Theorem opprdomnb 20692

Description: A class is a domain if and only if its opposite is a domain, biconditional form of opprdomn 20693. (Contributed by SN, 15-Jun-2015.)

Hypothesis

Ref	Expression
opprdomn.1	⊢ 𝑂 = (oppr‘𝑅)

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.

Step	Hyp	Ref	Expression
1		opprdomn.1	. . . 4 ⊢ 𝑂 = (oppr‘𝑅)
2	1	opprnzrb 20496	. . 3 ⊢ (𝑅 ∈ NzRing ↔ 𝑂 ∈ NzRing)
3		eqid 2741	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
4	1, 3	opprbas 20317	. . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑂)
5		eqid 2741	. . . . . . . . . 10 ⊢ (.r‘𝑅) = (.r‘𝑅)
6		eqid 2741	. . . . . . . . . 10 ⊢ (.r‘𝑂) = (.r‘𝑂)
7	3, 5, 1, 6	opprmul 20314	. . . . . . . . 9 ⊢ (𝑦(.r‘𝑂)𝑥) = (𝑥(.r‘𝑅)𝑦)
8	7	eqcomi 2750	. . . . . . . 8 ⊢ (𝑥(.r‘𝑅)𝑦) = (𝑦(.r‘𝑂)𝑥)
9		eqid 2741	. . . . . . . . 9 ⊢ (0g‘𝑅) = (0g‘𝑅)
10	1, 9	oppr0 20323	. . . . . . . 8 ⊢ (0g‘𝑅) = (0g‘𝑂)
11	8, 10	eqeq12i 2759	. . . . . . 7 ⊢ ((𝑥(.r‘𝑅)𝑦) = (0g‘𝑅) ↔ (𝑦(.r‘𝑂)𝑥) = (0g‘𝑂))
12	10	eqeq2i 2754	. . . . . . . . 9 ⊢ (𝑥 = (0g‘𝑅) ↔ 𝑥 = (0g‘𝑂))
13	10	eqeq2i 2754	. . . . . . . . 9 ⊢ (𝑦 = (0g‘𝑅) ↔ 𝑦 = (0g‘𝑂))
14	12, 13	orbi12i 921	. . . . . . . 8 ⊢ ((𝑥 = (0g‘𝑅) ∨ 𝑦 = (0g‘𝑅)) ↔ (𝑥 = (0g‘𝑂) ∨ 𝑦 = (0g‘𝑂)))
15		orcom 877	. . . . . . . 8 ⊢ ((𝑥 = (0g‘𝑂) ∨ 𝑦 = (0g‘𝑂)) ↔ (𝑦 = (0g‘𝑂) ∨ 𝑥 = (0g‘𝑂)))
16	14, 15	bitri 277	. . . . . . 7 ⊢ ((𝑥 = (0g‘𝑅) ∨ 𝑦 = (0g‘𝑅)) ↔ (𝑦 = (0g‘𝑂) ∨ 𝑥 = (0g‘𝑂)))
17	11, 16	imbi12i 352	. . . . . 6 ⊢ (((𝑥(.r‘𝑅)𝑦) = (0g‘𝑅) → (𝑥 = (0g‘𝑅) ∨ 𝑦 = (0g‘𝑅))) ↔ ((𝑦(.r‘𝑂)𝑥) = (0g‘𝑂) → (𝑦 = (0g‘𝑂) ∨ 𝑥 = (0g‘𝑂))))
18	4, 17	raleqbii 3313	. . . . 5 ⊢ (∀𝑦 ∈ (Base‘𝑅)((𝑥(.r‘𝑅)𝑦) = (0g‘𝑅) → (𝑥 = (0g‘𝑅) ∨ 𝑦 = (0g‘𝑅))) ↔ ∀𝑦 ∈ (Base‘𝑂)((𝑦(.r‘𝑂)𝑥) = (0g‘𝑂) → (𝑦 = (0g‘𝑂) ∨ 𝑥 = (0g‘𝑂))))
19	4, 18	raleqbii 3313	. . . 4 ⊢ (∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(.r‘𝑅)𝑦) = (0g‘𝑅) → (𝑥 = (0g‘𝑅) ∨ 𝑦 = (0g‘𝑅))) ↔ ∀𝑥 ∈ (Base‘𝑂)∀𝑦 ∈ (Base‘𝑂)((𝑦(.r‘𝑂)𝑥) = (0g‘𝑂) → (𝑦 = (0g‘𝑂) ∨ 𝑥 = (0g‘𝑂))))
20		ralcom 3269	. . . 4 ⊢ (∀𝑥 ∈ (Base‘𝑂)∀𝑦 ∈ (Base‘𝑂)((𝑦(.r‘𝑂)𝑥) = (0g‘𝑂) → (𝑦 = (0g‘𝑂) ∨ 𝑥 = (0g‘𝑂))) ↔ ∀𝑦 ∈ (Base‘𝑂)∀𝑥 ∈ (Base‘𝑂)((𝑦(.r‘𝑂)𝑥) = (0g‘𝑂) → (𝑦 = (0g‘𝑂) ∨ 𝑥 = (0g‘𝑂))))
21	19, 20	bitri 277	. . 3 ⊢ (∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(.r‘𝑅)𝑦) = (0g‘𝑅) → (𝑥 = (0g‘𝑅) ∨ 𝑦 = (0g‘𝑅))) ↔ ∀𝑦 ∈ (Base‘𝑂)∀𝑥 ∈ (Base‘𝑂)((𝑦(.r‘𝑂)𝑥) = (0g‘𝑂) → (𝑦 = (0g‘𝑂) ∨ 𝑥 = (0g‘𝑂))))
22	2, 21	anbi12i 635	. 2 ⊢ ((𝑅 ∈ NzRing ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(.r‘𝑅)𝑦) = (0g‘𝑅) → (𝑥 = (0g‘𝑅) ∨ 𝑦 = (0g‘𝑅)))) ↔ (𝑂 ∈ NzRing ∧ ∀𝑦 ∈ (Base‘𝑂)∀𝑥 ∈ (Base‘𝑂)((𝑦(.r‘𝑂)𝑥) = (0g‘𝑂) → (𝑦 = (0g‘𝑂) ∨ 𝑥 = (0g‘𝑂)))))
23	3, 5, 9	isdomn 20680	. 2 ⊢ (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(.r‘𝑅)𝑦) = (0g‘𝑅) → (𝑥 = (0g‘𝑅) ∨ 𝑦 = (0g‘𝑅)))))
24		eqid 2741	. . 3 ⊢ (Base‘𝑂) = (Base‘𝑂)
25		eqid 2741	. . 3 ⊢ (0g‘𝑂) = (0g‘𝑂)
26	24, 6, 25	isdomn 20680	. 2 ⊢ (𝑂 ∈ Domn ↔ (𝑂 ∈ NzRing ∧ ∀𝑦 ∈ (Base‘𝑂)∀𝑥 ∈ (Base‘𝑂)((𝑦(.r‘𝑂)𝑥) = (0g‘𝑂) → (𝑦 = (0g‘𝑂) ∨ 𝑥 = (0g‘𝑂)))))
27	22, 23, 26	3bitr4i 305	1 ⊢ (𝑅 ∈ Domn ↔ 𝑂 ∈ Domn)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 397   ∨ wo 854   = wceq 1548   ∈ wcel 2121  ∀wral 3055  ‘cfv 6488  (class class class)co 7359  Basecbs 17174  ._rcmulr 17216  0_gc0g 17397  opp_rcoppr 20310  NzRingcnzr 20487  Domncdomn 20667

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-sep 5220  ax-nul 5230  ax-pow 5296  ax-pr 5364  ax-un 7681  ax-cnex 11090  ax-resscn 11091  ax-1cn 11092  ax-icn 11093  ax-addcl 11094  ax-addrcl 11095  ax-mulcl 11096  ax-mulrcl 11097  ax-mulcom 11098  ax-addass 11099  ax-mulass 11100  ax-distr 11101  ax-i2m1 11102  ax-1ne0 11103  ax-1rid 11104  ax-rnegex 11105  ax-rrecex 11106  ax-cnre 11107  ax-pre-lttri 11108  ax-pre-lttrn 11109  ax-pre-ltadd 11110  ax-pre-mulgt0 11111

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-nel 3041  df-ral 3056  df-rex 3066  df-rmo 3346  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3725  df-csb 3833  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3904  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-iun 4925  df-br 5075  df-opab 5137  df-mpt 5156  df-tr 5182  df-id 5515  df-eprel 5520  df-po 5528  df-so 5529  df-fr 5573  df-we 5575  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6255  df-ord 6316  df-on 6317  df-lim 6318  df-suc 6319  df-iota 6444  df-fun 6490  df-fn 6491  df-f 6492  df-f1 6493  df-fo 6494  df-f1o 6495  df-fv 6496  df-riota 7316  df-ov 7362  df-oprab 7363  df-mpo 7364  df-om 7810  df-2nd 7934  df-tpos 8168  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8343  df-er 8637  df-en 8888  df-dom 8889  df-sdom 8890  df-pnf 11177  df-mnf 11178  df-xr 11179  df-ltxr 11180  df-le 11181  df-sub 11375  df-neg 11376  df-nn 12170  df-2 12239  df-3 12240  df-sets 17129  df-slot 17147  df-ndx 17159  df-base 17175  df-plusg 17228  df-mulr 17229  df-0g 17399  df-mgm 18603  df-sgrp 18682  df-mnd 18698  df-grp 18907  df-minusg 18908  df-cmn 19751  df-abl 19752  df-mgp 20116  df-rng 20128  df-ur 20157  df-ring 20210  df-oppr 20311  df-nzr 20488  df-domn 20670

This theorem is referenced by:  opprdomn  20693  isdomn4r  20694