Mathbox for Jeff Madsen < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dmnrngo Structured version   Visualization version   GIF version

Theorem dmnrngo 35488
 Description: A domain is a ring. (Contributed by Jeff Madsen, 6-Jan-2011.)
Assertion
Ref Expression
dmnrngo (𝑅 ∈ Dmn → 𝑅 ∈ RingOps)

Proof of Theorem dmnrngo
StepHypRef Expression
1 dmncrng 35487 . 2 (𝑅 ∈ Dmn → 𝑅 ∈ CRingOps)
2 crngorngo 35431 . 2 (𝑅 ∈ CRingOps → 𝑅 ∈ RingOps)
31, 2syl 17 1 (𝑅 ∈ Dmn → 𝑅 ∈ RingOps)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2112  RingOpscrngo 35325  CRingOpsccring 35424  Dmncdmn 35478 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-rab 3118  df-v 3446  df-un 3889  df-in 3891  df-ss 3901  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-iota 6287  df-fv 6336  df-crngo 35425  df-prrngo 35479  df-dmn 35480 This theorem is referenced by:  dmncan1  35507
 Copyright terms: Public domain W3C validator