df-psr 16418  df-mpl 16420
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