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Theorem mplval 16222
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 5925 . . . . . 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 5909 . . . . . . . 8  |-  ( ( i  =  I  /\  r  =  R )  ->  ( i mPwSer  r )  =  ( I mPwSer  R
) )
64, 5sylan9eqr 2370 . . . . . . 7  |-  ( ( ( i  =  I  /\  r  =  R )  /\  s  =  ( i mPwSer  r ) )  ->  s  =  ( I mPwSer  R ) )
7 mplval.s . . . . . . 7  |-  S  =  ( I mPwSer  R )
86, 7syl6eqr 2366 . . . . . 6  |-  ( ( ( i  =  I  /\  r  =  R )  /\  s  =  ( i mPwSer  r ) )  ->  s  =  S )
98fveq2d 5567 . . . . . . . . 9  |-  ( ( ( i  =  I  /\  r  =  R )  /\  s  =  ( i mPwSer  r ) )  ->  ( Base `  s )  =  (
Base `  S )
)
10 mplval.b . . . . . . . . 9  |-  B  =  ( Base `  S
)
119, 10syl6eqr 2366 . . . . . . . 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 5567 . . . . . . . . . . . . 13  |-  ( ( ( i  =  I  /\  r  =  R )  /\  s  =  ( i mPwSer  r ) )  ->  ( 0g `  r )  =  ( 0g `  R ) )
14 mplval.z . . . . . . . . . . . . 13  |-  .0.  =  ( 0g `  R )
1513, 14syl6eqr 2366 . . . . . . . . . . . 12  |-  ( ( ( i  =  I  /\  r  =  R )  /\  s  =  ( i mPwSer  r ) )  ->  ( 0g `  r )  =  .0.  )
1615sneqd 3687 . . . . . . . . . . 11  |-  ( ( ( i  =  I  /\  r  =  R )  /\  s  =  ( i mPwSer  r ) )  ->  { ( 0g `  r ) }  =  {  .0.  }
)
1716difeq2d 3328 . . . . . . . . . 10  |-  ( ( ( i  =  I  /\  r  =  R )  /\  s  =  ( i mPwSer  r ) )  ->  ( _V  \  { ( 0g `  r ) } )  =  ( _V  \  {  .0.  } ) )
1817imaeq2d 5049 . . . . . . . . 9  |-  ( ( ( i  =  I  /\  r  =  R )  /\  s  =  ( i mPwSer  r ) )  ->  ( `' f " ( _V  \  { ( 0g `  r ) } ) )  =  ( `' f " ( _V 
\  {  .0.  }
) ) )
1918eleq1d 2382 . . . . . . . 8  |-  ( ( ( i  =  I  /\  r  =  R )  /\  s  =  ( i mPwSer  r ) )  ->  ( ( `' f " ( _V  \  { ( 0g
`  r ) } ) )  e.  Fin  <->  ( `' f " ( _V  \  {  .0.  }
) )  e.  Fin ) )
2011, 19rabeqbidv 2817 . . . . . . 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 2366 . . . . . 6  |-  ( ( ( i  =  I  /\  r  =  R )  /\  s  =  ( i mPwSer  r ) )  ->  { f  e.  ( Base `  s
)  |  ( `' f " ( _V 
\  { ( 0g
`  r ) } ) )  e.  Fin }  =  U )
238, 22oveq12d 5918 . . . . 5  |-  ( ( ( i  =  I  /\  r  =  R )  /\  s  =  ( i mPwSer  r ) )  ->  ( ss  {
f  e.  ( Base `  s )  |  ( `' f " ( _V  \  { ( 0g
`  r ) } ) )  e.  Fin } )  =  ( Ss  U ) )
243, 23csbied 3157 . . . 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 16149 . . . 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 5925 . . . 4  |-  ( Ss  U )  e.  _V
2724, 25, 26ovmpt2a 6020 . . 3  |-  ( ( I  e.  _V  /\  R  e.  _V )  ->  ( I mPoly  R )  =  ( Ss  U ) )
28 reldmmpl 16221 . . . . . 6  |-  Rel  dom mPoly
2928ovprc 5927 . . . . 5  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( I mPoly  R )  =  (/) )
30 ress0 13249 . . . . 5  |-  ( (/)s  U )  =  (/)
3129, 30syl6eqr 2366 . . . 4  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( I mPoly  R )  =  ( (/)s  U ) )
32 reldmpsr 16158 . . . . . . 7  |-  Rel  dom mPwSer
3332ovprc 5927 . . . . . 6  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( I mPwSer  R )  =  (/) )
347, 33syl5eq 2360 . . . . 5  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  S  =  (/) )
3534oveq1d 5915 . . . 4  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( Ss  U )  =  (
(/)s  U ) )
3631, 35eqtr4d 2351 . . 3  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( I mPoly  R )  =  ( Ss  U ) )
3727, 36pm2.61i 156 . 2  |-  ( I mPoly 
R )  =  ( Ss  U )
381, 37eqtri 2336 1  |-  P  =  ( Ss  U )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 358    = wceq 1633    e. wcel 1701   {crab 2581   _Vcvv 2822   [_csb 3115    \ cdif 3183   (/)c0 3489   {csn 3674   `'ccnv 4725   "cima 4729   ` cfv 5292  (class class class)co 5900   Fincfn 6906   Basecbs 13195   ↾s cress 13196   0gc0g 13449   mPwSer cmps 16136   mPoly cmpl 16138
This theorem is referenced by:  mplbas  16223  mplval2  16225
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-slot 13199  df-base 13200  df-ress 13202  df-psr 16147  df-mpl 16149
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