Theorem opprringbg 14043

Description: Bidirectional form of opprring 14042. (Contributed by Mario Carneiro, 6-Dec-2014.)

Hypothesis

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

Assertion

Ref	Expression
opprringbg	⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Ring ↔ 𝑂 ∈ Ring))

Proof of Theorem opprringbg

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

Step	Hyp	Ref	Expression
1		opprbas.1	. . 3 ⊢ 𝑂 = (oppr‘𝑅)
2	1	opprring 14042	. 2 ⊢ (𝑅 ∈ Ring → 𝑂 ∈ Ring)
3		eqid 2229	. . . . . 6 ⊢ (oppr‘𝑂) = (oppr‘𝑂)
4	3	opprring 14042	. . . . 5 ⊢ (𝑂 ∈ Ring → (oppr‘𝑂) ∈ Ring)
5	4	adantl 277	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑂 ∈ Ring) → (oppr‘𝑂) ∈ Ring)
6		eqidd 2230	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑂 ∈ Ring) → (Base‘𝑅) = (Base‘𝑅))
7		eqid 2229	. . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅)
8	1, 7	opprbasg 14038	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → (Base‘𝑅) = (Base‘𝑂))
9		eqid 2229	. . . . . . 7 ⊢ (Base‘𝑂) = (Base‘𝑂)
10	3, 9	opprbasg 14038	. . . . . 6 ⊢ (𝑂 ∈ Ring → (Base‘𝑂) = (Base‘(oppr‘𝑂)))
11	8, 10	sylan9eq 2282	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑂 ∈ Ring) → (Base‘𝑅) = (Base‘(oppr‘𝑂)))
12		eqid 2229	. . . . . . . 8 ⊢ (+g‘𝑅) = (+g‘𝑅)
13	1, 12	oppraddg 14039	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → (+g‘𝑅) = (+g‘𝑂))
14		eqid 2229	. . . . . . . 8 ⊢ (+g‘𝑂) = (+g‘𝑂)
15	3, 14	oppraddg 14039	. . . . . . 7 ⊢ (𝑂 ∈ Ring → (+g‘𝑂) = (+g‘(oppr‘𝑂)))
16	13, 15	sylan9eq 2282	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑂 ∈ Ring) → (+g‘𝑅) = (+g‘(oppr‘𝑂)))
17	16	oveqdr 6029	. . . . 5 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑂 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥(+g‘𝑅)𝑦) = (𝑥(+g‘(oppr‘𝑂))𝑦))
18		eqid 2229	. . . . . . . . 9 ⊢ (.r‘𝑂) = (.r‘𝑂)
19		eqid 2229	. . . . . . . . 9 ⊢ (.r‘(oppr‘𝑂)) = (.r‘(oppr‘𝑂))
20	9, 18, 3, 19	opprmulg 14034	. . . . . . . 8 ⊢ ((𝑂 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(.r‘(oppr‘𝑂))𝑦) = (𝑦(.r‘𝑂)𝑥))
21	20	3adant1l 1254	. . . . . . 7 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑂 ∈ Ring) ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(.r‘(oppr‘𝑂))𝑦) = (𝑦(.r‘𝑂)𝑥))
22		simp1l 1045	. . . . . . . 8 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑂 ∈ Ring) ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑅 ∈ 𝑉)
23		simp3 1023	. . . . . . . 8 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑂 ∈ Ring) ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑦 ∈ (Base‘𝑅))
24		simp2 1022	. . . . . . . 8 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑂 ∈ Ring) ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑥 ∈ (Base‘𝑅))
25		eqid 2229	. . . . . . . . 9 ⊢ (.r‘𝑅) = (.r‘𝑅)
26	7, 25, 1, 18	opprmulg 14034	. . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑦(.r‘𝑂)𝑥) = (𝑥(.r‘𝑅)𝑦))
27	22, 23, 24, 26	syl3anc 1271	. . . . . . 7 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑂 ∈ Ring) ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑦(.r‘𝑂)𝑥) = (𝑥(.r‘𝑅)𝑦))
28	21, 27	eqtr2d 2263	. . . . . 6 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑂 ∈ Ring) ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(.r‘𝑅)𝑦) = (𝑥(.r‘(oppr‘𝑂))𝑦))
29	28	3expb 1228	. . . . 5 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑂 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥(.r‘𝑅)𝑦) = (𝑥(.r‘(oppr‘𝑂))𝑦))
30	6, 11, 17, 29	ringpropd 14001	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑂 ∈ Ring) → (𝑅 ∈ Ring ↔ (oppr‘𝑂) ∈ Ring))
31	5, 30	mpbird 167	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑂 ∈ Ring) → 𝑅 ∈ Ring)
32	31	ex 115	. 2 ⊢ (𝑅 ∈ 𝑉 → (𝑂 ∈ Ring → 𝑅 ∈ Ring))
33	2, 32	impbid2 143	1 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Ring ↔ 𝑂 ∈ Ring))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1002   = wceq 1395   ∈ wcel 2200  ‘cfv 5318  (class class class)co 6001  Basecbs 13032  +_gcplusg 13110  ._rcmulr 13111  Ringcrg 13959  opp_rcoppr 14030

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-addcom 8099  ax-addass 8101  ax-i2m1 8104  ax-0lt1 8105  ax-0id 8107  ax-rnegex 8108  ax-pre-ltirr 8111  ax-pre-lttrn 8113  ax-pre-ltadd 8115

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-tpos 6391  df-pnf 8183  df-mnf 8184  df-ltxr 8186  df-inn 9111  df-2 9169  df-3 9170  df-ndx 13035  df-slot 13036  df-base 13038  df-sets 13039  df-plusg 13123  df-mulr 13124  df-0g 13291  df-mgm 13389  df-sgrp 13435  df-mnd 13450  df-grp 13536  df-mgp 13884  df-ur 13923  df-ring 13961  df-oppr 14031

This theorem is referenced by:  rhmopp  14140  opprnzrbg  14149