Theorem dmncrng 37388

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

Step	Hyp	Ref	Expression
1		isdmn2 37387	. 2 ⊢ (𝑅 ∈ Dmn ↔ (𝑅 ∈ PrRing ∧ 𝑅 ∈ CRingOps))
2	1	simprbi 496	1 ⊢ (𝑅 ∈ Dmn → 𝑅 ∈ CRingOps)

Syntax hints:   → wi 4   ∈ wcel 2105  CRingOpsccring 37325  PrRingcprrng 37378  Dmncdmn 37379

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-crngo 37326  df-prrngo 37380  df-dmn 37381

This theorem is referenced by:  dmnrngo  37389  dmncan2  37409