Theorem dmnrngo 35337

Description: A domain is a ring. (Contributed by Jeff Madsen, 6-Jan-2011.)

Assertion

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

Proof of Theorem dmnrngo

Step	Hyp	Ref	Expression
1		dmncrng 35336	. 2 ⊢ (𝑅 ∈ Dmn → 𝑅 ∈ CRingOps)
2		crngorngo 35280	. 2 ⊢ (𝑅 ∈ CRingOps → 𝑅 ∈ RingOps)
3	1, 2	syl 17	1 ⊢ (𝑅 ∈ Dmn → 𝑅 ∈ RingOps)

Syntax hints:   → wi 4   ∈ wcel 2114  RingOpscrngo 35174  CRingOpsccring 35273  Dmncdmn 35327

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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-iota 6316  df-fv 6365  df-crngo 35274  df-prrngo 35328  df-dmn 35329

This theorem is referenced by:  dmncan1  35356