Theorem isdmn2 33854

Description: The predicate "is a domain". (Contributed by Jeff Madsen, 10-Jun-2010.)

Assertion

Ref	Expression
isdmn2	|- ( R e. Dmn <-> ( R e. PrRing /\ R e. CRingOps ) )

Proof of Theorem isdmn2

Step	Hyp	Ref	Expression
1		isdmn 33853	. 2 |- ( R e. Dmn <-> ( R e. PrRing /\ R e. Com2 ) )
2		prrngorngo 33850	. . . 4 |- ( R e. PrRing -> R e. RingOps )
3		iscrngo 33795	. . . . 5 |- ( R e. CRingOps <-> ( R e. RingOps /\ R e. Com2 ) )
4	3	baibr 945	. . . 4 |- ( R e. RingOps -> ( R e. Com2 <-> R e. CRingOps ) )
5	2, 4	syl 17	. . 3 |- ( R e. PrRing -> ( R e. Com2 <-> R e. CRingOps ) )
6	5	pm5.32i 669	. 2 |- ( ( R e. PrRing /\ R e. Com2 ) <-> ( R e. PrRing /\ R e. CRingOps ) )
7	1, 6	bitri 264	1 |- ( R e. Dmn <-> ( R e. PrRing /\ R e. CRingOps ) )

Syntax hints:   <-> wb 196   /\ wa 384   e. wcel 1990   RingOpscrngo 33693   Com2ccm2 33788  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:  dmncrng  33855  flddmn  33857  isdmn3  33873