Theorem opprrngbg 14321

Description: A set is a non-unital ring if and only if its opposite is a non-unital ring. Bidirectional form of opprrng 14320. (Contributed by AV, 15-Feb-2025.)

Hypothesis

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

Assertion

Ref	Expression
opprrngbg	⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rng ↔ 𝑂 ∈ Rng))

Proof of Theorem opprrngbg

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	opprrng 14320	. 2 ⊢ (𝑅 ∈ Rng → 𝑂 ∈ Rng)
3		eqid 2234	. . . 4 ⊢ (oppr‘𝑂) = (oppr‘𝑂)
4	3	opprrng 14320	. . 3 ⊢ (𝑂 ∈ Rng → (oppr‘𝑂) ∈ Rng)
5		eqid 2234	. . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅)
6	5	a1i 9	. . . 4 ⊢ (𝑅 ∈ 𝑉 → (Base‘𝑅) = (Base‘𝑅))
7	1, 5	opprbasg 14318	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → (Base‘𝑅) = (Base‘𝑂))
8	1	opprex 14316	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → 𝑂 ∈ V)
9		eqid 2234	. . . . . . 7 ⊢ (Base‘𝑂) = (Base‘𝑂)
10	3, 9	opprbasg 14318	. . . . . 6 ⊢ (𝑂 ∈ V → (Base‘𝑂) = (Base‘(oppr‘𝑂)))
11	8, 10	syl 14	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → (Base‘𝑂) = (Base‘(oppr‘𝑂)))
12	7, 11	eqtrd 2267	. . . 4 ⊢ (𝑅 ∈ 𝑉 → (Base‘𝑅) = (Base‘(oppr‘𝑂)))
13		eqid 2234	. . . . . . 7 ⊢ (+g‘𝑅) = (+g‘𝑅)
14	1, 13	oppraddg 14319	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → (+g‘𝑅) = (+g‘𝑂))
15		eqid 2234	. . . . . . . 8 ⊢ (+g‘𝑂) = (+g‘𝑂)
16	3, 15	oppraddg 14319	. . . . . . 7 ⊢ (𝑂 ∈ V → (+g‘𝑂) = (+g‘(oppr‘𝑂)))
17	8, 16	syl 14	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → (+g‘𝑂) = (+g‘(oppr‘𝑂)))
18	14, 17	eqtrd 2267	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → (+g‘𝑅) = (+g‘(oppr‘𝑂)))
19	18	oveqdr 6086	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥(+g‘𝑅)𝑦) = (𝑥(+g‘(oppr‘𝑂))𝑦))
20		vex 2818	. . . . . . . 8 ⊢ 𝑥 ∈ V
21	20	a1i 9	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → 𝑥 ∈ V)
22		vex 2818	. . . . . . . 8 ⊢ 𝑦 ∈ V
23	22	a1i 9	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → 𝑦 ∈ V)
24		eqid 2234	. . . . . . . 8 ⊢ (.r‘𝑂) = (.r‘𝑂)
25		eqid 2234	. . . . . . . 8 ⊢ (.r‘(oppr‘𝑂)) = (.r‘(oppr‘𝑂))
26	9, 24, 3, 25	opprmulg 14314	. . . . . . 7 ⊢ ((𝑂 ∈ V ∧ 𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥(.r‘(oppr‘𝑂))𝑦) = (𝑦(.r‘𝑂)𝑥))
27	8, 21, 23, 26	syl3anc 1274	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → (𝑥(.r‘(oppr‘𝑂))𝑦) = (𝑦(.r‘𝑂)𝑥))
28	27	adantr 276	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥(.r‘(oppr‘𝑂))𝑦) = (𝑦(.r‘𝑂)𝑥))
29		simpl 109	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → 𝑅 ∈ 𝑉)
30		simprr 533	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → 𝑦 ∈ (Base‘𝑅))
31		simprl 531	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → 𝑥 ∈ (Base‘𝑅))
32		eqid 2234	. . . . . . 7 ⊢ (.r‘𝑅) = (.r‘𝑅)
33	5, 32, 1, 24	opprmulg 14314	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑦(.r‘𝑂)𝑥) = (𝑥(.r‘𝑅)𝑦))
34	29, 30, 31, 33	syl3anc 1274	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑦(.r‘𝑂)𝑥) = (𝑥(.r‘𝑅)𝑦))
35	28, 34	eqtr2d 2268	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥(.r‘𝑅)𝑦) = (𝑥(.r‘(oppr‘𝑂))𝑦))
36	6, 12, 19, 35	rngpropd 14194	. . 3 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rng ↔ (oppr‘𝑂) ∈ Rng))
37	4, 36	imbitrrid 156	. 2 ⊢ (𝑅 ∈ 𝑉 → (𝑂 ∈ Rng → 𝑅 ∈ Rng))
38	2, 37	impbid2 143	1 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rng ↔ 𝑂 ∈ Rng))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398   ∈ wcel 2205  Vcvv 2815  ‘cfv 5357  (class class class)co 6058  Basecbs 13296  +_gcplusg 13374  ._rcmulr 13375  Rngcrng 14171  opp_rcoppr 14310

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-pre-ltirr 8255  ax-pre-lttrn 8257  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-tpos 6489  df-pnf 8326  df-mnf 8327  df-ltxr 8329  df-inn 9255  df-2 9313  df-3 9314  df-ndx 13299  df-slot 13300  df-base 13302  df-sets 13303  df-plusg 13387  df-mulr 13388  df-0g 13555  df-mgm 13619  df-sgrp 13665  df-mnd 13678  df-grp 13758  df-cmn 14039  df-abl 14040  df-mgp 14160  df-rng 14172  df-oppr 14311

This theorem is referenced by:  opprsubrngg  14457