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Theorem domnnzr 19276
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 2620 . . 3 (Base‘𝑅) = (Base‘𝑅)
2 eqid 2620 . . 3 (.r𝑅) = (.r𝑅)
3 eqid 2620 . . 3 (0g𝑅) = (0g𝑅)
41, 2, 3isdomn 19275 . 2 (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(.r𝑅)𝑦) = (0g𝑅) → (𝑥 = (0g𝑅) ∨ 𝑦 = (0g𝑅)))))
54simplbi 476 1 (𝑅 ∈ Domn → 𝑅 ∈ NzRing)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 383   = wceq 1481  wcel 1988  wral 2909  cfv 5876  (class class class)co 6635  Basecbs 15838  .rcmulr 15923  0gc0g 16081  NzRingcnzr 19238  Domncdomn 19261
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-nul 4780
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-iota 5839  df-fv 5884  df-ov 6638  df-domn 19265
This theorem is referenced by:  domnring  19277  opprdomn  19282  abvn0b  19283  fidomndrng  19288  domnchr  19861  znidomb  19891  nrgdomn  22456  ply1domn  23864  fta1glem1  23906  fta1glem2  23907  fta1b  23910  lgsqrlem4  25055  idomrootle  37592  deg1mhm  37604
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