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Theorem mdegfval 19448
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 5867 . . . . . . . . 9  |-  ( ( i  =  I  /\  r  =  R )  ->  ( i mPoly  r )  =  ( I mPoly  R
) )
3 mdegval.p . . . . . . . . 9  |-  P  =  ( I mPoly  R )
42, 3syl6eqr 2333 . . . . . . . 8  |-  ( ( i  =  I  /\  r  =  R )  ->  ( i mPoly  r )  =  P )
54fveq2d 5529 . . . . . . 7  |-  ( ( i  =  I  /\  r  =  R )  ->  ( Base `  (
i mPoly  r ) )  =  ( Base `  P
) )
6 mdegval.b . . . . . . 7  |-  B  =  ( Base `  P
)
75, 6syl6eqr 2333 . . . . . 6  |-  ( ( i  =  I  /\  r  =  R )  ->  ( Base `  (
i mPoly  r ) )  =  B )
8 fveq2 5525 . . . . . . . . . . . . . 14  |-  ( r  =  R  ->  ( 0g `  r )  =  ( 0g `  R
) )
9 mdegval.z . . . . . . . . . . . . . 14  |-  .0.  =  ( 0g `  R )
108, 9syl6eqr 2333 . . . . . . . . . . . . 13  |-  ( r  =  R  ->  ( 0g `  r )  =  .0.  )
1110sneqd 3653 . . . . . . . . . . . 12  |-  ( r  =  R  ->  { ( 0g `  r ) }  =  {  .0.  } )
1211difeq2d 3294 . . . . . . . . . . 11  |-  ( r  =  R  ->  ( _V  \  { ( 0g
`  r ) } )  =  ( _V 
\  {  .0.  }
) )
1312imaeq2d 5012 . . . . . . . . . 10  |-  ( r  =  R  ->  ( `' f " ( _V  \  { ( 0g
`  r ) } ) )  =  ( `' f " ( _V  \  {  .0.  }
) ) )
14 mpteq1 4100 . . . . . . . . . 10  |-  ( ( `' f " ( _V  \  { ( 0g
`  r ) } ) )  =  ( `' f " ( _V  \  {  .0.  }
) )  ->  (
h  e.  ( `' f " ( _V 
\  { ( 0g
`  r ) } ) )  |->  (fld  gsumg  h ) )  =  ( h  e.  ( `' f " ( _V  \  {  .0.  }
) )  |->  (fld  gsumg  h ) ) )
1513, 14syl 15 . . . . . . . . 9  |-  ( r  =  R  ->  (
h  e.  ( `' f " ( _V 
\  { ( 0g
`  r ) } ) )  |->  (fld  gsumg  h ) )  =  ( h  e.  ( `' f " ( _V  \  {  .0.  }
) )  |->  (fld  gsumg  h ) ) )
1615rneqd 4906 . . . . . . . 8  |-  ( r  =  R  ->  ran  ( h  e.  ( `' f " ( _V  \  { ( 0g
`  r ) } ) )  |->  (fld  gsumg  h ) )  =  ran  ( h  e.  ( `' f "
( _V  \  {  .0.  } ) )  |->  (fld  gsumg  h ) ) )
1716supeq1d 7199 . . . . . . 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* ,  <  ) )
1817adantl 452 . . . . . 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* ,  <  ) )
197, 18mpteq12dv 4098 . . . . 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* ,  <  ) ) )
20 df-mdeg 19441 . . . . 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* ,  <  ) ) )
21 fvex 5539 . . . . . . 7  |-  ( Base `  P )  e.  _V
226, 21eqeltri 2353 . . . . . 6  |-  B  e. 
_V
2322mptex 5746 . . . . 5  |-  ( f  e.  B  |->  sup ( ran  ( h  e.  ( `' f " ( _V  \  {  .0.  }
) )  |->  (fld  gsumg  h ) ) , 
RR* ,  <  ) )  e.  _V
2419, 20, 23ovmpt2a 5978 . . . 4  |-  ( ( I  e.  _V  /\  R  e.  _V )  ->  ( I mDeg  R )  =  ( f  e.  B  |->  sup ( ran  (
h  e.  ( `' f " ( _V 
\  {  .0.  }
) )  |->  (fld  gsumg  h ) ) , 
RR* ,  <  ) ) )
25 mdegval.h . . . . . . . . . 10  |-  H  =  ( h  e.  A  |->  (fld 
gsumg  h ) )
2625reseq1i 4951 . . . . . . . . 9  |-  ( H  |`  ( `' f "
( _V  \  {  .0.  } ) ) )  =  ( ( h  e.  A  |->  (fld  gsumg  h ) )  |`  ( `' f " ( _V  \  {  .0.  }
) ) )
27 cnvimass 5033 . . . . . . . . . . 11  |-  ( `' f " ( _V 
\  {  .0.  }
) )  C_  dom  f
28 eqid 2283 . . . . . . . . . . . . 13  |-  ( Base `  R )  =  (
Base `  R )
29 mdegval.a . . . . . . . . . . . . 13  |-  A  =  { m  e.  ( NN0  ^m  I )  |  ( `' m " NN )  e.  Fin }
30 simpr 447 . . . . . . . . . . . . 13  |-  ( ( ( I  e.  _V  /\  R  e.  _V )  /\  f  e.  B
)  ->  f  e.  B )
313, 28, 6, 29, 30mplelf 16178 . . . . . . . . . . . 12  |-  ( ( ( I  e.  _V  /\  R  e.  _V )  /\  f  e.  B
)  ->  f : A
--> ( Base `  R
) )
32 fdm 5393 . . . . . . . . . . . 12  |-  ( f : A --> ( Base `  R )  ->  dom  f  =  A )
3331, 32syl 15 . . . . . . . . . . 11  |-  ( ( ( I  e.  _V  /\  R  e.  _V )  /\  f  e.  B
)  ->  dom  f  =  A )
3427, 33syl5sseq 3226 . . . . . . . . . 10  |-  ( ( ( I  e.  _V  /\  R  e.  _V )  /\  f  e.  B
)  ->  ( `' f " ( _V  \  {  .0.  } ) ) 
C_  A )
35 resmpt 5000 . . . . . . . . . 10  |-  ( ( `' f " ( _V  \  {  .0.  }
) )  C_  A  ->  ( ( h  e.  A  |->  (fld 
gsumg  h ) )  |`  ( `' f " ( _V  \  {  .0.  }
) ) )  =  ( h  e.  ( `' f " ( _V  \  {  .0.  }
) )  |->  (fld  gsumg  h ) ) )
3634, 35syl 15 . . . . . . . . 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 ) ) )
3726, 36syl5req 2328 . . . . . . . 8  |-  ( ( ( I  e.  _V  /\  R  e.  _V )  /\  f  e.  B
)  ->  ( h  e.  ( `' f "
( _V  \  {  .0.  } ) )  |->  (fld  gsumg  h ) )  =  ( H  |`  ( `' f "
( _V  \  {  .0.  } ) ) ) )
3837rneqd 4906 . . . . . . 7  |-  ( ( ( I  e.  _V  /\  R  e.  _V )  /\  f  e.  B
)  ->  ran  ( h  e.  ( `' f
" ( _V  \  {  .0.  } ) ) 
|->  (fld 
gsumg  h ) )  =  ran  ( H  |`  ( `' f " ( _V  \  {  .0.  }
) ) ) )
39 df-ima 4702 . . . . . . 7  |-  ( H
" ( `' f
" ( _V  \  {  .0.  } ) ) )  =  ran  ( H  |`  ( `' f
" ( _V  \  {  .0.  } ) ) )
4038, 39syl6eqr 2333 . . . . . 6  |-  ( ( ( I  e.  _V  /\  R  e.  _V )  /\  f  e.  B
)  ->  ran  ( h  e.  ( `' f
" ( _V  \  {  .0.  } ) ) 
|->  (fld 
gsumg  h ) )  =  ( H " ( `' f " ( _V  \  {  .0.  }
) ) ) )
4140supeq1d 7199 . . . . 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* ,  <  ) )
4241mpteq2dva 4106 . . . 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* ,  <  ) ) )
4324, 42eqtrd 2315 . . 3  |-  ( ( I  e.  _V  /\  R  e.  _V )  ->  ( I mDeg  R )  =  ( f  e.  B  |->  sup ( ( H
" ( `' f
" ( _V  \  {  .0.  } ) ) ) ,  RR* ,  <  ) ) )
44 reldmmdeg 19443 . . . . . 6  |-  Rel  dom mDeg
4544ovprc 5885 . . . . 5  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( I mDeg  R )  =  (/) )
46 mpt0 5371 . . . . 5  |-  ( f  e.  (/)  |->  sup ( ( H
" ( `' f
" ( _V  \  {  .0.  } ) ) ) ,  RR* ,  <  ) )  =  (/)
4745, 46syl6eqr 2333 . . . 4  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( I mDeg  R )  =  ( f  e.  (/)  |->  sup ( ( H
" ( `' f
" ( _V  \  {  .0.  } ) ) ) ,  RR* ,  <  ) ) )
48 reldmmpl 16172 . . . . . . . . 9  |-  Rel  dom mPoly
4948ovprc 5885 . . . . . . . 8  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( I mPoly  R )  =  (/) )
503, 49syl5eq 2327 . . . . . . 7  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  P  =  (/) )
5150fveq2d 5529 . . . . . 6  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( Base `  P
)  =  ( Base `  (/) ) )
52 base0 13185 . . . . . 6  |-  (/)  =  (
Base `  (/) )
5351, 6, 523eqtr4g 2340 . . . . 5  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  B  =  (/) )
54 mpteq1 4100 . . . . 5  |-  ( B  =  (/)  ->  ( f  e.  B  |->  sup (
( H " ( `' f " ( _V  \  {  .0.  }
) ) ) , 
RR* ,  <  ) )  =  ( f  e.  (/)  |->  sup ( ( H
" ( `' f
" ( _V  \  {  .0.  } ) ) ) ,  RR* ,  <  ) ) )
5553, 54syl 15 . . . 4  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( f  e.  B  |->  sup ( ( H
" ( `' f
" ( _V  \  {  .0.  } ) ) ) ,  RR* ,  <  ) )  =  ( f  e.  (/)  |->  sup ( ( H
" ( `' f
" ( _V  \  {  .0.  } ) ) ) ,  RR* ,  <  ) ) )
5647, 55eqtr4d 2318 . . 3  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( I mDeg  R )  =  ( f  e.  B  |->  sup ( ( H
" ( `' f
" ( _V  \  {  .0.  } ) ) ) ,  RR* ,  <  ) ) )
5743, 56pm2.61i 156 . 2  |-  ( I mDeg 
R )  =  ( f  e.  B  |->  sup ( ( H "
( `' f "
( _V  \  {  .0.  } ) ) ) ,  RR* ,  <  )
)
581, 57eqtri 2303 1  |-  D  =  ( f  e.  B  |->  sup ( ( H
" ( `' f
" ( _V  \  {  .0.  } ) ) ) ,  RR* ,  <  ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 358    = wceq 1623    e. wcel 1684   {crab 2547   _Vcvv 2788    \ cdif 3149    C_ wss 3152   (/)c0 3455   {csn 3640    e. cmpt 4077   `'ccnv 4688   dom cdm 4689   ran crn 4690    |` cres 4691   "cima 4692   -->wf 5251   ` cfv 5255  (class class class)co 5858    ^m cmap 6772   Fincfn 6863   supcsup 7193   RR*cxr 8866    < clt 8867   NNcn 9746   NN0cn0 9965   Basecbs 13148   0gc0g 13400    gsumg cgsu 13401   mPoly cmpl 16089  ℂfldccnfld 16377   mDeg cmdg 19439
This theorem is referenced by:  mdegval  19449  mdegxrf  19454  mdegpropd  19470
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-tset 13227  df-psr 16098  df-mpl 16100  df-mdeg 19441
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