Theorem dmncrng 33855

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

Assertion

Ref	Expression
dmncrng	|- ( R e. Dmn -> R e. CRingOps )

Proof of Theorem dmncrng

Step	Hyp	Ref	Expression
1		isdmn2 33854	. 2 |- ( R e. Dmn <-> ( R e. PrRing /\ R e. CRingOps ) )
2	1	simprbi 480	1 |- ( R e. Dmn -> R e. CRingOps )

Syntax hints:   -> wi 4   e. wcel 1990  CRingOpsccring 33792   PrRingcprrng 33845   Dmncdmn 33846

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-crngo 33793  df-prrngo 33847  df-dmn 33848

This theorem is referenced by:  dmnrngo  33856  dmncan2  33876