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Theorem isdmn2 32807
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 32806 . 2 (𝑅 ∈ Dmn ↔ (𝑅 ∈ PrRing ∧ 𝑅 ∈ Com2))
2 prrngorngo 32803 . . . 4 (𝑅 ∈ PrRing → 𝑅 ∈ RingOps)
3 iscrngo 32748 . . . . 5 (𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ 𝑅 ∈ Com2))
43baibr 942 . . . 4 (𝑅 ∈ RingOps → (𝑅 ∈ Com2 ↔ 𝑅 ∈ CRingOps))
52, 4syl 17 . . 3 (𝑅 ∈ PrRing → (𝑅 ∈ Com2 ↔ 𝑅 ∈ CRingOps))
65pm5.32i 666 . 2 ((𝑅 ∈ PrRing ∧ 𝑅 ∈ Com2) ↔ (𝑅 ∈ PrRing ∧ 𝑅 ∈ CRingOps))
71, 6bitri 262 1 (𝑅 ∈ Dmn ↔ (𝑅 ∈ PrRing ∧ 𝑅 ∈ CRingOps))
Colors of variables: wff setvar class
Syntax hints:  wb 194  wa 382  wcel 1976  RingOpscrngo 32646  Com2ccm2 32741  CRingOpsccring 32745  PrRingcprrng 32798  Dmncdmn 32799
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-rex 2901  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-iota 5753  df-fv 5797  df-crngo 32746  df-prrngo 32800  df-dmn 32801
This theorem is referenced by:  dmncrng  32808  flddmn  32810  isdmn3  32826
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