Theorem opprringbg 13957

Description: Bidirectional form of opprring 13956. (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 13956	. 2 ⊢ (𝑅 ∈ Ring → 𝑂 ∈ Ring)
3		eqid 2207	. . . . . 6 ⊢ (oppr‘𝑂) = (oppr‘𝑂)
4	3	opprring 13956	. . . . 5 ⊢ (𝑂 ∈ Ring → (oppr‘𝑂) ∈ Ring)
5	4	adantl 277	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑂 ∈ Ring) → (oppr‘𝑂) ∈ Ring)
6		eqidd 2208	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑂 ∈ Ring) → (Base‘𝑅) = (Base‘𝑅))
7		eqid 2207	. . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅)
8	1, 7	opprbasg 13952	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → (Base‘𝑅) = (Base‘𝑂))
9		eqid 2207	. . . . . . 7 ⊢ (Base‘𝑂) = (Base‘𝑂)
10	3, 9	opprbasg 13952	. . . . . 6 ⊢ (𝑂 ∈ Ring → (Base‘𝑂) = (Base‘(oppr‘𝑂)))
11	8, 10	sylan9eq 2260	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑂 ∈ Ring) → (Base‘𝑅) = (Base‘(oppr‘𝑂)))
12		eqid 2207	. . . . . . . 8 ⊢ (+g‘𝑅) = (+g‘𝑅)
13	1, 12	oppraddg 13953	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → (+g‘𝑅) = (+g‘𝑂))
14		eqid 2207	. . . . . . . 8 ⊢ (+g‘𝑂) = (+g‘𝑂)
15	3, 14	oppraddg 13953	. . . . . . 7 ⊢ (𝑂 ∈ Ring → (+g‘𝑂) = (+g‘(oppr‘𝑂)))
16	13, 15	sylan9eq 2260	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑂 ∈ Ring) → (+g‘𝑅) = (+g‘(oppr‘𝑂)))
17	16	oveqdr 5995	. . . . 5 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑂 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥(+g‘𝑅)𝑦) = (𝑥(+g‘(oppr‘𝑂))𝑦))
18		eqid 2207	. . . . . . . . 9 ⊢ (.r‘𝑂) = (.r‘𝑂)
19		eqid 2207	. . . . . . . . 9 ⊢ (.r‘(oppr‘𝑂)) = (.r‘(oppr‘𝑂))
20	9, 18, 3, 19	opprmulg 13948	. . . . . . . 8 ⊢ ((𝑂 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(.r‘(oppr‘𝑂))𝑦) = (𝑦(.r‘𝑂)𝑥))
21	20	3adant1l 1233	. . . . . . 7 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑂 ∈ Ring) ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(.r‘(oppr‘𝑂))𝑦) = (𝑦(.r‘𝑂)𝑥))
22		simp1l 1024	. . . . . . . 8 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑂 ∈ Ring) ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑅 ∈ 𝑉)
23		simp3 1002	. . . . . . . 8 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑂 ∈ Ring) ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑦 ∈ (Base‘𝑅))
24		simp2 1001	. . . . . . . 8 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑂 ∈ Ring) ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑥 ∈ (Base‘𝑅))
25		eqid 2207	. . . . . . . . 9 ⊢ (.r‘𝑅) = (.r‘𝑅)
26	7, 25, 1, 18	opprmulg 13948	. . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑦(.r‘𝑂)𝑥) = (𝑥(.r‘𝑅)𝑦))
27	22, 23, 24, 26	syl3anc 1250	. . . . . . 7 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑂 ∈ Ring) ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑦(.r‘𝑂)𝑥) = (𝑥(.r‘𝑅)𝑦))
28	21, 27	eqtr2d 2241	. . . . . 6 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑂 ∈ Ring) ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(.r‘𝑅)𝑦) = (𝑥(.r‘(oppr‘𝑂))𝑦))
29	28	3expb 1207	. . . . 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 13915	. . . 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 981   = wceq 1373   ∈ wcel 2178  ‘cfv 5290  (class class class)co 5967  Basecbs 12947  +_gcplusg 13024  ._rcmulr 13025  Ringcrg 13873  opp_rcoppr 13944

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-addcom 8060  ax-addass 8062  ax-i2m1 8065  ax-0lt1 8066  ax-0id 8068  ax-rnegex 8069  ax-pre-ltirr 8072  ax-pre-lttrn 8074  ax-pre-ltadd 8076

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-tpos 6354  df-pnf 8144  df-mnf 8145  df-ltxr 8147  df-inn 9072  df-2 9130  df-3 9131  df-ndx 12950  df-slot 12951  df-base 12953  df-sets 12954  df-plusg 13037  df-mulr 13038  df-0g 13205  df-mgm 13303  df-sgrp 13349  df-mnd 13364  df-grp 13450  df-mgp 13798  df-ur 13837  df-ring 13875  df-oppr 13945

This theorem is referenced by:  rhmopp  14053  opprnzrbg  14062