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Theorem dmncrng 34180
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 34179 . 2 (𝑅 ∈ Dmn ↔ (𝑅 ∈ PrRing ∧ 𝑅 ∈ CRingOps))
21simprbi 478 1 (𝑅 ∈ Dmn → 𝑅 ∈ CRingOps)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2144  CRingOpsccring 34117  PrRingcprrng 34170  Dmncdmn 34171
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-rex 3066  df-rab 3069  df-v 3351  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-iota 5994  df-fv 6039  df-crngo 34118  df-prrngo 34172  df-dmn 34173
This theorem is referenced by:  dmnrngo  34181  dmncan2  34201
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