Theorem opprdomnbg 13754

Description: A class is a domain if and only if its opposite is a domain, biconditional form of opprdomn 13755. (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 13665	. . 3 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ NzRing ↔ 𝑂 ∈ NzRing))
3		eqid 2193	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
4	1, 3	opprbasg 13555	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → (Base‘𝑅) = (Base‘𝑂))
5		vex 2763	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
6		vex 2763	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
7		eqid 2193	. . . . . . . . . . 11 ⊢ (.r‘𝑅) = (.r‘𝑅)
8		eqid 2193	. . . . . . . . . . 11 ⊢ (.r‘𝑂) = (.r‘𝑂)
9	3, 7, 1, 8	opprmulg 13551	. . . . . . . . . 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 2199	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → (𝑥(.r‘𝑅)𝑦) = (𝑦(.r‘𝑂)𝑥))
12		eqid 2193	. . . . . . . . 9 ⊢ (0g‘𝑅) = (0g‘𝑅)
13	1, 12	oppr0g 13561	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → (0g‘𝑅) = (0g‘𝑂))
14	11, 13	eqeq12d 2208	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → ((𝑥(.r‘𝑅)𝑦) = (0g‘𝑅) ↔ (𝑦(.r‘𝑂)𝑥) = (0g‘𝑂)))
15	13	eqeq2d 2205	. . . . . . . . 9 ⊢ (𝑅 ∈ 𝑉 → (𝑥 = (0g‘𝑅) ↔ 𝑥 = (0g‘𝑂)))
16	13	eqeq2d 2205	. . . . . . . . 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 2706	. . . . 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 2706	. . . 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 2657	. . . 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 13749	. 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 2193	. . 3 ⊢ (Base‘𝑂) = (Base‘𝑂)
28		eqid 2193	. . 3 ⊢ (0g‘𝑂) = (0g‘𝑂)
29	27, 8, 28	isdomn 13749	. 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 2164  ∀wral 2472  Vcvv 2760  ‘cfv 5246  (class class class)co 5910  Basecbs 12608  ._rcmulr 12686  0_gc0g 12857  opp_rcoppr 13547  NzRingcnzr 13659  Domncdomn 13736

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4462  ax-setind 4565  ax-cnex 7953  ax-resscn 7954  ax-1cn 7955  ax-1re 7956  ax-icn 7957  ax-addcl 7958  ax-addrcl 7959  ax-mulcl 7960  ax-addcom 7962  ax-addass 7964  ax-i2m1 7967  ax-0lt1 7968  ax-0id 7970  ax-rnegex 7971  ax-pre-ltirr 7974  ax-pre-lttrn 7976  ax-pre-ltadd 7978

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4322  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-iota 5207  df-fun 5248  df-fn 5249  df-fv 5254  df-riota 5865  df-ov 5913  df-oprab 5914  df-mpo 5915  df-tpos 6289  df-pnf 8046  df-mnf 8047  df-ltxr 8049  df-inn 8973  df-2 9031  df-3 9032  df-ndx 12611  df-slot 12612  df-base 12614  df-sets 12615  df-plusg 12698  df-mulr 12699  df-0g 12859  df-mgm 12929  df-sgrp 12975  df-mnd 12988  df-grp 13065  df-mgp 13401  df-ur 13440  df-ring 13478  df-oppr 13548  df-nzr 13660  df-domn 13739

This theorem is referenced by:  opprdomn  13755