Theorem opprlring 14614

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 14611	. 2 ⊢ (𝑅 ∈ LRing → 𝑅 ∈ Ring)
2		lringring 14611	. . 3 ⊢ (𝑂 ∈ LRing → 𝑂 ∈ Ring)
3		opprlring.1	. . . 4 ⊢ 𝑂 = (oppr‘𝑅)
4	3	opprringb 14496	. . 3 ⊢ (𝑅 ∈ Ring ↔ 𝑂 ∈ Ring)
5	2, 4	sylibr 134	. 2 ⊢ (𝑂 ∈ LRing → 𝑅 ∈ Ring)
6	3	opprnzrbg 14602	. . . 4 ⊢ (𝑅 ∈ Ring → (𝑅 ∈ NzRing ↔ 𝑂 ∈ NzRing))
7		eqid 2238	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
8	3, 7	opprbasg 14490	. . . . 5 ⊢ (𝑅 ∈ Ring → (Base‘𝑅) = (Base‘𝑂))
9		eqid 2238	. . . . . . . . . 10 ⊢ (+g‘𝑅) = (+g‘𝑅)
10	3, 9	oppraddg 14491	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → (+g‘𝑅) = (+g‘𝑂))
11	10	oveqd 6105	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → (𝑥(+g‘𝑅)𝑦) = (𝑥(+g‘𝑂)𝑦))
12		eqid 2238	. . . . . . . . 9 ⊢ (1r‘𝑅) = (1r‘𝑅)
13	3, 12	oppr1g 14498	. . . . . . . 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 14527	. . . . . . . . 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 14609	. . 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 14609	. . 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 5380  (class class class)co 6088  Basecbs 13411  +_gcplusg 13491  1_rcur 14370  Ringcrg 14408  opp_rcoppr 14482  Unitcui 14503  NzRingcnzr 14596  LRingclring 14607

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 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-cnex 8277  ax-resscn 8278  ax-1cn 8279  ax-1re 8280  ax-icn 8281  ax-addcl 8282  ax-addrcl 8283  ax-mulcl 8284  ax-addcom 8286  ax-addass 8288  ax-i2m1 8291  ax-0lt1 8292  ax-0id 8294  ax-rnegex 8295  ax-pre-ltirr 8298  ax-pre-lttrn 8300  ax-pre-ltadd 8302

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 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-tpos 6520  df-pnf 8369  df-mnf 8370  df-ltxr 8372  df-inn 9315  df-2 9373  df-3 9374  df-ndx 13414  df-slot 13415  df-base 13417  df-sets 13418  df-plusg 13504  df-mulr 13505  df-0g 13672  df-mgm 13736  df-sgrp 13777  df-mnd 13790  df-grp 13868  df-minusg 13869  df-cmn 14180  df-abl 14181  df-mgp 14326  df-ur 14371  df-srg 14376  df-ring 14410  df-oppr 14483  df-dvdsr 14505  df-unit 14506  df-nzr 14597  df-lring 14608

This theorem is used by:  opprdrng  14730