Theorem opprrngbg 13428

Description: A set is a non-unital ring if and only if its opposite is a non-unital ring. Bidirectional form of opprrng 13427. (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 13427	. 2 ⊢ (𝑅 ∈ Rng → 𝑂 ∈ Rng)
3		eqid 2189	. . . 4 ⊢ (oppr‘𝑂) = (oppr‘𝑂)
4	3	opprrng 13427	. . 3 ⊢ (𝑂 ∈ Rng → (oppr‘𝑂) ∈ Rng)
5		eqid 2189	. . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅)
6	5	a1i 9	. . . 4 ⊢ (𝑅 ∈ 𝑉 → (Base‘𝑅) = (Base‘𝑅))
7	1, 5	opprbasg 13425	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → (Base‘𝑅) = (Base‘𝑂))
8	1	opprex 13423	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → 𝑂 ∈ V)
9		eqid 2189	. . . . . . 7 ⊢ (Base‘𝑂) = (Base‘𝑂)
10	3, 9	opprbasg 13425	. . . . . 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 2222	. . . 4 ⊢ (𝑅 ∈ 𝑉 → (Base‘𝑅) = (Base‘(oppr‘𝑂)))
13		eqid 2189	. . . . . . 7 ⊢ (+g‘𝑅) = (+g‘𝑅)
14	1, 13	oppraddg 13426	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → (+g‘𝑅) = (+g‘𝑂))
15		eqid 2189	. . . . . . . 8 ⊢ (+g‘𝑂) = (+g‘𝑂)
16	3, 15	oppraddg 13426	. . . . . . 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 2222	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → (+g‘𝑅) = (+g‘(oppr‘𝑂)))
19	18	oveqdr 5924	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥(+g‘𝑅)𝑦) = (𝑥(+g‘(oppr‘𝑂))𝑦))
20		vex 2755	. . . . . . . 8 ⊢ 𝑥 ∈ V
21	20	a1i 9	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → 𝑥 ∈ V)
22		vex 2755	. . . . . . . 8 ⊢ 𝑦 ∈ V
23	22	a1i 9	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → 𝑦 ∈ V)
24		eqid 2189	. . . . . . . 8 ⊢ (.r‘𝑂) = (.r‘𝑂)
25		eqid 2189	. . . . . . . 8 ⊢ (.r‘(oppr‘𝑂)) = (.r‘(oppr‘𝑂))
26	9, 24, 3, 25	opprmulg 13421	. . . . . . 7 ⊢ ((𝑂 ∈ V ∧ 𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥(.r‘(oppr‘𝑂))𝑦) = (𝑦(.r‘𝑂)𝑥))
27	8, 21, 23, 26	syl3anc 1249	. . . . . 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 531	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → 𝑦 ∈ (Base‘𝑅))
31		simprl 529	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → 𝑥 ∈ (Base‘𝑅))
32		eqid 2189	. . . . . . 7 ⊢ (.r‘𝑅) = (.r‘𝑅)
33	5, 32, 1, 24	opprmulg 13421	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑦(.r‘𝑂)𝑥) = (𝑥(.r‘𝑅)𝑦))
34	29, 30, 31, 33	syl3anc 1249	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑦(.r‘𝑂)𝑥) = (𝑥(.r‘𝑅)𝑦))
35	28, 34	eqtr2d 2223	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥(.r‘𝑅)𝑦) = (𝑥(.r‘(oppr‘𝑂))𝑦))
36	6, 12, 19, 35	rngpropd 13309	. . 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 1364   ∈ wcel 2160  Vcvv 2752  ‘cfv 5235  (class class class)co 5896  Basecbs 12512  +_gcplusg 12589  ._rcmulr 12590  Rngcrng 13286  opp_rcoppr 13417

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7932  ax-resscn 7933  ax-1cn 7934  ax-1re 7935  ax-icn 7936  ax-addcl 7937  ax-addrcl 7938  ax-mulcl 7939  ax-addcom 7941  ax-addass 7943  ax-i2m1 7946  ax-0lt1 7947  ax-0id 7949  ax-rnegex 7950  ax-pre-ltirr 7953  ax-pre-lttrn 7955  ax-pre-ltadd 7957

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-fv 5243  df-riota 5852  df-ov 5899  df-oprab 5900  df-mpo 5901  df-tpos 6270  df-pnf 8024  df-mnf 8025  df-ltxr 8027  df-inn 8950  df-2 9008  df-3 9009  df-ndx 12515  df-slot 12516  df-base 12518  df-sets 12519  df-plusg 12602  df-mulr 12603  df-0g 12763  df-mgm 12832  df-sgrp 12865  df-mnd 12878  df-grp 12948  df-cmn 13225  df-abl 13226  df-mgp 13275  df-rng 13287  df-oppr 13418

This theorem is referenced by:  opprsubrngg  13558