Theorem isdmn2 38168

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

Assertion

Ref	Expression
isdmn2	⊢ (𝑅 ∈ Dmn ↔ (𝑅 ∈ PrRing ∧ 𝑅 ∈ CRingOps))

Proof of Theorem isdmn2

Step	Hyp	Ref	Expression
1		isdmn 38167	. 2 ⊢ (𝑅 ∈ Dmn ↔ (𝑅 ∈ PrRing ∧ 𝑅 ∈ Com2))
2		prrngorngo 38164	. . . 4 ⊢ (𝑅 ∈ PrRing → 𝑅 ∈ RingOps)
3		iscrngo 38109	. . . . 5 ⊢ (𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ 𝑅 ∈ Com2))
4	3	baibr 536	. . . 4 ⊢ (𝑅 ∈ RingOps → (𝑅 ∈ Com2 ↔ 𝑅 ∈ CRingOps))
5	2, 4	syl 17	. . 3 ⊢ (𝑅 ∈ PrRing → (𝑅 ∈ Com2 ↔ 𝑅 ∈ CRingOps))
6	5	pm5.32i 574	. 2 ⊢ ((𝑅 ∈ PrRing ∧ 𝑅 ∈ Com2) ↔ (𝑅 ∈ PrRing ∧ 𝑅 ∈ CRingOps))
7	1, 6	bitri 275	1 ⊢ (𝑅 ∈ Dmn ↔ (𝑅 ∈ PrRing ∧ 𝑅 ∈ CRingOps))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∈ wcel 2113  RingOpscrngo 38007  Com2ccm2 38102  CRingOpsccring 38106  PrRingcprrng 38159  Dmncdmn 38160

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-iota 6445  df-fv 6497  df-crngo 38107  df-prrngo 38161  df-dmn 38162

This theorem is referenced by:  dmncrng  38169  flddmn  38171  isdmn3  38187