Theorem dmnrngo 38992

Description: Obsolete theorem, use idomringd 20979 instead. A domain is a ring. (Contributed by Jeff Madsen, 6-Jan-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
dmnrngo	⊢ (𝑅 ∈ Dmn → 𝑅 ∈ RingOps)

Proof of Theorem dmnrngo

Step	Hyp	Ref	Expression
1		dmncrng 38991	. 2 ⊢ (𝑅 ∈ Dmn → 𝑅 ∈ CRingOps)
2		crngorngo 38935	. 2 ⊢ (𝑅 ∈ CRingOps → 𝑅 ∈ RingOps)
3	1, 2	syl 18	1 ⊢ (𝑅 ∈ Dmn → 𝑅 ∈ RingOps)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2144  RingOpscrngo 38829  CRingOpsccring 38928  Dmncdmn 38982

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-iota 6494  df-fv 6546  df-crngo 38929  df-prrngo 38983  df-dmn 38984

This theorem is used by:  dmncan1  39011