Theorem reldmevls 22373

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 22363	. 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 7546	1 ⊢ Rel dom evalSub

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570  Vcvv 3451  ⦋csb 3847  {csn 4584   ↦ cmpt 5186   × cxp 5649  dom cdm 5651   ∘ ccom 5655  Rel wrel 5656  ‘cfv 6531  ℩crio 7368  (class class class)co 7412   ↑_m cmap 8831  Basecbs 17367   ↾_s cress 17388   ↑_s cpws 17597  CRingccrg 20440   RingHom crh 20679  SubRingcsubrg 20801  algSccascl 22140   mVar cmvr 22193   mPoly cmpl 22194   evalSub ces 22361

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 2213  ax-ext 2733  ax-sep 5249  ax-pr 5391

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-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-xp 5657  df-rel 5658  df-dm 5661  df-oprab 7416  df-mpo 7417  df-evls 22363

This theorem is used by:  mpfrcl  22374  evlval  22389