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Theorem isdomn2 20668
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 2736 . . 3 (.r𝑅) = (.r𝑅)
3 isdomn2.z . . 3 0 = (0g𝑅)
41, 2, 3isdomn 20663 . 2 (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ ∀𝑥𝐵𝑦𝐵 ((𝑥(.r𝑅)𝑦) = 0 → (𝑥 = 0𝑦 = 0 ))))
5 dfss3 3919 . . . 4 ((𝐵 ∖ { 0 }) ⊆ 𝐸 ↔ ∀𝑥 ∈ (𝐵 ∖ { 0 })𝑥𝐸)
6 isdomn2.t . . . . . . . . 9 𝐸 = (RLReg‘𝑅)
76, 1, 2, 3isrrg 20657 . . . . . . . 8 (𝑥𝐸 ↔ (𝑥𝐵 ∧ ∀𝑦𝐵 ((𝑥(.r𝑅)𝑦) = 0𝑦 = 0 )))
87baib 536 . . . . . . 7 (𝑥𝐵 → (𝑥𝐸 ↔ ∀𝑦𝐵 ((𝑥(.r𝑅)𝑦) = 0𝑦 = 0 )))
98imbi2d 340 . . . . . 6 (𝑥𝐵 → ((𝑥0𝑥𝐸) ↔ (𝑥0 → ∀𝑦𝐵 ((𝑥(.r𝑅)𝑦) = 0𝑦 = 0 ))))
109ralbiia 3090 . . . . 5 (∀𝑥𝐵 (𝑥0𝑥𝐸) ↔ ∀𝑥𝐵 (𝑥0 → ∀𝑦𝐵 ((𝑥(.r𝑅)𝑦) = 0𝑦 = 0 )))
11 eldifsn 4733 . . . . . . . 8 (𝑥 ∈ (𝐵 ∖ { 0 }) ↔ (𝑥𝐵𝑥0 ))
1211imbi1i 349 . . . . . . 7 ((𝑥 ∈ (𝐵 ∖ { 0 }) → 𝑥𝐸) ↔ ((𝑥𝐵𝑥0 ) → 𝑥𝐸))
13 impexp 451 . . . . . . 7 (((𝑥𝐵𝑥0 ) → 𝑥𝐸) ↔ (𝑥𝐵 → (𝑥0𝑥𝐸)))
1412, 13bitri 274 . . . . . 6 ((𝑥 ∈ (𝐵 ∖ { 0 }) → 𝑥𝐸) ↔ (𝑥𝐵 → (𝑥0𝑥𝐸)))
1514ralbii2 3088 . . . . 5 (∀𝑥 ∈ (𝐵 ∖ { 0 })𝑥𝐸 ↔ ∀𝑥𝐵 (𝑥0𝑥𝐸))
16 con34b 315 . . . . . . . . 9 (((𝑥(.r𝑅)𝑦) = 0 → (𝑥 = 0𝑦 = 0 )) ↔ (¬ (𝑥 = 0𝑦 = 0 ) → ¬ (𝑥(.r𝑅)𝑦) = 0 ))
17 impexp 451 . . . . . . . . . 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 349 . . . . . . . . . 10 ((¬ (𝑥 = 0𝑦 = 0 ) → ¬ (𝑥(.r𝑅)𝑦) = 0 ) ↔ ((¬ 𝑥 = 0 ∧ ¬ 𝑦 = 0 ) → ¬ (𝑥(.r𝑅)𝑦) = 0 ))
20 df-ne 2941 . . . . . . . . . . 11 (𝑥0 ↔ ¬ 𝑥 = 0 )
21 con34b 315 . . . . . . . . . . 11 (((𝑥(.r𝑅)𝑦) = 0𝑦 = 0 ) ↔ (¬ 𝑦 = 0 → ¬ (𝑥(.r𝑅)𝑦) = 0 ))
2220, 21imbi12i 350 . . . . . . . . . 10 ((𝑥0 → ((𝑥(.r𝑅)𝑦) = 0𝑦 = 0 )) ↔ (¬ 𝑥 = 0 → (¬ 𝑦 = 0 → ¬ (𝑥(.r𝑅)𝑦) = 0 )))
2317, 19, 223bitr4i 302 . . . . . . . . 9 ((¬ (𝑥 = 0𝑦 = 0 ) → ¬ (𝑥(.r𝑅)𝑦) = 0 ) ↔ (𝑥0 → ((𝑥(.r𝑅)𝑦) = 0𝑦 = 0 )))
2416, 23bitri 274 . . . . . . . 8 (((𝑥(.r𝑅)𝑦) = 0 → (𝑥 = 0𝑦 = 0 )) ↔ (𝑥0 → ((𝑥(.r𝑅)𝑦) = 0𝑦 = 0 )))
2524ralbii 3092 . . . . . . 7 (∀𝑦𝐵 ((𝑥(.r𝑅)𝑦) = 0 → (𝑥 = 0𝑦 = 0 )) ↔ ∀𝑦𝐵 (𝑥0 → ((𝑥(.r𝑅)𝑦) = 0𝑦 = 0 )))
26 r19.21v 3172 . . . . . . 7 (∀𝑦𝐵 (𝑥0 → ((𝑥(.r𝑅)𝑦) = 0𝑦 = 0 )) ↔ (𝑥0 → ∀𝑦𝐵 ((𝑥(.r𝑅)𝑦) = 0𝑦 = 0 )))
2725, 26bitri 274 . . . . . 6 (∀𝑦𝐵 ((𝑥(.r𝑅)𝑦) = 0 → (𝑥 = 0𝑦 = 0 )) ↔ (𝑥0 → ∀𝑦𝐵 ((𝑥(.r𝑅)𝑦) = 0𝑦 = 0 )))
2827ralbii 3092 . . . . 5 (∀𝑥𝐵𝑦𝐵 ((𝑥(.r𝑅)𝑦) = 0 → (𝑥 = 0𝑦 = 0 )) ↔ ∀𝑥𝐵 (𝑥0 → ∀𝑦𝐵 ((𝑥(.r𝑅)𝑦) = 0𝑦 = 0 )))
2910, 15, 283bitr4i 302 . . . 4 (∀𝑥 ∈ (𝐵 ∖ { 0 })𝑥𝐸 ↔ ∀𝑥𝐵𝑦𝐵 ((𝑥(.r𝑅)𝑦) = 0 → (𝑥 = 0𝑦 = 0 )))
305, 29bitr2i 275 . . 3 (∀𝑥𝐵𝑦𝐵 ((𝑥(.r𝑅)𝑦) = 0 → (𝑥 = 0𝑦 = 0 )) ↔ (𝐵 ∖ { 0 }) ⊆ 𝐸)
3130anbi2i 623 . 2 ((𝑅 ∈ NzRing ∧ ∀𝑥𝐵𝑦𝐵 ((𝑥(.r𝑅)𝑦) = 0 → (𝑥 = 0𝑦 = 0 ))) ↔ (𝑅 ∈ NzRing ∧ (𝐵 ∖ { 0 }) ⊆ 𝐸))
324, 31bitri 274 1 (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ (𝐵 ∖ { 0 }) ⊆ 𝐸))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wo 844   = wceq 1540  wcel 2105  wne 2940  wral 3061  cdif 3894  wss 3897  {csn 4572  cfv 6473  (class class class)co 7329  Basecbs 17001  .rcmulr 17052  0gc0g 17239  NzRingcnzr 20626  RLRegcrlreg 20648  Domncdomn 20649
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5240  ax-nul 5247  ax-pr 5369
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-sbc 3727  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4269  df-if 4473  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4852  df-br 5090  df-opab 5152  df-mpt 5173  df-id 5512  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-iota 6425  df-fun 6475  df-fv 6481  df-ov 7332  df-rlreg 20652  df-domn 20653
This theorem is referenced by:  domnrrg  20669  drngdomn  20672
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