Theorem opprsubgg 13846

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 2206	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → (Base‘𝑅) = (Base‘𝑅))
2		opprbas.1	. . . . . 6 ⊢ 𝑂 = (oppr‘𝑅)
3		eqid 2205	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
4	2, 3	opprbasg 13837	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → (Base‘𝑅) = (Base‘𝑂))
5		eqid 2205	. . . . . . 7 ⊢ (+g‘𝑅) = (+g‘𝑅)
6	2, 5	oppraddg 13838	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → (+g‘𝑅) = (+g‘𝑂))
7	6	oveqdr 5972	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥(+g‘𝑅)𝑦) = (𝑥(+g‘𝑂)𝑦))
8	1, 4, 7	grppropd 13349	. . . 4 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Grp ↔ 𝑂 ∈ Grp))
9		eqidd 2206	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → (Base‘(𝑅 ↾s 𝑥)) = (Base‘(𝑅 ↾s 𝑥)))
10		eqidd 2206	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ↾s 𝑥) = (𝑅 ↾s 𝑥))
11		id 19	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ 𝑉)
12		vex 2775	. . . . . . . 8 ⊢ 𝑥 ∈ V
13	12	a1i 9	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → 𝑥 ∈ V)
14	10, 1, 11, 13	ressbasd 12899	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → (𝑥 ∩ (Base‘𝑅)) = (Base‘(𝑅 ↾s 𝑥)))
15		eqidd 2206	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → (𝑂 ↾s 𝑥) = (𝑂 ↾s 𝑥))
16	2	opprex 13835	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → 𝑂 ∈ V)
17	15, 4, 16, 13	ressbasd 12899	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → (𝑥 ∩ (Base‘𝑅)) = (Base‘(𝑂 ↾s 𝑥)))
18	14, 17	eqtr3d 2240	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → (Base‘(𝑅 ↾s 𝑥)) = (Base‘(𝑂 ↾s 𝑥)))
19		eqidd 2206	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → (+g‘𝑅) = (+g‘𝑅))
20	10, 19, 13, 11	ressplusgd 12961	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → (+g‘𝑅) = (+g‘(𝑅 ↾s 𝑥)))
21	15, 6, 13, 16	ressplusgd 12961	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → (+g‘𝑅) = (+g‘(𝑂 ↾s 𝑥)))
22	20, 21	eqtr3d 2240	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → (+g‘(𝑅 ↾s 𝑥)) = (+g‘(𝑂 ↾s 𝑥)))
23	22	oveqdr 5972	. . . . 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 13349	. . . 4 ⊢ (𝑅 ∈ 𝑉 → ((𝑅 ↾s 𝑥) ∈ Grp ↔ (𝑂 ↾s 𝑥) ∈ Grp))
25	8, 24	3anbi13d 1327	. . 3 ⊢ (𝑅 ∈ 𝑉 → ((𝑅 ∈ Grp ∧ 𝑥 ⊆ (Base‘𝑅) ∧ (𝑅 ↾s 𝑥) ∈ Grp) ↔ (𝑂 ∈ Grp ∧ 𝑥 ⊆ (Base‘𝑅) ∧ (𝑂 ↾s 𝑥) ∈ Grp)))
26	3	issubg 13509	. . . 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 2205	. . . . 5 ⊢ (Base‘𝑂) = (Base‘𝑂)
29	28	issubg 13509	. . . 4 ⊢ (𝑥 ∈ (SubGrp‘𝑂) ↔ (𝑂 ∈ Grp ∧ 𝑥 ⊆ (Base‘𝑂) ∧ (𝑂 ↾s 𝑥) ∈ Grp))
30	4	sseq2d 3223	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → (𝑥 ⊆ (Base‘𝑅) ↔ 𝑥 ⊆ (Base‘𝑂)))
31	30	3anbi2d 1330	. . . 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 2203	1 ⊢ (𝑅 ∈ 𝑉 → (SubGrp‘𝑅) = (SubGrp‘𝑂))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 981   = wceq 1373   ∈ wcel 2176  Vcvv 2772   ∩ cin 3165   ⊆ wss 3166  ‘cfv 5271  (class class class)co 5944  Basecbs 12832   ↾_s cress 12833  +_gcplusg 12909  Grpcgrp 13332  SubGrpcsubg 13503  opp_rcoppr 13829

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-addcom 8025  ax-addass 8027  ax-i2m1 8030  ax-0lt1 8031  ax-0id 8033  ax-rnegex 8034  ax-pre-ltirr 8037  ax-pre-lttrn 8039  ax-pre-ltadd 8041

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-tpos 6331  df-pnf 8109  df-mnf 8110  df-ltxr 8112  df-inn 9037  df-2 9095  df-3 9096  df-ndx 12835  df-slot 12836  df-base 12838  df-sets 12839  df-iress 12840  df-plusg 12922  df-mulr 12923  df-0g 13090  df-mgm 13188  df-sgrp 13234  df-mnd 13249  df-grp 13335  df-subg 13506  df-oppr 13830

This theorem is referenced by:  opprsubrngg  13973  isridlrng  14244  isridl  14266