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Theorem opprdomnb 20717
Description: A class is a domain if and only if its opposite is a domain, biconditional form of opprdomn 20718. (Contributed by SN, 15-Jun-2015.)
Hypothesis
Ref Expression
opprdomn.1 𝑂 = (oppr𝑅)
Assertion
Ref Expression
opprdomnb (𝑅 ∈ Domn ↔ 𝑂 ∈ Domn)

Proof of Theorem opprdomnb
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opprdomn.1 . . . 4 𝑂 = (oppr𝑅)
21opprnzrb 20521 . . 3 (𝑅 ∈ NzRing ↔ 𝑂 ∈ NzRing)
3 eqid 2737 . . . . . 6 (Base‘𝑅) = (Base‘𝑅)
41, 3opprbas 20341 . . . . 5 (Base‘𝑅) = (Base‘𝑂)
5 eqid 2737 . . . . . . . . . 10 (.r𝑅) = (.r𝑅)
6 eqid 2737 . . . . . . . . . 10 (.r𝑂) = (.r𝑂)
73, 5, 1, 6opprmul 20337 . . . . . . . . 9 (𝑦(.r𝑂)𝑥) = (𝑥(.r𝑅)𝑦)
87eqcomi 2746 . . . . . . . 8 (𝑥(.r𝑅)𝑦) = (𝑦(.r𝑂)𝑥)
9 eqid 2737 . . . . . . . . 9 (0g𝑅) = (0g𝑅)
101, 9oppr0 20349 . . . . . . . 8 (0g𝑅) = (0g𝑂)
118, 10eqeq12i 2755 . . . . . . 7 ((𝑥(.r𝑅)𝑦) = (0g𝑅) ↔ (𝑦(.r𝑂)𝑥) = (0g𝑂))
1210eqeq2i 2750 . . . . . . . . 9 (𝑥 = (0g𝑅) ↔ 𝑥 = (0g𝑂))
1310eqeq2i 2750 . . . . . . . . 9 (𝑦 = (0g𝑅) ↔ 𝑦 = (0g𝑂))
1412, 13orbi12i 915 . . . . . . . 8 ((𝑥 = (0g𝑅) ∨ 𝑦 = (0g𝑅)) ↔ (𝑥 = (0g𝑂) ∨ 𝑦 = (0g𝑂)))
15 orcom 871 . . . . . . . 8 ((𝑥 = (0g𝑂) ∨ 𝑦 = (0g𝑂)) ↔ (𝑦 = (0g𝑂) ∨ 𝑥 = (0g𝑂)))
1614, 15bitri 275 . . . . . . 7 ((𝑥 = (0g𝑅) ∨ 𝑦 = (0g𝑅)) ↔ (𝑦 = (0g𝑂) ∨ 𝑥 = (0g𝑂)))
1711, 16imbi12i 350 . . . . . 6 (((𝑥(.r𝑅)𝑦) = (0g𝑅) → (𝑥 = (0g𝑅) ∨ 𝑦 = (0g𝑅))) ↔ ((𝑦(.r𝑂)𝑥) = (0g𝑂) → (𝑦 = (0g𝑂) ∨ 𝑥 = (0g𝑂))))
184, 17raleqbii 3344 . . . . 5 (∀𝑦 ∈ (Base‘𝑅)((𝑥(.r𝑅)𝑦) = (0g𝑅) → (𝑥 = (0g𝑅) ∨ 𝑦 = (0g𝑅))) ↔ ∀𝑦 ∈ (Base‘𝑂)((𝑦(.r𝑂)𝑥) = (0g𝑂) → (𝑦 = (0g𝑂) ∨ 𝑥 = (0g𝑂))))
194, 18raleqbii 3344 . . . 4 (∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(.r𝑅)𝑦) = (0g𝑅) → (𝑥 = (0g𝑅) ∨ 𝑦 = (0g𝑅))) ↔ ∀𝑥 ∈ (Base‘𝑂)∀𝑦 ∈ (Base‘𝑂)((𝑦(.r𝑂)𝑥) = (0g𝑂) → (𝑦 = (0g𝑂) ∨ 𝑥 = (0g𝑂))))
20 ralcom 3289 . . . 4 (∀𝑥 ∈ (Base‘𝑂)∀𝑦 ∈ (Base‘𝑂)((𝑦(.r𝑂)𝑥) = (0g𝑂) → (𝑦 = (0g𝑂) ∨ 𝑥 = (0g𝑂))) ↔ ∀𝑦 ∈ (Base‘𝑂)∀𝑥 ∈ (Base‘𝑂)((𝑦(.r𝑂)𝑥) = (0g𝑂) → (𝑦 = (0g𝑂) ∨ 𝑥 = (0g𝑂))))
2119, 20bitri 275 . . 3 (∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(.r𝑅)𝑦) = (0g𝑅) → (𝑥 = (0g𝑅) ∨ 𝑦 = (0g𝑅))) ↔ ∀𝑦 ∈ (Base‘𝑂)∀𝑥 ∈ (Base‘𝑂)((𝑦(.r𝑂)𝑥) = (0g𝑂) → (𝑦 = (0g𝑂) ∨ 𝑥 = (0g𝑂))))
222, 21anbi12i 628 . 2 ((𝑅 ∈ NzRing ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(.r𝑅)𝑦) = (0g𝑅) → (𝑥 = (0g𝑅) ∨ 𝑦 = (0g𝑅)))) ↔ (𝑂 ∈ NzRing ∧ ∀𝑦 ∈ (Base‘𝑂)∀𝑥 ∈ (Base‘𝑂)((𝑦(.r𝑂)𝑥) = (0g𝑂) → (𝑦 = (0g𝑂) ∨ 𝑥 = (0g𝑂)))))
233, 5, 9isdomn 20705 . 2 (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(.r𝑅)𝑦) = (0g𝑅) → (𝑥 = (0g𝑅) ∨ 𝑦 = (0g𝑅)))))
24 eqid 2737 . . 3 (Base‘𝑂) = (Base‘𝑂)
25 eqid 2737 . . 3 (0g𝑂) = (0g𝑂)
2624, 6, 25isdomn 20705 . 2 (𝑂 ∈ Domn ↔ (𝑂 ∈ NzRing ∧ ∀𝑦 ∈ (Base‘𝑂)∀𝑥 ∈ (Base‘𝑂)((𝑦(.r𝑂)𝑥) = (0g𝑂) → (𝑦 = (0g𝑂) ∨ 𝑥 = (0g𝑂)))))
2722, 23, 263bitr4i 303 1 (𝑅 ∈ Domn ↔ 𝑂 ∈ Domn)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 848   = wceq 1540  wcel 2108  wral 3061  cfv 6561  (class class class)co 7431  Basecbs 17247  .rcmulr 17298  0gc0g 17484  opprcoppr 20333  NzRingcnzr 20512  Domncdomn 20692
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-tpos 8251  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-plusg 17310  df-mulr 17311  df-0g 17486  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-grp 18954  df-minusg 18955  df-cmn 19800  df-abl 19801  df-mgp 20138  df-rng 20150  df-ur 20179  df-ring 20232  df-oppr 20334  df-nzr 20513  df-domn 20695
This theorem is referenced by:  opprdomn  20718  isdomn4r  20719
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