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Theorem reldmevls 20225
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 20214 . 2 evalSub = (𝑖 ∈ V, 𝑠 ∈ CRing ↦ (Base‘𝑠) / 𝑏(𝑟 ∈ (SubRing‘𝑠) ↦ (𝑖 mPoly (𝑠s 𝑟)) / 𝑤(𝑓 ∈ (𝑤 RingHom (𝑠s (𝑏m 𝑖)))((𝑓 ∘ (algSc‘𝑤)) = (𝑥𝑟 ↦ ((𝑏m 𝑖) × {𝑥})) ∧ (𝑓 ∘ (𝑖 mVar (𝑠s 𝑟))) = (𝑥𝑖 ↦ (𝑔 ∈ (𝑏m 𝑖) ↦ (𝑔𝑥)))))))
21reldmmpo 7274 1 Rel dom evalSub
Colors of variables: wff setvar class
Syntax hints:  wa 396   = wceq 1528  Vcvv 3492  csb 3880  {csn 4557  cmpt 5137   × cxp 5546  dom cdm 5548  ccom 5552  Rel wrel 5553  cfv 6348  crio 7102  (class class class)co 7145  m cmap 8395  Basecbs 16471  s cress 16472  s cpws 16708  CRingccrg 19227   RingHom crh 19393  SubRingcsubrg 19460  algSccascl 20012   mVar cmvr 20060   mPoly cmpl 20061   evalSub ces 20212
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-xp 5554  df-rel 5555  df-dm 5558  df-oprab 7149  df-mpo 7150  df-evls 20214
This theorem is referenced by:  mpfrcl  20226  evlval  20236
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