Theorem domnnzr 20623

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 2733	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
2		eqid 2733	. . 3 ⊢ (.r‘𝑅) = (.r‘𝑅)
3		eqid 2733	. . 3 ⊢ (0g‘𝑅) = (0g‘𝑅)
4	1, 2, 3	isdomn 20622	. 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 1541   ∈ wcel 2113  ∀wral 3048  ‘cfv 6486  (class class class)co 7352  Basecbs 17122  ._rcmulr 17164  0_gc0g 17345  NzRingcnzr 20429  Domncdomn 20609

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705  ax-nul 5246

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ne 2930  df-ral 3049  df-rab 3397  df-v 3439  df-sbc 3738  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-iota 6442  df-fv 6494  df-ov 7355  df-domn 20612

This theorem is referenced by:  domnring  20624  isdomn4  20633  fidomndrng  20690  abvn0b  20753  domnchr  21471  znidomb  21500  nrgdomn  24587  ply1domn  26057  fta1glem1  26101  fta1glem2  26102  fta1b  26105  idomrootle  26106  lgsqrlem4  27288  domnprodn0  33249  subrdom  33258  fracfld  33281  qsidomlem1  33424  1arithufdlem1  33516  ply1dg1rt  33550  assafld  33671  idomnnzpownz  42246  idomnnzgmulnz  42247  deg1gprod  42254  deg1pow  42255  domnexpgn0cl  42642  fiabv  42655  deg1mhm  43318