MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mdegfval Unicode version

Theorem mdegfval 19464
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 5883 . . . . . . . . 9  |-  ( ( i  =  I  /\  r  =  R )  ->  ( i mPoly  r )  =  ( I mPoly  R
) )
3 mdegval.p . . . . . . . . 9  |-  P  =  ( I mPoly  R )
42, 3syl6eqr 2346 . . . . . . . 8  |-  ( ( i  =  I  /\  r  =  R )  ->  ( i mPoly  r )  =  P )
54fveq2d 5545 . . . . . . 7  |-  ( ( i  =  I  /\  r  =  R )  ->  ( Base `  (
i mPoly  r ) )  =  ( Base `  P
) )
6 mdegval.b . . . . . . 7  |-  B  =  ( Base `  P
)
75, 6syl6eqr 2346 . . . . . 6  |-  ( ( i  =  I  /\  r  =  R )  ->  ( Base `  (
i mPoly  r ) )  =  B )
8 fveq2 5541 . . . . . . . . . . . . . 14  |-  ( r  =  R  ->  ( 0g `  r )  =  ( 0g `  R
) )
9 mdegval.z . . . . . . . . . . . . . 14  |-  .0.  =  ( 0g `  R )
108, 9syl6eqr 2346 . . . . . . . . . . . . 13  |-  ( r  =  R  ->  ( 0g `  r )  =  .0.  )
1110sneqd 3666 . . . . . . . . . . . 12  |-  ( r  =  R  ->  { ( 0g `  r ) }  =  {  .0.  } )
1211difeq2d 3307 . . . . . . . . . . 11  |-  ( r  =  R  ->  ( _V  \  { ( 0g
`  r ) } )  =  ( _V 
\  {  .0.  }
) )
1312imaeq2d 5028 . . . . . . . . . 10  |-  ( r  =  R  ->  ( `' f " ( _V  \  { ( 0g
`  r ) } ) )  =  ( `' f " ( _V  \  {  .0.  }
) ) )
14 mpteq1 4116 . . . . . . . . . 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 4922 . . . . . . . 8  |-  ( r  =  R  ->  ran  ( h  e.  ( `' f " ( _V  \  { ( 0g
`  r ) } ) )  |->  (fld  gsumg  h ) )  =  ran  ( h  e.  ( `' f "
( _V  \  {  .0.  } ) )  |->  (fld  gsumg  h ) ) )
1716supeq1d 7215 . . . . . . 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 4114 . . . . 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 19457 . . . . 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 5555 . . . . . . 7  |-  ( Base `  P )  e.  _V
226, 21eqeltri 2366 . . . . . 6  |-  B  e. 
_V
2322mptex 5762 . . . . 5  |-  ( f  e.  B  |->  sup ( ran  ( h  e.  ( `' f " ( _V  \  {  .0.  }
) )  |->  (fld  gsumg  h ) ) , 
RR* ,  <  ) )  e.  _V
2419, 20, 23ovmpt2a 5994 . . . 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 4967 . . . . . . . . 9  |-  ( H  |`  ( `' f "
( _V  \  {  .0.  } ) ) )  =  ( ( h  e.  A  |->  (fld  gsumg  h ) )  |`  ( `' f " ( _V  \  {  .0.  }
) ) )
27 cnvimass 5049 . . . . . . . . . . 11  |-  ( `' f " ( _V 
\  {  .0.  }
) )  C_  dom  f
28 eqid 2296 . . . . . . . . . . . . 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 16194 . . . . . . . . . . . 12  |-  ( ( ( I  e.  _V  /\  R  e.  _V )  /\  f  e.  B
)  ->  f : A
--> ( Base `  R
) )
32 fdm 5409 . . . . . . . . . . . 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 3239 . . . . . . . . . 10  |-  ( ( ( I  e.  _V  /\  R  e.  _V )  /\  f  e.  B
)  ->  ( `' f " ( _V  \  {  .0.  } ) ) 
C_  A )
35 resmpt 5016 . . . . . . . . . 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 2341 . . . . . . . 8  |-  ( ( ( I  e.  _V  /\  R  e.  _V )  /\  f  e.  B
)  ->  ( h  e.  ( `' f "
( _V  \  {  .0.  } ) )  |->  (fld  gsumg  h ) )  =  ( H  |`  ( `' f "
( _V  \  {  .0.  } ) ) ) )
3837rneqd 4922 . . . . . . 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 4718 . . . . . . 7  |-  ( H
" ( `' f
" ( _V  \  {  .0.  } ) ) )  =  ran  ( H  |`  ( `' f
" ( _V  \  {  .0.  } ) ) )
4038, 39syl6eqr 2346 . . . . . 6  |-  ( ( ( I  e.  _V  /\  R  e.  _V )  /\  f  e.  B
)  ->  ran  ( h  e.  ( `' f
" ( _V  \  {  .0.  } ) ) 
|->  (fld 
gsumg  h ) )  =  ( H " ( `' f " ( _V  \  {  .0.  }
) ) ) )
4140supeq1d 7215 . . . . 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 4122 . . . 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 2328 . . 3  |-  ( ( I  e.  _V  /\  R  e.  _V )  ->  ( I mDeg  R )  =  ( f  e.  B  |->  sup ( ( H
" ( `' f
" ( _V  \  {  .0.  } ) ) ) ,  RR* ,  <  ) ) )
44 reldmmdeg 19459 . . . . . 6  |-  Rel  dom mDeg
4544ovprc 5901 . . . . 5  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( I mDeg  R )  =  (/) )
46 mpt0 5387 . . . . 5  |-  ( f  e.  (/)  |->  sup ( ( H
" ( `' f
" ( _V  \  {  .0.  } ) ) ) ,  RR* ,  <  ) )  =  (/)
4745, 46syl6eqr 2346 . . . 4  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( I mDeg  R )  =  ( f  e.  (/)  |->  sup ( ( H
" ( `' f
" ( _V  \  {  .0.  } ) ) ) ,  RR* ,  <  ) ) )
48 reldmmpl 16188 . . . . . . . . 9  |-  Rel  dom mPoly
4948ovprc 5901 . . . . . . . 8  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( I mPoly  R )  =  (/) )
503, 49syl5eq 2340 . . . . . . 7  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  P  =  (/) )
5150fveq2d 5545 . . . . . 6  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( Base `  P
)  =  ( Base `  (/) ) )
52 base0 13201 . . . . . 6  |-  (/)  =  (
Base `  (/) )
5351, 6, 523eqtr4g 2353 . . . . 5  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  B  =  (/) )
54 mpteq1 4116 . . . . 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 2331 . . 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 2316 1  |-  D  =  ( f  e.  B  |->  sup ( ( H
" ( `' f
" ( _V  \  {  .0.  } ) ) ) ,  RR* ,  <  ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 358    = wceq 1632    e. wcel 1696   {crab 2560   _Vcvv 2801    \ cdif 3162    C_ wss 3165   (/)c0 3468   {csn 3653    e. cmpt 4093   `'ccnv 4704   dom cdm 4705   ran crn 4706    |` cres 4707   "cima 4708   -->wf 5267   ` cfv 5271  (class class class)co 5874    ^m cmap 6788   Fincfn 6879   supcsup 7209   RR*cxr 8882    < clt 8883   NNcn 9762   NN0cn0 9981   Basecbs 13164   0gc0g 13416    gsumg cgsu 13417   mPoly cmpl 16105  ℂfldccnfld 16393   mDeg cmdg 19455
This theorem is referenced by:  mdegval  19465  mdegxrf  19470  mdegpropd  19486
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-sca 13240  df-vsca 13241  df-tset 13243  df-psr 16114  df-mpl 16116  df-mdeg 19457
  Copyright terms: Public domain W3C validator