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

Theorem reldmevls 19417
Description: Well-behaved binary operation property of evalSub. (Contributed by Stefan O'Rear, 19-Mar-2015.)
Assertion
Ref Expression
reldmevls  |-  Rel  dom evalSub

Proof of Theorem reldmevls
Dummy variables  b 
f  g  i  r  s  w  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-evls 16117 . 2  |- evalSub  =  ( i  e.  _V , 
s  e.  CRing  |->  [_ ( Base `  s )  / 
b ]_ ( r  e.  (SubRing `  s )  |-> 
[_ ( i mPoly  (
ss  r ) )  /  w ]_ ( iota_ f  e.  ( w RingHom  ( s  ^s  ( b  ^m  i
) ) ) ( ( f  o.  (algSc `  w ) )  =  ( x  e.  r 
|->  ( ( b  ^m  i )  X.  {
x } ) )  /\  ( f  o.  ( i mVar  ( ss  r ) ) )  =  ( x  e.  i 
|->  ( g  e.  ( b  ^m  i ) 
|->  ( g `  x
) ) ) ) ) ) )
21reldmmpt2 5971 1  |-  Rel  dom evalSub
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1632   _Vcvv 2801   [_csb 3094   {csn 3653    e. cmpt 4093    X. cxp 4703   dom cdm 4705    o. ccom 4709   Rel wrel 4710   ` cfv 5271  (class class class)co 5874   iota_crio 6313    ^m cmap 6788   Basecbs 13164   ↾s cress 13165    ^s cpws 13363   CRingccrg 15354   RingHom crh 15510  SubRingcsubrg 15557  algSccascl 16068   mVar cmvr 16104   mPoly cmpl 16105   evalSub ces 16106
This theorem is referenced by:  mpfrcl  19418  evlval  19424
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-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-dm 4715  df-oprab 5878  df-mpt2 5879  df-evls 16117
  Copyright terms: Public domain W3C validator