Theorem dmncrng 38057

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 38056	. 2 ⊢ (𝑅 ∈ Dmn ↔ (𝑅 ∈ PrRing ∧ 𝑅 ∈ CRingOps))
2	1	simprbi 496	1 ⊢ (𝑅 ∈ Dmn → 𝑅 ∈ CRingOps)

Syntax hints:   → wi 4   ∈ wcel 2109  CRingOpsccring 37994  PrRingcprrng 38047  Dmncdmn 38048

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 2702

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 2709  df-cleq 2722  df-clel 2804  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-iota 6467  df-fv 6522  df-crngo 37995  df-prrngo 38049  df-dmn 38050

This theorem is referenced by:  dmnrngo  38058  dmncan2  38078