Theorem opprsubgg 13583

Description: Being a subgroup is a symmetric property. (Contributed by Mario Carneiro, 6-Dec-2014.)

Hypothesis

Ref	Expression
opprbas.1	⊢ 𝑂 = (oppr‘𝑅)

Assertion

Ref	Expression
opprsubgg	⊢ (𝑅 ∈ 𝑉 → (SubGrp‘𝑅) = (SubGrp‘𝑂))

Proof of Theorem opprsubgg

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

Step	Hyp	Ref	Expression
1		eqidd 2194	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → (Base‘𝑅) = (Base‘𝑅))
2		opprbas.1	. . . . . 6 ⊢ 𝑂 = (oppr‘𝑅)
3		eqid 2193	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
4	2, 3	opprbasg 13574	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → (Base‘𝑅) = (Base‘𝑂))
5		eqid 2193	. . . . . . 7 ⊢ (+g‘𝑅) = (+g‘𝑅)
6	2, 5	oppraddg 13575	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → (+g‘𝑅) = (+g‘𝑂))
7	6	oveqdr 5947	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥(+g‘𝑅)𝑦) = (𝑥(+g‘𝑂)𝑦))
8	1, 4, 7	grppropd 13092	. . . 4 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Grp ↔ 𝑂 ∈ Grp))
9		eqidd 2194	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → (Base‘(𝑅 ↾s 𝑥)) = (Base‘(𝑅 ↾s 𝑥)))
10		eqidd 2194	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ↾s 𝑥) = (𝑅 ↾s 𝑥))
11		id 19	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ 𝑉)
12		vex 2763	. . . . . . . 8 ⊢ 𝑥 ∈ V
13	12	a1i 9	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → 𝑥 ∈ V)
14	10, 1, 11, 13	ressbasd 12688	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → (𝑥 ∩ (Base‘𝑅)) = (Base‘(𝑅 ↾s 𝑥)))
15		eqidd 2194	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → (𝑂 ↾s 𝑥) = (𝑂 ↾s 𝑥))
16	2	opprex 13572	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → 𝑂 ∈ V)
17	15, 4, 16, 13	ressbasd 12688	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → (𝑥 ∩ (Base‘𝑅)) = (Base‘(𝑂 ↾s 𝑥)))
18	14, 17	eqtr3d 2228	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → (Base‘(𝑅 ↾s 𝑥)) = (Base‘(𝑂 ↾s 𝑥)))
19		eqidd 2194	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → (+g‘𝑅) = (+g‘𝑅))
20	10, 19, 13, 11	ressplusgd 12749	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → (+g‘𝑅) = (+g‘(𝑅 ↾s 𝑥)))
21	15, 6, 13, 16	ressplusgd 12749	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → (+g‘𝑅) = (+g‘(𝑂 ↾s 𝑥)))
22	20, 21	eqtr3d 2228	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → (+g‘(𝑅 ↾s 𝑥)) = (+g‘(𝑂 ↾s 𝑥)))
23	22	oveqdr 5947	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑧 ∈ (Base‘(𝑅 ↾s 𝑥)) ∧ 𝑤 ∈ (Base‘(𝑅 ↾s 𝑥)))) → (𝑧(+g‘(𝑅 ↾s 𝑥))𝑤) = (𝑧(+g‘(𝑂 ↾s 𝑥))𝑤))
24	9, 18, 23	grppropd 13092	. . . 4 ⊢ (𝑅 ∈ 𝑉 → ((𝑅 ↾s 𝑥) ∈ Grp ↔ (𝑂 ↾s 𝑥) ∈ Grp))
25	8, 24	3anbi13d 1325	. . 3 ⊢ (𝑅 ∈ 𝑉 → ((𝑅 ∈ Grp ∧ 𝑥 ⊆ (Base‘𝑅) ∧ (𝑅 ↾s 𝑥) ∈ Grp) ↔ (𝑂 ∈ Grp ∧ 𝑥 ⊆ (Base‘𝑅) ∧ (𝑂 ↾s 𝑥) ∈ Grp)))
26	3	issubg 13246	. . . 4 ⊢ (𝑥 ∈ (SubGrp‘𝑅) ↔ (𝑅 ∈ Grp ∧ 𝑥 ⊆ (Base‘𝑅) ∧ (𝑅 ↾s 𝑥) ∈ Grp))
27	26	a1i 9	. . 3 ⊢ (𝑅 ∈ 𝑉 → (𝑥 ∈ (SubGrp‘𝑅) ↔ (𝑅 ∈ Grp ∧ 𝑥 ⊆ (Base‘𝑅) ∧ (𝑅 ↾s 𝑥) ∈ Grp)))
28		eqid 2193	. . . . 5 ⊢ (Base‘𝑂) = (Base‘𝑂)
29	28	issubg 13246	. . . 4 ⊢ (𝑥 ∈ (SubGrp‘𝑂) ↔ (𝑂 ∈ Grp ∧ 𝑥 ⊆ (Base‘𝑂) ∧ (𝑂 ↾s 𝑥) ∈ Grp))
30	4	sseq2d 3210	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → (𝑥 ⊆ (Base‘𝑅) ↔ 𝑥 ⊆ (Base‘𝑂)))
31	30	3anbi2d 1328	. . . 4 ⊢ (𝑅 ∈ 𝑉 → ((𝑂 ∈ Grp ∧ 𝑥 ⊆ (Base‘𝑅) ∧ (𝑂 ↾s 𝑥) ∈ Grp) ↔ (𝑂 ∈ Grp ∧ 𝑥 ⊆ (Base‘𝑂) ∧ (𝑂 ↾s 𝑥) ∈ Grp)))
32	29, 31	bitr4id 199	. . 3 ⊢ (𝑅 ∈ 𝑉 → (𝑥 ∈ (SubGrp‘𝑂) ↔ (𝑂 ∈ Grp ∧ 𝑥 ⊆ (Base‘𝑅) ∧ (𝑂 ↾s 𝑥) ∈ Grp)))
33	25, 27, 32	3bitr4d 220	. 2 ⊢ (𝑅 ∈ 𝑉 → (𝑥 ∈ (SubGrp‘𝑅) ↔ 𝑥 ∈ (SubGrp‘𝑂)))
34	33	eqrdv 2191	1 ⊢ (𝑅 ∈ 𝑉 → (SubGrp‘𝑅) = (SubGrp‘𝑂))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 980   = wceq 1364   ∈ wcel 2164  Vcvv 2760   ∩ cin 3153   ⊆ wss 3154  ‘cfv 5255  (class class class)co 5919  Basecbs 12621   ↾_s cress 12622  +_gcplusg 12698  Grpcgrp 13075  SubGrpcsubg 13240  opp_rcoppr 13566

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 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-addcom 7974  ax-addass 7976  ax-i2m1 7979  ax-0lt1 7980  ax-0id 7982  ax-rnegex 7983  ax-pre-ltirr 7986  ax-pre-lttrn 7988  ax-pre-ltadd 7990

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 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-tpos 6300  df-pnf 8058  df-mnf 8059  df-ltxr 8061  df-inn 8985  df-2 9043  df-3 9044  df-ndx 12624  df-slot 12625  df-base 12627  df-sets 12628  df-iress 12629  df-plusg 12711  df-mulr 12712  df-0g 12872  df-mgm 12942  df-sgrp 12988  df-mnd 13001  df-grp 13078  df-subg 13243  df-oppr 13567

This theorem is referenced by:  opprsubrngg  13710  isridlrng  13981  isridl  14003