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Theorem reldmevls 21647
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 π‘Ÿ 𝑠 𝑀 π‘₯ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-evls 21635 . 2 evalSub = (𝑖 ∈ V, 𝑠 ∈ CRing ↦ ⦋(Baseβ€˜π‘ ) / π‘β¦Œ(π‘Ÿ ∈ (SubRingβ€˜π‘ ) ↦ ⦋(𝑖 mPoly (𝑠 β†Ύs π‘Ÿ)) / π‘€β¦Œ(℩𝑓 ∈ (𝑀 RingHom (𝑠 ↑s (𝑏 ↑m 𝑖)))((𝑓 ∘ (algScβ€˜π‘€)) = (π‘₯ ∈ π‘Ÿ ↦ ((𝑏 ↑m 𝑖) Γ— {π‘₯})) ∧ (𝑓 ∘ (𝑖 mVar (𝑠 β†Ύs π‘Ÿ))) = (π‘₯ ∈ 𝑖 ↦ (𝑔 ∈ (𝑏 ↑m 𝑖) ↦ (π‘”β€˜π‘₯)))))))
21reldmmpo 7543 1 Rel dom evalSub
Colors of variables: wff setvar class
Syntax hints:   ∧ wa 397   = wceq 1542  Vcvv 3475  β¦‹csb 3894  {csn 4629   ↦ cmpt 5232   Γ— cxp 5675  dom cdm 5677   ∘ ccom 5681  Rel wrel 5682  β€˜cfv 6544  β„©crio 7364  (class class class)co 7409   ↑m cmap 8820  Basecbs 17144   β†Ύs cress 17173   ↑s cpws 17392  CRingccrg 20057   RingHom crh 20248  SubRingcsubrg 20315  algSccascl 21407   mVar cmvr 21458   mPoly cmpl 21459   evalSub ces 21633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-xp 5683  df-rel 5684  df-dm 5687  df-oprab 7413  df-mpo 7414  df-evls 21635
This theorem is referenced by:  mpfrcl  21648  evlval  21658
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