Theorem domnnzr 19656

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 2825	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
2		eqid 2825	. . 3 ⊢ (.r‘𝑅) = (.r‘𝑅)
3		eqid 2825	. . 3 ⊢ (0g‘𝑅) = (0g‘𝑅)
4	1, 2, 3	isdomn 19655	. 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 493	1 ⊢ (𝑅 ∈ Domn → 𝑅 ∈ NzRing)

Syntax hints:   → wi 4   ∨ wo 880   = wceq 1658   ∈ wcel 2166  ∀wral 3117  ‘cfv 6123  (class class class)co 6905  Basecbs 16222  ._rcmulr 16306  0_gc0g 16453  NzRingcnzr 19618  Domncdomn 19641

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-nul 5013

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-iota 6086  df-fv 6131  df-ov 6908  df-domn 19645

This theorem is referenced by:  domnring  19657  opprdomn  19662  abvn0b  19663  fidomndrng  19668  domnchr  20240  znidomb  20269  nrgdomn  22845  ply1domn  24282  fta1glem1  24324  fta1glem2  24325  fta1b  24328  lgsqrlem4  25487  idomrootle  38616  deg1mhm  38628