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Theorem reldmevls 19401
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 16101 . 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 5955 1  |-  Rel  dom evalSub
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1623   _Vcvv 2788   [_csb 3081   {csn 3640    e. cmpt 4077    X. cxp 4687   dom cdm 4689    o. ccom 4693   Rel wrel 4694   ` cfv 5255  (class class class)co 5858   iota_crio 6297    ^m cmap 6772   Basecbs 13148   ↾s cress 13149    ^s cpws 13347   CRingccrg 15338   RingHom crh 15494  SubRingcsubrg 15541  algSccascl 16052   mVar cmvr 16088   mPoly cmpl 16089   evalSub ces 16090
This theorem is referenced by:  mpfrcl  19402  evlval  19408
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-evls 16101
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