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Theorem domnnzr 20911
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 20910 . 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 3062  cfv 6544  (class class class)co 7409  Basecbs 17144  .rcmulr 17198  0gc0g 17385  NzRingcnzr 20291  Domncdomn 20896
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 5307
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 2942  df-ral 3063  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-iota 6496  df-fv 6552  df-ov 7412  df-domn 20900
This theorem is referenced by:  domnring  20912  isdomn4  20918  opprdomn  20919  abvn0b  20920  fidomndrng  20926  domnchr  21084  znidomb  21117  nrgdomn  24188  ply1domn  25641  fta1glem1  25683  fta1glem2  25684  fta1b  25687  lgsqrlem4  26852  qsidomlem1  32571  idomrootle  41937  deg1mhm  41949
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