Theorem opprrngbg 14356

Description: A set is a non-unital ring if and only if its opposite is a non-unital ring. Bidirectional form of opprrng 14355. (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 14355	. 2 ⊢ (𝑅 ∈ Rng → 𝑂 ∈ Rng)
3		eqid 2238	. . . 4 ⊢ (oppr‘𝑂) = (oppr‘𝑂)
4	3	opprrng 14355	. . 3 ⊢ (𝑂 ∈ Rng → (oppr‘𝑂) ∈ Rng)
5		eqid 2238	. . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅)
6	5	a1i 9	. . . 4 ⊢ (𝑅 ∈ 𝑉 → (Base‘𝑅) = (Base‘𝑅))
7	1, 5	opprbasg 14353	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → (Base‘𝑅) = (Base‘𝑂))
8	1	opprex 14351	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → 𝑂 ∈ V)
9		eqid 2238	. . . . . . 7 ⊢ (Base‘𝑂) = (Base‘𝑂)
10	3, 9	opprbasg 14353	. . . . . 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 2271	. . . 4 ⊢ (𝑅 ∈ 𝑉 → (Base‘𝑅) = (Base‘(oppr‘𝑂)))
13		eqid 2238	. . . . . . 7 ⊢ (+g‘𝑅) = (+g‘𝑅)
14	1, 13	oppraddg 14354	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → (+g‘𝑅) = (+g‘𝑂))
15		eqid 2238	. . . . . . . 8 ⊢ (+g‘𝑂) = (+g‘𝑂)
16	3, 15	oppraddg 14354	. . . . . . 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 2271	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → (+g‘𝑅) = (+g‘(oppr‘𝑂)))
19	18	oveqdr 6103	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥(+g‘𝑅)𝑦) = (𝑥(+g‘(oppr‘𝑂))𝑦))
20		vex 2824	. . . . . . . 8 ⊢ 𝑥 ∈ V
21	20	a1i 9	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → 𝑥 ∈ V)
22		vex 2824	. . . . . . . 8 ⊢ 𝑦 ∈ V
23	22	a1i 9	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → 𝑦 ∈ V)
24		eqid 2238	. . . . . . . 8 ⊢ (.r‘𝑂) = (.r‘𝑂)
25		eqid 2238	. . . . . . . 8 ⊢ (.r‘(oppr‘𝑂)) = (.r‘(oppr‘𝑂))
26	9, 24, 3, 25	opprmulg 14349	. . . . . . 7 ⊢ ((𝑂 ∈ V ∧ 𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥(.r‘(oppr‘𝑂))𝑦) = (𝑦(.r‘𝑂)𝑥))
27	8, 21, 23, 26	syl3anc 1278	. . . . . 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 537	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → 𝑦 ∈ (Base‘𝑅))
31		simprl 535	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → 𝑥 ∈ (Base‘𝑅))
32		eqid 2238	. . . . . . 7 ⊢ (.r‘𝑅) = (.r‘𝑅)
33	5, 32, 1, 24	opprmulg 14349	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑦(.r‘𝑂)𝑥) = (𝑥(.r‘𝑅)𝑦))
34	29, 30, 31, 33	syl3anc 1278	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑦(.r‘𝑂)𝑥) = (𝑥(.r‘𝑅)𝑦))
35	28, 34	eqtr2d 2272	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥(.r‘𝑅)𝑦) = (𝑥(.r‘(oppr‘𝑂))𝑦))
36	6, 12, 19, 35	rngpropd 14229	. . 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 1402   ∈ wcel 2209  Vcvv 2821  ‘cfv 5372  (class class class)co 6075  Basecbs 13330  +_gcplusg 13408  ._rcmulr 13409  Rngcrng 14206  opp_rcoppr 14345

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 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 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-pre-ltirr 8281  ax-pre-lttrn 8283  ax-pre-ltadd 8285

This theorem 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-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 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-tpos 6506  df-pnf 8352  df-mnf 8353  df-ltxr 8355  df-inn 9284  df-2 9342  df-3 9343  df-ndx 13333  df-slot 13334  df-base 13336  df-sets 13337  df-plusg 13421  df-mulr 13422  df-0g 13589  df-mgm 13653  df-sgrp 13694  df-mnd 13707  df-grp 13785  df-cmn 14066  df-abl 14067  df-mgp 14195  df-rng 14207  df-oppr 14346

This theorem is referenced by:  opprsubrngg  14492