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Theorem dmncrng 38042
Description: A domain is a commutative ring. (Contributed by Jeff Madsen, 6-Jan-2011.)
Assertion
Ref Expression
dmncrng (𝑅 ∈ Dmn → 𝑅 ∈ CRingOps)

Proof of Theorem dmncrng
StepHypRef Expression
1 isdmn2 38041 . 2 (𝑅 ∈ Dmn ↔ (𝑅 ∈ PrRing ∧ 𝑅 ∈ CRingOps))
21simprbi 496 1 (𝑅 ∈ Dmn → 𝑅 ∈ CRingOps)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2105  CRingOpsccring 37979  PrRingcprrng 38032  Dmncdmn 38033
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-iota 6515  df-fv 6570  df-crngo 37980  df-prrngo 38034  df-dmn 38035
This theorem is referenced by:  dmnrngo  38043  dmncan2  38063
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