Theorem opprlring 14506

Description: The opposite of a local ring is also a local ring. (Contributed by NM, 18-Oct-2014.)

Hypothesis

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

Assertion

Ref	Expression
opprlring	⊢ (𝑅 ∈ LRing ↔ 𝑂 ∈ LRing)

Proof of Theorem opprlring

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lringring 14503	. 2 ⊢ (𝑅 ∈ LRing → 𝑅 ∈ Ring)
2		lringring 14503	. . 3 ⊢ (𝑂 ∈ LRing → 𝑂 ∈ Ring)
3		opprlring.1	. . . 4 ⊢ 𝑂 = (oppr‘𝑅)
4	3	opprringb 14388	. . 3 ⊢ (𝑅 ∈ Ring ↔ 𝑂 ∈ Ring)
5	2, 4	sylibr 134	. 2 ⊢ (𝑂 ∈ LRing → 𝑅 ∈ Ring)
6	3	opprnzrbg 14494	. . . 4 ⊢ (𝑅 ∈ Ring → (𝑅 ∈ NzRing ↔ 𝑂 ∈ NzRing))
7		eqid 2238	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
8	3, 7	opprbasg 14382	. . . . 5 ⊢ (𝑅 ∈ Ring → (Base‘𝑅) = (Base‘𝑂))
9		eqid 2238	. . . . . . . . . 10 ⊢ (+g‘𝑅) = (+g‘𝑅)
10	3, 9	oppraddg 14383	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → (+g‘𝑅) = (+g‘𝑂))
11	10	oveqd 6102	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → (𝑥(+g‘𝑅)𝑦) = (𝑥(+g‘𝑂)𝑦))
12		eqid 2238	. . . . . . . . 9 ⊢ (1r‘𝑅) = (1r‘𝑅)
13	3, 12	oppr1g 14390	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) = (1r‘𝑂))
14	11, 13	eqeq12d 2253	. . . . . . 7 ⊢ (𝑅 ∈ Ring → ((𝑥(+g‘𝑅)𝑦) = (1r‘𝑅) ↔ (𝑥(+g‘𝑂)𝑦) = (1r‘𝑂)))
15		eqidd 2239	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → (Unit‘𝑅) = (Unit‘𝑅))
16	3	a1i 9	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → 𝑂 = (oppr‘𝑅))
17		id 19	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Ring)
18	15, 16, 17	opprunitd 14419	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → (Unit‘𝑅) = (Unit‘𝑂))
19	18	eleq2d 2308	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → (𝑥 ∈ (Unit‘𝑅) ↔ 𝑥 ∈ (Unit‘𝑂)))
20	18	eleq2d 2308	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → (𝑦 ∈ (Unit‘𝑅) ↔ 𝑦 ∈ (Unit‘𝑂)))
21	19, 20	orbi12d 805	. . . . . . 7 ⊢ (𝑅 ∈ Ring → ((𝑥 ∈ (Unit‘𝑅) ∨ 𝑦 ∈ (Unit‘𝑅)) ↔ (𝑥 ∈ (Unit‘𝑂) ∨ 𝑦 ∈ (Unit‘𝑂))))
22	14, 21	imbi12d 234	. . . . . 6 ⊢ (𝑅 ∈ Ring → (((𝑥(+g‘𝑅)𝑦) = (1r‘𝑅) → (𝑥 ∈ (Unit‘𝑅) ∨ 𝑦 ∈ (Unit‘𝑅))) ↔ ((𝑥(+g‘𝑂)𝑦) = (1r‘𝑂) → (𝑥 ∈ (Unit‘𝑂) ∨ 𝑦 ∈ (Unit‘𝑂)))))
23	8, 22	raleqbidv 2765	. . . . 5 ⊢ (𝑅 ∈ Ring → (∀𝑦 ∈ (Base‘𝑅)((𝑥(+g‘𝑅)𝑦) = (1r‘𝑅) → (𝑥 ∈ (Unit‘𝑅) ∨ 𝑦 ∈ (Unit‘𝑅))) ↔ ∀𝑦 ∈ (Base‘𝑂)((𝑥(+g‘𝑂)𝑦) = (1r‘𝑂) → (𝑥 ∈ (Unit‘𝑂) ∨ 𝑦 ∈ (Unit‘𝑂)))))
24	8, 23	raleqbidv 2765	. . . 4 ⊢ (𝑅 ∈ Ring → (∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(+g‘𝑅)𝑦) = (1r‘𝑅) → (𝑥 ∈ (Unit‘𝑅) ∨ 𝑦 ∈ (Unit‘𝑅))) ↔ ∀𝑥 ∈ (Base‘𝑂)∀𝑦 ∈ (Base‘𝑂)((𝑥(+g‘𝑂)𝑦) = (1r‘𝑂) → (𝑥 ∈ (Unit‘𝑂) ∨ 𝑦 ∈ (Unit‘𝑂)))))
25	6, 24	anbi12d 477	. . 3 ⊢ (𝑅 ∈ Ring → ((𝑅 ∈ NzRing ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(+g‘𝑅)𝑦) = (1r‘𝑅) → (𝑥 ∈ (Unit‘𝑅) ∨ 𝑦 ∈ (Unit‘𝑅)))) ↔ (𝑂 ∈ NzRing ∧ ∀𝑥 ∈ (Base‘𝑂)∀𝑦 ∈ (Base‘𝑂)((𝑥(+g‘𝑂)𝑦) = (1r‘𝑂) → (𝑥 ∈ (Unit‘𝑂) ∨ 𝑦 ∈ (Unit‘𝑂))))))
26		eqid 2238	. . . 4 ⊢ (Unit‘𝑅) = (Unit‘𝑅)
27	7, 9, 12, 26	islring 14501	. . 3 ⊢ (𝑅 ∈ LRing ↔ (𝑅 ∈ NzRing ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(+g‘𝑅)𝑦) = (1r‘𝑅) → (𝑥 ∈ (Unit‘𝑅) ∨ 𝑦 ∈ (Unit‘𝑅)))))
28		eqid 2238	. . . 4 ⊢ (Base‘𝑂) = (Base‘𝑂)
29		eqid 2238	. . . 4 ⊢ (+g‘𝑂) = (+g‘𝑂)
30		eqid 2238	. . . 4 ⊢ (1r‘𝑂) = (1r‘𝑂)
31		eqid 2238	. . . 4 ⊢ (Unit‘𝑂) = (Unit‘𝑂)
32	28, 29, 30, 31	islring 14501	. . 3 ⊢ (𝑂 ∈ LRing ↔ (𝑂 ∈ NzRing ∧ ∀𝑥 ∈ (Base‘𝑂)∀𝑦 ∈ (Base‘𝑂)((𝑥(+g‘𝑂)𝑦) = (1r‘𝑂) → (𝑥 ∈ (Unit‘𝑂) ∨ 𝑦 ∈ (Unit‘𝑂)))))
33	25, 27, 32	3bitr4g 223	. 2 ⊢ (𝑅 ∈ Ring → (𝑅 ∈ LRing ↔ 𝑂 ∈ LRing))
34	1, 5, 33	pm5.21nii 716	1 ⊢ (𝑅 ∈ LRing ↔ 𝑂 ∈ LRing)

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 720   = wceq 1402   ∈ wcel 2209  ∀wral 2528  ‘cfv 5377  (class class class)co 6085  Basecbs 13354  +_gcplusg 13433  1_rcur 14264  Ringcrg 14302  opp_rcoppr 14374  Unitcui 14395  NzRingcnzr 14488  LRingclring 14499

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-pre-ltirr 8291  ax-pre-lttrn 8293  ax-pre-ltadd 8295

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-tpos 6516  df-pnf 8362  df-mnf 8363  df-ltxr 8365  df-inn 9306  df-2 9364  df-3 9365  df-ndx 13357  df-slot 13358  df-base 13360  df-sets 13361  df-plusg 13446  df-mulr 13447  df-0g 13614  df-mgm 13678  df-sgrp 13719  df-mnd 13732  df-grp 13810  df-minusg 13811  df-cmn 14091  df-abl 14092  df-mgp 14220  df-ur 14265  df-srg 14270  df-ring 14304  df-oppr 14375  df-dvdsr 14397  df-unit 14398  df-nzr 14489  df-lring 14500

This theorem is used by:  opprdrng  14622