Theorem isdmn2 38054

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 38053	. 2 ⊢ (𝑅 ∈ Dmn ↔ (𝑅 ∈ PrRing ∧ 𝑅 ∈ Com2))
2		prrngorngo 38050	. . . 4 ⊢ (𝑅 ∈ PrRing → 𝑅 ∈ RingOps)
3		iscrngo 37995	. . . . 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 2109  RingOpscrngo 37893  Com2ccm2 37988  CRingOpsccring 37992  PrRingcprrng 38045  Dmncdmn 38046

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 2701

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 2708  df-cleq 2721  df-clel 2803  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-iota 6442  df-fv 6494  df-crngo 37993  df-prrngo 38047  df-dmn 38048

This theorem is referenced by:  dmncrng  38055  flddmn  38057  isdmn3  38073