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Theorem isdmn2 37528
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
StepHypRef Expression
1 isdmn 37527 . 2 (𝑅 ∈ Dmn ↔ (𝑅 ∈ PrRing ∧ 𝑅 ∈ Com2))
2 prrngorngo 37524 . . . 4 (𝑅 ∈ PrRing → 𝑅 ∈ RingOps)
3 iscrngo 37469 . . . . 5 (𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ 𝑅 ∈ Com2))
43baibr 536 . . . 4 (𝑅 ∈ RingOps → (𝑅 ∈ Com2 ↔ 𝑅 ∈ CRingOps))
52, 4syl 17 . . 3 (𝑅 ∈ PrRing → (𝑅 ∈ Com2 ↔ 𝑅 ∈ CRingOps))
65pm5.32i 574 . 2 ((𝑅 ∈ PrRing ∧ 𝑅 ∈ Com2) ↔ (𝑅 ∈ PrRing ∧ 𝑅 ∈ CRingOps))
71, 6bitri 275 1 (𝑅 ∈ Dmn ↔ (𝑅 ∈ PrRing ∧ 𝑅 ∈ CRingOps))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395  wcel 2099  RingOpscrngo 37367  Com2ccm2 37462  CRingOpsccring 37466  PrRingcprrng 37519  Dmncdmn 37520
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-iota 6500  df-fv 6556  df-crngo 37467  df-prrngo 37521  df-dmn 37522
This theorem is referenced by:  dmncrng  37529  flddmn  37531  isdmn3  37547
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