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Theorem reldmevls 19930
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 16412 . 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 6173 1  |-  Rel  dom evalSub
Colors of variables: wff set class
Syntax hints:    /\ wa 359    = wceq 1652   _Vcvv 2948   [_csb 3243   {csn 3806    e. cmpt 4258    X. cxp 4868   dom cdm 4870    o. ccom 4874   Rel wrel 4875   ` cfv 5446  (class class class)co 6073   iota_crio 6534    ^m cmap 7010   Basecbs 13461   ↾s cress 13462    ^s cpws 13662   CRingccrg 15653   RingHom crh 15809  SubRingcsubrg 15856  algSccascl 16363   mVar cmvr 16399   mPoly cmpl 16400   evalSub ces 16401
This theorem is referenced by:  mpfrcl  19931  evlval  19937
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-xp 4876  df-rel 4877  df-dm 4880  df-oprab 6077  df-mpt2 6078  df-evls 16412
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