Theorem opprdomnbg 14223

Description: A class is a domain if and only if its opposite is a domain, biconditional form of opprdomn 14224. (Contributed by SN, 15-Jun-2015.)

Hypothesis

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

Assertion

Ref	Expression
opprdomnbg	⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Domn ↔ 𝑂 ∈ Domn))

Proof of Theorem opprdomnbg

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	opprnzrbg 14134	. . 3 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ NzRing ↔ 𝑂 ∈ NzRing))
3		eqid 2229	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
4	1, 3	opprbasg 14024	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → (Base‘𝑅) = (Base‘𝑂))
5		vex 2802	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
6		vex 2802	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
7		eqid 2229	. . . . . . . . . . 11 ⊢ (.r‘𝑅) = (.r‘𝑅)
8		eqid 2229	. . . . . . . . . . 11 ⊢ (.r‘𝑂) = (.r‘𝑂)
9	3, 7, 1, 8	opprmulg 14020	. . . . . . . . . 10 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑦 ∈ V ∧ 𝑥 ∈ V) → (𝑦(.r‘𝑂)𝑥) = (𝑥(.r‘𝑅)𝑦))
10	5, 6, 9	mp3an23 1363	. . . . . . . . 9 ⊢ (𝑅 ∈ 𝑉 → (𝑦(.r‘𝑂)𝑥) = (𝑥(.r‘𝑅)𝑦))
11	10	eqcomd 2235	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → (𝑥(.r‘𝑅)𝑦) = (𝑦(.r‘𝑂)𝑥))
12		eqid 2229	. . . . . . . . 9 ⊢ (0g‘𝑅) = (0g‘𝑅)
13	1, 12	oppr0g 14030	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → (0g‘𝑅) = (0g‘𝑂))
14	11, 13	eqeq12d 2244	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → ((𝑥(.r‘𝑅)𝑦) = (0g‘𝑅) ↔ (𝑦(.r‘𝑂)𝑥) = (0g‘𝑂)))
15	13	eqeq2d 2241	. . . . . . . . 9 ⊢ (𝑅 ∈ 𝑉 → (𝑥 = (0g‘𝑅) ↔ 𝑥 = (0g‘𝑂)))
16	13	eqeq2d 2241	. . . . . . . . 9 ⊢ (𝑅 ∈ 𝑉 → (𝑦 = (0g‘𝑅) ↔ 𝑦 = (0g‘𝑂)))
17	15, 16	orbi12d 798	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → ((𝑥 = (0g‘𝑅) ∨ 𝑦 = (0g‘𝑅)) ↔ (𝑥 = (0g‘𝑂) ∨ 𝑦 = (0g‘𝑂))))
18		orcom 733	. . . . . . . 8 ⊢ ((𝑥 = (0g‘𝑂) ∨ 𝑦 = (0g‘𝑂)) ↔ (𝑦 = (0g‘𝑂) ∨ 𝑥 = (0g‘𝑂)))
19	17, 18	bitrdi 196	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → ((𝑥 = (0g‘𝑅) ∨ 𝑦 = (0g‘𝑅)) ↔ (𝑦 = (0g‘𝑂) ∨ 𝑥 = (0g‘𝑂))))
20	14, 19	imbi12d 234	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → (((𝑥(.r‘𝑅)𝑦) = (0g‘𝑅) → (𝑥 = (0g‘𝑅) ∨ 𝑦 = (0g‘𝑅))) ↔ ((𝑦(.r‘𝑂)𝑥) = (0g‘𝑂) → (𝑦 = (0g‘𝑂) ∨ 𝑥 = (0g‘𝑂)))))
21	4, 20	raleqbidv 2744	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → (∀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)))))
22	4, 21	raleqbidv 2744	. . . 4 ⊢ (𝑅 ∈ 𝑉 → (∀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)))))
23		ralcom 2694	. . . 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))))
24	22, 23	bitrdi 196	. . 3 ⊢ (𝑅 ∈ 𝑉 → (∀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)))))
25	2, 24	anbi12d 473	. 2 ⊢ (𝑅 ∈ 𝑉 → ((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))))))
26	3, 7, 12	isdomn 14218	. 2 ⊢ (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(.r‘𝑅)𝑦) = (0g‘𝑅) → (𝑥 = (0g‘𝑅) ∨ 𝑦 = (0g‘𝑅)))))
27		eqid 2229	. . 3 ⊢ (Base‘𝑂) = (Base‘𝑂)
28		eqid 2229	. . 3 ⊢ (0g‘𝑂) = (0g‘𝑂)
29	27, 8, 28	isdomn 14218	. 2 ⊢ (𝑂 ∈ Domn ↔ (𝑂 ∈ NzRing ∧ ∀𝑦 ∈ (Base‘𝑂)∀𝑥 ∈ (Base‘𝑂)((𝑦(.r‘𝑂)𝑥) = (0g‘𝑂) → (𝑦 = (0g‘𝑂) ∨ 𝑥 = (0g‘𝑂)))))
30	25, 26, 29	3bitr4g 223	1 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Domn ↔ 𝑂 ∈ Domn))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 713   = wceq 1395   ∈ wcel 2200  ∀wral 2508  Vcvv 2799  ‘cfv 5314  (class class class)co 5994  Basecbs 13018  ._rcmulr 13097  0_gc0g 13275  opp_rcoppr 14016  NzRingcnzr 14128  Domncdomn 14205

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4521  ax-setind 4626  ax-cnex 8078  ax-resscn 8079  ax-1cn 8080  ax-1re 8081  ax-icn 8082  ax-addcl 8083  ax-addrcl 8084  ax-mulcl 8085  ax-addcom 8087  ax-addass 8089  ax-i2m1 8092  ax-0lt1 8093  ax-0id 8095  ax-rnegex 8096  ax-pre-ltirr 8099  ax-pre-lttrn 8101  ax-pre-ltadd 8103

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4381  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-ima 4729  df-iota 5274  df-fun 5316  df-fn 5317  df-fv 5322  df-riota 5947  df-ov 5997  df-oprab 5998  df-mpo 5999  df-tpos 6381  df-pnf 8171  df-mnf 8172  df-ltxr 8174  df-inn 9099  df-2 9157  df-3 9158  df-ndx 13021  df-slot 13022  df-base 13024  df-sets 13025  df-plusg 13109  df-mulr 13110  df-0g 13277  df-mgm 13375  df-sgrp 13421  df-mnd 13436  df-grp 13522  df-mgp 13870  df-ur 13909  df-ring 13947  df-oppr 14017  df-nzr 14129  df-domn 14208

This theorem is referenced by:  opprdomn  14224