Theorem opprsubgg 13580

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 e. 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 e. V -> ( Base ` R ) = ( Base ` R ) )
2		opprbas.1	. . . . . 6 |- O = (oppr ` R )
3		eqid 2193	. . . . . 6 |- ( Base ` R ) = ( Base ` R )
4	2, 3	opprbasg 13571	. . . . 5 |- ( R e. V -> ( Base ` R ) = ( Base ` O ) )
5		eqid 2193	. . . . . . 7 |- ( +g ` R ) = ( +g ` R )
6	2, 5	oppraddg 13572	. . . . . 6 |- ( R e. V -> ( +g ` R ) = ( +g ` O ) )
7	6	oveqdr 5946	. . . . 5 |- ( ( R e. V /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) -> ( x ( +g ` R ) y ) = ( x ( +g ` O ) y ) )
8	1, 4, 7	grppropd 13089	. . . 4 |- ( R e. V -> ( R e. Grp <-> O e. Grp ) )
9		eqidd 2194	. . . . 5 |- ( R e. V -> ( Base ` ( R ↾s x ) ) = ( Base ` ( R ↾s x ) ) )
10		eqidd 2194	. . . . . . 7 |- ( R e. V -> ( R ↾s x ) = ( R ↾s x ) )
11		id 19	. . . . . . 7 |- ( R e. V -> R e. V )
12		vex 2763	. . . . . . . 8 |- x e. _V
13	12	a1i 9	. . . . . . 7 |- ( R e. V -> x e. _V )
14	10, 1, 11, 13	ressbasd 12685	. . . . . 6 |- ( R e. V -> ( x i^i ( Base ` R ) ) = ( Base ` ( R ↾s x ) ) )
15		eqidd 2194	. . . . . . 7 |- ( R e. V -> ( O ↾s x ) = ( O ↾s x ) )
16	2	opprex 13569	. . . . . . 7 |- ( R e. V -> O e. _V )
17	15, 4, 16, 13	ressbasd 12685	. . . . . 6 |- ( R e. V -> ( x i^i ( Base ` R ) ) = ( Base ` ( O ↾s x ) ) )
18	14, 17	eqtr3d 2228	. . . . 5 |- ( R e. V -> ( Base ` ( R ↾s x ) ) = ( Base ` ( O ↾s x ) ) )
19		eqidd 2194	. . . . . . . 8 |- ( R e. V -> ( +g ` R ) = ( +g ` R ) )
20	10, 19, 13, 11	ressplusgd 12746	. . . . . . 7 |- ( R e. V -> ( +g ` R ) = ( +g ` ( R ↾s x ) ) )
21	15, 6, 13, 16	ressplusgd 12746	. . . . . . 7 |- ( R e. V -> ( +g ` R ) = ( +g ` ( O ↾s x ) ) )
22	20, 21	eqtr3d 2228	. . . . . 6 |- ( R e. V -> ( +g ` ( R ↾s x ) ) = ( +g ` ( O ↾s x ) ) )
23	22	oveqdr 5946	. . . . 5 |- ( ( R e. V /\ ( z e. ( Base ` ( R ↾s x ) ) /\ w e. ( Base ` ( R ↾s x ) ) ) ) -> ( z ( +g ` ( R ↾s x ) ) w ) = ( z ( +g ` ( O ↾s x ) ) w ) )
24	9, 18, 23	grppropd 13089	. . . 4 |- ( R e. V -> ( ( R ↾s x ) e. Grp <-> ( O ↾s x ) e. Grp ) )
25	8, 24	3anbi13d 1325	. . 3 |- ( R e. V -> ( ( R e. Grp /\ x C_ ( Base ` R ) /\ ( R ↾s x ) e. Grp ) <-> ( O e. Grp /\ x C_ ( Base ` R ) /\ ( O ↾s x ) e. Grp ) ) )
26	3	issubg 13243	. . . 4 |- ( x e. (SubGrp ` R ) <-> ( R e. Grp /\ x C_ ( Base ` R ) /\ ( R ↾s x ) e. Grp ) )
27	26	a1i 9	. . 3 |- ( R e. V -> ( x e. (SubGrp ` R ) <-> ( R e. Grp /\ x C_ ( Base ` R ) /\ ( R ↾s x ) e. Grp ) ) )
28		eqid 2193	. . . . 5 |- ( Base ` O ) = ( Base ` O )
29	28	issubg 13243	. . . 4 |- ( x e. (SubGrp ` O ) <-> ( O e. Grp /\ x C_ ( Base ` O ) /\ ( O ↾s x ) e. Grp ) )
30	4	sseq2d 3209	. . . . 5 |- ( R e. V -> ( x C_ ( Base ` R ) <-> x C_ ( Base ` O ) ) )
31	30	3anbi2d 1328	. . . 4 |- ( R e. V -> ( ( O e. Grp /\ x C_ ( Base ` R ) /\ ( O ↾s x ) e. Grp ) <-> ( O e. Grp /\ x C_ ( Base ` O ) /\ ( O ↾s x ) e. Grp ) ) )
32	29, 31	bitr4id 199	. . 3 |- ( R e. V -> ( x e. (SubGrp ` O ) <-> ( O e. Grp /\ x C_ ( Base ` R ) /\ ( O ↾s x ) e. Grp ) ) )
33	25, 27, 32	3bitr4d 220	. 2 |- ( R e. V -> ( x e. (SubGrp ` R ) <-> x e. (SubGrp ` O ) ) )
34	33	eqrdv 2191	1 |- ( R e. V -> (SubGrp ` R ) = (SubGrp ` O ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   = wceq 1364   e. wcel 2164   _Vcvv 2760   i^i cin 3152   C_ wss 3153   ` cfv 5254  (class class class)co 5918   Basecbs 12618   ↾_s cress 12619   +g cplusg 12695   Grpcgrp 13072  SubGrpcsubg 13237  opp_rcoppr 13563

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 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-i2m1 7977  ax-0lt1 7978  ax-0id 7980  ax-rnegex 7981  ax-pre-ltirr 7984  ax-pre-lttrn 7986  ax-pre-ltadd 7988

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 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-tpos 6298  df-pnf 8056  df-mnf 8057  df-ltxr 8059  df-inn 8983  df-2 9041  df-3 9042  df-ndx 12621  df-slot 12622  df-base 12624  df-sets 12625  df-iress 12626  df-plusg 12708  df-mulr 12709  df-0g 12869  df-mgm 12939  df-sgrp 12985  df-mnd 12998  df-grp 13075  df-subg 13240  df-oppr 13564

This theorem is referenced by:  opprsubrngg  13707  isridlrng  13978  isridl  14000