Theorem dmncrng 38809

Description: Obsolete theorem, use idomcringd 20891 instead. A domain is a commutative ring. (Contributed by Jeff Madsen, 6-Jan-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
dmncrng	⊢ (𝑅 ∈ Dmn → 𝑅 ∈ CRingOps)

Proof of Theorem dmncrng

Step	Hyp	Ref	Expression
1		isdmn2 38808	. 2 ⊢ (𝑅 ∈ Dmn ↔ (𝑅 ∈ PrRing ∧ 𝑅 ∈ CRingOps))
2	1	simprbi 503	1 ⊢ (𝑅 ∈ Dmn → 𝑅 ∈ CRingOps)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145  CRingOpsccring 38746  PrRingcprrng 38799  Dmncdmn 38800

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-iota 6489  df-fv 6541  df-crngo 38747  df-prrngo 38801  df-dmn 38802

This theorem is used by:  dmnrngo  38810  dmncan2  38830