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Theorem isdomn2 19501
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 2760 . . 3 (.r𝑅) = (.r𝑅)
3 isdomn2.z . . 3 0 = (0g𝑅)
41, 2, 3isdomn 19496 . 2 (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ ∀𝑥𝐵𝑦𝐵 ((𝑥(.r𝑅)𝑦) = 0 → (𝑥 = 0𝑦 = 0 ))))
5 dfss3 3733 . . . 4 ((𝐵 ∖ { 0 }) ⊆ 𝐸 ↔ ∀𝑥 ∈ (𝐵 ∖ { 0 })𝑥𝐸)
6 isdomn2.t . . . . . . . . 9 𝐸 = (RLReg‘𝑅)
76, 1, 2, 3isrrg 19490 . . . . . . . 8 (𝑥𝐸 ↔ (𝑥𝐵 ∧ ∀𝑦𝐵 ((𝑥(.r𝑅)𝑦) = 0𝑦 = 0 )))
87baib 982 . . . . . . 7 (𝑥𝐵 → (𝑥𝐸 ↔ ∀𝑦𝐵 ((𝑥(.r𝑅)𝑦) = 0𝑦 = 0 )))
98imbi2d 329 . . . . . 6 (𝑥𝐵 → ((𝑥0𝑥𝐸) ↔ (𝑥0 → ∀𝑦𝐵 ((𝑥(.r𝑅)𝑦) = 0𝑦 = 0 ))))
109ralbiia 3117 . . . . 5 (∀𝑥𝐵 (𝑥0𝑥𝐸) ↔ ∀𝑥𝐵 (𝑥0 → ∀𝑦𝐵 ((𝑥(.r𝑅)𝑦) = 0𝑦 = 0 )))
11 eldifsn 4462 . . . . . . . 8 (𝑥 ∈ (𝐵 ∖ { 0 }) ↔ (𝑥𝐵𝑥0 ))
1211imbi1i 338 . . . . . . 7 ((𝑥 ∈ (𝐵 ∖ { 0 }) → 𝑥𝐸) ↔ ((𝑥𝐵𝑥0 ) → 𝑥𝐸))
13 impexp 461 . . . . . . 7 (((𝑥𝐵𝑥0 ) → 𝑥𝐸) ↔ (𝑥𝐵 → (𝑥0𝑥𝐸)))
1412, 13bitri 264 . . . . . 6 ((𝑥 ∈ (𝐵 ∖ { 0 }) → 𝑥𝐸) ↔ (𝑥𝐵 → (𝑥0𝑥𝐸)))
1514ralbii2 3116 . . . . 5 (∀𝑥 ∈ (𝐵 ∖ { 0 })𝑥𝐸 ↔ ∀𝑥𝐵 (𝑥0𝑥𝐸))
16 con34b 305 . . . . . . . . 9 (((𝑥(.r𝑅)𝑦) = 0 → (𝑥 = 0𝑦 = 0 )) ↔ (¬ (𝑥 = 0𝑦 = 0 ) → ¬ (𝑥(.r𝑅)𝑦) = 0 ))
17 impexp 461 . . . . . . . . . 10 (((¬ 𝑥 = 0 ∧ ¬ 𝑦 = 0 ) → ¬ (𝑥(.r𝑅)𝑦) = 0 ) ↔ (¬ 𝑥 = 0 → (¬ 𝑦 = 0 → ¬ (𝑥(.r𝑅)𝑦) = 0 )))
18 ioran 512 . . . . . . . . . . 11 (¬ (𝑥 = 0𝑦 = 0 ) ↔ (¬ 𝑥 = 0 ∧ ¬ 𝑦 = 0 ))
1918imbi1i 338 . . . . . . . . . 10 ((¬ (𝑥 = 0𝑦 = 0 ) → ¬ (𝑥(.r𝑅)𝑦) = 0 ) ↔ ((¬ 𝑥 = 0 ∧ ¬ 𝑦 = 0 ) → ¬ (𝑥(.r𝑅)𝑦) = 0 ))
20 df-ne 2933 . . . . . . . . . . 11 (𝑥0 ↔ ¬ 𝑥 = 0 )
21 con34b 305 . . . . . . . . . . 11 (((𝑥(.r𝑅)𝑦) = 0𝑦 = 0 ) ↔ (¬ 𝑦 = 0 → ¬ (𝑥(.r𝑅)𝑦) = 0 ))
2220, 21imbi12i 339 . . . . . . . . . 10 ((𝑥0 → ((𝑥(.r𝑅)𝑦) = 0𝑦 = 0 )) ↔ (¬ 𝑥 = 0 → (¬ 𝑦 = 0 → ¬ (𝑥(.r𝑅)𝑦) = 0 )))
2317, 19, 223bitr4i 292 . . . . . . . . 9 ((¬ (𝑥 = 0𝑦 = 0 ) → ¬ (𝑥(.r𝑅)𝑦) = 0 ) ↔ (𝑥0 → ((𝑥(.r𝑅)𝑦) = 0𝑦 = 0 )))
2416, 23bitri 264 . . . . . . . 8 (((𝑥(.r𝑅)𝑦) = 0 → (𝑥 = 0𝑦 = 0 )) ↔ (𝑥0 → ((𝑥(.r𝑅)𝑦) = 0𝑦 = 0 )))
2524ralbii 3118 . . . . . . 7 (∀𝑦𝐵 ((𝑥(.r𝑅)𝑦) = 0 → (𝑥 = 0𝑦 = 0 )) ↔ ∀𝑦𝐵 (𝑥0 → ((𝑥(.r𝑅)𝑦) = 0𝑦 = 0 )))
26 r19.21v 3098 . . . . . . 7 (∀𝑦𝐵 (𝑥0 → ((𝑥(.r𝑅)𝑦) = 0𝑦 = 0 )) ↔ (𝑥0 → ∀𝑦𝐵 ((𝑥(.r𝑅)𝑦) = 0𝑦 = 0 )))
2725, 26bitri 264 . . . . . 6 (∀𝑦𝐵 ((𝑥(.r𝑅)𝑦) = 0 → (𝑥 = 0𝑦 = 0 )) ↔ (𝑥0 → ∀𝑦𝐵 ((𝑥(.r𝑅)𝑦) = 0𝑦 = 0 )))
2827ralbii 3118 . . . . 5 (∀𝑥𝐵𝑦𝐵 ((𝑥(.r𝑅)𝑦) = 0 → (𝑥 = 0𝑦 = 0 )) ↔ ∀𝑥𝐵 (𝑥0 → ∀𝑦𝐵 ((𝑥(.r𝑅)𝑦) = 0𝑦 = 0 )))
2910, 15, 283bitr4i 292 . . . 4 (∀𝑥 ∈ (𝐵 ∖ { 0 })𝑥𝐸 ↔ ∀𝑥𝐵𝑦𝐵 ((𝑥(.r𝑅)𝑦) = 0 → (𝑥 = 0𝑦 = 0 )))
305, 29bitr2i 265 . . 3 (∀𝑥𝐵𝑦𝐵 ((𝑥(.r𝑅)𝑦) = 0 → (𝑥 = 0𝑦 = 0 )) ↔ (𝐵 ∖ { 0 }) ⊆ 𝐸)
3130anbi2i 732 . 2 ((𝑅 ∈ NzRing ∧ ∀𝑥𝐵𝑦𝐵 ((𝑥(.r𝑅)𝑦) = 0 → (𝑥 = 0𝑦 = 0 ))) ↔ (𝑅 ∈ NzRing ∧ (𝐵 ∖ { 0 }) ⊆ 𝐸))
324, 31bitri 264 1 (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ (𝐵 ∖ { 0 }) ⊆ 𝐸))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383   = wceq 1632  wcel 2139  wne 2932  wral 3050  cdif 3712  wss 3715  {csn 4321  cfv 6049  (class class class)co 6813  Basecbs 16059  .rcmulr 16144  0gc0g 16302  NzRingcnzr 19459  RLRegcrlreg 19481  Domncdomn 19482
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-iota 6012  df-fun 6051  df-fv 6057  df-ov 6816  df-rlreg 19485  df-domn 19486
This theorem is referenced by:  domnrrg  19502  drngdomn  19505
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