Theorem opprdomnbg 13908

Description: A class is a domain if and only if its opposite is a domain, biconditional form of opprdomn 13909. (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 13819	. . 3 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ NzRing ↔ 𝑂 ∈ NzRing))
3		eqid 2196	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
4	1, 3	opprbasg 13709	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → (Base‘𝑅) = (Base‘𝑂))
5		vex 2766	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
6		vex 2766	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
7		eqid 2196	. . . . . . . . . . 11 ⊢ (.r‘𝑅) = (.r‘𝑅)
8		eqid 2196	. . . . . . . . . . 11 ⊢ (.r‘𝑂) = (.r‘𝑂)
9	3, 7, 1, 8	opprmulg 13705	. . . . . . . . . 10 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑦 ∈ V ∧ 𝑥 ∈ V) → (𝑦(.r‘𝑂)𝑥) = (𝑥(.r‘𝑅)𝑦))
10	5, 6, 9	mp3an23 1340	. . . . . . . . 9 ⊢ (𝑅 ∈ 𝑉 → (𝑦(.r‘𝑂)𝑥) = (𝑥(.r‘𝑅)𝑦))
11	10	eqcomd 2202	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → (𝑥(.r‘𝑅)𝑦) = (𝑦(.r‘𝑂)𝑥))
12		eqid 2196	. . . . . . . . 9 ⊢ (0g‘𝑅) = (0g‘𝑅)
13	1, 12	oppr0g 13715	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → (0g‘𝑅) = (0g‘𝑂))
14	11, 13	eqeq12d 2211	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → ((𝑥(.r‘𝑅)𝑦) = (0g‘𝑅) ↔ (𝑦(.r‘𝑂)𝑥) = (0g‘𝑂)))
15	13	eqeq2d 2208	. . . . . . . . 9 ⊢ (𝑅 ∈ 𝑉 → (𝑥 = (0g‘𝑅) ↔ 𝑥 = (0g‘𝑂)))
16	13	eqeq2d 2208	. . . . . . . . 9 ⊢ (𝑅 ∈ 𝑉 → (𝑦 = (0g‘𝑅) ↔ 𝑦 = (0g‘𝑂)))
17	15, 16	orbi12d 794	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → ((𝑥 = (0g‘𝑅) ∨ 𝑦 = (0g‘𝑅)) ↔ (𝑥 = (0g‘𝑂) ∨ 𝑦 = (0g‘𝑂))))
18		orcom 729	. . . . . . . 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 2709	. . . . 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 2709	. . . 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 2660	. . . 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 13903	. 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 2196	. . 3 ⊢ (Base‘𝑂) = (Base‘𝑂)
28		eqid 2196	. . 3 ⊢ (0g‘𝑂) = (0g‘𝑂)
29	27, 8, 28	isdomn 13903	. 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 709   = wceq 1364   ∈ wcel 2167  ∀wral 2475  Vcvv 2763  ‘cfv 5259  (class class class)co 5925  Basecbs 12705  ._rcmulr 12783  0_gc0g 12960  opp_rcoppr 13701  NzRingcnzr 13813  Domncdomn 13890

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7989  ax-resscn 7990  ax-1cn 7991  ax-1re 7992  ax-icn 7993  ax-addcl 7994  ax-addrcl 7995  ax-mulcl 7996  ax-addcom 7998  ax-addass 8000  ax-i2m1 8003  ax-0lt1 8004  ax-0id 8006  ax-rnegex 8007  ax-pre-ltirr 8010  ax-pre-lttrn 8012  ax-pre-ltadd 8014

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-tpos 6312  df-pnf 8082  df-mnf 8083  df-ltxr 8085  df-inn 9010  df-2 9068  df-3 9069  df-ndx 12708  df-slot 12709  df-base 12711  df-sets 12712  df-plusg 12795  df-mulr 12796  df-0g 12962  df-mgm 13060  df-sgrp 13106  df-mnd 13121  df-grp 13207  df-mgp 13555  df-ur 13594  df-ring 13632  df-oppr 13702  df-nzr 13814  df-domn 13893

This theorem is referenced by:  opprdomn  13909