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Theorem domnnzr 20781
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 2733 . . 3 (Base‘𝑅) = (Base‘𝑅)
2 eqid 2733 . . 3 (.r𝑅) = (.r𝑅)
3 eqid 2733 . . 3 (0g𝑅) = (0g𝑅)
41, 2, 3isdomn 20780 . 2 (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(.r𝑅)𝑦) = (0g𝑅) → (𝑥 = (0g𝑅) ∨ 𝑦 = (0g𝑅)))))
54simplbi 499 1 (𝑅 ∈ Domn → 𝑅 ∈ NzRing)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 846   = wceq 1542  wcel 2107  wral 3061  cfv 6497  (class class class)co 7358  Basecbs 17088  .rcmulr 17139  0gc0g 17326  NzRingcnzr 20743  Domncdomn 20766
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-nul 5264
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-ral 3062  df-rab 3407  df-v 3446  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-iota 6449  df-fv 6505  df-ov 7361  df-domn 20770
This theorem is referenced by:  domnring  20782  opprdomn  20787  abvn0b  20788  fidomndrng  20794  domnchr  20951  znidomb  20984  nrgdomn  24051  ply1domn  25504  fta1glem1  25546  fta1glem2  25547  fta1b  25550  lgsqrlem4  26713  qsidomlem1  32273  isdomn4  40670  idomrootle  41565  deg1mhm  41577
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