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Theorem isdomn2 20915
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 2733 . . 3 (.rβ€˜π‘…) = (.rβ€˜π‘…)
3 isdomn2.z . . 3 0 = (0gβ€˜π‘…)
41, 2, 3isdomn 20910 . 2 (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 ((π‘₯(.rβ€˜π‘…)𝑦) = 0 β†’ (π‘₯ = 0 ∨ 𝑦 = 0 ))))
5 dfss3 3971 . . . 4 ((𝐡 βˆ– { 0 }) βŠ† 𝐸 ↔ βˆ€π‘₯ ∈ (𝐡 βˆ– { 0 })π‘₯ ∈ 𝐸)
6 isdomn2.t . . . . . . . . 9 𝐸 = (RLRegβ€˜π‘…)
76, 1, 2, 3isrrg 20904 . . . . . . . 8 (π‘₯ ∈ 𝐸 ↔ (π‘₯ ∈ 𝐡 ∧ βˆ€π‘¦ ∈ 𝐡 ((π‘₯(.rβ€˜π‘…)𝑦) = 0 β†’ 𝑦 = 0 )))
87baib 537 . . . . . . 7 (π‘₯ ∈ 𝐡 β†’ (π‘₯ ∈ 𝐸 ↔ βˆ€π‘¦ ∈ 𝐡 ((π‘₯(.rβ€˜π‘…)𝑦) = 0 β†’ 𝑦 = 0 )))
98imbi2d 341 . . . . . 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 3092 . . . . 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 4791 . . . . . . . 8 (π‘₯ ∈ (𝐡 βˆ– { 0 }) ↔ (π‘₯ ∈ 𝐡 ∧ π‘₯ β‰  0 ))
1211imbi1i 350 . . . . . . 7 ((π‘₯ ∈ (𝐡 βˆ– { 0 }) β†’ π‘₯ ∈ 𝐸) ↔ ((π‘₯ ∈ 𝐡 ∧ π‘₯ β‰  0 ) β†’ π‘₯ ∈ 𝐸))
13 impexp 452 . . . . . . 7 (((π‘₯ ∈ 𝐡 ∧ π‘₯ β‰  0 ) β†’ π‘₯ ∈ 𝐸) ↔ (π‘₯ ∈ 𝐡 β†’ (π‘₯ β‰  0 β†’ π‘₯ ∈ 𝐸)))
1412, 13bitri 275 . . . . . 6 ((π‘₯ ∈ (𝐡 βˆ– { 0 }) β†’ π‘₯ ∈ 𝐸) ↔ (π‘₯ ∈ 𝐡 β†’ (π‘₯ β‰  0 β†’ π‘₯ ∈ 𝐸)))
1514ralbii2 3090 . . . . 5 (βˆ€π‘₯ ∈ (𝐡 βˆ– { 0 })π‘₯ ∈ 𝐸 ↔ βˆ€π‘₯ ∈ 𝐡 (π‘₯ β‰  0 β†’ π‘₯ ∈ 𝐸))
16 con34b 316 . . . . . . . . 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 452 . . . . . . . . . 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 983 . . . . . . . . . . 11 (Β¬ (π‘₯ = 0 ∨ 𝑦 = 0 ) ↔ (Β¬ π‘₯ = 0 ∧ Β¬ 𝑦 = 0 ))
1918imbi1i 350 . . . . . . . . . 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 2942 . . . . . . . . . . 11 (π‘₯ β‰  0 ↔ Β¬ π‘₯ = 0 )
21 con34b 316 . . . . . . . . . . 11 (((π‘₯(.rβ€˜π‘…)𝑦) = 0 β†’ 𝑦 = 0 ) ↔ (Β¬ 𝑦 = 0 β†’ Β¬ (π‘₯(.rβ€˜π‘…)𝑦) = 0 ))
2220, 21imbi12i 351 . . . . . . . . . 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 303 . . . . . . . . 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 275 . . . . . . . 8 (((π‘₯(.rβ€˜π‘…)𝑦) = 0 β†’ (π‘₯ = 0 ∨ 𝑦 = 0 )) ↔ (π‘₯ β‰  0 β†’ ((π‘₯(.rβ€˜π‘…)𝑦) = 0 β†’ 𝑦 = 0 )))
2524ralbii 3094 . . . . . . 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 3180 . . . . . . 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 275 . . . . . 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 3094 . . . . 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 303 . . . 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 276 . . 3 (βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 ((π‘₯(.rβ€˜π‘…)𝑦) = 0 β†’ (π‘₯ = 0 ∨ 𝑦 = 0 )) ↔ (𝐡 βˆ– { 0 }) βŠ† 𝐸)
3130anbi2i 624 . 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 275 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 397   ∨ wo 846   = wceq 1542   ∈ wcel 2107   β‰  wne 2941  βˆ€wral 3062   βˆ– cdif 3946   βŠ† wss 3949  {csn 4629  β€˜cfv 6544  (class class class)co 7409  Basecbs 17144  .rcmulr 17198  0gc0g 17385  NzRingcnzr 20291  RLRegcrlreg 20895  Domncdomn 20896
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552  df-ov 7412  df-rlreg 20899  df-domn 20900
This theorem is referenced by:  domnrrg  20916  drngdomn  20921
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