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Theorem isdomn 19213
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 fvex 6158 . . . 4 (Base‘𝑟) ∈ V
21a1i 11 . . 3 (𝑟 = 𝑅 → (Base‘𝑟) ∈ V)
3 fveq2 6148 . . . 4 (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
4 isdomn.b . . . 4 𝐵 = (Base‘𝑅)
53, 4syl6eqr 2673 . . 3 (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
6 fvex 6158 . . . . 5 (0g𝑟) ∈ V
76a1i 11 . . . 4 ((𝑟 = 𝑅𝑏 = 𝐵) → (0g𝑟) ∈ V)
8 fveq2 6148 . . . . . 6 (𝑟 = 𝑅 → (0g𝑟) = (0g𝑅))
98adantr 481 . . . . 5 ((𝑟 = 𝑅𝑏 = 𝐵) → (0g𝑟) = (0g𝑅))
10 isdomn.z . . . . 5 0 = (0g𝑅)
119, 10syl6eqr 2673 . . . 4 ((𝑟 = 𝑅𝑏 = 𝐵) → (0g𝑟) = 0 )
12 simplr 791 . . . . 5 (((𝑟 = 𝑅𝑏 = 𝐵) ∧ 𝑧 = 0 ) → 𝑏 = 𝐵)
13 fveq2 6148 . . . . . . . . . 10 (𝑟 = 𝑅 → (.r𝑟) = (.r𝑅))
14 isdomn.t . . . . . . . . . 10 · = (.r𝑅)
1513, 14syl6eqr 2673 . . . . . . . . 9 (𝑟 = 𝑅 → (.r𝑟) = · )
1615oveqdr 6628 . . . . . . . 8 ((𝑟 = 𝑅𝑏 = 𝐵) → (𝑥(.r𝑟)𝑦) = (𝑥 · 𝑦))
17 id 22 . . . . . . . 8 (𝑧 = 0𝑧 = 0 )
1816, 17eqeqan12d 2637 . . . . . . 7 (((𝑟 = 𝑅𝑏 = 𝐵) ∧ 𝑧 = 0 ) → ((𝑥(.r𝑟)𝑦) = 𝑧 ↔ (𝑥 · 𝑦) = 0 ))
19 eqeq2 2632 . . . . . . . . 9 (𝑧 = 0 → (𝑥 = 𝑧𝑥 = 0 ))
20 eqeq2 2632 . . . . . . . . 9 (𝑧 = 0 → (𝑦 = 𝑧𝑦 = 0 ))
2119, 20orbi12d 745 . . . . . . . 8 (𝑧 = 0 → ((𝑥 = 𝑧𝑦 = 𝑧) ↔ (𝑥 = 0𝑦 = 0 )))
2221adantl 482 . . . . . . 7 (((𝑟 = 𝑅𝑏 = 𝐵) ∧ 𝑧 = 0 ) → ((𝑥 = 𝑧𝑦 = 𝑧) ↔ (𝑥 = 0𝑦 = 0 )))
2318, 22imbi12d 334 . . . . . 6 (((𝑟 = 𝑅𝑏 = 𝐵) ∧ 𝑧 = 0 ) → (((𝑥(.r𝑟)𝑦) = 𝑧 → (𝑥 = 𝑧𝑦 = 𝑧)) ↔ ((𝑥 · 𝑦) = 0 → (𝑥 = 0𝑦 = 0 ))))
2412, 23raleqbidv 3141 . . . . 5 (((𝑟 = 𝑅𝑏 = 𝐵) ∧ 𝑧 = 0 ) → (∀𝑦𝑏 ((𝑥(.r𝑟)𝑦) = 𝑧 → (𝑥 = 𝑧𝑦 = 𝑧)) ↔ ∀𝑦𝐵 ((𝑥 · 𝑦) = 0 → (𝑥 = 0𝑦 = 0 ))))
2512, 24raleqbidv 3141 . . . 4 (((𝑟 = 𝑅𝑏 = 𝐵) ∧ 𝑧 = 0 ) → (∀𝑥𝑏𝑦𝑏 ((𝑥(.r𝑟)𝑦) = 𝑧 → (𝑥 = 𝑧𝑦 = 𝑧)) ↔ ∀𝑥𝐵𝑦𝐵 ((𝑥 · 𝑦) = 0 → (𝑥 = 0𝑦 = 0 ))))
267, 11, 25sbcied2 3455 . . 3 ((𝑟 = 𝑅𝑏 = 𝐵) → ([(0g𝑟) / 𝑧]𝑥𝑏𝑦𝑏 ((𝑥(.r𝑟)𝑦) = 𝑧 → (𝑥 = 𝑧𝑦 = 𝑧)) ↔ ∀𝑥𝐵𝑦𝐵 ((𝑥 · 𝑦) = 0 → (𝑥 = 0𝑦 = 0 ))))
272, 5, 26sbcied2 3455 . 2 (𝑟 = 𝑅 → ([(Base‘𝑟) / 𝑏][(0g𝑟) / 𝑧]𝑥𝑏𝑦𝑏 ((𝑥(.r𝑟)𝑦) = 𝑧 → (𝑥 = 𝑧𝑦 = 𝑧)) ↔ ∀𝑥𝐵𝑦𝐵 ((𝑥 · 𝑦) = 0 → (𝑥 = 0𝑦 = 0 ))))
28 df-domn 19203 . 2 Domn = {𝑟 ∈ NzRing ∣ [(Base‘𝑟) / 𝑏][(0g𝑟) / 𝑧]𝑥𝑏𝑦𝑏 ((𝑥(.r𝑟)𝑦) = 𝑧 → (𝑥 = 𝑧𝑦 = 𝑧))}
2927, 28elrab2 3348 1 (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ ∀𝑥𝐵𝑦𝐵 ((𝑥 · 𝑦) = 0 → (𝑥 = 0𝑦 = 0 ))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 383  wa 384   = wceq 1480  wcel 1987  wral 2907  Vcvv 3186  [wsbc 3417  cfv 5847  (class class class)co 6604  Basecbs 15781  .rcmulr 15863  0gc0g 16021  NzRingcnzr 19176  Domncdomn 19199
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-nul 4749
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-iota 5810  df-fv 5855  df-ov 6607  df-domn 19203
This theorem is referenced by:  domnnzr  19214  domneq0  19216  isdomn2  19218  opprdomn  19220  abvn0b  19221  znfld  19828  ply1domn  23787  fta1b  23833  isdomn3  37260
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