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Theorem domnnzr 20910
Description: A domain is a nonzero ring. (Contributed by Mario Carneiro, 28-Mar-2015.)
Assertion
Ref Expression
domnnzr (𝑅 ∈ Domn → 𝑅 ∈ NzRing)

Proof of Theorem domnnzr
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2732 . . 3 (Base‘𝑅) = (Base‘𝑅)
2 eqid 2732 . . 3 (.r𝑅) = (.r𝑅)
3 eqid 2732 . . 3 (0g𝑅) = (0g𝑅)
41, 2, 3isdomn 20909 . 2 (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(.r𝑅)𝑦) = (0g𝑅) → (𝑥 = (0g𝑅) ∨ 𝑦 = (0g𝑅)))))
54simplbi 498 1 (𝑅 ∈ Domn → 𝑅 ∈ NzRing)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 845   = wceq 1541  wcel 2106  wral 3061  cfv 6543  (class class class)co 7408  Basecbs 17143  .rcmulr 17197  0gc0g 17384  NzRingcnzr 20290  Domncdomn 20895
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-nul 5306
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-ov 7411  df-domn 20899
This theorem is referenced by:  domnring  20911  isdomn4  20917  opprdomn  20918  abvn0b  20919  fidomndrng  20925  domnchr  21083  znidomb  21116  nrgdomn  24187  ply1domn  25640  fta1glem1  25682  fta1glem2  25683  fta1b  25686  lgsqrlem4  26849  qsidomlem1  32566  idomrootle  41927  deg1mhm  41939
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