Theorem dmnrngo 38042

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 38041	. 2 ⊢ (𝑅 ∈ Dmn → 𝑅 ∈ CRingOps)
2		crngorngo 37985	. 2 ⊢ (𝑅 ∈ CRingOps → 𝑅 ∈ RingOps)
3	1, 2	syl 17	1 ⊢ (𝑅 ∈ Dmn → 𝑅 ∈ RingOps)

Syntax hints:   → wi 4   ∈ wcel 2108  RingOpscrngo 37879  CRingOpsccring 37978  Dmncdmn 38032

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2728  df-clel 2815  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4906  df-br 5142  df-iota 6512  df-fv 6567  df-crngo 37979  df-prrngo 38033  df-dmn 38034

This theorem is referenced by:  dmncan1  38061