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Theorem dmnrngo 38004
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 38003 . 2 (𝑅 ∈ Dmn → 𝑅 ∈ CRingOps)
2 crngorngo 37947 . 2 (𝑅 ∈ CRingOps → 𝑅 ∈ RingOps)
31, 2syl 17 1 (𝑅 ∈ Dmn → 𝑅 ∈ RingOps)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2104  RingOpscrngo 37841  CRingOpsccring 37940  Dmncdmn 37994
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-ext 2704
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1087  df-tru 1538  df-fal 1548  df-ex 1775  df-sb 2061  df-clab 2711  df-cleq 2725  df-clel 2812  df-rab 3433  df-v 3479  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4915  df-br 5150  df-iota 6510  df-fv 6566  df-crngo 37941  df-prrngo 37995  df-dmn 37996
This theorem is referenced by:  dmncan1  38023
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