Theorem opprlring 14445

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 14442	. 2 ⊢ (𝑅 ∈ LRing → 𝑅 ∈ Ring)
2		lringring 14442	. . 3 ⊢ (𝑂 ∈ LRing → 𝑂 ∈ Ring)
3		opprlring.1	. . . 4 ⊢ 𝑂 = (oppr‘𝑅)
4	3	opprringb 14327	. . 3 ⊢ (𝑅 ∈ Ring ↔ 𝑂 ∈ Ring)
5	2, 4	sylibr 134	. 2 ⊢ (𝑂 ∈ LRing → 𝑅 ∈ Ring)
6	3	opprnzrbg 14433	. . . 4 ⊢ (𝑅 ∈ Ring → (𝑅 ∈ NzRing ↔ 𝑂 ∈ NzRing))
7		eqid 2234	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
8	3, 7	opprbasg 14321	. . . . 5 ⊢ (𝑅 ∈ Ring → (Base‘𝑅) = (Base‘𝑂))
9		eqid 2234	. . . . . . . . . 10 ⊢ (+g‘𝑅) = (+g‘𝑅)
10	3, 9	oppraddg 14322	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → (+g‘𝑅) = (+g‘𝑂))
11	10	oveqd 6075	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → (𝑥(+g‘𝑅)𝑦) = (𝑥(+g‘𝑂)𝑦))
12		eqid 2234	. . . . . . . . 9 ⊢ (1r‘𝑅) = (1r‘𝑅)
13	3, 12	oppr1g 14329	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) = (1r‘𝑂))
14	11, 13	eqeq12d 2249	. . . . . . 7 ⊢ (𝑅 ∈ Ring → ((𝑥(+g‘𝑅)𝑦) = (1r‘𝑅) ↔ (𝑥(+g‘𝑂)𝑦) = (1r‘𝑂)))
15		eqidd 2235	. . . . . . . . . 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 14358	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → (Unit‘𝑅) = (Unit‘𝑂))
19	18	eleq2d 2304	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → (𝑥 ∈ (Unit‘𝑅) ↔ 𝑥 ∈ (Unit‘𝑂)))
20	18	eleq2d 2304	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → (𝑦 ∈ (Unit‘𝑅) ↔ 𝑦 ∈ (Unit‘𝑂)))
21	19, 20	orbi12d 801	. . . . . . 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 2759	. . . . 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 2759	. . . 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 473	. . 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 2234	. . . 4 ⊢ (Unit‘𝑅) = (Unit‘𝑅)
27	7, 9, 12, 26	islring 14440	. . 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 2234	. . . 4 ⊢ (Base‘𝑂) = (Base‘𝑂)
29		eqid 2234	. . . 4 ⊢ (+g‘𝑂) = (+g‘𝑂)
30		eqid 2234	. . . 4 ⊢ (1r‘𝑂) = (1r‘𝑂)
31		eqid 2234	. . . 4 ⊢ (Unit‘𝑂) = (Unit‘𝑂)
32	28, 29, 30, 31	islring 14440	. . 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 712	1 ⊢ (𝑅 ∈ LRing ↔ 𝑂 ∈ LRing)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 716   = wceq 1398   ∈ wcel 2205  ∀wral 2522  ‘cfv 5357  (class class class)co 6058  Basecbs 13299  +_gcplusg 13377  1_rcur 14205  Ringcrg 14242  opp_rcoppr 14313  Unitcui 14334  NzRingcnzr 14427  LRingclring 14438

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-pre-ltirr 8255  ax-pre-lttrn 8257  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-tpos 6489  df-pnf 8326  df-mnf 8327  df-ltxr 8329  df-inn 9258  df-2 9316  df-3 9317  df-ndx 13302  df-slot 13303  df-base 13305  df-sets 13306  df-plusg 13390  df-mulr 13391  df-0g 13558  df-mgm 13622  df-sgrp 13668  df-mnd 13681  df-grp 13761  df-minusg 13762  df-cmn 14042  df-abl 14043  df-mgp 14163  df-ur 14206  df-srg 14210  df-ring 14244  df-oppr 14314  df-dvdsr 14336  df-unit 14337  df-nzr 14428  df-lring 14439

This theorem is referenced by:  opprdrng  14561