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Theorem isdomn 20060
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 6660 . . 3 (𝑟 = 𝑅 → (Base‘𝑟) ∈ V)
2 fveq2 6645 . . . 4 (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
3 isdomn.b . . . 4 𝐵 = (Base‘𝑅)
42, 3eqtr4di 2851 . . 3 (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
5 fvexd 6660 . . . 4 ((𝑟 = 𝑅𝑏 = 𝐵) → (0g𝑟) ∈ V)
6 fveq2 6645 . . . . . 6 (𝑟 = 𝑅 → (0g𝑟) = (0g𝑅))
76adantr 484 . . . . 5 ((𝑟 = 𝑅𝑏 = 𝐵) → (0g𝑟) = (0g𝑅))
8 isdomn.z . . . . 5 0 = (0g𝑅)
97, 8eqtr4di 2851 . . . 4 ((𝑟 = 𝑅𝑏 = 𝐵) → (0g𝑟) = 0 )
10 simplr 768 . . . . 5 (((𝑟 = 𝑅𝑏 = 𝐵) ∧ 𝑧 = 0 ) → 𝑏 = 𝐵)
11 fveq2 6645 . . . . . . . . . 10 (𝑟 = 𝑅 → (.r𝑟) = (.r𝑅))
12 isdomn.t . . . . . . . . . 10 · = (.r𝑅)
1311, 12eqtr4di 2851 . . . . . . . . 9 (𝑟 = 𝑅 → (.r𝑟) = · )
1413oveqdr 7163 . . . . . . . 8 ((𝑟 = 𝑅𝑏 = 𝐵) → (𝑥(.r𝑟)𝑦) = (𝑥 · 𝑦))
15 id 22 . . . . . . . 8 (𝑧 = 0𝑧 = 0 )
1614, 15eqeqan12d 2815 . . . . . . 7 (((𝑟 = 𝑅𝑏 = 𝐵) ∧ 𝑧 = 0 ) → ((𝑥(.r𝑟)𝑦) = 𝑧 ↔ (𝑥 · 𝑦) = 0 ))
17 eqeq2 2810 . . . . . . . . 9 (𝑧 = 0 → (𝑥 = 𝑧𝑥 = 0 ))
18 eqeq2 2810 . . . . . . . . 9 (𝑧 = 0 → (𝑦 = 𝑧𝑦 = 0 ))
1917, 18orbi12d 916 . . . . . . . 8 (𝑧 = 0 → ((𝑥 = 𝑧𝑦 = 𝑧) ↔ (𝑥 = 0𝑦 = 0 )))
2019adantl 485 . . . . . . 7 (((𝑟 = 𝑅𝑏 = 𝐵) ∧ 𝑧 = 0 ) → ((𝑥 = 𝑧𝑦 = 𝑧) ↔ (𝑥 = 0𝑦 = 0 )))
2116, 20imbi12d 348 . . . . . 6 (((𝑟 = 𝑅𝑏 = 𝐵) ∧ 𝑧 = 0 ) → (((𝑥(.r𝑟)𝑦) = 𝑧 → (𝑥 = 𝑧𝑦 = 𝑧)) ↔ ((𝑥 · 𝑦) = 0 → (𝑥 = 0𝑦 = 0 ))))
2210, 21raleqbidv 3354 . . . . 5 (((𝑟 = 𝑅𝑏 = 𝐵) ∧ 𝑧 = 0 ) → (∀𝑦𝑏 ((𝑥(.r𝑟)𝑦) = 𝑧 → (𝑥 = 𝑧𝑦 = 𝑧)) ↔ ∀𝑦𝐵 ((𝑥 · 𝑦) = 0 → (𝑥 = 0𝑦 = 0 ))))
2310, 22raleqbidv 3354 . . . 4 (((𝑟 = 𝑅𝑏 = 𝐵) ∧ 𝑧 = 0 ) → (∀𝑥𝑏𝑦𝑏 ((𝑥(.r𝑟)𝑦) = 𝑧 → (𝑥 = 𝑧𝑦 = 𝑧)) ↔ ∀𝑥𝐵𝑦𝐵 ((𝑥 · 𝑦) = 0 → (𝑥 = 0𝑦 = 0 ))))
245, 9, 23sbcied2 3763 . . 3 ((𝑟 = 𝑅𝑏 = 𝐵) → ([(0g𝑟) / 𝑧]𝑥𝑏𝑦𝑏 ((𝑥(.r𝑟)𝑦) = 𝑧 → (𝑥 = 𝑧𝑦 = 𝑧)) ↔ ∀𝑥𝐵𝑦𝐵 ((𝑥 · 𝑦) = 0 → (𝑥 = 0𝑦 = 0 ))))
251, 4, 24sbcied2 3763 . 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 20050 . 2 Domn = {𝑟 ∈ NzRing ∣ [(Base‘𝑟) / 𝑏][(0g𝑟) / 𝑧]𝑥𝑏𝑦𝑏 ((𝑥(.r𝑟)𝑦) = 𝑧 → (𝑥 = 𝑧𝑦 = 𝑧))}
2725, 26elrab2 3631 1 (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ ∀𝑥𝐵𝑦𝐵 ((𝑥 · 𝑦) = 0 → (𝑥 = 0𝑦 = 0 ))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wo 844   = wceq 1538  wcel 2111  wral 3106  Vcvv 3441  [wsbc 3720  cfv 6324  (class class class)co 7135  Basecbs 16475  .rcmulr 16558  0gc0g 16705  NzRingcnzr 20023  Domncdomn 20046
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-nul 5174
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-iota 6283  df-fv 6332  df-ov 7138  df-domn 20050
This theorem is referenced by:  domnnzr  20061  domneq0  20063  isdomn2  20065  opprdomn  20067  abvn0b  20068  znfld  20252  ply1domn  24724  fta1b  24770  prmidl0  31034  qsidomlem2  31037  isdomn3  40143
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