Theorem domnnzr 20615

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 2729	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
2		eqid 2729	. . 3 ⊢ (.r‘𝑅) = (.r‘𝑅)
3		eqid 2729	. . 3 ⊢ (0g‘𝑅) = (0g‘𝑅)
4	1, 2, 3	isdomn 20614	. 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 847   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ‘cfv 6511  (class class class)co 7387  Basecbs 17179  ._rcmulr 17221  0_gc0g 17402  NzRingcnzr 20421  Domncdomn 20601

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-nul 5261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rab 3406  df-v 3449  df-sbc 3754  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-iota 6464  df-fv 6519  df-ov 7390  df-domn 20604

This theorem is referenced by:  domnring  20616  isdomn4  20625  fidomndrng  20682  abvn0b  20745  domnchr  21442  znidomb  21471  nrgdomn  24559  ply1domn  26029  fta1glem1  26073  fta1glem2  26074  fta1b  26077  idomrootle  26078  lgsqrlem4  27260  domnprodn0  33226  subrdom  33235  fracfld  33258  qsidomlem1  33423  1arithufdlem1  33515  ply1dg1rt  33548  assafld  33633  idomnnzpownz  42120  idomnnzgmulnz  42121  deg1gprod  42128  deg1pow  42129  domnexpgn0cl  42511  fiabv  42524  deg1mhm  43189