Theorem dmncrng 38023

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

Assertion

Ref	Expression
dmncrng	⊢ (𝑅 ∈ Dmn → 𝑅 ∈ CRingOps)

Proof of Theorem dmncrng

Step	Hyp	Ref	Expression
1		isdmn2 38022	. 2 ⊢ (𝑅 ∈ Dmn ↔ (𝑅 ∈ PrRing ∧ 𝑅 ∈ CRingOps))
2	1	simprbi 496	1 ⊢ (𝑅 ∈ Dmn → 𝑅 ∈ CRingOps)

Syntax hints:   → wi 4   ∈ wcel 2109  CRingOpsccring 37960  PrRingcprrng 38013  Dmncdmn 38014

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 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-iota 6452  df-fv 6507  df-crngo 37961  df-prrngo 38015  df-dmn 38016

This theorem is referenced by:  dmnrngo  38024  dmncan2  38044