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Theorem mdegfval 19975
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 6082 . . . . . . . . 9  |-  ( ( i  =  I  /\  r  =  R )  ->  ( i mPoly  r )  =  ( I mPoly  R
) )
3 mdegval.p . . . . . . . . 9  |-  P  =  ( I mPoly  R )
42, 3syl6eqr 2485 . . . . . . . 8  |-  ( ( i  =  I  /\  r  =  R )  ->  ( i mPoly  r )  =  P )
54fveq2d 5724 . . . . . . 7  |-  ( ( i  =  I  /\  r  =  R )  ->  ( Base `  (
i mPoly  r ) )  =  ( Base `  P
) )
6 mdegval.b . . . . . . 7  |-  B  =  ( Base `  P
)
75, 6syl6eqr 2485 . . . . . 6  |-  ( ( i  =  I  /\  r  =  R )  ->  ( Base `  (
i mPoly  r ) )  =  B )
8 fveq2 5720 . . . . . . . . . . . . . 14  |-  ( r  =  R  ->  ( 0g `  r )  =  ( 0g `  R
) )
9 mdegval.z . . . . . . . . . . . . . 14  |-  .0.  =  ( 0g `  R )
108, 9syl6eqr 2485 . . . . . . . . . . . . 13  |-  ( r  =  R  ->  ( 0g `  r )  =  .0.  )
1110sneqd 3819 . . . . . . . . . . . 12  |-  ( r  =  R  ->  { ( 0g `  r ) }  =  {  .0.  } )
1211difeq2d 3457 . . . . . . . . . . 11  |-  ( r  =  R  ->  ( _V  \  { ( 0g
`  r ) } )  =  ( _V 
\  {  .0.  }
) )
1312imaeq2d 5195 . . . . . . . . . 10  |-  ( r  =  R  ->  ( `' f " ( _V  \  { ( 0g
`  r ) } ) )  =  ( `' f " ( _V  \  {  .0.  }
) ) )
1413mpteq1d 4282 . . . . . . . . 9  |-  ( r  =  R  ->  (
h  e.  ( `' f " ( _V 
\  { ( 0g
`  r ) } ) )  |->  (fld  gsumg  h ) )  =  ( h  e.  ( `' f " ( _V  \  {  .0.  }
) )  |->  (fld  gsumg  h ) ) )
1514rneqd 5089 . . . . . . . 8  |-  ( r  =  R  ->  ran  ( h  e.  ( `' f " ( _V  \  { ( 0g
`  r ) } ) )  |->  (fld  gsumg  h ) )  =  ran  ( h  e.  ( `' f "
( _V  \  {  .0.  } ) )  |->  (fld  gsumg  h ) ) )
1615supeq1d 7443 . . . . . . 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 4279 . . . . 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 19968 . . . . 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 5734 . . . . . . 7  |-  ( Base `  P )  e.  _V
216, 20eqeltri 2505 . . . . . 6  |-  B  e. 
_V
2221mptex 5958 . . . . 5  |-  ( f  e.  B  |->  sup ( ran  ( h  e.  ( `' f " ( _V  \  {  .0.  }
) )  |->  (fld  gsumg  h ) ) , 
RR* ,  <  ) )  e.  _V
2318, 19, 22ovmpt2a 6196 . . . 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 5134 . . . . . . . . 9  |-  ( H  |`  ( `' f "
( _V  \  {  .0.  } ) ) )  =  ( ( h  e.  A  |->  (fld  gsumg  h ) )  |`  ( `' f " ( _V  \  {  .0.  }
) ) )
26 cnvimass 5216 . . . . . . . . . . 11  |-  ( `' f " ( _V 
\  {  .0.  }
) )  C_  dom  f
27 eqid 2435 . . . . . . . . . . . . 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 16487 . . . . . . . . . . . 12  |-  ( ( ( I  e.  _V  /\  R  e.  _V )  /\  f  e.  B
)  ->  f : A
--> ( Base `  R
) )
31 fdm 5587 . . . . . . . . . . . 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 3388 . . . . . . . . . 10  |-  ( ( ( I  e.  _V  /\  R  e.  _V )  /\  f  e.  B
)  ->  ( `' f " ( _V  \  {  .0.  } ) ) 
C_  A )
34 resmpt 5183 . . . . . . . . . 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 2480 . . . . . . . 8  |-  ( ( ( I  e.  _V  /\  R  e.  _V )  /\  f  e.  B
)  ->  ( h  e.  ( `' f "
( _V  \  {  .0.  } ) )  |->  (fld  gsumg  h ) )  =  ( H  |`  ( `' f "
( _V  \  {  .0.  } ) ) ) )
3736rneqd 5089 . . . . . . 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 4883 . . . . . . 7  |-  ( H
" ( `' f
" ( _V  \  {  .0.  } ) ) )  =  ran  ( H  |`  ( `' f
" ( _V  \  {  .0.  } ) ) )
3937, 38syl6eqr 2485 . . . . . 6  |-  ( ( ( I  e.  _V  /\  R  e.  _V )  /\  f  e.  B
)  ->  ran  ( h  e.  ( `' f
" ( _V  \  {  .0.  } ) ) 
|->  (fld 
gsumg  h ) )  =  ( H " ( `' f " ( _V  \  {  .0.  }
) ) ) )
4039supeq1d 7443 . . . . 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 4287 . . . 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 2467 . . 3  |-  ( ( I  e.  _V  /\  R  e.  _V )  ->  ( I mDeg  R )  =  ( f  e.  B  |->  sup ( ( H
" ( `' f
" ( _V  \  {  .0.  } ) ) ) ,  RR* ,  <  ) ) )
43 reldmmdeg 19970 . . . . . 6  |-  Rel  dom mDeg
4443ovprc 6100 . . . . 5  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( I mDeg  R )  =  (/) )
45 mpt0 5564 . . . . 5  |-  ( f  e.  (/)  |->  sup ( ( H
" ( `' f
" ( _V  \  {  .0.  } ) ) ) ,  RR* ,  <  ) )  =  (/)
4644, 45syl6eqr 2485 . . . 4  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( I mDeg  R )  =  ( f  e.  (/)  |->  sup ( ( H
" ( `' f
" ( _V  \  {  .0.  } ) ) ) ,  RR* ,  <  ) ) )
47 reldmmpl 16481 . . . . . . . . 9  |-  Rel  dom mPoly
4847ovprc 6100 . . . . . . . 8  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( I mPoly  R )  =  (/) )
493, 48syl5eq 2479 . . . . . . 7  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  P  =  (/) )
5049fveq2d 5724 . . . . . 6  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( Base `  P
)  =  ( Base `  (/) ) )
51 base0 13496 . . . . . 6  |-  (/)  =  (
Base `  (/) )
5250, 6, 513eqtr4g 2492 . . . . 5  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  B  =  (/) )
5352mpteq1d 4282 . . . 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 2470 . . 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 2455 1  |-  D  =  ( f  e.  B  |->  sup ( ( H
" ( `' f
" ( _V  \  {  .0.  } ) ) ) ,  RR* ,  <  ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 359    = wceq 1652    e. wcel 1725   {crab 2701   _Vcvv 2948    \ cdif 3309    C_ wss 3312   (/)c0 3620   {csn 3806    e. cmpt 4258   `'ccnv 4869   dom cdm 4870   ran crn 4871    |` cres 4872   "cima 4873   -->wf 5442   ` cfv 5446  (class class class)co 6073    ^m cmap 7010   Fincfn 7101   supcsup 7437   RR*cxr 9109    < clt 9110   NNcn 9990   NN0cn0 10211   Basecbs 13459   0gc0g 13713    gsumg cgsu 13714   mPoly cmpl 16398  ℂfldccnfld 16693   mDeg cmdg 19966
This theorem is referenced by:  mdegval  19976  mdegxrf  19981  mdegpropd  19997
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9036  ax-resscn 9037  ax-1cn 9038  ax-icn 9039  ax-addcl 9040  ax-addrcl 9041  ax-mulcl 9042  ax-mulrcl 9043  ax-mulcom 9044  ax-addass 9045  ax-mulass 9046  ax-distr 9047  ax-i2m1 9048  ax-1ne0 9049  ax-1rid 9050  ax-rnegex 9051  ax-rrecex 9052  ax-cnre 9053  ax-pre-lttri 9054  ax-pre-lttrn 9055  ax-pre-ltadd 9056  ax-pre-mulgt0 9057
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-pnf 9112  df-mnf 9113  df-xr 9114  df-ltxr 9115  df-le 9116  df-sub 9283  df-neg 9284  df-nn 9991  df-2 10048  df-3 10049  df-4 10050  df-5 10051  df-6 10052  df-7 10053  df-8 10054  df-9 10055  df-n0 10212  df-z 10273  df-uz 10479  df-fz 11034  df-struct 13461  df-ndx 13462  df-slot 13463  df-base 13464  df-sets 13465  df-ress 13466  df-plusg 13532  df-mulr 13533  df-sca 13535  df-vsca 13536  df-tset 13538  df-psr 16407  df-mpl 16409  df-mdeg 19968
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