Theorem dmnrngo 38009

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 38008	. 2 ⊢ (𝑅 ∈ Dmn → 𝑅 ∈ CRingOps)
2		crngorngo 37952	. 2 ⊢ (𝑅 ∈ CRingOps → 𝑅 ∈ RingOps)
3	1, 2	syl 17	1 ⊢ (𝑅 ∈ Dmn → 𝑅 ∈ RingOps)

Syntax hints:   → wi 4   ∈ wcel 2108  RingOpscrngo 37846  CRingOpsccring 37945  Dmncdmn 37999

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-iota 6520  df-fv 6576  df-crngo 37946  df-prrngo 38000  df-dmn 38001

This theorem is referenced by:  dmncan1  38028