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Theorem isdomn2 20069
Description: A ring is a domain iff all nonzero elements are nonzero-divisors. (Contributed by Mario Carneiro, 28-Mar-2015.)
Hypotheses
Ref Expression
isdomn2.b 𝐵 = (Base‘𝑅)
isdomn2.t 𝐸 = (RLReg‘𝑅)
isdomn2.z 0 = (0g𝑅)
Assertion
Ref Expression
isdomn2 (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ (𝐵 ∖ { 0 }) ⊆ 𝐸))

Proof of Theorem isdomn2
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isdomn2.b . . 3 𝐵 = (Base‘𝑅)
2 eqid 2801 . . 3 (.r𝑅) = (.r𝑅)
3 isdomn2.z . . 3 0 = (0g𝑅)
41, 2, 3isdomn 20064 . 2 (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ ∀𝑥𝐵𝑦𝐵 ((𝑥(.r𝑅)𝑦) = 0 → (𝑥 = 0𝑦 = 0 ))))
5 dfss3 3906 . . . 4 ((𝐵 ∖ { 0 }) ⊆ 𝐸 ↔ ∀𝑥 ∈ (𝐵 ∖ { 0 })𝑥𝐸)
6 isdomn2.t . . . . . . . . 9 𝐸 = (RLReg‘𝑅)
76, 1, 2, 3isrrg 20058 . . . . . . . 8 (𝑥𝐸 ↔ (𝑥𝐵 ∧ ∀𝑦𝐵 ((𝑥(.r𝑅)𝑦) = 0𝑦 = 0 )))
87baib 539 . . . . . . 7 (𝑥𝐵 → (𝑥𝐸 ↔ ∀𝑦𝐵 ((𝑥(.r𝑅)𝑦) = 0𝑦 = 0 )))
98imbi2d 344 . . . . . 6 (𝑥𝐵 → ((𝑥0𝑥𝐸) ↔ (𝑥0 → ∀𝑦𝐵 ((𝑥(.r𝑅)𝑦) = 0𝑦 = 0 ))))
109ralbiia 3135 . . . . 5 (∀𝑥𝐵 (𝑥0𝑥𝐸) ↔ ∀𝑥𝐵 (𝑥0 → ∀𝑦𝐵 ((𝑥(.r𝑅)𝑦) = 0𝑦 = 0 )))
11 eldifsn 4683 . . . . . . . 8 (𝑥 ∈ (𝐵 ∖ { 0 }) ↔ (𝑥𝐵𝑥0 ))
1211imbi1i 353 . . . . . . 7 ((𝑥 ∈ (𝐵 ∖ { 0 }) → 𝑥𝐸) ↔ ((𝑥𝐵𝑥0 ) → 𝑥𝐸))
13 impexp 454 . . . . . . 7 (((𝑥𝐵𝑥0 ) → 𝑥𝐸) ↔ (𝑥𝐵 → (𝑥0𝑥𝐸)))
1412, 13bitri 278 . . . . . 6 ((𝑥 ∈ (𝐵 ∖ { 0 }) → 𝑥𝐸) ↔ (𝑥𝐵 → (𝑥0𝑥𝐸)))
1514ralbii2 3134 . . . . 5 (∀𝑥 ∈ (𝐵 ∖ { 0 })𝑥𝐸 ↔ ∀𝑥𝐵 (𝑥0𝑥𝐸))
16 con34b 319 . . . . . . . . 9 (((𝑥(.r𝑅)𝑦) = 0 → (𝑥 = 0𝑦 = 0 )) ↔ (¬ (𝑥 = 0𝑦 = 0 ) → ¬ (𝑥(.r𝑅)𝑦) = 0 ))
17 impexp 454 . . . . . . . . . 10 (((¬ 𝑥 = 0 ∧ ¬ 𝑦 = 0 ) → ¬ (𝑥(.r𝑅)𝑦) = 0 ) ↔ (¬ 𝑥 = 0 → (¬ 𝑦 = 0 → ¬ (𝑥(.r𝑅)𝑦) = 0 )))
18 ioran 981 . . . . . . . . . . 11 (¬ (𝑥 = 0𝑦 = 0 ) ↔ (¬ 𝑥 = 0 ∧ ¬ 𝑦 = 0 ))
1918imbi1i 353 . . . . . . . . . 10 ((¬ (𝑥 = 0𝑦 = 0 ) → ¬ (𝑥(.r𝑅)𝑦) = 0 ) ↔ ((¬ 𝑥 = 0 ∧ ¬ 𝑦 = 0 ) → ¬ (𝑥(.r𝑅)𝑦) = 0 ))
20 df-ne 2991 . . . . . . . . . . 11 (𝑥0 ↔ ¬ 𝑥 = 0 )
21 con34b 319 . . . . . . . . . . 11 (((𝑥(.r𝑅)𝑦) = 0𝑦 = 0 ) ↔ (¬ 𝑦 = 0 → ¬ (𝑥(.r𝑅)𝑦) = 0 ))
2220, 21imbi12i 354 . . . . . . . . . 10 ((𝑥0 → ((𝑥(.r𝑅)𝑦) = 0𝑦 = 0 )) ↔ (¬ 𝑥 = 0 → (¬ 𝑦 = 0 → ¬ (𝑥(.r𝑅)𝑦) = 0 )))
2317, 19, 223bitr4i 306 . . . . . . . . 9 ((¬ (𝑥 = 0𝑦 = 0 ) → ¬ (𝑥(.r𝑅)𝑦) = 0 ) ↔ (𝑥0 → ((𝑥(.r𝑅)𝑦) = 0𝑦 = 0 )))
2416, 23bitri 278 . . . . . . . 8 (((𝑥(.r𝑅)𝑦) = 0 → (𝑥 = 0𝑦 = 0 )) ↔ (𝑥0 → ((𝑥(.r𝑅)𝑦) = 0𝑦 = 0 )))
2524ralbii 3136 . . . . . . 7 (∀𝑦𝐵 ((𝑥(.r𝑅)𝑦) = 0 → (𝑥 = 0𝑦 = 0 )) ↔ ∀𝑦𝐵 (𝑥0 → ((𝑥(.r𝑅)𝑦) = 0𝑦 = 0 )))
26 r19.21v 3145 . . . . . . 7 (∀𝑦𝐵 (𝑥0 → ((𝑥(.r𝑅)𝑦) = 0𝑦 = 0 )) ↔ (𝑥0 → ∀𝑦𝐵 ((𝑥(.r𝑅)𝑦) = 0𝑦 = 0 )))
2725, 26bitri 278 . . . . . 6 (∀𝑦𝐵 ((𝑥(.r𝑅)𝑦) = 0 → (𝑥 = 0𝑦 = 0 )) ↔ (𝑥0 → ∀𝑦𝐵 ((𝑥(.r𝑅)𝑦) = 0𝑦 = 0 )))
2827ralbii 3136 . . . . 5 (∀𝑥𝐵𝑦𝐵 ((𝑥(.r𝑅)𝑦) = 0 → (𝑥 = 0𝑦 = 0 )) ↔ ∀𝑥𝐵 (𝑥0 → ∀𝑦𝐵 ((𝑥(.r𝑅)𝑦) = 0𝑦 = 0 )))
2910, 15, 283bitr4i 306 . . . 4 (∀𝑥 ∈ (𝐵 ∖ { 0 })𝑥𝐸 ↔ ∀𝑥𝐵𝑦𝐵 ((𝑥(.r𝑅)𝑦) = 0 → (𝑥 = 0𝑦 = 0 )))
305, 29bitr2i 279 . . 3 (∀𝑥𝐵𝑦𝐵 ((𝑥(.r𝑅)𝑦) = 0 → (𝑥 = 0𝑦 = 0 )) ↔ (𝐵 ∖ { 0 }) ⊆ 𝐸)
3130anbi2i 625 . 2 ((𝑅 ∈ NzRing ∧ ∀𝑥𝐵𝑦𝐵 ((𝑥(.r𝑅)𝑦) = 0 → (𝑥 = 0𝑦 = 0 ))) ↔ (𝑅 ∈ NzRing ∧ (𝐵 ∖ { 0 }) ⊆ 𝐸))
324, 31bitri 278 1 (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ (𝐵 ∖ { 0 }) ⊆ 𝐸))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wo 844   = wceq 1538  wcel 2112  wne 2990  wral 3109  cdif 3881  wss 3884  {csn 4528  cfv 6328  (class class class)co 7139  Basecbs 16479  .rcmulr 16562  0gc0g 16709  NzRingcnzr 20027  RLRegcrlreg 20049  Domncdomn 20050
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 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298
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 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-iota 6287  df-fun 6330  df-fv 6336  df-ov 7142  df-rlreg 20053  df-domn 20054
This theorem is referenced by:  domnrrg  20070  drngdomn  20073
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