Theorem domnnzr 21115

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.

Step	Hyp	Ref	Expression
1		eqid 2731	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
2		eqid 2731	. . 3 ⊢ (.r‘𝑅) = (.r‘𝑅)
3		eqid 2731	. . 3 ⊢ (0g‘𝑅) = (0g‘𝑅)
4	1, 2, 3	isdomn 21114	. 2 ⊢ (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(.r‘𝑅)𝑦) = (0g‘𝑅) → (𝑥 = (0g‘𝑅) ∨ 𝑦 = (0g‘𝑅)))))
5	4	simplbi 497	1 ⊢ (𝑅 ∈ Domn → 𝑅 ∈ NzRing)

Syntax hints:   → wi 4   ∨ wo 844   = wceq 1540   ∈ wcel 2105  ∀wral 3060  ‘cfv 6543  (class class class)co 7412  Basecbs 17151  ._rcmulr 17205  0_gc0g 17392  NzRingcnzr 20407  Domncdomn 21100

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702  ax-nul 5306

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rab 3432  df-v 3475  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-ov 7415  df-domn 21104

This theorem is referenced by:  domnring  21116  isdomn4  21122  opprdomn  21123  abvn0b  21124  fidomndrng  21130  domnchr  21307  znidomb  21340  nrgdomn  24421  ply1domn  25890  fta1glem1  25932  fta1glem2  25933  fta1b  25936  lgsqrlem4  27103  qsidomlem1  32860  idomrootle  42252  deg1mhm  42264