Theorem opprsubgg 13879

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 13870	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → (Base‘𝑅) = (Base‘𝑂))
5		eqid 2205	. . . . . . 7 ⊢ (+g‘𝑅) = (+g‘𝑅)
6	2, 5	oppraddg 13871	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → (+g‘𝑅) = (+g‘𝑂))
7	6	oveqdr 5974	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥(+g‘𝑅)𝑦) = (𝑥(+g‘𝑂)𝑦))
8	1, 4, 7	grppropd 13382	. . . 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 12932	. . . . . 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 13868	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → 𝑂 ∈ V)
17	15, 4, 16, 13	ressbasd 12932	. . . . . 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 12994	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → (+g‘𝑅) = (+g‘(𝑅 ↾s 𝑥)))
21	15, 6, 13, 16	ressplusgd 12994	. . . . . . 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 5974	. . . . 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 13382	. . . 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 13542	. . . 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 13542	. . . 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 5272  (class class class)co 5946  Basecbs 12865   ↾_s cress 12866  +_gcplusg 12942  Grpcgrp 13365  SubGrpcsubg 13536  opp_rcoppr 13862

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 4163  ax-nul 4171  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-setind 4586  ax-cnex 8018  ax-resscn 8019  ax-1cn 8020  ax-1re 8021  ax-icn 8022  ax-addcl 8023  ax-addrcl 8024  ax-mulcl 8025  ax-addcom 8027  ax-addass 8029  ax-i2m1 8032  ax-0lt1 8033  ax-0id 8035  ax-rnegex 8036  ax-pre-ltirr 8039  ax-pre-lttrn 8041  ax-pre-ltadd 8043

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 4046  df-opab 4107  df-mpt 4108  df-id 4341  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689  df-iota 5233  df-fun 5274  df-fn 5275  df-fv 5280  df-riota 5901  df-ov 5949  df-oprab 5950  df-mpo 5951  df-tpos 6333  df-pnf 8111  df-mnf 8112  df-ltxr 8114  df-inn 9039  df-2 9097  df-3 9098  df-ndx 12868  df-slot 12869  df-base 12871  df-sets 12872  df-iress 12873  df-plusg 12955  df-mulr 12956  df-0g 13123  df-mgm 13221  df-sgrp 13267  df-mnd 13282  df-grp 13368  df-subg 13539  df-oppr 13863

This theorem is referenced by:  opprsubrngg  14006  isridlrng  14277  isridl  14299