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Theorem isdomn 19617
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 6426 . . 3 (𝑟 = 𝑅 → (Base‘𝑟) ∈ V)
2 fveq2 6411 . . . 4 (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
3 isdomn.b . . . 4 𝐵 = (Base‘𝑅)
42, 3syl6eqr 2851 . . 3 (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
5 fvexd 6426 . . . 4 ((𝑟 = 𝑅𝑏 = 𝐵) → (0g𝑟) ∈ V)
6 fveq2 6411 . . . . . 6 (𝑟 = 𝑅 → (0g𝑟) = (0g𝑅))
76adantr 473 . . . . 5 ((𝑟 = 𝑅𝑏 = 𝐵) → (0g𝑟) = (0g𝑅))
8 isdomn.z . . . . 5 0 = (0g𝑅)
97, 8syl6eqr 2851 . . . 4 ((𝑟 = 𝑅𝑏 = 𝐵) → (0g𝑟) = 0 )
10 simplr 786 . . . . 5 (((𝑟 = 𝑅𝑏 = 𝐵) ∧ 𝑧 = 0 ) → 𝑏 = 𝐵)
11 fveq2 6411 . . . . . . . . . 10 (𝑟 = 𝑅 → (.r𝑟) = (.r𝑅))
12 isdomn.t . . . . . . . . . 10 · = (.r𝑅)
1311, 12syl6eqr 2851 . . . . . . . . 9 (𝑟 = 𝑅 → (.r𝑟) = · )
1413oveqdr 6906 . . . . . . . 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 943 . . . . . . . 8 (𝑧 = 0 → ((𝑥 = 𝑧𝑦 = 𝑧) ↔ (𝑥 = 0𝑦 = 0 )))
2019adantl 474 . . . . . . 7 (((𝑟 = 𝑅𝑏 = 𝐵) ∧ 𝑧 = 0 ) → ((𝑥 = 𝑧𝑦 = 𝑧) ↔ (𝑥 = 0𝑦 = 0 )))
2116, 20imbi12d 336 . . . . . 6 (((𝑟 = 𝑅𝑏 = 𝐵) ∧ 𝑧 = 0 ) → (((𝑥(.r𝑟)𝑦) = 𝑧 → (𝑥 = 𝑧𝑦 = 𝑧)) ↔ ((𝑥 · 𝑦) = 0 → (𝑥 = 0𝑦 = 0 ))))
2210, 21raleqbidv 3335 . . . . 5 (((𝑟 = 𝑅𝑏 = 𝐵) ∧ 𝑧 = 0 ) → (∀𝑦𝑏 ((𝑥(.r𝑟)𝑦) = 𝑧 → (𝑥 = 𝑧𝑦 = 𝑧)) ↔ ∀𝑦𝐵 ((𝑥 · 𝑦) = 0 → (𝑥 = 0𝑦 = 0 ))))
2310, 22raleqbidv 3335 . . . 4 (((𝑟 = 𝑅𝑏 = 𝐵) ∧ 𝑧 = 0 ) → (∀𝑥𝑏𝑦𝑏 ((𝑥(.r𝑟)𝑦) = 𝑧 → (𝑥 = 𝑧𝑦 = 𝑧)) ↔ ∀𝑥𝐵𝑦𝐵 ((𝑥 · 𝑦) = 0 → (𝑥 = 0𝑦 = 0 ))))
245, 9, 23sbcied2 3671 . . 3 ((𝑟 = 𝑅𝑏 = 𝐵) → ([(0g𝑟) / 𝑧]𝑥𝑏𝑦𝑏 ((𝑥(.r𝑟)𝑦) = 𝑧 → (𝑥 = 𝑧𝑦 = 𝑧)) ↔ ∀𝑥𝐵𝑦𝐵 ((𝑥 · 𝑦) = 0 → (𝑥 = 0𝑦 = 0 ))))
251, 4, 24sbcied2 3671 . 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 19607 . 2 Domn = {𝑟 ∈ NzRing ∣ [(Base‘𝑟) / 𝑏][(0g𝑟) / 𝑧]𝑥𝑏𝑦𝑏 ((𝑥(.r𝑟)𝑦) = 𝑧 → (𝑥 = 𝑧𝑦 = 𝑧))}
2725, 26elrab2 3560 1 (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ ∀𝑥𝐵𝑦𝐵 ((𝑥 · 𝑦) = 0 → (𝑥 = 0𝑦 = 0 ))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  wo 874   = wceq 1653  wcel 2157  wral 3089  Vcvv 3385  [wsbc 3633  cfv 6101  (class class class)co 6878  Basecbs 16184  .rcmulr 16268  0gc0g 16415  NzRingcnzr 19580  Domncdomn 19603
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-nul 4983
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-iota 6064  df-fv 6109  df-ov 6881  df-domn 19607
This theorem is referenced by:  domnnzr  19618  domneq0  19620  isdomn2  19622  opprdomn  19624  abvn0b  19625  znfld  20230  ply1domn  24224  fta1b  24270  isdomn3  38567
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