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Theorem domnnzr 19989
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 2825 . . 3 (Base‘𝑅) = (Base‘𝑅)
2 eqid 2825 . . 3 (.r𝑅) = (.r𝑅)
3 eqid 2825 . . 3 (0g𝑅) = (0g𝑅)
41, 2, 3isdomn 19988 . 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 843   = wceq 1530  wcel 2107  wral 3142  cfv 6351  (class class class)co 7151  Basecbs 16475  .rcmulr 16558  0gc0g 16705  NzRingcnzr 19951  Domncdomn 19974
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  ax-nul 5206
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-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  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-ov 7154  df-domn 19978
This theorem is referenced by:  domnring  19990  opprdomn  19995  abvn0b  19996  fidomndrng  20001  domnchr  20595  znidomb  20624  nrgdomn  23195  ply1domn  24632  fta1glem1  24674  fta1glem2  24675  fta1b  24678  lgsqrlem4  25839  qsidomlem1  30869  idomrootle  39656  deg1mhm  39668
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