Theorem domnnzr 19510

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 2771	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
2		eqid 2771	. . 3 ⊢ (.r‘𝑅) = (.r‘𝑅)
3		eqid 2771	. . 3 ⊢ (0g‘𝑅) = (0g‘𝑅)
4	1, 2, 3	isdomn 19509	. 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 485	1 ⊢ (𝑅 ∈ Domn → 𝑅 ∈ NzRing)

Syntax hints:   → wi 4   ∨ wo 836   = wceq 1631   ∈ wcel 2145  ∀wral 3061  ‘cfv 6031  (class class class)co 6793  Basecbs 16064  ._rcmulr 16150  0_gc0g 16308  NzRingcnzr 19472  Domncdomn 19495

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-nul 4923

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-iota 5994  df-fv 6039  df-ov 6796  df-domn 19499

This theorem is referenced by:  domnring  19511  opprdomn  19516  abvn0b  19517  fidomndrng  19522  domnchr  20095  znidomb  20125  nrgdomn  22695  ply1domn  24103  fta1glem1  24145  fta1glem2  24146  fta1b  24149  lgsqrlem4  25295  idomrootle  38299  deg1mhm  38311