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Theorem opprringbg 13323
Description: Bidirectional form of opprring 13322. (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 13322 . 2 (𝑅 ∈ Ring → 𝑂 ∈ Ring)
3 eqid 2187 . . . . . 6 (oppr𝑂) = (oppr𝑂)
43opprring 13322 . . . . 5 (𝑂 ∈ Ring → (oppr𝑂) ∈ Ring)
54adantl 277 . . . 4 ((𝑅𝑉𝑂 ∈ Ring) → (oppr𝑂) ∈ Ring)
6 eqidd 2188 . . . . 5 ((𝑅𝑉𝑂 ∈ Ring) → (Base‘𝑅) = (Base‘𝑅))
7 eqid 2187 . . . . . . 7 (Base‘𝑅) = (Base‘𝑅)
81, 7opprbasg 13318 . . . . . 6 (𝑅𝑉 → (Base‘𝑅) = (Base‘𝑂))
9 eqid 2187 . . . . . . 7 (Base‘𝑂) = (Base‘𝑂)
103, 9opprbasg 13318 . . . . . 6 (𝑂 ∈ Ring → (Base‘𝑂) = (Base‘(oppr𝑂)))
118, 10sylan9eq 2240 . . . . 5 ((𝑅𝑉𝑂 ∈ Ring) → (Base‘𝑅) = (Base‘(oppr𝑂)))
12 eqid 2187 . . . . . . . 8 (+g𝑅) = (+g𝑅)
131, 12oppraddg 13319 . . . . . . 7 (𝑅𝑉 → (+g𝑅) = (+g𝑂))
14 eqid 2187 . . . . . . . 8 (+g𝑂) = (+g𝑂)
153, 14oppraddg 13319 . . . . . . 7 (𝑂 ∈ Ring → (+g𝑂) = (+g‘(oppr𝑂)))
1613, 15sylan9eq 2240 . . . . . 6 ((𝑅𝑉𝑂 ∈ Ring) → (+g𝑅) = (+g‘(oppr𝑂)))
1716oveqdr 5916 . . . . 5 (((𝑅𝑉𝑂 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥(+g𝑅)𝑦) = (𝑥(+g‘(oppr𝑂))𝑦))
18 eqid 2187 . . . . . . . . 9 (.r𝑂) = (.r𝑂)
19 eqid 2187 . . . . . . . . 9 (.r‘(oppr𝑂)) = (.r‘(oppr𝑂))
209, 18, 3, 19opprmulg 13314 . . . . . . . 8 ((𝑂 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(.r‘(oppr𝑂))𝑦) = (𝑦(.r𝑂)𝑥))
21203adant1l 1231 . . . . . . 7 (((𝑅𝑉𝑂 ∈ Ring) ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(.r‘(oppr𝑂))𝑦) = (𝑦(.r𝑂)𝑥))
22 simp1l 1022 . . . . . . . 8 (((𝑅𝑉𝑂 ∈ Ring) ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑅𝑉)
23 simp3 1000 . . . . . . . 8 (((𝑅𝑉𝑂 ∈ Ring) ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑦 ∈ (Base‘𝑅))
24 simp2 999 . . . . . . . 8 (((𝑅𝑉𝑂 ∈ Ring) ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑥 ∈ (Base‘𝑅))
25 eqid 2187 . . . . . . . . 9 (.r𝑅) = (.r𝑅)
267, 25, 1, 18opprmulg 13314 . . . . . . . 8 ((𝑅𝑉𝑦 ∈ (Base‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑦(.r𝑂)𝑥) = (𝑥(.r𝑅)𝑦))
2722, 23, 24, 26syl3anc 1248 . . . . . . 7 (((𝑅𝑉𝑂 ∈ Ring) ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑦(.r𝑂)𝑥) = (𝑥(.r𝑅)𝑦))
2821, 27eqtr2d 2221 . . . . . 6 (((𝑅𝑉𝑂 ∈ Ring) ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(.r𝑅)𝑦) = (𝑥(.r‘(oppr𝑂))𝑦))
29283expb 1205 . . . . 5 (((𝑅𝑉𝑂 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥(.r𝑅)𝑦) = (𝑥(.r‘(oppr𝑂))𝑦))
306, 11, 17, 29ringpropd 13285 . . . 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 979   = wceq 1363  wcel 2158  cfv 5228  (class class class)co 5888  Basecbs 12475  +gcplusg 12550  .rcmulr 12551  Ringcrg 13243  opprcoppr 13310
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7915  ax-resscn 7916  ax-1cn 7917  ax-1re 7918  ax-icn 7919  ax-addcl 7920  ax-addrcl 7921  ax-mulcl 7922  ax-addcom 7924  ax-addass 7926  ax-i2m1 7929  ax-0lt1 7930  ax-0id 7932  ax-rnegex 7933  ax-pre-ltirr 7936  ax-pre-lttrn 7938  ax-pre-ltadd 7940
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-tpos 6259  df-pnf 8007  df-mnf 8008  df-ltxr 8010  df-inn 8933  df-2 8991  df-3 8992  df-ndx 12478  df-slot 12479  df-base 12481  df-sets 12482  df-plusg 12563  df-mulr 12564  df-0g 12724  df-mgm 12793  df-sgrp 12826  df-mnd 12839  df-grp 12901  df-mgp 13171  df-ur 13207  df-ring 13245  df-oppr 13311
This theorem is referenced by: (None)
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