Theorem opprrngbg 14041

Description: A set is a non-unital ring if and only if its opposite is a non-unital ring. Bidirectional form of opprrng 14040. (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 14040	. 2 ⊢ (𝑅 ∈ Rng → 𝑂 ∈ Rng)
3		eqid 2229	. . . 4 ⊢ (oppr‘𝑂) = (oppr‘𝑂)
4	3	opprrng 14040	. . 3 ⊢ (𝑂 ∈ Rng → (oppr‘𝑂) ∈ Rng)
5		eqid 2229	. . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅)
6	5	a1i 9	. . . 4 ⊢ (𝑅 ∈ 𝑉 → (Base‘𝑅) = (Base‘𝑅))
7	1, 5	opprbasg 14038	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → (Base‘𝑅) = (Base‘𝑂))
8	1	opprex 14036	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → 𝑂 ∈ V)
9		eqid 2229	. . . . . . 7 ⊢ (Base‘𝑂) = (Base‘𝑂)
10	3, 9	opprbasg 14038	. . . . . 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 2262	. . . 4 ⊢ (𝑅 ∈ 𝑉 → (Base‘𝑅) = (Base‘(oppr‘𝑂)))
13		eqid 2229	. . . . . . 7 ⊢ (+g‘𝑅) = (+g‘𝑅)
14	1, 13	oppraddg 14039	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → (+g‘𝑅) = (+g‘𝑂))
15		eqid 2229	. . . . . . . 8 ⊢ (+g‘𝑂) = (+g‘𝑂)
16	3, 15	oppraddg 14039	. . . . . . 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 2262	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → (+g‘𝑅) = (+g‘(oppr‘𝑂)))
19	18	oveqdr 6029	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥(+g‘𝑅)𝑦) = (𝑥(+g‘(oppr‘𝑂))𝑦))
20		vex 2802	. . . . . . . 8 ⊢ 𝑥 ∈ V
21	20	a1i 9	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → 𝑥 ∈ V)
22		vex 2802	. . . . . . . 8 ⊢ 𝑦 ∈ V
23	22	a1i 9	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → 𝑦 ∈ V)
24		eqid 2229	. . . . . . . 8 ⊢ (.r‘𝑂) = (.r‘𝑂)
25		eqid 2229	. . . . . . . 8 ⊢ (.r‘(oppr‘𝑂)) = (.r‘(oppr‘𝑂))
26	9, 24, 3, 25	opprmulg 14034	. . . . . . 7 ⊢ ((𝑂 ∈ V ∧ 𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥(.r‘(oppr‘𝑂))𝑦) = (𝑦(.r‘𝑂)𝑥))
27	8, 21, 23, 26	syl3anc 1271	. . . . . 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 2229	. . . . . . 7 ⊢ (.r‘𝑅) = (.r‘𝑅)
33	5, 32, 1, 24	opprmulg 14034	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑦(.r‘𝑂)𝑥) = (𝑥(.r‘𝑅)𝑦))
34	29, 30, 31, 33	syl3anc 1271	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑦(.r‘𝑂)𝑥) = (𝑥(.r‘𝑅)𝑦))
35	28, 34	eqtr2d 2263	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥(.r‘𝑅)𝑦) = (𝑥(.r‘(oppr‘𝑂))𝑦))
36	6, 12, 19, 35	rngpropd 13918	. . 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 1395   ∈ wcel 2200  Vcvv 2799  ‘cfv 5318  (class class class)co 6001  Basecbs 13032  +_gcplusg 13110  ._rcmulr 13111  Rngcrng 13895  opp_rcoppr 14030

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-addcom 8099  ax-addass 8101  ax-i2m1 8104  ax-0lt1 8105  ax-0id 8107  ax-rnegex 8108  ax-pre-ltirr 8111  ax-pre-lttrn 8113  ax-pre-ltadd 8115

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-tpos 6391  df-pnf 8183  df-mnf 8184  df-ltxr 8186  df-inn 9111  df-2 9169  df-3 9170  df-ndx 13035  df-slot 13036  df-base 13038  df-sets 13039  df-plusg 13123  df-mulr 13124  df-0g 13291  df-mgm 13389  df-sgrp 13435  df-mnd 13450  df-grp 13536  df-cmn 13823  df-abl 13824  df-mgp 13884  df-rng 13896  df-oppr 14031

This theorem is referenced by:  opprsubrngg  14175