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Theorem isdomn2 21115
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 𝐡 = (Baseβ€˜π‘…)
isdomn2.t 𝐸 = (RLRegβ€˜π‘…)
isdomn2.z 0 = (0gβ€˜π‘…)
Assertion
Ref Expression
isdomn2 (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ (𝐡 βˆ– { 0 }) βŠ† 𝐸))
Proof of Theorem isdomn2
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isdomn2.b . . 3 𝐡 = (Baseβ€˜π‘…)
2 eqid 2732 . . 3 (.rβ€˜π‘…) = (.rβ€˜π‘…)
3 isdomn2.z . . 3 0 = (0gβ€˜π‘…)
41, 2, 3isdomn 21110 . 2 (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 ((π‘₯(.rβ€˜π‘…)𝑦) = 0 β†’ (π‘₯ = 0 ∨ 𝑦 = 0 ))))
5 dfss3 3970 . . . 4 ((𝐡 βˆ– { 0 }) βŠ† 𝐸 ↔ βˆ€π‘₯ ∈ (𝐡 βˆ– { 0 })π‘₯ ∈ 𝐸)
6 isdomn2.t . . . . . . . . 9 𝐸 = (RLRegβ€˜π‘…)
76, 1, 2, 3isrrg 21104 . . . . . . . 8 (π‘₯ ∈ 𝐸 ↔ (π‘₯ ∈ 𝐡 ∧ βˆ€π‘¦ ∈ 𝐡 ((π‘₯(.rβ€˜π‘…)𝑦) = 0 β†’ 𝑦 = 0 )))
87baib 536 . . . . . . 7 (π‘₯ ∈ 𝐡 β†’ (π‘₯ ∈ 𝐸 ↔ βˆ€π‘¦ ∈ 𝐡 ((π‘₯(.rβ€˜π‘…)𝑦) = 0 β†’ 𝑦 = 0 )))
98imbi2d 340 . . . . . 6 (π‘₯ ∈ 𝐡 β†’ ((π‘₯ β‰  0 β†’ π‘₯ ∈ 𝐸) ↔ (π‘₯ β‰  0 β†’ βˆ€π‘¦ ∈ 𝐡 ((π‘₯(.rβ€˜π‘…)𝑦) = 0 β†’ 𝑦 = 0 ))))
109ralbiia 3091 . . . . 5 (βˆ€π‘₯ ∈ 𝐡 (π‘₯ β‰  0 β†’ π‘₯ ∈ 𝐸) ↔ βˆ€π‘₯ ∈ 𝐡 (π‘₯ β‰  0 β†’ βˆ€π‘¦ ∈ 𝐡 ((π‘₯(.rβ€˜π‘…)𝑦) = 0 β†’ 𝑦 = 0 )))
11 eldifsn 4790 . . . . . . . 8 (π‘₯ ∈ (𝐡 βˆ– { 0 }) ↔ (π‘₯ ∈ 𝐡 ∧ π‘₯ β‰  0 ))
1211imbi1i 349 . . . . . . 7 ((π‘₯ ∈ (𝐡 βˆ– { 0 }) β†’ π‘₯ ∈ 𝐸) ↔ ((π‘₯ ∈ 𝐡 ∧ π‘₯ β‰  0 ) β†’ π‘₯ ∈ 𝐸))
13 impexp 451 . . . . . . 7 (((π‘₯ ∈ 𝐡 ∧ π‘₯ β‰  0 ) β†’ π‘₯ ∈ 𝐸) ↔ (π‘₯ ∈ 𝐡 β†’ (π‘₯ β‰  0 β†’ π‘₯ ∈ 𝐸)))
1412, 13bitri 274 . . . . . 6 ((π‘₯ ∈ (𝐡 βˆ– { 0 }) β†’ π‘₯ ∈ 𝐸) ↔ (π‘₯ ∈ 𝐡 β†’ (π‘₯ β‰  0 β†’ π‘₯ ∈ 𝐸)))
1514ralbii2 3089 . . . . 5 (βˆ€π‘₯ ∈ (𝐡 βˆ– { 0 })π‘₯ ∈ 𝐸 ↔ βˆ€π‘₯ ∈ 𝐡 (π‘₯ β‰  0 β†’ π‘₯ ∈ 𝐸))
16 con34b 315 . . . . . . . . 9 (((π‘₯(.rβ€˜π‘…)𝑦) = 0 β†’ (π‘₯ = 0 ∨ 𝑦 = 0 )) ↔ (Β¬ (π‘₯ = 0 ∨ 𝑦 = 0 ) β†’ Β¬ (π‘₯(.rβ€˜π‘…)𝑦) = 0 ))
17 impexp 451 . . . . . . . . . 10 (((Β¬ π‘₯ = 0 ∧ Β¬ 𝑦 = 0 ) β†’ Β¬ (π‘₯(.rβ€˜π‘…)𝑦) = 0 ) ↔ (Β¬ π‘₯ = 0 β†’ (Β¬ 𝑦 = 0 β†’ Β¬ (π‘₯(.rβ€˜π‘…)𝑦) = 0 )))
18 ioran 982 . . . . . . . . . . 11 (Β¬ (π‘₯ = 0 ∨ 𝑦 = 0 ) ↔ (Β¬ π‘₯ = 0 ∧ Β¬ 𝑦 = 0 ))
1918imbi1i 349 . . . . . . . . . 10 ((Β¬ (π‘₯ = 0 ∨ 𝑦 = 0 ) β†’ Β¬ (π‘₯(.rβ€˜π‘…)𝑦) = 0 ) ↔ ((Β¬ π‘₯ = 0 ∧ Β¬ 𝑦 = 0 ) β†’ Β¬ (π‘₯(.rβ€˜π‘…)𝑦) = 0 ))
20 df-ne 2941 . . . . . . . . . . 11 (π‘₯ β‰  0 ↔ Β¬ π‘₯ = 0 )
21 con34b 315 . . . . . . . . . . 11 (((π‘₯(.rβ€˜π‘…)𝑦) = 0 β†’ 𝑦 = 0 ) ↔ (Β¬ 𝑦 = 0 β†’ Β¬ (π‘₯(.rβ€˜π‘…)𝑦) = 0 ))
2220, 21imbi12i 350 . . . . . . . . . 10 ((π‘₯ β‰  0 β†’ ((π‘₯(.rβ€˜π‘…)𝑦) = 0 β†’ 𝑦 = 0 )) ↔ (Β¬ π‘₯ = 0 β†’ (Β¬ 𝑦 = 0 β†’ Β¬ (π‘₯(.rβ€˜π‘…)𝑦) = 0 )))
2317, 19, 223bitr4i 302 . . . . . . . . 9 ((Β¬ (π‘₯ = 0 ∨ 𝑦 = 0 ) β†’ Β¬ (π‘₯(.rβ€˜π‘…)𝑦) = 0 ) ↔ (π‘₯ β‰  0 β†’ ((π‘₯(.rβ€˜π‘…)𝑦) = 0 β†’ 𝑦 = 0 )))
2416, 23bitri 274 . . . . . . . 8 (((π‘₯(.rβ€˜π‘…)𝑦) = 0 β†’ (π‘₯ = 0 ∨ 𝑦 = 0 )) ↔ (π‘₯ β‰  0 β†’ ((π‘₯(.rβ€˜π‘…)𝑦) = 0 β†’ 𝑦 = 0 )))
2524ralbii 3093 . . . . . . 7 (βˆ€π‘¦ ∈ 𝐡 ((π‘₯(.rβ€˜π‘…)𝑦) = 0 β†’ (π‘₯ = 0 ∨ 𝑦 = 0 )) ↔ βˆ€π‘¦ ∈ 𝐡 (π‘₯ β‰  0 β†’ ((π‘₯(.rβ€˜π‘…)𝑦) = 0 β†’ 𝑦 = 0 )))
26 r19.21v 3179 . . . . . . 7 (βˆ€π‘¦ ∈ 𝐡 (π‘₯ β‰  0 β†’ ((π‘₯(.rβ€˜π‘…)𝑦) = 0 β†’ 𝑦 = 0 )) ↔ (π‘₯ β‰  0 β†’ βˆ€π‘¦ ∈ 𝐡 ((π‘₯(.rβ€˜π‘…)𝑦) = 0 β†’ 𝑦 = 0 )))
2725, 26bitri 274 . . . . . 6 (βˆ€π‘¦ ∈ 𝐡 ((π‘₯(.rβ€˜π‘…)𝑦) = 0 β†’ (π‘₯ = 0 ∨ 𝑦 = 0 )) ↔ (π‘₯ β‰  0 β†’ βˆ€π‘¦ ∈ 𝐡 ((π‘₯(.rβ€˜π‘…)𝑦) = 0 β†’ 𝑦 = 0 )))
2827ralbii 3093 . . . . 5 (βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 ((π‘₯(.rβ€˜π‘…)𝑦) = 0 β†’ (π‘₯ = 0 ∨ 𝑦 = 0 )) ↔ βˆ€π‘₯ ∈ 𝐡 (π‘₯ β‰  0 β†’ βˆ€π‘¦ ∈ 𝐡 ((π‘₯(.rβ€˜π‘…)𝑦) = 0 β†’ 𝑦 = 0 )))
2910, 15, 283bitr4i 302 . . . 4 (βˆ€π‘₯ ∈ (𝐡 βˆ– { 0 })π‘₯ ∈ 𝐸 ↔ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 ((π‘₯(.rβ€˜π‘…)𝑦) = 0 β†’ (π‘₯ = 0 ∨ 𝑦 = 0 )))
305, 29bitr2i 275 . . 3 (βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 ((π‘₯(.rβ€˜π‘…)𝑦) = 0 β†’ (π‘₯ = 0 ∨ 𝑦 = 0 )) ↔ (𝐡 βˆ– { 0 }) βŠ† 𝐸)
3130anbi2i 623 . 2 ((𝑅 ∈ NzRing ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 ((π‘₯(.rβ€˜π‘…)𝑦) = 0 β†’ (π‘₯ = 0 ∨ 𝑦 = 0 ))) ↔ (𝑅 ∈ NzRing ∧ (𝐡 βˆ– { 0 }) βŠ† 𝐸))
324, 31bitri 274 1 (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ (𝐡 βˆ– { 0 }) βŠ† 𝐸))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 845   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  βˆ€wral 3061   βˆ– cdif 3945   βŠ† wss 3948  {csn 4628  β€˜cfv 6543  (class class class)co 7411  Basecbs 17148  .rcmulr 17202  0gc0g 17389  NzRingcnzr 20403  RLRegcrlreg 21095  Domncdomn 21096
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7414  df-rlreg 21099  df-domn 21100
This theorem is referenced by:  domnrrg  21116  drngdomn  21121
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