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Theorem isdomn 20685
Description: Expand definition of a domain. (Contributed by Mario Carneiro, 28-Mar-2015.)
Hypotheses
Ref Expression
isdomn.b 𝐵 = (Base‘𝑅)
isdomn.t · = (.r𝑅)
isdomn.z 0 = (0g𝑅)
Assertion
Ref Expression
isdomn (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ ∀𝑥𝐵𝑦𝐵 ((𝑥 · 𝑦) = 0 → (𝑥 = 0𝑦 = 0 ))))
Distinct variable groups:   𝑥,𝐵,𝑦   𝑥,𝑅,𝑦   𝑥, 0 ,𝑦
Allowed substitution hints:   · (𝑥,𝑦)

Proof of Theorem isdomn
Dummy variables 𝑏 𝑟 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvexd 6918 . . 3 (𝑟 = 𝑅 → (Base‘𝑟) ∈ V)
2 fveq2 6903 . . . 4 (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
3 isdomn.b . . . 4 𝐵 = (Base‘𝑅)
42, 3eqtr4di 2784 . . 3 (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
5 fvexd 6918 . . . 4 ((𝑟 = 𝑅𝑏 = 𝐵) → (0g𝑟) ∈ V)
6 fveq2 6903 . . . . . 6 (𝑟 = 𝑅 → (0g𝑟) = (0g𝑅))
76adantr 479 . . . . 5 ((𝑟 = 𝑅𝑏 = 𝐵) → (0g𝑟) = (0g𝑅))
8 isdomn.z . . . . 5 0 = (0g𝑅)
97, 8eqtr4di 2784 . . . 4 ((𝑟 = 𝑅𝑏 = 𝐵) → (0g𝑟) = 0 )
10 simplr 767 . . . . 5 (((𝑟 = 𝑅𝑏 = 𝐵) ∧ 𝑧 = 0 ) → 𝑏 = 𝐵)
11 fveq2 6903 . . . . . . . . . 10 (𝑟 = 𝑅 → (.r𝑟) = (.r𝑅))
12 isdomn.t . . . . . . . . . 10 · = (.r𝑅)
1311, 12eqtr4di 2784 . . . . . . . . 9 (𝑟 = 𝑅 → (.r𝑟) = · )
1413oveqdr 7454 . . . . . . . 8 ((𝑟 = 𝑅𝑏 = 𝐵) → (𝑥(.r𝑟)𝑦) = (𝑥 · 𝑦))
15 id 22 . . . . . . . 8 (𝑧 = 0𝑧 = 0 )
1614, 15eqeqan12d 2740 . . . . . . 7 (((𝑟 = 𝑅𝑏 = 𝐵) ∧ 𝑧 = 0 ) → ((𝑥(.r𝑟)𝑦) = 𝑧 ↔ (𝑥 · 𝑦) = 0 ))
17 eqeq2 2738 . . . . . . . . 9 (𝑧 = 0 → (𝑥 = 𝑧𝑥 = 0 ))
18 eqeq2 2738 . . . . . . . . 9 (𝑧 = 0 → (𝑦 = 𝑧𝑦 = 0 ))
1917, 18orbi12d 916 . . . . . . . 8 (𝑧 = 0 → ((𝑥 = 𝑧𝑦 = 𝑧) ↔ (𝑥 = 0𝑦 = 0 )))
2019adantl 480 . . . . . . 7 (((𝑟 = 𝑅𝑏 = 𝐵) ∧ 𝑧 = 0 ) → ((𝑥 = 𝑧𝑦 = 𝑧) ↔ (𝑥 = 0𝑦 = 0 )))
2116, 20imbi12d 343 . . . . . 6 (((𝑟 = 𝑅𝑏 = 𝐵) ∧ 𝑧 = 0 ) → (((𝑥(.r𝑟)𝑦) = 𝑧 → (𝑥 = 𝑧𝑦 = 𝑧)) ↔ ((𝑥 · 𝑦) = 0 → (𝑥 = 0𝑦 = 0 ))))
2210, 21raleqbidv 3330 . . . . 5 (((𝑟 = 𝑅𝑏 = 𝐵) ∧ 𝑧 = 0 ) → (∀𝑦𝑏 ((𝑥(.r𝑟)𝑦) = 𝑧 → (𝑥 = 𝑧𝑦 = 𝑧)) ↔ ∀𝑦𝐵 ((𝑥 · 𝑦) = 0 → (𝑥 = 0𝑦 = 0 ))))
2310, 22raleqbidv 3330 . . . 4 (((𝑟 = 𝑅𝑏 = 𝐵) ∧ 𝑧 = 0 ) → (∀𝑥𝑏𝑦𝑏 ((𝑥(.r𝑟)𝑦) = 𝑧 → (𝑥 = 𝑧𝑦 = 𝑧)) ↔ ∀𝑥𝐵𝑦𝐵 ((𝑥 · 𝑦) = 0 → (𝑥 = 0𝑦 = 0 ))))
245, 9, 23sbcied2 3824 . . 3 ((𝑟 = 𝑅𝑏 = 𝐵) → ([(0g𝑟) / 𝑧]𝑥𝑏𝑦𝑏 ((𝑥(.r𝑟)𝑦) = 𝑧 → (𝑥 = 𝑧𝑦 = 𝑧)) ↔ ∀𝑥𝐵𝑦𝐵 ((𝑥 · 𝑦) = 0 → (𝑥 = 0𝑦 = 0 ))))
251, 4, 24sbcied2 3824 . 2 (𝑟 = 𝑅 → ([(Base‘𝑟) / 𝑏][(0g𝑟) / 𝑧]𝑥𝑏𝑦𝑏 ((𝑥(.r𝑟)𝑦) = 𝑧 → (𝑥 = 𝑧𝑦 = 𝑧)) ↔ ∀𝑥𝐵𝑦𝐵 ((𝑥 · 𝑦) = 0 → (𝑥 = 0𝑦 = 0 ))))
26 df-domn 20675 . 2 Domn = {𝑟 ∈ NzRing ∣ [(Base‘𝑟) / 𝑏][(0g𝑟) / 𝑧]𝑥𝑏𝑦𝑏 ((𝑥(.r𝑟)𝑦) = 𝑧 → (𝑥 = 𝑧𝑦 = 𝑧))}
2725, 26elrab2 3684 1 (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ ∀𝑥𝐵𝑦𝐵 ((𝑥 · 𝑦) = 0 → (𝑥 = 0𝑦 = 0 ))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wo 845   = wceq 1534  wcel 2099  wral 3051  Vcvv 3462  [wsbc 3776  cfv 6556  (class class class)co 7426  Basecbs 17215  .rcmulr 17269  0gc0g 17456  NzRingcnzr 20496  Domncdomn 20672
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-nul 5313
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ne 2931  df-ral 3052  df-rab 3420  df-v 3464  df-sbc 3777  df-dif 3950  df-un 3952  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4916  df-br 5156  df-iota 6508  df-fv 6564  df-ov 7429  df-domn 20675
This theorem is referenced by:  domnnzr  20686  domneq0  20688  isdomn2  20691  isdomn2OLD  20692  isdomn3  20695  isdomn4  20696  opprdomnb  20697  abvn0b  20817  znfld  21560  ply1domn  26154  fta1b  26202  subrdom  33139  prmidl0  33327  qsidomlem2  33330
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