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Theorem domnnzr 20779
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 2765 . . 3 (Base‘𝑅) = (Base‘𝑅)
2 eqid 2765 . . 3 (.r𝑅) = (.r𝑅)
3 eqid 2765 . . 3 (0g𝑅) = (0g𝑅)
41, 2, 3isdomn 20778 . 2 (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(.r𝑅)𝑦) = (0g𝑅) → (𝑥 = (0g𝑅) ∨ 𝑦 = (0g𝑅)))))
54simplbi 501 1 (𝑅 ∈ Domn → 𝑅 ∈ NzRing)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 860   = wceq 1563  wcel 2145  wral 3079  cfv 6525  (class class class)co 7400  Basecbs 17257  .rcmulr 17299  0gc0g 17480  NzRingcnzr 20583  Domncdomn 20765
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-nul 5260
This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3080  df-rab 3418  df-v 3459  df-sbc 3748  df-dif 3910  df-un 3912  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5105  df-iota 6481  df-fv 6533  df-ov 7403  df-domn 20768
This theorem is referenced by:  domnring  20780  isdomn4  20788  fidomndrng  20843  abvn0b  20905  qsidomlem1  21437  domnchr  21639  znidomb  21668  nrgdomn  24785  ply1domn  26238  fta1glem1  26282  fta1glem2  26283  fta1b  26286  idomrootle  26287  lgsqrlem4  27467  domnprodn0  33506  domnprodeq0  33507  subrdom  33513  ricdomn1  33517  fracfld  33539  1arithufdlem1  33746  ply1dg1rt  33782  deg1prod  33785  mplidomlem  33829  vietadeg1  33880  assafld  33939  idomnnzpownz  42756  idomnnzgmulnz  42757  deg1gprod  42764  deg1pow  42765  domnexpgn0cl  43148  fiabv  43161  deg1mhm  43784
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