Theorem opprringbg 14469

Description: Bidirectional form of opprring 14468. (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 14468	. 2 ⊢ (𝑅 ∈ Ring → 𝑂 ∈ Ring)
3		eqid 2238	. . . . . 6 ⊢ (oppr‘𝑂) = (oppr‘𝑂)
4	3	opprring 14468	. . . . 5 ⊢ (𝑂 ∈ Ring → (oppr‘𝑂) ∈ Ring)
5	4	adantl 277	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑂 ∈ Ring) → (oppr‘𝑂) ∈ Ring)
6		eqidd 2239	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑂 ∈ Ring) → (Base‘𝑅) = (Base‘𝑅))
7		eqid 2238	. . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅)
8	1, 7	opprbasg 14464	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → (Base‘𝑅) = (Base‘𝑂))
9		eqid 2238	. . . . . . 7 ⊢ (Base‘𝑂) = (Base‘𝑂)
10	3, 9	opprbasg 14464	. . . . . 6 ⊢ (𝑂 ∈ Ring → (Base‘𝑂) = (Base‘(oppr‘𝑂)))
11	8, 10	sylan9eq 2291	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑂 ∈ Ring) → (Base‘𝑅) = (Base‘(oppr‘𝑂)))
12		eqid 2238	. . . . . . . 8 ⊢ (+g‘𝑅) = (+g‘𝑅)
13	1, 12	oppraddg 14465	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → (+g‘𝑅) = (+g‘𝑂))
14		eqid 2238	. . . . . . . 8 ⊢ (+g‘𝑂) = (+g‘𝑂)
15	3, 14	oppraddg 14465	. . . . . . 7 ⊢ (𝑂 ∈ Ring → (+g‘𝑂) = (+g‘(oppr‘𝑂)))
16	13, 15	sylan9eq 2291	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑂 ∈ Ring) → (+g‘𝑅) = (+g‘(oppr‘𝑂)))
17	16	oveqdr 6113	. . . . 5 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑂 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥(+g‘𝑅)𝑦) = (𝑥(+g‘(oppr‘𝑂))𝑦))
18		eqid 2238	. . . . . . . . 9 ⊢ (.r‘𝑂) = (.r‘𝑂)
19		eqid 2238	. . . . . . . . 9 ⊢ (.r‘(oppr‘𝑂)) = (.r‘(oppr‘𝑂))
20	9, 18, 3, 19	opprmulg 14460	. . . . . . . 8 ⊢ ((𝑂 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(.r‘(oppr‘𝑂))𝑦) = (𝑦(.r‘𝑂)𝑥))
21	20	3adant1l 1261	. . . . . . 7 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑂 ∈ Ring) ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(.r‘(oppr‘𝑂))𝑦) = (𝑦(.r‘𝑂)𝑥))
22		simp1l 1052	. . . . . . . 8 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑂 ∈ Ring) ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑅 ∈ 𝑉)
23		simp3 1030	. . . . . . . 8 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑂 ∈ Ring) ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑦 ∈ (Base‘𝑅))
24		simp2 1029	. . . . . . . 8 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑂 ∈ Ring) ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑥 ∈ (Base‘𝑅))
25		eqid 2238	. . . . . . . . 9 ⊢ (.r‘𝑅) = (.r‘𝑅)
26	7, 25, 1, 18	opprmulg 14460	. . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑦(.r‘𝑂)𝑥) = (𝑥(.r‘𝑅)𝑦))
27	22, 23, 24, 26	syl3anc 1278	. . . . . . 7 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑂 ∈ Ring) ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑦(.r‘𝑂)𝑥) = (𝑥(.r‘𝑅)𝑦))
28	21, 27	eqtr2d 2272	. . . . . 6 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑂 ∈ Ring) ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(.r‘𝑅)𝑦) = (𝑥(.r‘(oppr‘𝑂))𝑦))
29	28	3expb 1235	. . . . 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 14427	. . . 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))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1009   = wceq 1402   ∈ wcel 2209  ‘cfv 5377  (class class class)co 6085  Basecbs 13404  +_gcplusg 13484  ._rcmulr 13485  Ringcrg 14384  opp_rcoppr 14456

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-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8271  ax-resscn 8272  ax-1cn 8273  ax-1re 8274  ax-icn 8275  ax-addcl 8276  ax-addrcl 8277  ax-mulcl 8278  ax-addcom 8280  ax-addass 8282  ax-i2m1 8285  ax-0lt1 8286  ax-0id 8288  ax-rnegex 8289  ax-pre-ltirr 8292  ax-pre-lttrn 8294  ax-pre-ltadd 8296

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-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-tpos 6516  df-pnf 8363  df-mnf 8364  df-ltxr 8366  df-inn 9308  df-2 9366  df-3 9367  df-ndx 13407  df-slot 13408  df-base 13410  df-sets 13411  df-plusg 13497  df-mulr 13498  df-0g 13665  df-mgm 13729  df-sgrp 13770  df-mnd 13783  df-grp 13861  df-mgp 14302  df-ur 14347  df-ring 14386  df-oppr 14457

This theorem is used by:  opprringb  14470  rhmopp  14567  opprnzrbg  14576