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Theorem mplval 18700
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 6342 . . . . . 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 6323 . . . . . . . 8  |-  ( ( i  =  I  /\  r  =  R )  ->  ( i mPwSer  r )  =  ( I mPwSer  R
) )
64, 5sylan9eqr 2517 . . . . . . 7  |-  ( ( ( i  =  I  /\  r  =  R )  /\  s  =  ( i mPwSer  r ) )  ->  s  =  ( I mPwSer  R ) )
7 mplval.s . . . . . . 7  |-  S  =  ( I mPwSer  R )
86, 7syl6eqr 2513 . . . . . 6  |-  ( ( ( i  =  I  /\  r  =  R )  /\  s  =  ( i mPwSer  r ) )  ->  s  =  S )
98fveq2d 5891 . . . . . . . . 9  |-  ( ( ( i  =  I  /\  r  =  R )  /\  s  =  ( i mPwSer  r ) )  ->  ( Base `  s )  =  (
Base `  S )
)
10 mplval.b . . . . . . . . 9  |-  B  =  ( Base `  S
)
119, 10syl6eqr 2513 . . . . . . . 8  |-  ( ( ( i  =  I  /\  r  =  R )  /\  s  =  ( i mPwSer  r ) )  ->  ( Base `  s )  =  B )
12 simplr 767 . . . . . . . . . . 11  |-  ( ( ( i  =  I  /\  r  =  R )  /\  s  =  ( i mPwSer  r ) )  ->  r  =  R )
1312fveq2d 5891 . . . . . . . . . 10  |-  ( ( ( i  =  I  /\  r  =  R )  /\  s  =  ( i mPwSer  r ) )  ->  ( 0g `  r )  =  ( 0g `  R ) )
14 mplval.z . . . . . . . . . 10  |-  .0.  =  ( 0g `  R )
1513, 14syl6eqr 2513 . . . . . . . . 9  |-  ( ( ( i  =  I  /\  r  =  R )  /\  s  =  ( i mPwSer  r ) )  ->  ( 0g `  r )  =  .0.  )
1615breq2d 4427 . . . . . . . 8  |-  ( ( ( i  =  I  /\  r  =  R )  /\  s  =  ( i mPwSer  r ) )  ->  ( f finSupp  ( 0g `  r )  <-> 
f finSupp  .0.  ) )
1711, 16rabeqbidv 3051 . . . . . . 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 2513 . . . . . 6  |-  ( ( ( i  =  I  /\  r  =  R )  /\  s  =  ( i mPwSer  r ) )  ->  { f  e.  ( Base `  s
)  |  f finSupp  ( 0g `  r ) }  =  U )
208, 19oveq12d 6332 . . . . 5  |-  ( ( ( i  =  I  /\  r  =  R )  /\  s  =  ( i mPwSer  r ) )  ->  ( ss  {
f  e.  ( Base `  s )  |  f finSupp 
( 0g `  r
) } )  =  ( Ss  U ) )
213, 20csbied 3401 . . . 4  |-  ( ( i  =  I  /\  r  =  R )  ->  [_ ( i mPwSer  r
)  /  s ]_ ( ss  { f  e.  (
Base `  s )  |  f finSupp  ( 0g `  r ) } )  =  ( Ss  U ) )
22 df-mpl 18630 . . . 4  |- mPoly  =  ( i  e.  _V , 
r  e.  _V  |->  [_ ( i mPwSer  r )  /  s ]_ (
ss 
{ f  e.  (
Base `  s )  |  f finSupp  ( 0g `  r ) } ) )
23 ovex 6342 . . . 4  |-  ( Ss  U )  e.  _V
2421, 22, 23ovmpt2a 6453 . . 3  |-  ( ( I  e.  _V  /\  R  e.  _V )  ->  ( I mPoly  R )  =  ( Ss  U ) )
25 reldmmpl 18699 . . . . . 6  |-  Rel  dom mPoly
2625ovprc 6344 . . . . 5  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( I mPoly  R )  =  (/) )
27 ress0 15231 . . . . 5  |-  ( (/)s  U )  =  (/)
2826, 27syl6eqr 2513 . . . 4  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( I mPoly  R )  =  ( (/)s  U ) )
29 reldmpsr 18633 . . . . . . 7  |-  Rel  dom mPwSer
3029ovprc 6344 . . . . . 6  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( I mPwSer  R )  =  (/) )
317, 30syl5eq 2507 . . . . 5  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  S  =  (/) )
3231oveq1d 6329 . . . 4  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( Ss  U )  =  (
(/)s  U ) )
3328, 32eqtr4d 2498 . . 3  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( I mPoly  R )  =  ( Ss  U ) )
3424, 33pm2.61i 169 . 2  |-  ( I mPoly 
R )  =  ( Ss  U )
351, 34eqtri 2483 1  |-  P  =  ( Ss  U )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 375    = wceq 1454    e. wcel 1897   {crab 2752   _Vcvv 3056   [_csb 3374   (/)c0 3742   class class class wbr 4415   ` cfv 5600  (class class class)co 6314   finSupp cfsupp 7908   Basecbs 15169   ↾s cress 15170   0gc0g 15386   mPwSer cmps 18623   mPoly cmpl 18625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-nul 3743  df-if 3893  df-sn 3980  df-pr 3982  df-op 3986  df-uni 4212  df-br 4416  df-opab 4475  df-mpt 4476  df-id 4767  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-iota 5564  df-fun 5602  df-fv 5608  df-ov 6317  df-oprab 6318  df-mpt2 6319  df-slot 15173  df-base 15174  df-ress 15176  df-psr 18628  df-mpl 18630
This theorem is referenced by:  mplbas  18701  mplval2  18703
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