Theorem domnnzr 20917

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 2732	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
2		eqid 2732	. . 3 ⊢ (.r‘𝑅) = (.r‘𝑅)
3		eqid 2732	. . 3 ⊢ (0g‘𝑅) = (0g‘𝑅)
4	1, 2, 3	isdomn 20916	. 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 498	1 ⊢ (𝑅 ∈ Domn → 𝑅 ∈ NzRing)

Syntax hints:   → wi 4   ∨ wo 845   = wceq 1541   ∈ wcel 2106  ∀wral 3061  ‘cfv 6543  (class class class)co 7411  Basecbs 17146  ._rcmulr 17200  0_gc0g 17387  NzRingcnzr 20295  Domncdomn 20902

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-nul 5306

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rab 3433  df-v 3476  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 7414  df-domn 20906

This theorem is referenced by:  domnring  20918  isdomn4  20924  opprdomn  20925  abvn0b  20926  fidomndrng  20932  domnchr  21090  znidomb  21123  nrgdomn  24195  ply1domn  25648  fta1glem1  25690  fta1glem2  25691  fta1b  25694  lgsqrlem4  26859  qsidomlem1  32616  idomrootle  42019  deg1mhm  42031