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

Theorem reldmmdeg 19443
Description: Multivariate degree is a binary operation. (Contributed by Stefan O'Rear, 28-Mar-2015.)
Assertion
Ref Expression
reldmmdeg  |-  Rel  dom mDeg

Proof of Theorem reldmmdeg
Dummy variables  i 
r  h  f are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-mdeg 19441 . 2  |- 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* ,  <  ) ) )
21reldmmpt2 5955 1  |-  Rel  dom mDeg
Colors of variables: wff set class
Syntax hints:   _Vcvv 2788    \ cdif 3149   {csn 3640    e. cmpt 4077   `'ccnv 4688   dom cdm 4689   ran crn 4690   "cima 4692   Rel wrel 4694   ` cfv 5255  (class class class)co 5858   supcsup 7193   RR*cxr 8866    < clt 8867   Basecbs 13148   0gc0g 13400    gsumg cgsu 13401   mPoly cmpl 16089  ℂfldccnfld 16377   mDeg cmdg 19439
This theorem is referenced by:  mdegfval  19448  deg1fval  19466
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-dm 4699  df-oprab 5862  df-mpt2 5863  df-mdeg 19441
  Copyright terms: Public domain W3C validator