Theorem opprsubrngg 14356

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 14348	. . . 4 ⊢ (𝑥 ∈ (SubRng‘𝑅) → 𝑅 ∈ Rng)
2	1	a1i 9	. . 3 ⊢ (𝑅 ∈ 𝑉 → (𝑥 ∈ (SubRng‘𝑅) → 𝑅 ∈ Rng))
3		subrngrcl 14348	. . . 4 ⊢ (𝑥 ∈ (SubRng‘𝑂) → 𝑂 ∈ Rng)
4		opprsubrng.o	. . . . 5 ⊢ 𝑂 = (oppr‘𝑅)
5	4	opprrngbg 14222	. . . 4 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rng ↔ 𝑂 ∈ Rng))
6	3, 5	imbitrrid 156	. . 3 ⊢ (𝑅 ∈ 𝑉 → (𝑥 ∈ (SubRng‘𝑂) → 𝑅 ∈ Rng))
7	4	opprsubgg 14228	. . . . . . 7 ⊢ (𝑅 ∈ Rng → (SubGrp‘𝑅) = (SubGrp‘𝑂))
8	7	eleq2d 2302	. . . . . 6 ⊢ (𝑅 ∈ Rng → (𝑥 ∈ (SubGrp‘𝑅) ↔ 𝑥 ∈ (SubGrp‘𝑂)))
9		ralcom 2706	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝑥 ∀𝑦 ∈ 𝑥 (𝑧(.r‘𝑅)𝑦) ∈ 𝑥 ↔ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑧(.r‘𝑅)𝑦) ∈ 𝑥)
10		vex 2816	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
11		vex 2816	. . . . . . . . . 10 ⊢ 𝑧 ∈ V
12		eqid 2232	. . . . . . . . . . 11 ⊢ (Base‘𝑅) = (Base‘𝑅)
13		eqid 2232	. . . . . . . . . . 11 ⊢ (.r‘𝑅) = (.r‘𝑅)
14		eqid 2232	. . . . . . . . . . 11 ⊢ (.r‘𝑂) = (.r‘𝑂)
15	12, 13, 4, 14	opprmulg 14215	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Rng ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑦(.r‘𝑂)𝑧) = (𝑧(.r‘𝑅)𝑦))
16	10, 11, 15	mp3an23 1366	. . . . . . . . 9 ⊢ (𝑅 ∈ Rng → (𝑦(.r‘𝑂)𝑧) = (𝑧(.r‘𝑅)𝑦))
17	16	eleq1d 2301	. . . . . . . 8 ⊢ (𝑅 ∈ Rng → ((𝑦(.r‘𝑂)𝑧) ∈ 𝑥 ↔ (𝑧(.r‘𝑅)𝑦) ∈ 𝑥))
18	17	2ralbidv 2566	. . . . . . 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 14355	. . . . 5 ⊢ (𝑅 ∈ Rng → (𝑥 ∈ (SubRng‘𝑅) ↔ (𝑥 ∈ (SubGrp‘𝑅) ∧ ∀𝑧 ∈ 𝑥 ∀𝑦 ∈ 𝑥 (𝑧(.r‘𝑅)𝑦) ∈ 𝑥)))
22	4	opprrng 14221	. . . . . 6 ⊢ (𝑅 ∈ Rng → 𝑂 ∈ Rng)
23		eqid 2232	. . . . . . 7 ⊢ (Base‘𝑂) = (Base‘𝑂)
24	23, 14	issubrng2 14355	. . . . . 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 713	. 2 ⊢ (𝑅 ∈ 𝑉 → (𝑥 ∈ (SubRng‘𝑅) ↔ 𝑥 ∈ (SubRng‘𝑂)))
29	28	eqrdv 2230	1 ⊢ (𝑅 ∈ 𝑉 → (SubRng‘𝑅) = (SubRng‘𝑂))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398   ∈ wcel 2203  ∀wral 2520  Vcvv 2813  ‘cfv 5352  (class class class)co 6050  Basecbs 13212  ._rcmulr 13291  SubGrpcsubg 13884  Rngcrng 14076  opp_rcoppr 14211  SubRngcsubrng 14342

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-i2m1 8232  ax-0lt1 8233  ax-0id 8235  ax-rnegex 8236  ax-pre-ltirr 8239  ax-pre-lttrn 8241  ax-pre-ltadd 8243

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-tpos 6476  df-pnf 8310  df-mnf 8311  df-ltxr 8313  df-inn 9238  df-2 9296  df-3 9297  df-ndx 13215  df-slot 13216  df-base 13218  df-sets 13219  df-iress 13220  df-plusg 13303  df-mulr 13304  df-0g 13471  df-mgm 13569  df-sgrp 13615  df-mnd 13630  df-grp 13716  df-subg 13887  df-cmn 14003  df-abl 14004  df-mgp 14065  df-rng 14077  df-oppr 14212  df-subrng 14343

This theorem is referenced by: (None)