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Theorem isdmn2 35201
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 35200 . 2 (𝑅 ∈ Dmn ↔ (𝑅 ∈ PrRing ∧ 𝑅 ∈ Com2))
2 prrngorngo 35197 . . . 4 (𝑅 ∈ PrRing → 𝑅 ∈ RingOps)
3 iscrngo 35142 . . . . 5 (𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ 𝑅 ∈ Com2))
43baibr 537 . . . 4 (𝑅 ∈ RingOps → (𝑅 ∈ Com2 ↔ 𝑅 ∈ CRingOps))
52, 4syl 17 . . 3 (𝑅 ∈ PrRing → (𝑅 ∈ Com2 ↔ 𝑅 ∈ CRingOps))
65pm5.32i 575 . 2 ((𝑅 ∈ PrRing ∧ 𝑅 ∈ Com2) ↔ (𝑅 ∈ PrRing ∧ 𝑅 ∈ CRingOps))
71, 6bitri 276 1 (𝑅 ∈ Dmn ↔ (𝑅 ∈ PrRing ∧ 𝑅 ∈ CRingOps))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396  wcel 2107  RingOpscrngo 35040  Com2ccm2 35135  CRingOpsccring 35139  PrRingcprrng 35192  Dmncdmn 35193
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-rex 3148  df-rab 3151  df-v 3501  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-iota 6311  df-fv 6359  df-crngo 35140  df-prrngo 35194  df-dmn 35195
This theorem is referenced by:  dmncrng  35202  flddmn  35204  isdmn3  35220
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