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Theorem mdegfval 19854
Description: Value of the multivariate degree function. (Contributed by Stefan O'Rear, 19-Mar-2015.)
Hypotheses
Ref Expression
mdegval.d  |-  D  =  ( I mDeg  R )
mdegval.p  |-  P  =  ( I mPoly  R )
mdegval.b  |-  B  =  ( Base `  P
)
mdegval.z  |-  .0.  =  ( 0g `  R )
mdegval.a  |-  A  =  { m  e.  ( NN0  ^m  I )  |  ( `' m " NN )  e.  Fin }
mdegval.h  |-  H  =  ( h  e.  A  |->  (fld 
gsumg  h ) )
Assertion
Ref Expression
mdegfval  |-  D  =  ( f  e.  B  |->  sup ( ( H
" ( `' f
" ( _V  \  {  .0.  } ) ) ) ,  RR* ,  <  ) )
Distinct variable groups:    A, h    B, f    f, I    m, I    R, f    .0. , h    f, h
Allowed substitution hints:    A( f, m)    B( h, m)    D( f, h, m)    P( f, h, m)    R( h, m)    H( f, h, m)    I( h)    .0. ( f, m)

Proof of Theorem mdegfval
Dummy variables  i 
r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mdegval.d . 2  |-  D  =  ( I mDeg  R )
2 oveq12 6031 . . . . . . . . 9  |-  ( ( i  =  I  /\  r  =  R )  ->  ( i mPoly  r )  =  ( I mPoly  R
) )
3 mdegval.p . . . . . . . . 9  |-  P  =  ( I mPoly  R )
42, 3syl6eqr 2439 . . . . . . . 8  |-  ( ( i  =  I  /\  r  =  R )  ->  ( i mPoly  r )  =  P )
54fveq2d 5674 . . . . . . 7  |-  ( ( i  =  I  /\  r  =  R )  ->  ( Base `  (
i mPoly  r ) )  =  ( Base `  P
) )
6 mdegval.b . . . . . . 7  |-  B  =  ( Base `  P
)
75, 6syl6eqr 2439 . . . . . 6  |-  ( ( i  =  I  /\  r  =  R )  ->  ( Base `  (
i mPoly  r ) )  =  B )
8 fveq2 5670 . . . . . . . . . . . . . 14  |-  ( r  =  R  ->  ( 0g `  r )  =  ( 0g `  R
) )
9 mdegval.z . . . . . . . . . . . . . 14  |-  .0.  =  ( 0g `  R )
108, 9syl6eqr 2439 . . . . . . . . . . . . 13  |-  ( r  =  R  ->  ( 0g `  r )  =  .0.  )
1110sneqd 3772 . . . . . . . . . . . 12  |-  ( r  =  R  ->  { ( 0g `  r ) }  =  {  .0.  } )
1211difeq2d 3410 . . . . . . . . . . 11  |-  ( r  =  R  ->  ( _V  \  { ( 0g
`  r ) } )  =  ( _V 
\  {  .0.  }
) )
1312imaeq2d 5145 . . . . . . . . . 10  |-  ( r  =  R  ->  ( `' f " ( _V  \  { ( 0g
`  r ) } ) )  =  ( `' f " ( _V  \  {  .0.  }
) ) )
1413mpteq1d 4233 . . . . . . . . 9  |-  ( r  =  R  ->  (
h  e.  ( `' f " ( _V 
\  { ( 0g
`  r ) } ) )  |->  (fld  gsumg  h ) )  =  ( h  e.  ( `' f " ( _V  \  {  .0.  }
) )  |->  (fld  gsumg  h ) ) )
1514rneqd 5039 . . . . . . . 8  |-  ( r  =  R  ->  ran  ( h  e.  ( `' f " ( _V  \  { ( 0g
`  r ) } ) )  |->  (fld  gsumg  h ) )  =  ran  ( h  e.  ( `' f "
( _V  \  {  .0.  } ) )  |->  (fld  gsumg  h ) ) )
1615supeq1d 7388 . . . . . . 7  |-  ( r  =  R  ->  sup ( ran  ( h  e.  ( `' f "
( _V  \  {
( 0g `  r
) } ) ) 
|->  (fld 
gsumg  h ) ) , 
RR* ,  <  )  =  sup ( ran  (
h  e.  ( `' f " ( _V 
\  {  .0.  }
) )  |->  (fld  gsumg  h ) ) , 
RR* ,  <  ) )
1716adantl 453 . . . . . 6  |-  ( ( i  =  I  /\  r  =  R )  ->  sup ( ran  (
h  e.  ( `' f " ( _V 
\  { ( 0g
`  r ) } ) )  |->  (fld  gsumg  h ) ) , 
RR* ,  <  )  =  sup ( ran  (
h  e.  ( `' f " ( _V 
\  {  .0.  }
) )  |->  (fld  gsumg  h ) ) , 
RR* ,  <  ) )
187, 17mpteq12dv 4230 . . . . 5  |-  ( ( i  =  I  /\  r  =  R )  ->  ( f  e.  (
Base `  ( i mPoly  r ) )  |->  sup ( ran  ( h  e.  ( `' f
" ( _V  \  { ( 0g `  r ) } ) )  |->  (fld 
gsumg  h ) ) , 
RR* ,  <  ) )  =  ( f  e.  B  |->  sup ( ran  (
h  e.  ( `' f " ( _V 
\  {  .0.  }
) )  |->  (fld  gsumg  h ) ) , 
RR* ,  <  ) ) )
19 df-mdeg 19847 . . . . 5  |- mDeg  =  ( i  e.  _V , 
r  e.  _V  |->  ( f  e.  ( Base `  ( i mPoly  r ) )  |->  sup ( ran  (
h  e.  ( `' f " ( _V 
\  { ( 0g
`  r ) } ) )  |->  (fld  gsumg  h ) ) , 
RR* ,  <  ) ) )
20 fvex 5684 . . . . . . 7  |-  ( Base `  P )  e.  _V
216, 20eqeltri 2459 . . . . . 6  |-  B  e. 
_V
2221mptex 5907 . . . . 5  |-  ( f  e.  B  |->  sup ( ran  ( h  e.  ( `' f " ( _V  \  {  .0.  }
) )  |->  (fld  gsumg  h ) ) , 
RR* ,  <  ) )  e.  _V
2318, 19, 22ovmpt2a 6145 . . . 4  |-  ( ( I  e.  _V  /\  R  e.  _V )  ->  ( I mDeg  R )  =  ( f  e.  B  |->  sup ( ran  (
h  e.  ( `' f " ( _V 
\  {  .0.  }
) )  |->  (fld  gsumg  h ) ) , 
RR* ,  <  ) ) )
24 mdegval.h . . . . . . . . . 10  |-  H  =  ( h  e.  A  |->  (fld 
gsumg  h ) )
2524reseq1i 5084 . . . . . . . . 9  |-  ( H  |`  ( `' f "
( _V  \  {  .0.  } ) ) )  =  ( ( h  e.  A  |->  (fld  gsumg  h ) )  |`  ( `' f " ( _V  \  {  .0.  }
) ) )
26 cnvimass 5166 . . . . . . . . . . 11  |-  ( `' f " ( _V 
\  {  .0.  }
) )  C_  dom  f
27 eqid 2389 . . . . . . . . . . . . 13  |-  ( Base `  R )  =  (
Base `  R )
28 mdegval.a . . . . . . . . . . . . 13  |-  A  =  { m  e.  ( NN0  ^m  I )  |  ( `' m " NN )  e.  Fin }
29 simpr 448 . . . . . . . . . . . . 13  |-  ( ( ( I  e.  _V  /\  R  e.  _V )  /\  f  e.  B
)  ->  f  e.  B )
303, 27, 6, 28, 29mplelf 16426 . . . . . . . . . . . 12  |-  ( ( ( I  e.  _V  /\  R  e.  _V )  /\  f  e.  B
)  ->  f : A
--> ( Base `  R
) )
31 fdm 5537 . . . . . . . . . . . 12  |-  ( f : A --> ( Base `  R )  ->  dom  f  =  A )
3230, 31syl 16 . . . . . . . . . . 11  |-  ( ( ( I  e.  _V  /\  R  e.  _V )  /\  f  e.  B
)  ->  dom  f  =  A )
3326, 32syl5sseq 3341 . . . . . . . . . 10  |-  ( ( ( I  e.  _V  /\  R  e.  _V )  /\  f  e.  B
)  ->  ( `' f " ( _V  \  {  .0.  } ) ) 
C_  A )
34 resmpt 5133 . . . . . . . . . 10  |-  ( ( `' f " ( _V  \  {  .0.  }
) )  C_  A  ->  ( ( h  e.  A  |->  (fld 
gsumg  h ) )  |`  ( `' f " ( _V  \  {  .0.  }
) ) )  =  ( h  e.  ( `' f " ( _V  \  {  .0.  }
) )  |->  (fld  gsumg  h ) ) )
3533, 34syl 16 . . . . . . . . 9  |-  ( ( ( I  e.  _V  /\  R  e.  _V )  /\  f  e.  B
)  ->  ( (
h  e.  A  |->  (fld  gsumg  h ) )  |`  ( `' f " ( _V  \  {  .0.  } ) ) )  =  ( h  e.  ( `' f
" ( _V  \  {  .0.  } ) ) 
|->  (fld 
gsumg  h ) ) )
3625, 35syl5req 2434 . . . . . . . 8  |-  ( ( ( I  e.  _V  /\  R  e.  _V )  /\  f  e.  B
)  ->  ( h  e.  ( `' f "
( _V  \  {  .0.  } ) )  |->  (fld  gsumg  h ) )  =  ( H  |`  ( `' f "
( _V  \  {  .0.  } ) ) ) )
3736rneqd 5039 . . . . . . 7  |-  ( ( ( I  e.  _V  /\  R  e.  _V )  /\  f  e.  B
)  ->  ran  ( h  e.  ( `' f
" ( _V  \  {  .0.  } ) ) 
|->  (fld 
gsumg  h ) )  =  ran  ( H  |`  ( `' f " ( _V  \  {  .0.  }
) ) ) )
38 df-ima 4833 . . . . . . 7  |-  ( H
" ( `' f
" ( _V  \  {  .0.  } ) ) )  =  ran  ( H  |`  ( `' f
" ( _V  \  {  .0.  } ) ) )
3937, 38syl6eqr 2439 . . . . . 6  |-  ( ( ( I  e.  _V  /\  R  e.  _V )  /\  f  e.  B
)  ->  ran  ( h  e.  ( `' f
" ( _V  \  {  .0.  } ) ) 
|->  (fld 
gsumg  h ) )  =  ( H " ( `' f " ( _V  \  {  .0.  }
) ) ) )
4039supeq1d 7388 . . . . 5  |-  ( ( ( I  e.  _V  /\  R  e.  _V )  /\  f  e.  B
)  ->  sup ( ran  ( h  e.  ( `' f " ( _V  \  {  .0.  }
) )  |->  (fld  gsumg  h ) ) , 
RR* ,  <  )  =  sup ( ( H
" ( `' f
" ( _V  \  {  .0.  } ) ) ) ,  RR* ,  <  ) )
4140mpteq2dva 4238 . . . 4  |-  ( ( I  e.  _V  /\  R  e.  _V )  ->  ( f  e.  B  |->  sup ( ran  (
h  e.  ( `' f " ( _V 
\  {  .0.  }
) )  |->  (fld  gsumg  h ) ) , 
RR* ,  <  ) )  =  ( f  e.  B  |->  sup ( ( H
" ( `' f
" ( _V  \  {  .0.  } ) ) ) ,  RR* ,  <  ) ) )
4223, 41eqtrd 2421 . . 3  |-  ( ( I  e.  _V  /\  R  e.  _V )  ->  ( I mDeg  R )  =  ( f  e.  B  |->  sup ( ( H
" ( `' f
" ( _V  \  {  .0.  } ) ) ) ,  RR* ,  <  ) ) )
43 reldmmdeg 19849 . . . . . 6  |-  Rel  dom mDeg
4443ovprc 6049 . . . . 5  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( I mDeg  R )  =  (/) )
45 mpt0 5514 . . . . 5  |-  ( f  e.  (/)  |->  sup ( ( H
" ( `' f
" ( _V  \  {  .0.  } ) ) ) ,  RR* ,  <  ) )  =  (/)
4644, 45syl6eqr 2439 . . . 4  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( I mDeg  R )  =  ( f  e.  (/)  |->  sup ( ( H
" ( `' f
" ( _V  \  {  .0.  } ) ) ) ,  RR* ,  <  ) ) )
47 reldmmpl 16420 . . . . . . . . 9  |-  Rel  dom mPoly
4847ovprc 6049 . . . . . . . 8  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( I mPoly  R )  =  (/) )
493, 48syl5eq 2433 . . . . . . 7  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  P  =  (/) )
5049fveq2d 5674 . . . . . 6  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( Base `  P
)  =  ( Base `  (/) ) )
51 base0 13435 . . . . . 6  |-  (/)  =  (
Base `  (/) )
5250, 6, 513eqtr4g 2446 . . . . 5  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  B  =  (/) )
5352mpteq1d 4233 . . . 4  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( f  e.  B  |->  sup ( ( H
" ( `' f
" ( _V  \  {  .0.  } ) ) ) ,  RR* ,  <  ) )  =  ( f  e.  (/)  |->  sup ( ( H
" ( `' f
" ( _V  \  {  .0.  } ) ) ) ,  RR* ,  <  ) ) )
5446, 53eqtr4d 2424 . . 3  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( I mDeg  R )  =  ( f  e.  B  |->  sup ( ( H
" ( `' f
" ( _V  \  {  .0.  } ) ) ) ,  RR* ,  <  ) ) )
5542, 54pm2.61i 158 . 2  |-  ( I mDeg 
R )  =  ( f  e.  B  |->  sup ( ( H "
( `' f "
( _V  \  {  .0.  } ) ) ) ,  RR* ,  <  )
)
561, 55eqtri 2409 1  |-  D  =  ( f  e.  B  |->  sup ( ( H
" ( `' f
" ( _V  \  {  .0.  } ) ) ) ,  RR* ,  <  ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 359    = wceq 1649    e. wcel 1717   {crab 2655   _Vcvv 2901    \ cdif 3262    C_ wss 3265   (/)c0 3573   {csn 3759    e. cmpt 4209   `'ccnv 4819   dom cdm 4820   ran crn 4821    |` cres 4822   "cima 4823   -->wf 5392   ` cfv 5396  (class class class)co 6022    ^m cmap 6956   Fincfn 7047   supcsup 7382   RR*cxr 9054    < clt 9055   NNcn 9934   NN0cn0 10155   Basecbs 13398   0gc0g 13652    gsumg cgsu 13653   mPoly cmpl 16337  ℂfldccnfld 16628   mDeg cmdg 19845
This theorem is referenced by:  mdegval  19855  mdegxrf  19860  mdegpropd  19876
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-int 3995  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-of 6246  df-1st 6290  df-2nd 6291  df-riota 6487  df-recs 6571  df-rdg 6606  df-1o 6662  df-oadd 6666  df-er 6843  df-map 6958  df-en 7048  df-dom 7049  df-sdom 7050  df-fin 7051  df-sup 7383  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-nn 9935  df-2 9992  df-3 9993  df-4 9994  df-5 9995  df-6 9996  df-7 9997  df-8 9998  df-9 9999  df-n0 10156  df-z 10217  df-uz 10423  df-fz 10978  df-struct 13400  df-ndx 13401  df-slot 13402  df-base 13403  df-sets 13404  df-ress 13405  df-plusg 13471  df-mulr 13472  df-sca 13474  df-vsca 13475  df-tset 13477  df-psr 16346  df-mpl 16348  df-mdeg 19847
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