Theorem dmnrngo 36142

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 36141	. 2 ⊢ (𝑅 ∈ Dmn → 𝑅 ∈ CRingOps)
2		crngorngo 36085	. 2 ⊢ (𝑅 ∈ CRingOps → 𝑅 ∈ RingOps)
3	1, 2	syl 17	1 ⊢ (𝑅 ∈ Dmn → 𝑅 ∈ RingOps)

Syntax hints:   → wi 4   ∈ wcel 2108  RingOpscrngo 35979  CRingOpsccring 36078  Dmncdmn 36132

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-iota 6376  df-fv 6426  df-crngo 36079  df-prrngo 36133  df-dmn 36134

This theorem is referenced by:  dmncan1  36161