Theorem reldmevls 21997

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 21987	. 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 7525	1 ⊢ Rel dom evalSub

Syntax hints:   ∧ wa 395   = wceq 1540  Vcvv 3450  ⦋csb 3864  {csn 4591   ↦ cmpt 5190   × cxp 5638  dom cdm 5640   ∘ ccom 5644  Rel wrel 5645  ‘cfv 6513  ℩crio 7345  (class class class)co 7389   ↑_m cmap 8801  Basecbs 17185   ↾_s cress 17206   ↑_s cpws 17415  CRingccrg 20149   RingHom crh 20384  SubRingcsubrg 20484  algSccascl 21767   mVar cmvr 21820   mPoly cmpl 21821   evalSub ces 21985

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5253  ax-nul 5263  ax-pr 5389

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-rab 3409  df-v 3452  df-dif 3919  df-un 3921  df-ss 3933  df-nul 4299  df-if 4491  df-sn 4592  df-pr 4594  df-op 4598  df-br 5110  df-opab 5172  df-xp 5646  df-rel 5647  df-dm 5650  df-oprab 7393  df-mpo 7394  df-evls 21987

This theorem is referenced by:  mpfrcl  21998  evlval  22008