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Theorem mplval 17952
Description: Value of the set of multivariate polynomials. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 25-Jun-2019.)
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 finSupp  .0.  }
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 6320 . . . . . 6  |-  ( i mPwSer 
r )  e.  _V
32a1i 11 . . . . 5  |-  ( ( i  =  I  /\  r  =  R )  ->  ( i mPwSer  r )  e.  _V )
4 id 22 . . . . . . . 8  |-  ( s  =  ( i mPwSer  r
)  ->  s  =  ( i mPwSer  r )
)
5 oveq12 6304 . . . . . . . 8  |-  ( ( i  =  I  /\  r  =  R )  ->  ( i mPwSer  r )  =  ( I mPwSer  R
) )
64, 5sylan9eqr 2530 . . . . . . 7  |-  ( ( ( i  =  I  /\  r  =  R )  /\  s  =  ( i mPwSer  r ) )  ->  s  =  ( I mPwSer  R ) )
7 mplval.s . . . . . . 7  |-  S  =  ( I mPwSer  R )
86, 7syl6eqr 2526 . . . . . 6  |-  ( ( ( i  =  I  /\  r  =  R )  /\  s  =  ( i mPwSer  r ) )  ->  s  =  S )
98fveq2d 5876 . . . . . . . . 9  |-  ( ( ( i  =  I  /\  r  =  R )  /\  s  =  ( i mPwSer  r ) )  ->  ( Base `  s )  =  (
Base `  S )
)
10 mplval.b . . . . . . . . 9  |-  B  =  ( Base `  S
)
119, 10syl6eqr 2526 . . . . . . . 8  |-  ( ( ( i  =  I  /\  r  =  R )  /\  s  =  ( i mPwSer  r ) )  ->  ( Base `  s )  =  B )
12 simplr 754 . . . . . . . . . . 11  |-  ( ( ( i  =  I  /\  r  =  R )  /\  s  =  ( i mPwSer  r ) )  ->  r  =  R )
1312fveq2d 5876 . . . . . . . . . 10  |-  ( ( ( i  =  I  /\  r  =  R )  /\  s  =  ( i mPwSer  r ) )  ->  ( 0g `  r )  =  ( 0g `  R ) )
14 mplval.z . . . . . . . . . 10  |-  .0.  =  ( 0g `  R )
1513, 14syl6eqr 2526 . . . . . . . . 9  |-  ( ( ( i  =  I  /\  r  =  R )  /\  s  =  ( i mPwSer  r ) )  ->  ( 0g `  r )  =  .0.  )
1615breq2d 4465 . . . . . . . 8  |-  ( ( ( i  =  I  /\  r  =  R )  /\  s  =  ( i mPwSer  r ) )  ->  ( f finSupp  ( 0g `  r )  <-> 
f finSupp  .0.  ) )
1711, 16rabeqbidv 3113 . . . . . . 7  |-  ( ( ( i  =  I  /\  r  =  R )  /\  s  =  ( i mPwSer  r ) )  ->  { f  e.  ( Base `  s
)  |  f finSupp  ( 0g `  r ) }  =  { f  e.  B  |  f finSupp  .0.  } )
18 mplval.u . . . . . . 7  |-  U  =  { f  e.  B  |  f finSupp  .0.  }
1917, 18syl6eqr 2526 . . . . . 6  |-  ( ( ( i  =  I  /\  r  =  R )  /\  s  =  ( i mPwSer  r ) )  ->  { f  e.  ( Base `  s
)  |  f finSupp  ( 0g `  r ) }  =  U )
208, 19oveq12d 6313 . . . . 5  |-  ( ( ( i  =  I  /\  r  =  R )  /\  s  =  ( i mPwSer  r ) )  ->  ( ss  {
f  e.  ( Base `  s )  |  f finSupp 
( 0g `  r
) } )  =  ( Ss  U ) )
213, 20csbied 3467 . . . 4  |-  ( ( i  =  I  /\  r  =  R )  ->  [_ ( i mPwSer  r
)  /  s ]_ ( ss  { f  e.  (
Base `  s )  |  f finSupp  ( 0g `  r ) } )  =  ( Ss  U ) )
22 df-mpl 17875 . . . 4  |- mPoly  =  ( i  e.  _V , 
r  e.  _V  |->  [_ ( i mPwSer  r )  /  s ]_ (
ss 
{ f  e.  (
Base `  s )  |  f finSupp  ( 0g `  r ) } ) )
23 ovex 6320 . . . 4  |-  ( Ss  U )  e.  _V
2421, 22, 23ovmpt2a 6428 . . 3  |-  ( ( I  e.  _V  /\  R  e.  _V )  ->  ( I mPoly  R )  =  ( Ss  U ) )
25 reldmmpl 17951 . . . . . 6  |-  Rel  dom mPoly
2625ovprc 6322 . . . . 5  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( I mPoly  R )  =  (/) )
27 ress0 14565 . . . . 5  |-  ( (/)s  U )  =  (/)
2826, 27syl6eqr 2526 . . . 4  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( I mPoly  R )  =  ( (/)s  U ) )
29 reldmpsr 17878 . . . . . . 7  |-  Rel  dom mPwSer
3029ovprc 6322 . . . . . 6  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( I mPwSer  R )  =  (/) )
317, 30syl5eq 2520 . . . . 5  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  S  =  (/) )
3231oveq1d 6310 . . . 4  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( Ss  U )  =  (
(/)s  U ) )
3328, 32eqtr4d 2511 . . 3  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( I mPoly  R )  =  ( Ss  U ) )
3424, 33pm2.61i 164 . 2  |-  ( I mPoly 
R )  =  ( Ss  U )
351, 34eqtri 2496 1  |-  P  =  ( Ss  U )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 369    = wceq 1379    e. wcel 1767   {crab 2821   _Vcvv 3118   [_csb 3440   (/)c0 3790   class class class wbr 4453   ` cfv 5594  (class class class)co 6295   finSupp cfsupp 7841   Basecbs 14506   ↾s cress 14507   0gc0g 14711   mPwSer cmps 17868   mPoly cmpl 17870
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-slot 14510  df-base 14511  df-ress 14513  df-psr 17873  df-mpl 17875
This theorem is referenced by:  mplvalOLD  17953  mplbas  17954  mplval2  17958
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