Theorem opprsubgg 14087

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 2230	. . . . 5 |- ( R e. V -> ( Base ` R ) = ( Base ` R ) )
2		opprbas.1	. . . . . 6 |- O = (oppr ` R )
3		eqid 2229	. . . . . 6 |- ( Base ` R ) = ( Base ` R )
4	2, 3	opprbasg 14078	. . . . 5 |- ( R e. V -> ( Base ` R ) = ( Base ` O ) )
5		eqid 2229	. . . . . . 7 |- ( +g ` R ) = ( +g ` R )
6	2, 5	oppraddg 14079	. . . . . 6 |- ( R e. V -> ( +g ` R ) = ( +g ` O ) )
7	6	oveqdr 6041	. . . . 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 13590	. . . 4 |- ( R e. V -> ( R e. Grp <-> O e. Grp ) )
9		eqidd 2230	. . . . 5 |- ( R e. V -> ( Base ` ( R ↾s x ) ) = ( Base ` ( R ↾s x ) ) )
10		eqidd 2230	. . . . . . 7 |- ( R e. V -> ( R ↾s x ) = ( R ↾s x ) )
11		id 19	. . . . . . 7 |- ( R e. V -> R e. V )
12		vex 2803	. . . . . . . 8 |- x e. _V
13	12	a1i 9	. . . . . . 7 |- ( R e. V -> x e. _V )
14	10, 1, 11, 13	ressbasd 13140	. . . . . 6 |- ( R e. V -> ( x i^i ( Base ` R ) ) = ( Base ` ( R ↾s x ) ) )
15		eqidd 2230	. . . . . . 7 |- ( R e. V -> ( O ↾s x ) = ( O ↾s x ) )
16	2	opprex 14076	. . . . . . 7 |- ( R e. V -> O e. _V )
17	15, 4, 16, 13	ressbasd 13140	. . . . . 6 |- ( R e. V -> ( x i^i ( Base ` R ) ) = ( Base ` ( O ↾s x ) ) )
18	14, 17	eqtr3d 2264	. . . . 5 |- ( R e. V -> ( Base ` ( R ↾s x ) ) = ( Base ` ( O ↾s x ) ) )
19		eqidd 2230	. . . . . . . 8 |- ( R e. V -> ( +g ` R ) = ( +g ` R ) )
20	10, 19, 13, 11	ressplusgd 13202	. . . . . . 7 |- ( R e. V -> ( +g ` R ) = ( +g ` ( R ↾s x ) ) )
21	15, 6, 13, 16	ressplusgd 13202	. . . . . . 7 |- ( R e. V -> ( +g ` R ) = ( +g ` ( O ↾s x ) ) )
22	20, 21	eqtr3d 2264	. . . . . 6 |- ( R e. V -> ( +g ` ( R ↾s x ) ) = ( +g ` ( O ↾s x ) ) )
23	22	oveqdr 6041	. . . . 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 13590	. . . 4 |- ( R e. V -> ( ( R ↾s x ) e. Grp <-> ( O ↾s x ) e. Grp ) )
25	8, 24	3anbi13d 1348	. . 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 13750	. . . 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 2229	. . . . 5 |- ( Base ` O ) = ( Base ` O )
29	28	issubg 13750	. . . 4 |- ( x e. (SubGrp ` O ) <-> ( O e. Grp /\ x C_ ( Base ` O ) /\ ( O ↾s x ) e. Grp ) )
30	4	sseq2d 3255	. . . . 5 |- ( R e. V -> ( x C_ ( Base ` R ) <-> x C_ ( Base ` O ) ) )
31	30	3anbi2d 1351	. . . 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 2227	1 |- ( R e. V -> (SubGrp ` R ) = (SubGrp ` O ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200   _Vcvv 2800   i^i cin 3197   C_ wss 3198   ` cfv 5324  (class class class)co 6013   Basecbs 13072   ↾_s cress 13073   +g cplusg 13150   Grpcgrp 13573  SubGrpcsubg 13744  opp_rcoppr 14070

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-addcom 8122  ax-addass 8124  ax-i2m1 8127  ax-0lt1 8128  ax-0id 8130  ax-rnegex 8131  ax-pre-ltirr 8134  ax-pre-lttrn 8136  ax-pre-ltadd 8138

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-tpos 6406  df-pnf 8206  df-mnf 8207  df-ltxr 8209  df-inn 9134  df-2 9192  df-3 9193  df-ndx 13075  df-slot 13076  df-base 13078  df-sets 13079  df-iress 13080  df-plusg 13163  df-mulr 13164  df-0g 13331  df-mgm 13429  df-sgrp 13475  df-mnd 13490  df-grp 13576  df-subg 13747  df-oppr 14071

This theorem is referenced by:  opprsubrngg  14215  isridlrng  14486  isridl  14508