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Theorem mdegfval 21531
Description: Value of the multivariate degree function. (Contributed by Stefan O'Rear, 19-Mar-2015.) (Revised by AV, 25-Jun-2019.)
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 supp  .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 6100 . . . . . . . . 9  |-  ( ( i  =  I  /\  r  =  R )  ->  ( i mPoly  r )  =  ( I mPoly  R
) )
3 mdegval.p . . . . . . . . 9  |-  P  =  ( I mPoly  R )
42, 3syl6eqr 2493 . . . . . . . 8  |-  ( ( i  =  I  /\  r  =  R )  ->  ( i mPoly  r )  =  P )
54fveq2d 5695 . . . . . . 7  |-  ( ( i  =  I  /\  r  =  R )  ->  ( Base `  (
i mPoly  r ) )  =  ( Base `  P
) )
6 mdegval.b . . . . . . 7  |-  B  =  ( Base `  P
)
75, 6syl6eqr 2493 . . . . . 6  |-  ( ( i  =  I  /\  r  =  R )  ->  ( Base `  (
i mPoly  r ) )  =  B )
8 fveq2 5691 . . . . . . . . . . . 12  |-  ( r  =  R  ->  ( 0g `  r )  =  ( 0g `  R
) )
9 mdegval.z . . . . . . . . . . . 12  |-  .0.  =  ( 0g `  R )
108, 9syl6eqr 2493 . . . . . . . . . . 11  |-  ( r  =  R  ->  ( 0g `  r )  =  .0.  )
1110oveq2d 6107 . . . . . . . . . 10  |-  ( r  =  R  ->  (
f supp  ( 0g `  r ) )  =  ( f supp  .0.  )
)
1211mpteq1d 4373 . . . . . . . . 9  |-  ( r  =  R  ->  (
h  e.  ( f supp  ( 0g `  r
) )  |->  (fld  gsumg  h ) )  =  ( h  e.  ( f supp  .0.  )  |->  (fld  gsumg  h ) ) )
1312rneqd 5067 . . . . . . . 8  |-  ( r  =  R  ->  ran  ( h  e.  (
f supp  ( 0g `  r ) )  |->  (fld  gsumg  h ) )  =  ran  (
h  e.  ( f supp 
.0.  )  |->  (fld  gsumg  h ) ) )
1413supeq1d 7696 . . . . . . 7  |-  ( r  =  R  ->  sup ( ran  ( h  e.  ( f supp  ( 0g
`  r ) ) 
|->  (fld 
gsumg  h ) ) , 
RR* ,  <  )  =  sup ( ran  (
h  e.  ( f supp 
.0.  )  |->  (fld  gsumg  h ) ) , 
RR* ,  <  ) )
1514adantl 466 . . . . . 6  |-  ( ( i  =  I  /\  r  =  R )  ->  sup ( ran  (
h  e.  ( f supp  ( 0g `  r
) )  |->  (fld  gsumg  h ) ) , 
RR* ,  <  )  =  sup ( ran  (
h  e.  ( f supp 
.0.  )  |->  (fld  gsumg  h ) ) , 
RR* ,  <  ) )
167, 15mpteq12dv 4370 . . . . 5  |-  ( ( i  =  I  /\  r  =  R )  ->  ( f  e.  (
Base `  ( i mPoly  r ) )  |->  sup ( ran  ( h  e.  ( f supp  ( 0g `  r ) ) 
|->  (fld 
gsumg  h ) ) , 
RR* ,  <  ) )  =  ( f  e.  B  |->  sup ( ran  (
h  e.  ( f supp 
.0.  )  |->  (fld  gsumg  h ) ) , 
RR* ,  <  ) ) )
17 df-mdeg 21524 . . . . 5  |- mDeg  =  ( i  e.  _V , 
r  e.  _V  |->  ( f  e.  ( Base `  ( i mPoly  r ) )  |->  sup ( ran  (
h  e.  ( f supp  ( 0g `  r
) )  |->  (fld  gsumg  h ) ) , 
RR* ,  <  ) ) )
18 fvex 5701 . . . . . . 7  |-  ( Base `  P )  e.  _V
196, 18eqeltri 2513 . . . . . 6  |-  B  e. 
_V
2019mptex 5948 . . . . 5  |-  ( f  e.  B  |->  sup ( ran  ( h  e.  ( f supp  .0.  )  |->  (fld  gsumg  h ) ) ,  RR* ,  <  ) )  e.  _V
2116, 17, 20ovmpt2a 6221 . . . 4  |-  ( ( I  e.  _V  /\  R  e.  _V )  ->  ( I mDeg  R )  =  ( f  e.  B  |->  sup ( ran  (
h  e.  ( f supp 
.0.  )  |->  (fld  gsumg  h ) ) , 
RR* ,  <  ) ) )
22 mdegval.h . . . . . . . . . 10  |-  H  =  ( h  e.  A  |->  (fld 
gsumg  h ) )
2322reseq1i 5106 . . . . . . . . 9  |-  ( H  |`  ( f supp  .0.  )
)  =  ( ( h  e.  A  |->  (fld  gsumg  h ) )  |`  ( f supp  .0.  ) )
24 suppssdm 6703 . . . . . . . . . . 11  |-  ( f supp 
.0.  )  C_  dom  f
25 eqid 2443 . . . . . . . . . . . . 13  |-  ( Base `  R )  =  (
Base `  R )
26 mdegval.a . . . . . . . . . . . . 13  |-  A  =  { m  e.  ( NN0  ^m  I )  |  ( `' m " NN )  e.  Fin }
27 simpr 461 . . . . . . . . . . . . 13  |-  ( ( ( I  e.  _V  /\  R  e.  _V )  /\  f  e.  B
)  ->  f  e.  B )
283, 25, 6, 26, 27mplelf 17509 . . . . . . . . . . . 12  |-  ( ( ( I  e.  _V  /\  R  e.  _V )  /\  f  e.  B
)  ->  f : A
--> ( Base `  R
) )
29 fdm 5563 . . . . . . . . . . . 12  |-  ( f : A --> ( Base `  R )  ->  dom  f  =  A )
3028, 29syl 16 . . . . . . . . . . 11  |-  ( ( ( I  e.  _V  /\  R  e.  _V )  /\  f  e.  B
)  ->  dom  f  =  A )
3124, 30syl5sseq 3404 . . . . . . . . . 10  |-  ( ( ( I  e.  _V  /\  R  e.  _V )  /\  f  e.  B
)  ->  ( f supp  .0.  )  C_  A )
32 resmpt 5156 . . . . . . . . . 10  |-  ( ( f supp  .0.  )  C_  A  ->  ( ( h  e.  A  |->  (fld  gsumg  h ) )  |`  ( f supp  .0.  )
)  =  ( h  e.  ( f supp  .0.  )  |->  (fld 
gsumg  h ) ) )
3331, 32syl 16 . . . . . . . . 9  |-  ( ( ( I  e.  _V  /\  R  e.  _V )  /\  f  e.  B
)  ->  ( (
h  e.  A  |->  (fld  gsumg  h ) )  |`  ( f supp  .0.  ) )  =  ( h  e.  ( f supp 
.0.  )  |->  (fld  gsumg  h ) ) )
3423, 33syl5req 2488 . . . . . . . 8  |-  ( ( ( I  e.  _V  /\  R  e.  _V )  /\  f  e.  B
)  ->  ( h  e.  ( f supp  .0.  )  |->  (fld 
gsumg  h ) )  =  ( H  |`  (
f supp  .0.  ) )
)
3534rneqd 5067 . . . . . . 7  |-  ( ( ( I  e.  _V  /\  R  e.  _V )  /\  f  e.  B
)  ->  ran  ( h  e.  ( f supp  .0.  )  |->  (fld 
gsumg  h ) )  =  ran  ( H  |`  ( f supp  .0.  )
) )
36 df-ima 4853 . . . . . . 7  |-  ( H
" ( f supp  .0.  ) )  =  ran  ( H  |`  ( f supp 
.0.  ) )
3735, 36syl6eqr 2493 . . . . . 6  |-  ( ( ( I  e.  _V  /\  R  e.  _V )  /\  f  e.  B
)  ->  ran  ( h  e.  ( f supp  .0.  )  |->  (fld 
gsumg  h ) )  =  ( H " (
f supp  .0.  ) )
)
3837supeq1d 7696 . . . . 5  |-  ( ( ( I  e.  _V  /\  R  e.  _V )  /\  f  e.  B
)  ->  sup ( ran  ( h  e.  ( f supp  .0.  )  |->  (fld  gsumg  h ) ) ,  RR* ,  <  )  =  sup ( ( H " ( f supp 
.0.  ) ) , 
RR* ,  <  ) )
3938mpteq2dva 4378 . . . 4  |-  ( ( I  e.  _V  /\  R  e.  _V )  ->  ( f  e.  B  |->  sup ( ran  (
h  e.  ( f supp 
.0.  )  |->  (fld  gsumg  h ) ) , 
RR* ,  <  ) )  =  ( f  e.  B  |->  sup ( ( H
" ( f supp  .0.  ) ) ,  RR* ,  <  ) ) )
4021, 39eqtrd 2475 . . 3  |-  ( ( I  e.  _V  /\  R  e.  _V )  ->  ( I mDeg  R )  =  ( f  e.  B  |->  sup ( ( H
" ( f supp  .0.  ) ) ,  RR* ,  <  ) ) )
41 reldmmdeg 21526 . . . . . 6  |-  Rel  dom mDeg
4241ovprc 6118 . . . . 5  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( I mDeg  R )  =  (/) )
43 mpt0 5538 . . . . 5  |-  ( f  e.  (/)  |->  sup ( ( H
" ( f supp  .0.  ) ) ,  RR* ,  <  ) )  =  (/)
4442, 43syl6eqr 2493 . . . 4  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( I mDeg  R )  =  ( f  e.  (/)  |->  sup ( ( H
" ( f supp  .0.  ) ) ,  RR* ,  <  ) ) )
45 reldmmpl 17500 . . . . . . . . 9  |-  Rel  dom mPoly
4645ovprc 6118 . . . . . . . 8  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( I mPoly  R )  =  (/) )
473, 46syl5eq 2487 . . . . . . 7  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  P  =  (/) )
4847fveq2d 5695 . . . . . 6  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( Base `  P
)  =  ( Base `  (/) ) )
49 base0 14213 . . . . . 6  |-  (/)  =  (
Base `  (/) )
5048, 6, 493eqtr4g 2500 . . . . 5  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  B  =  (/) )
5150mpteq1d 4373 . . . 4  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( f  e.  B  |->  sup ( ( H
" ( f supp  .0.  ) ) ,  RR* ,  <  ) )  =  ( f  e.  (/)  |->  sup ( ( H "
( f supp  .0.  )
) ,  RR* ,  <  ) ) )
5244, 51eqtr4d 2478 . . 3  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( I mDeg  R )  =  ( f  e.  B  |->  sup ( ( H
" ( f supp  .0.  ) ) ,  RR* ,  <  ) ) )
5340, 52pm2.61i 164 . 2  |-  ( I mDeg 
R )  =  ( f  e.  B  |->  sup ( ( H "
( f supp  .0.  )
) ,  RR* ,  <  ) )
541, 53eqtri 2463 1  |-  D  =  ( f  e.  B  |->  sup ( ( H
" ( f supp  .0.  ) ) ,  RR* ,  <  ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 369    = wceq 1369    e. wcel 1756   {crab 2719   _Vcvv 2972    C_ wss 3328   (/)c0 3637    e. cmpt 4350   `'ccnv 4839   dom cdm 4840   ran crn 4841    |` cres 4842   "cima 4843   -->wf 5414   ` cfv 5418  (class class class)co 6091   supp csupp 6690    ^m cmap 7214   Fincfn 7310   supcsup 7690   RR*cxr 9417    < clt 9418   NNcn 10322   NN0cn0 10579   Basecbs 14174   0gc0g 14378    gsumg cgsu 14379   mPoly cmpl 17420  ℂfldccnfld 17818   mDeg cmdg 21522
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-of 6320  df-om 6477  df-1st 6577  df-2nd 6578  df-supp 6691  df-recs 6832  df-rdg 6866  df-1o 6920  df-oadd 6924  df-er 7101  df-map 7216  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-fsupp 7621  df-sup 7691  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-2 10380  df-3 10381  df-4 10382  df-5 10383  df-6 10384  df-7 10385  df-8 10386  df-9 10387  df-n0 10580  df-z 10647  df-uz 10862  df-fz 11438  df-struct 14176  df-ndx 14177  df-slot 14178  df-base 14179  df-sets 14180  df-ress 14181  df-plusg 14251  df-mulr 14252  df-sca 14254  df-vsca 14255  df-tset 14257  df-psr 17423  df-mpl 17425  df-mdeg 21524
This theorem is referenced by:  mdegfvalOLD  21532  mdegval  21533  mdegxrf  21539  mdegpropd  21555
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