Theorem dmncrng 38630

Description: Obsolete theorem, use idomcringd 20811 instead. A domain is a commutative ring. (Contributed by Jeff Madsen, 6-Jan-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem dmncrng

Step	Hyp	Ref	Expression
1		isdmn2 38629	. 2 ⊢ (𝑅 ∈ Dmn ↔ (𝑅 ∈ PrRing ∧ 𝑅 ∈ CRingOps))
2	1	simprbi 502	1 ⊢ (𝑅 ∈ Dmn → 𝑅 ∈ CRingOps)

Syntax hints:   → wi 4   ∈ wcel 2149  CRingOpsccring 38567  PrRingcprrng 38620  Dmncdmn 38621

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3423  df-v 3463  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4491  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5112  df-iota 6493  df-fv 6545  df-crngo 38568  df-prrngo 38622  df-dmn 38623

This theorem is referenced by:  dmnrngo  38631  dmncan2  38651