Theorem reldmevls 22306

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.

Step	Hyp	Ref	Expression
1		df-evls 22296	. 2 ⊢ evalSub = (𝑖 ∈ V, 𝑠 ∈ CRing ↦ ⦋(Base‘𝑠) / 𝑏⦌(𝑟 ∈ (SubRing‘𝑠) ↦ ⦋(𝑖 mPoly (𝑠 ↾s 𝑟)) / 𝑤⦌(℩𝑓 ∈ (𝑤 RingHom (𝑠 ↑s (𝑏 ↑m 𝑖)))((𝑓 ∘ (algSc‘𝑤)) = (𝑥 ∈ 𝑟 ↦ ((𝑏 ↑m 𝑖) × {𝑥})) ∧ (𝑓 ∘ (𝑖 mVar (𝑠 ↾s 𝑟))) = (𝑥 ∈ 𝑖 ↦ (𝑔 ∈ (𝑏 ↑m 𝑖) ↦ (𝑔‘𝑥)))))))
2	1	reldmmpo 7551	1 ⊢ Rel dom evalSub

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570  Vcvv 3453  ⦋csb 3850  {csn 4587   ↦ cmpt 5190   × cxp 5657  dom cdm 5659   ∘ ccom 5663  Rel wrel 5664  ‘cfv 6537  ℩crio 7373  (class class class)co 7417   ↑_m cmap 8830  Basecbs 17307   ↾_s cress 17328   ↑_s cpws 17537  CRingccrg 20379   RingHom crh 20616  SubRingcsubrg 20737  algSccascl 22073   mVar cmvr 22126   mPoly cmpl 22127   evalSub ces 22294

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-br 5108  df-opab 5172  df-xp 5665  df-rel 5666  df-dm 5669  df-oprab 7421  df-mpo 7422  df-evls 22296

This theorem is used by:  mpfrcl  22307  evlval  22322