Theorem opprsubgg 14328

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 2235	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → (Base‘𝑅) = (Base‘𝑅))
2		opprbas.1	. . . . . 6 ⊢ 𝑂 = (oppr‘𝑅)
3		eqid 2234	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
4	2, 3	opprbasg 14318	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → (Base‘𝑅) = (Base‘𝑂))
5		eqid 2234	. . . . . . 7 ⊢ (+g‘𝑅) = (+g‘𝑅)
6	2, 5	oppraddg 14319	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → (+g‘𝑅) = (+g‘𝑂))
7	6	oveqdr 6086	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥(+g‘𝑅)𝑦) = (𝑥(+g‘𝑂)𝑦))
8	1, 4, 7	grppropd 13772	. . . 4 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Grp ↔ 𝑂 ∈ Grp))
9		eqidd 2235	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → (Base‘(𝑅 ↾s 𝑥)) = (Base‘(𝑅 ↾s 𝑥)))
10		eqidd 2235	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ↾s 𝑥) = (𝑅 ↾s 𝑥))
11		id 19	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ 𝑉)
12		vex 2818	. . . . . . . 8 ⊢ 𝑥 ∈ V
13	12	a1i 9	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → 𝑥 ∈ V)
14	10, 1, 11, 13	ressbasd 13364	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → (𝑥 ∩ (Base‘𝑅)) = (Base‘(𝑅 ↾s 𝑥)))
15		eqidd 2235	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → (𝑂 ↾s 𝑥) = (𝑂 ↾s 𝑥))
16	2	opprex 14316	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → 𝑂 ∈ V)
17	15, 4, 16, 13	ressbasd 13364	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → (𝑥 ∩ (Base‘𝑅)) = (Base‘(𝑂 ↾s 𝑥)))
18	14, 17	eqtr3d 2269	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → (Base‘(𝑅 ↾s 𝑥)) = (Base‘(𝑂 ↾s 𝑥)))
19		eqidd 2235	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → (+g‘𝑅) = (+g‘𝑅))
20	10, 19, 13, 11	ressplusgd 13426	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → (+g‘𝑅) = (+g‘(𝑅 ↾s 𝑥)))
21	15, 6, 13, 16	ressplusgd 13426	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → (+g‘𝑅) = (+g‘(𝑂 ↾s 𝑥)))
22	20, 21	eqtr3d 2269	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → (+g‘(𝑅 ↾s 𝑥)) = (+g‘(𝑂 ↾s 𝑥)))
23	22	oveqdr 6086	. . . . 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 13772	. . . 4 ⊢ (𝑅 ∈ 𝑉 → ((𝑅 ↾s 𝑥) ∈ Grp ↔ (𝑂 ↾s 𝑥) ∈ Grp))
25	8, 24	3anbi13d 1351	. . 3 ⊢ (𝑅 ∈ 𝑉 → ((𝑅 ∈ Grp ∧ 𝑥 ⊆ (Base‘𝑅) ∧ (𝑅 ↾s 𝑥) ∈ Grp) ↔ (𝑂 ∈ Grp ∧ 𝑥 ⊆ (Base‘𝑅) ∧ (𝑂 ↾s 𝑥) ∈ Grp)))
26	3	issubg 13926	. . . 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 2234	. . . . 5 ⊢ (Base‘𝑂) = (Base‘𝑂)
29	28	issubg 13926	. . . 4 ⊢ (𝑥 ∈ (SubGrp‘𝑂) ↔ (𝑂 ∈ Grp ∧ 𝑥 ⊆ (Base‘𝑂) ∧ (𝑂 ↾s 𝑥) ∈ Grp))
30	4	sseq2d 3272	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → (𝑥 ⊆ (Base‘𝑅) ↔ 𝑥 ⊆ (Base‘𝑂)))
31	30	3anbi2d 1354	. . . 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 2232	1 ⊢ (𝑅 ∈ 𝑉 → (SubGrp‘𝑅) = (SubGrp‘𝑂))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1005   = wceq 1398   ∈ wcel 2205  Vcvv 2815   ∩ cin 3213   ⊆ wss 3214  ‘cfv 5357  (class class class)co 6058  Basecbs 13296   ↾_s cress 13297  +_gcplusg 13374  Grpcgrp 13755  SubGrpcsubg 13920  opp_rcoppr 14310

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 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-pre-ltirr 8255  ax-pre-lttrn 8257  ax-pre-ltadd 8259

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 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-tpos 6489  df-pnf 8326  df-mnf 8327  df-ltxr 8329  df-inn 9255  df-2 9313  df-3 9314  df-ndx 13299  df-slot 13300  df-base 13302  df-sets 13303  df-iress 13304  df-plusg 13387  df-mulr 13388  df-0g 13555  df-mgm 13619  df-sgrp 13665  df-mnd 13678  df-grp 13758  df-subg 13923  df-oppr 14311

This theorem is referenced by:  opprsubrngg  14457  isridlrng  14756  isridl  14778