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Theorem domnnzr 19996
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 2818 . . 3 (Base‘𝑅) = (Base‘𝑅)
2 eqid 2818 . . 3 (.r𝑅) = (.r𝑅)
3 eqid 2818 . . 3 (0g𝑅) = (0g𝑅)
41, 2, 3isdomn 19995 . 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 841   = wceq 1528  wcel 2105  wral 3135  cfv 6348  (class class class)co 7145  Basecbs 16471  .rcmulr 16554  0gc0g 16701  NzRingcnzr 19958  Domncdomn 19981
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-nul 5201
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-iota 6307  df-fv 6356  df-ov 7148  df-domn 19985
This theorem is referenced by:  domnring  19997  opprdomn  20002  abvn0b  20003  fidomndrng  20008  domnchr  20607  znidomb  20636  nrgdomn  23207  ply1domn  24644  fta1glem1  24686  fta1glem2  24687  fta1b  24690  lgsqrlem4  25852  qsidomlem1  30882  idomrootle  39673  deg1mhm  39685
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