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Theorem opprringbg 13397
Description: Bidirectional form of opprring 13396. (Contributed by Mario Carneiro, 6-Dec-2014.)
Hypothesis
Ref Expression
opprbas.1 𝑂 = (oppr𝑅)
Assertion
Ref Expression
opprringbg (𝑅𝑉 → (𝑅 ∈ Ring ↔ 𝑂 ∈ Ring))

Proof of Theorem opprringbg
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opprbas.1 . . 3 𝑂 = (oppr𝑅)
21opprring 13396 . 2 (𝑅 ∈ Ring → 𝑂 ∈ Ring)
3 eqid 2189 . . . . . 6 (oppr𝑂) = (oppr𝑂)
43opprring 13396 . . . . 5 (𝑂 ∈ Ring → (oppr𝑂) ∈ Ring)
54adantl 277 . . . 4 ((𝑅𝑉𝑂 ∈ Ring) → (oppr𝑂) ∈ Ring)
6 eqidd 2190 . . . . 5 ((𝑅𝑉𝑂 ∈ Ring) → (Base‘𝑅) = (Base‘𝑅))
7 eqid 2189 . . . . . . 7 (Base‘𝑅) = (Base‘𝑅)
81, 7opprbasg 13392 . . . . . 6 (𝑅𝑉 → (Base‘𝑅) = (Base‘𝑂))
9 eqid 2189 . . . . . . 7 (Base‘𝑂) = (Base‘𝑂)
103, 9opprbasg 13392 . . . . . 6 (𝑂 ∈ Ring → (Base‘𝑂) = (Base‘(oppr𝑂)))
118, 10sylan9eq 2242 . . . . 5 ((𝑅𝑉𝑂 ∈ Ring) → (Base‘𝑅) = (Base‘(oppr𝑂)))
12 eqid 2189 . . . . . . . 8 (+g𝑅) = (+g𝑅)
131, 12oppraddg 13393 . . . . . . 7 (𝑅𝑉 → (+g𝑅) = (+g𝑂))
14 eqid 2189 . . . . . . . 8 (+g𝑂) = (+g𝑂)
153, 14oppraddg 13393 . . . . . . 7 (𝑂 ∈ Ring → (+g𝑂) = (+g‘(oppr𝑂)))
1613, 15sylan9eq 2242 . . . . . 6 ((𝑅𝑉𝑂 ∈ Ring) → (+g𝑅) = (+g‘(oppr𝑂)))
1716oveqdr 5919 . . . . 5 (((𝑅𝑉𝑂 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥(+g𝑅)𝑦) = (𝑥(+g‘(oppr𝑂))𝑦))
18 eqid 2189 . . . . . . . . 9 (.r𝑂) = (.r𝑂)
19 eqid 2189 . . . . . . . . 9 (.r‘(oppr𝑂)) = (.r‘(oppr𝑂))
209, 18, 3, 19opprmulg 13388 . . . . . . . 8 ((𝑂 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(.r‘(oppr𝑂))𝑦) = (𝑦(.r𝑂)𝑥))
21203adant1l 1232 . . . . . . 7 (((𝑅𝑉𝑂 ∈ Ring) ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(.r‘(oppr𝑂))𝑦) = (𝑦(.r𝑂)𝑥))
22 simp1l 1023 . . . . . . . 8 (((𝑅𝑉𝑂 ∈ Ring) ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑅𝑉)
23 simp3 1001 . . . . . . . 8 (((𝑅𝑉𝑂 ∈ Ring) ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑦 ∈ (Base‘𝑅))
24 simp2 1000 . . . . . . . 8 (((𝑅𝑉𝑂 ∈ Ring) ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑥 ∈ (Base‘𝑅))
25 eqid 2189 . . . . . . . . 9 (.r𝑅) = (.r𝑅)
267, 25, 1, 18opprmulg 13388 . . . . . . . 8 ((𝑅𝑉𝑦 ∈ (Base‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑦(.r𝑂)𝑥) = (𝑥(.r𝑅)𝑦))
2722, 23, 24, 26syl3anc 1249 . . . . . . 7 (((𝑅𝑉𝑂 ∈ Ring) ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑦(.r𝑂)𝑥) = (𝑥(.r𝑅)𝑦))
2821, 27eqtr2d 2223 . . . . . 6 (((𝑅𝑉𝑂 ∈ Ring) ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(.r𝑅)𝑦) = (𝑥(.r‘(oppr𝑂))𝑦))
29283expb 1206 . . . . 5 (((𝑅𝑉𝑂 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥(.r𝑅)𝑦) = (𝑥(.r‘(oppr𝑂))𝑦))
306, 11, 17, 29ringpropd 13359 . . . 4 ((𝑅𝑉𝑂 ∈ Ring) → (𝑅 ∈ Ring ↔ (oppr𝑂) ∈ Ring))
315, 30mpbird 167 . . 3 ((𝑅𝑉𝑂 ∈ Ring) → 𝑅 ∈ Ring)
3231ex 115 . 2 (𝑅𝑉 → (𝑂 ∈ Ring → 𝑅 ∈ Ring))
332, 32impbid2 143 1 (𝑅𝑉 → (𝑅 ∈ Ring ↔ 𝑂 ∈ Ring))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  w3a 980   = wceq 1364  wcel 2160  cfv 5231  (class class class)co 5891  Basecbs 12486  +gcplusg 12561  .rcmulr 12562  Ringcrg 13317  opprcoppr 13384
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 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-cnex 7921  ax-resscn 7922  ax-1cn 7923  ax-1re 7924  ax-icn 7925  ax-addcl 7926  ax-addrcl 7927  ax-mulcl 7928  ax-addcom 7930  ax-addass 7932  ax-i2m1 7935  ax-0lt1 7936  ax-0id 7938  ax-rnegex 7939  ax-pre-ltirr 7942  ax-pre-lttrn 7944  ax-pre-ltadd 7946
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 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-fv 5239  df-riota 5847  df-ov 5894  df-oprab 5895  df-mpo 5896  df-tpos 6264  df-pnf 8013  df-mnf 8014  df-ltxr 8016  df-inn 8939  df-2 8997  df-3 8998  df-ndx 12489  df-slot 12490  df-base 12492  df-sets 12493  df-plusg 12574  df-mulr 12575  df-0g 12735  df-mgm 12804  df-sgrp 12837  df-mnd 12850  df-grp 12920  df-mgp 13242  df-ur 13281  df-ring 13319  df-oppr 13385
This theorem is referenced by:  rhmopp  13493
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