Theorem opprsubrngg 14492

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 14484	. . . 4 ⊢ (𝑥 ∈ (SubRng‘𝑅) → 𝑅 ∈ Rng)
2	1	a1i 9	. . 3 ⊢ (𝑅 ∈ 𝑉 → (𝑥 ∈ (SubRng‘𝑅) → 𝑅 ∈ Rng))
3		subrngrcl 14484	. . . 4 ⊢ (𝑥 ∈ (SubRng‘𝑂) → 𝑂 ∈ Rng)
4		opprsubrng.o	. . . . 5 ⊢ 𝑂 = (oppr‘𝑅)
5	4	opprrngbg 14356	. . . 4 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rng ↔ 𝑂 ∈ Rng))
6	3, 5	imbitrrid 156	. . 3 ⊢ (𝑅 ∈ 𝑉 → (𝑥 ∈ (SubRng‘𝑂) → 𝑅 ∈ Rng))
7	4	opprsubgg 14363	. . . . . . 7 ⊢ (𝑅 ∈ Rng → (SubGrp‘𝑅) = (SubGrp‘𝑂))
8	7	eleq2d 2308	. . . . . 6 ⊢ (𝑅 ∈ Rng → (𝑥 ∈ (SubGrp‘𝑅) ↔ 𝑥 ∈ (SubGrp‘𝑂)))
9		ralcom 2714	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝑥 ∀𝑦 ∈ 𝑥 (𝑧(.r‘𝑅)𝑦) ∈ 𝑥 ↔ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑧(.r‘𝑅)𝑦) ∈ 𝑥)
10		vex 2824	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
11		vex 2824	. . . . . . . . . 10 ⊢ 𝑧 ∈ V
12		eqid 2238	. . . . . . . . . . 11 ⊢ (Base‘𝑅) = (Base‘𝑅)
13		eqid 2238	. . . . . . . . . . 11 ⊢ (.r‘𝑅) = (.r‘𝑅)
14		eqid 2238	. . . . . . . . . . 11 ⊢ (.r‘𝑂) = (.r‘𝑂)
15	12, 13, 4, 14	opprmulg 14349	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Rng ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑦(.r‘𝑂)𝑧) = (𝑧(.r‘𝑅)𝑦))
16	10, 11, 15	mp3an23 1370	. . . . . . . . 9 ⊢ (𝑅 ∈ Rng → (𝑦(.r‘𝑂)𝑧) = (𝑧(.r‘𝑅)𝑦))
17	16	eleq1d 2307	. . . . . . . 8 ⊢ (𝑅 ∈ Rng → ((𝑦(.r‘𝑂)𝑧) ∈ 𝑥 ↔ (𝑧(.r‘𝑅)𝑦) ∈ 𝑥))
18	17	2ralbidv 2574	. . . . . . 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 477	. . . . 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 14491	. . . . 5 ⊢ (𝑅 ∈ Rng → (𝑥 ∈ (SubRng‘𝑅) ↔ (𝑥 ∈ (SubGrp‘𝑅) ∧ ∀𝑧 ∈ 𝑥 ∀𝑦 ∈ 𝑥 (𝑧(.r‘𝑅)𝑦) ∈ 𝑥)))
22	4	opprrng 14355	. . . . . 6 ⊢ (𝑅 ∈ Rng → 𝑂 ∈ Rng)
23		eqid 2238	. . . . . . 7 ⊢ (Base‘𝑂) = (Base‘𝑂)
24	23, 14	issubrng2 14491	. . . . . 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 717	. 2 ⊢ (𝑅 ∈ 𝑉 → (𝑥 ∈ (SubRng‘𝑅) ↔ 𝑥 ∈ (SubRng‘𝑂)))
29	28	eqrdv 2236	1 ⊢ (𝑅 ∈ 𝑉 → (SubRng‘𝑅) = (SubRng‘𝑂))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1402   ∈ wcel 2209  ∀wral 2528  Vcvv 2821  ‘cfv 5372  (class class class)co 6075  Basecbs 13330  ._rcmulr 13409  SubGrpcsubg 13947  Rngcrng 14206  opp_rcoppr 14345  SubRngcsubrng 14478

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-pre-ltirr 8281  ax-pre-lttrn 8283  ax-pre-ltadd 8285

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-tpos 6506  df-pnf 8352  df-mnf 8353  df-ltxr 8355  df-inn 9284  df-2 9342  df-3 9343  df-ndx 13333  df-slot 13334  df-base 13336  df-sets 13337  df-iress 13338  df-plusg 13421  df-mulr 13422  df-0g 13589  df-mgm 13653  df-sgrp 13694  df-mnd 13707  df-grp 13785  df-subg 13950  df-cmn 14066  df-abl 14067  df-mgp 14195  df-rng 14207  df-oppr 14346  df-subrng 14479

This theorem is referenced by: (None)