MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isdomn2 Structured version   Visualization version   GIF version

Theorem isdomn2 19213
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 2626 . . 3 (.r𝑅) = (.r𝑅)
3 isdomn2.z . . 3 0 = (0g𝑅)
41, 2, 3isdomn 19208 . 2 (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ ∀𝑥𝐵𝑦𝐵 ((𝑥(.r𝑅)𝑦) = 0 → (𝑥 = 0𝑦 = 0 ))))
5 dfss3 3578 . . . 4 ((𝐵 ∖ { 0 }) ⊆ 𝐸 ↔ ∀𝑥 ∈ (𝐵 ∖ { 0 })𝑥𝐸)
6 isdomn2.t . . . . . . . . 9 𝐸 = (RLReg‘𝑅)
76, 1, 2, 3isrrg 19202 . . . . . . . 8 (𝑥𝐸 ↔ (𝑥𝐵 ∧ ∀𝑦𝐵 ((𝑥(.r𝑅)𝑦) = 0𝑦 = 0 )))
87baib 943 . . . . . . 7 (𝑥𝐵 → (𝑥𝐸 ↔ ∀𝑦𝐵 ((𝑥(.r𝑅)𝑦) = 0𝑦 = 0 )))
98imbi2d 330 . . . . . 6 (𝑥𝐵 → ((𝑥0𝑥𝐸) ↔ (𝑥0 → ∀𝑦𝐵 ((𝑥(.r𝑅)𝑦) = 0𝑦 = 0 ))))
109ralbiia 2978 . . . . 5 (∀𝑥𝐵 (𝑥0𝑥𝐸) ↔ ∀𝑥𝐵 (𝑥0 → ∀𝑦𝐵 ((𝑥(.r𝑅)𝑦) = 0𝑦 = 0 )))
11 eldifsn 4292 . . . . . . . 8 (𝑥 ∈ (𝐵 ∖ { 0 }) ↔ (𝑥𝐵𝑥0 ))
1211imbi1i 339 . . . . . . 7 ((𝑥 ∈ (𝐵 ∖ { 0 }) → 𝑥𝐸) ↔ ((𝑥𝐵𝑥0 ) → 𝑥𝐸))
13 impexp 462 . . . . . . 7 (((𝑥𝐵𝑥0 ) → 𝑥𝐸) ↔ (𝑥𝐵 → (𝑥0𝑥𝐸)))
1412, 13bitri 264 . . . . . 6 ((𝑥 ∈ (𝐵 ∖ { 0 }) → 𝑥𝐸) ↔ (𝑥𝐵 → (𝑥0𝑥𝐸)))
1514ralbii2 2977 . . . . 5 (∀𝑥 ∈ (𝐵 ∖ { 0 })𝑥𝐸 ↔ ∀𝑥𝐵 (𝑥0𝑥𝐸))
16 con34b 306 . . . . . . . . 9 (((𝑥(.r𝑅)𝑦) = 0 → (𝑥 = 0𝑦 = 0 )) ↔ (¬ (𝑥 = 0𝑦 = 0 ) → ¬ (𝑥(.r𝑅)𝑦) = 0 ))
17 impexp 462 . . . . . . . . . 10 (((¬ 𝑥 = 0 ∧ ¬ 𝑦 = 0 ) → ¬ (𝑥(.r𝑅)𝑦) = 0 ) ↔ (¬ 𝑥 = 0 → (¬ 𝑦 = 0 → ¬ (𝑥(.r𝑅)𝑦) = 0 )))
18 ioran 511 . . . . . . . . . . 11 (¬ (𝑥 = 0𝑦 = 0 ) ↔ (¬ 𝑥 = 0 ∧ ¬ 𝑦 = 0 ))
1918imbi1i 339 . . . . . . . . . 10 ((¬ (𝑥 = 0𝑦 = 0 ) → ¬ (𝑥(.r𝑅)𝑦) = 0 ) ↔ ((¬ 𝑥 = 0 ∧ ¬ 𝑦 = 0 ) → ¬ (𝑥(.r𝑅)𝑦) = 0 ))
20 df-ne 2797 . . . . . . . . . . 11 (𝑥0 ↔ ¬ 𝑥 = 0 )
21 con34b 306 . . . . . . . . . . 11 (((𝑥(.r𝑅)𝑦) = 0𝑦 = 0 ) ↔ (¬ 𝑦 = 0 → ¬ (𝑥(.r𝑅)𝑦) = 0 ))
2220, 21imbi12i 340 . . . . . . . . . 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 2979 . . . . . . 7 (∀𝑦𝐵 ((𝑥(.r𝑅)𝑦) = 0 → (𝑥 = 0𝑦 = 0 )) ↔ ∀𝑦𝐵 (𝑥0 → ((𝑥(.r𝑅)𝑦) = 0𝑦 = 0 )))
26 r19.21v 2959 . . . . . . 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 2979 . . . . 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 729 . 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 383  wa 384   = wceq 1480  wcel 1992  wne 2796  wral 2912  cdif 3557  wss 3560  {csn 4153  cfv 5850  (class class class)co 6605  Basecbs 15776  .rcmulr 15858  0gc0g 16016  NzRingcnzr 19171  RLRegcrlreg 19193  Domncdomn 19194
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 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872
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 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-iota 5813  df-fun 5852  df-fv 5858  df-ov 6608  df-rlreg 19197  df-domn 19198
This theorem is referenced by:  domnrrg  19214  drngdomn  19217
  Copyright terms: Public domain W3C validator