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Theorem reldmevls 19805
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 16347 . 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 6120 1  |-  Rel  dom evalSub
Colors of variables: wff set class
Syntax hints:    /\ wa 359    = wceq 1649   _Vcvv 2899   [_csb 3194   {csn 3757    e. cmpt 4207    X. cxp 4816   dom cdm 4818    o. ccom 4822   Rel wrel 4823   ` cfv 5394  (class class class)co 6020   iota_crio 6478    ^m cmap 6954   Basecbs 13396   ↾s cress 13397    ^s cpws 13597   CRingccrg 15588   RingHom crh 15744  SubRingcsubrg 15791  algSccascl 16298   mVar cmvr 16334   mPoly cmpl 16335   evalSub ces 16336
This theorem is referenced by:  mpfrcl  19806  evlval  19812
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-br 4154  df-opab 4208  df-xp 4824  df-rel 4825  df-dm 4828  df-oprab 6024  df-mpt2 6025  df-evls 16347
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