Theorem opprsubrngg 13558

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 13550	. . . 4 ⊢ (𝑥 ∈ (SubRng‘𝑅) → 𝑅 ∈ Rng)
2	1	a1i 9	. . 3 ⊢ (𝑅 ∈ 𝑉 → (𝑥 ∈ (SubRng‘𝑅) → 𝑅 ∈ Rng))
3		subrngrcl 13550	. . . 4 ⊢ (𝑥 ∈ (SubRng‘𝑂) → 𝑂 ∈ Rng)
4		opprsubrng.o	. . . . 5 ⊢ 𝑂 = (oppr‘𝑅)
5	4	opprrngbg 13428	. . . 4 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rng ↔ 𝑂 ∈ Rng))
6	3, 5	imbitrrid 156	. . 3 ⊢ (𝑅 ∈ 𝑉 → (𝑥 ∈ (SubRng‘𝑂) → 𝑅 ∈ Rng))
7	4	opprsubgg 13434	. . . . . . 7 ⊢ (𝑅 ∈ Rng → (SubGrp‘𝑅) = (SubGrp‘𝑂))
8	7	eleq2d 2259	. . . . . 6 ⊢ (𝑅 ∈ Rng → (𝑥 ∈ (SubGrp‘𝑅) ↔ 𝑥 ∈ (SubGrp‘𝑂)))
9		ralcom 2653	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝑥 ∀𝑦 ∈ 𝑥 (𝑧(.r‘𝑅)𝑦) ∈ 𝑥 ↔ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑧(.r‘𝑅)𝑦) ∈ 𝑥)
10		vex 2755	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
11		vex 2755	. . . . . . . . . 10 ⊢ 𝑧 ∈ V
12		eqid 2189	. . . . . . . . . . 11 ⊢ (Base‘𝑅) = (Base‘𝑅)
13		eqid 2189	. . . . . . . . . . 11 ⊢ (.r‘𝑅) = (.r‘𝑅)
14		eqid 2189	. . . . . . . . . . 11 ⊢ (.r‘𝑂) = (.r‘𝑂)
15	12, 13, 4, 14	opprmulg 13421	. . . . . . . . . 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 2258	. . . . . . . 8 ⊢ (𝑅 ∈ Rng → ((𝑦(.r‘𝑂)𝑧) ∈ 𝑥 ↔ (𝑧(.r‘𝑅)𝑦) ∈ 𝑥))
18	17	2ralbidv 2514	. . . . . . 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 13557	. . . . 5 ⊢ (𝑅 ∈ Rng → (𝑥 ∈ (SubRng‘𝑅) ↔ (𝑥 ∈ (SubGrp‘𝑅) ∧ ∀𝑧 ∈ 𝑥 ∀𝑦 ∈ 𝑥 (𝑧(.r‘𝑅)𝑦) ∈ 𝑥)))
22	4	opprrng 13427	. . . . . 6 ⊢ (𝑅 ∈ Rng → 𝑂 ∈ Rng)
23		eqid 2189	. . . . . . 7 ⊢ (Base‘𝑂) = (Base‘𝑂)
24	23, 14	issubrng2 13557	. . . . . 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 2187	1 ⊢ (𝑅 ∈ 𝑉 → (SubRng‘𝑅) = (SubRng‘𝑂))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364   ∈ wcel 2160  ∀wral 2468  Vcvv 2752  ‘cfv 5235  (class class class)co 5896  Basecbs 12512  ._rcmulr 12590  SubGrpcsubg 13106  Rngcrng 13286  opp_rcoppr 13417  SubRngcsubrng 13544

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7932  ax-resscn 7933  ax-1cn 7934  ax-1re 7935  ax-icn 7936  ax-addcl 7937  ax-addrcl 7938  ax-mulcl 7939  ax-addcom 7941  ax-addass 7943  ax-i2m1 7946  ax-0lt1 7947  ax-0id 7949  ax-rnegex 7950  ax-pre-ltirr 7953  ax-pre-lttrn 7955  ax-pre-ltadd 7957

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-fv 5243  df-riota 5852  df-ov 5899  df-oprab 5900  df-mpo 5901  df-tpos 6270  df-pnf 8024  df-mnf 8025  df-ltxr 8027  df-inn 8950  df-2 9008  df-3 9009  df-ndx 12515  df-slot 12516  df-base 12518  df-sets 12519  df-iress 12520  df-plusg 12602  df-mulr 12603  df-0g 12763  df-mgm 12832  df-sgrp 12865  df-mnd 12878  df-grp 12948  df-subg 13109  df-cmn 13225  df-abl 13226  df-mgp 13275  df-rng 13287  df-oppr 13418  df-subrng 13545

This theorem is referenced by: (None)