Theorem domnnzr 20622

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 2730	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
2		eqid 2730	. . 3 ⊢ (.r‘𝑅) = (.r‘𝑅)
3		eqid 2730	. . 3 ⊢ (0g‘𝑅) = (0g‘𝑅)
4	1, 2, 3	isdomn 20621	. 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 3045  ‘cfv 6514  (class class class)co 7390  Basecbs 17186  ._rcmulr 17228  0_gc0g 17409  NzRingcnzr 20428  Domncdomn 20608

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 2702  ax-nul 5264

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 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-rab 3409  df-v 3452  df-sbc 3757  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-iota 6467  df-fv 6522  df-ov 7393  df-domn 20611

This theorem is referenced by:  domnring  20623  isdomn4  20632  fidomndrng  20689  abvn0b  20752  domnchr  21449  znidomb  21478  nrgdomn  24566  ply1domn  26036  fta1glem1  26080  fta1glem2  26081  fta1b  26084  idomrootle  26085  lgsqrlem4  27267  domnprodn0  33233  subrdom  33242  fracfld  33265  qsidomlem1  33430  1arithufdlem1  33522  ply1dg1rt  33555  assafld  33640  idomnnzpownz  42127  idomnnzgmulnz  42128  deg1gprod  42135  deg1pow  42136  domnexpgn0cl  42518  fiabv  42531  deg1mhm  43196