Theorem opprsubrngg 13843

Description: Being a subring is a symmetric property. (Contributed by AV, 15-Feb-2025.)

Hypothesis

Ref	Expression
opprsubrng.o	⊢ 𝑂 = (oppr‘𝑅)

Assertion

Ref	Expression
opprsubrngg	⊢ (𝑅 ∈ 𝑉 → (SubRng‘𝑅) = (SubRng‘𝑂))

Proof of Theorem opprsubrngg

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		subrngrcl 13835	. . . 4 ⊢ (𝑥 ∈ (SubRng‘𝑅) → 𝑅 ∈ Rng)
2	1	a1i 9	. . 3 ⊢ (𝑅 ∈ 𝑉 → (𝑥 ∈ (SubRng‘𝑅) → 𝑅 ∈ Rng))
3		subrngrcl 13835	. . . 4 ⊢ (𝑥 ∈ (SubRng‘𝑂) → 𝑂 ∈ Rng)
4		opprsubrng.o	. . . . 5 ⊢ 𝑂 = (oppr‘𝑅)
5	4	opprrngbg 13710	. . . 4 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rng ↔ 𝑂 ∈ Rng))
6	3, 5	imbitrrid 156	. . 3 ⊢ (𝑅 ∈ 𝑉 → (𝑥 ∈ (SubRng‘𝑂) → 𝑅 ∈ Rng))
7	4	opprsubgg 13716	. . . . . . 7 ⊢ (𝑅 ∈ Rng → (SubGrp‘𝑅) = (SubGrp‘𝑂))
8	7	eleq2d 2266	. . . . . 6 ⊢ (𝑅 ∈ Rng → (𝑥 ∈ (SubGrp‘𝑅) ↔ 𝑥 ∈ (SubGrp‘𝑂)))
9		ralcom 2660	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝑥 ∀𝑦 ∈ 𝑥 (𝑧(.r‘𝑅)𝑦) ∈ 𝑥 ↔ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑧(.r‘𝑅)𝑦) ∈ 𝑥)
10		vex 2766	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
11		vex 2766	. . . . . . . . . 10 ⊢ 𝑧 ∈ V
12		eqid 2196	. . . . . . . . . . 11 ⊢ (Base‘𝑅) = (Base‘𝑅)
13		eqid 2196	. . . . . . . . . . 11 ⊢ (.r‘𝑅) = (.r‘𝑅)
14		eqid 2196	. . . . . . . . . . 11 ⊢ (.r‘𝑂) = (.r‘𝑂)
15	12, 13, 4, 14	opprmulg 13703	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Rng ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑦(.r‘𝑂)𝑧) = (𝑧(.r‘𝑅)𝑦))
16	10, 11, 15	mp3an23 1340	. . . . . . . . 9 ⊢ (𝑅 ∈ Rng → (𝑦(.r‘𝑂)𝑧) = (𝑧(.r‘𝑅)𝑦))
17	16	eleq1d 2265	. . . . . . . 8 ⊢ (𝑅 ∈ Rng → ((𝑦(.r‘𝑂)𝑧) ∈ 𝑥 ↔ (𝑧(.r‘𝑅)𝑦) ∈ 𝑥))
18	17	2ralbidv 2521	. . . . . . 7 ⊢ (𝑅 ∈ Rng → (∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑦(.r‘𝑂)𝑧) ∈ 𝑥 ↔ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑧(.r‘𝑅)𝑦) ∈ 𝑥))
19	9, 18	bitr4id 199	. . . . . 6 ⊢ (𝑅 ∈ Rng → (∀𝑧 ∈ 𝑥 ∀𝑦 ∈ 𝑥 (𝑧(.r‘𝑅)𝑦) ∈ 𝑥 ↔ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑦(.r‘𝑂)𝑧) ∈ 𝑥))
20	8, 19	anbi12d 473	. . . . 5 ⊢ (𝑅 ∈ Rng → ((𝑥 ∈ (SubGrp‘𝑅) ∧ ∀𝑧 ∈ 𝑥 ∀𝑦 ∈ 𝑥 (𝑧(.r‘𝑅)𝑦) ∈ 𝑥) ↔ (𝑥 ∈ (SubGrp‘𝑂) ∧ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑦(.r‘𝑂)𝑧) ∈ 𝑥)))
21	12, 13	issubrng2 13842	. . . . 5 ⊢ (𝑅 ∈ Rng → (𝑥 ∈ (SubRng‘𝑅) ↔ (𝑥 ∈ (SubGrp‘𝑅) ∧ ∀𝑧 ∈ 𝑥 ∀𝑦 ∈ 𝑥 (𝑧(.r‘𝑅)𝑦) ∈ 𝑥)))
22	4	opprrng 13709	. . . . . 6 ⊢ (𝑅 ∈ Rng → 𝑂 ∈ Rng)
23		eqid 2196	. . . . . . 7 ⊢ (Base‘𝑂) = (Base‘𝑂)
24	23, 14	issubrng2 13842	. . . . . 6 ⊢ (𝑂 ∈ Rng → (𝑥 ∈ (SubRng‘𝑂) ↔ (𝑥 ∈ (SubGrp‘𝑂) ∧ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑦(.r‘𝑂)𝑧) ∈ 𝑥)))
25	22, 24	syl 14	. . . . 5 ⊢ (𝑅 ∈ Rng → (𝑥 ∈ (SubRng‘𝑂) ↔ (𝑥 ∈ (SubGrp‘𝑂) ∧ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑦(.r‘𝑂)𝑧) ∈ 𝑥)))
26	20, 21, 25	3bitr4d 220	. . . 4 ⊢ (𝑅 ∈ Rng → (𝑥 ∈ (SubRng‘𝑅) ↔ 𝑥 ∈ (SubRng‘𝑂)))
27	26	a1i 9	. . 3 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rng → (𝑥 ∈ (SubRng‘𝑅) ↔ 𝑥 ∈ (SubRng‘𝑂))))
28	2, 6, 27	pm5.21ndd 706	. 2 ⊢ (𝑅 ∈ 𝑉 → (𝑥 ∈ (SubRng‘𝑅) ↔ 𝑥 ∈ (SubRng‘𝑂)))
29	28	eqrdv 2194	1 ⊢ (𝑅 ∈ 𝑉 → (SubRng‘𝑅) = (SubRng‘𝑂))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364   ∈ wcel 2167  ∀wral 2475  Vcvv 2763  ‘cfv 5259  (class class class)co 5925  Basecbs 12703  ._rcmulr 12781  SubGrpcsubg 13373  Rngcrng 13564  opp_rcoppr 13699  SubRngcsubrng 13829

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 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-addcom 7996  ax-addass 7998  ax-i2m1 8001  ax-0lt1 8002  ax-0id 8004  ax-rnegex 8005  ax-pre-ltirr 8008  ax-pre-lttrn 8010  ax-pre-ltadd 8012

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 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-tpos 6312  df-pnf 8080  df-mnf 8081  df-ltxr 8083  df-inn 9008  df-2 9066  df-3 9067  df-ndx 12706  df-slot 12707  df-base 12709  df-sets 12710  df-iress 12711  df-plusg 12793  df-mulr 12794  df-0g 12960  df-mgm 13058  df-sgrp 13104  df-mnd 13119  df-grp 13205  df-subg 13376  df-cmn 13492  df-abl 13493  df-mgp 13553  df-rng 13565  df-oppr 13700  df-subrng 13830

This theorem is referenced by: (None)