Theorem reldmevls 22007

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 21997	. 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 7487	1 ⊢ Rel dom evalSub

Syntax hints:   ∧ wa 395   = wceq 1540  Vcvv 3438  ⦋csb 3853  {csn 4579   ↦ cmpt 5176   × cxp 5621  dom cdm 5623   ∘ ccom 5627  Rel wrel 5628  ‘cfv 6486  ℩crio 7309  (class class class)co 7353   ↑_m cmap 8760  Basecbs 17138   ↾_s cress 17159   ↑_s cpws 17368  CRingccrg 20137   RingHom crh 20372  SubRingcsubrg 20472  algSccascl 21777   mVar cmvr 21830   mPoly cmpl 21831   evalSub ces 21995

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 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-br 5096  df-opab 5158  df-xp 5629  df-rel 5630  df-dm 5633  df-oprab 7357  df-mpo 7358  df-evls 21997

This theorem is referenced by:  mpfrcl  22008  evlval  22018