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Theorem dmncrng 35399
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 35398 . 2 (𝑅 ∈ Dmn ↔ (𝑅 ∈ PrRing ∧ 𝑅 ∈ CRingOps))
21simprbi 500 1 (𝑅 ∈ Dmn → 𝑅 ∈ CRingOps)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2115  CRingOpsccring 35336  PrRingcprrng 35389  Dmncdmn 35390
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796
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 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-rab 3141  df-v 3481  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4822  df-br 5050  df-iota 6297  df-fv 6346  df-crngo 35337  df-prrngo 35391  df-dmn 35392
This theorem is referenced by:  dmnrngo  35400  dmncan2  35420
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