Theorem opprrngbg 13634

Description: A set is a non-unital ring if and only if its opposite is a non-unital ring. Bidirectional form of opprrng 13633. (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 13633	. 2 ⊢ (𝑅 ∈ Rng → 𝑂 ∈ Rng)
3		eqid 2196	. . . 4 ⊢ (oppr‘𝑂) = (oppr‘𝑂)
4	3	opprrng 13633	. . 3 ⊢ (𝑂 ∈ Rng → (oppr‘𝑂) ∈ Rng)
5		eqid 2196	. . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅)
6	5	a1i 9	. . . 4 ⊢ (𝑅 ∈ 𝑉 → (Base‘𝑅) = (Base‘𝑅))
7	1, 5	opprbasg 13631	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → (Base‘𝑅) = (Base‘𝑂))
8	1	opprex 13629	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → 𝑂 ∈ V)
9		eqid 2196	. . . . . . 7 ⊢ (Base‘𝑂) = (Base‘𝑂)
10	3, 9	opprbasg 13631	. . . . . 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 2229	. . . 4 ⊢ (𝑅 ∈ 𝑉 → (Base‘𝑅) = (Base‘(oppr‘𝑂)))
13		eqid 2196	. . . . . . 7 ⊢ (+g‘𝑅) = (+g‘𝑅)
14	1, 13	oppraddg 13632	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → (+g‘𝑅) = (+g‘𝑂))
15		eqid 2196	. . . . . . . 8 ⊢ (+g‘𝑂) = (+g‘𝑂)
16	3, 15	oppraddg 13632	. . . . . . 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 2229	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → (+g‘𝑅) = (+g‘(oppr‘𝑂)))
19	18	oveqdr 5950	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥(+g‘𝑅)𝑦) = (𝑥(+g‘(oppr‘𝑂))𝑦))
20		vex 2766	. . . . . . . 8 ⊢ 𝑥 ∈ V
21	20	a1i 9	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → 𝑥 ∈ V)
22		vex 2766	. . . . . . . 8 ⊢ 𝑦 ∈ V
23	22	a1i 9	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → 𝑦 ∈ V)
24		eqid 2196	. . . . . . . 8 ⊢ (.r‘𝑂) = (.r‘𝑂)
25		eqid 2196	. . . . . . . 8 ⊢ (.r‘(oppr‘𝑂)) = (.r‘(oppr‘𝑂))
26	9, 24, 3, 25	opprmulg 13627	. . . . . . 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 2196	. . . . . . 7 ⊢ (.r‘𝑅) = (.r‘𝑅)
33	5, 32, 1, 24	opprmulg 13627	. . . . . 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 2230	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥(.r‘𝑅)𝑦) = (𝑥(.r‘(oppr‘𝑂))𝑦))
36	6, 12, 19, 35	rngpropd 13511	. . 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 2167  Vcvv 2763  ‘cfv 5258  (class class class)co 5922  Basecbs 12678  +_gcplusg 12755  ._rcmulr 12756  Rngcrng 13488  opp_rcoppr 13623

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-pre-ltirr 7991  ax-pre-lttrn 7993  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-tpos 6303  df-pnf 8063  df-mnf 8064  df-ltxr 8066  df-inn 8991  df-2 9049  df-3 9050  df-ndx 12681  df-slot 12682  df-base 12684  df-sets 12685  df-plusg 12768  df-mulr 12769  df-0g 12929  df-mgm 12999  df-sgrp 13045  df-mnd 13058  df-grp 13135  df-cmn 13416  df-abl 13417  df-mgp 13477  df-rng 13489  df-oppr 13624

This theorem is referenced by:  opprsubrngg  13767