Theorem opprsubgg 14217

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 2233	. . . . 5 |- ( R e. V -> ( Base ` R ) = ( Base ` R ) )
2		opprbas.1	. . . . . 6 |- O = (oppr ` R )
3		eqid 2232	. . . . . 6 |- ( Base ` R ) = ( Base ` R )
4	2, 3	opprbasg 14208	. . . . 5 |- ( R e. V -> ( Base ` R ) = ( Base ` O ) )
5		eqid 2232	. . . . . . 7 |- ( +g ` R ) = ( +g ` R )
6	2, 5	oppraddg 14209	. . . . . 6 |- ( R e. V -> ( +g ` R ) = ( +g ` O ) )
7	6	oveqdr 6077	. . . . 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 13719	. . . 4 |- ( R e. V -> ( R e. Grp <-> O e. Grp ) )
9		eqidd 2233	. . . . 5 |- ( R e. V -> ( Base ` ( R ↾s x ) ) = ( Base ` ( R ↾s x ) ) )
10		eqidd 2233	. . . . . . 7 |- ( R e. V -> ( R ↾s x ) = ( R ↾s x ) )
11		id 19	. . . . . . 7 |- ( R e. V -> R e. V )
12		vex 2815	. . . . . . . 8 |- x e. _V
13	12	a1i 9	. . . . . . 7 |- ( R e. V -> x e. _V )
14	10, 1, 11, 13	ressbasd 13269	. . . . . 6 |- ( R e. V -> ( x i^i ( Base ` R ) ) = ( Base ` ( R ↾s x ) ) )
15		eqidd 2233	. . . . . . 7 |- ( R e. V -> ( O ↾s x ) = ( O ↾s x ) )
16	2	opprex 14206	. . . . . . 7 |- ( R e. V -> O e. _V )
17	15, 4, 16, 13	ressbasd 13269	. . . . . 6 |- ( R e. V -> ( x i^i ( Base ` R ) ) = ( Base ` ( O ↾s x ) ) )
18	14, 17	eqtr3d 2267	. . . . 5 |- ( R e. V -> ( Base ` ( R ↾s x ) ) = ( Base ` ( O ↾s x ) ) )
19		eqidd 2233	. . . . . . . 8 |- ( R e. V -> ( +g ` R ) = ( +g ` R ) )
20	10, 19, 13, 11	ressplusgd 13331	. . . . . . 7 |- ( R e. V -> ( +g ` R ) = ( +g ` ( R ↾s x ) ) )
21	15, 6, 13, 16	ressplusgd 13331	. . . . . . 7 |- ( R e. V -> ( +g ` R ) = ( +g ` ( O ↾s x ) ) )
22	20, 21	eqtr3d 2267	. . . . . 6 |- ( R e. V -> ( +g ` ( R ↾s x ) ) = ( +g ` ( O ↾s x ) ) )
23	22	oveqdr 6077	. . . . 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 13719	. . . 4 |- ( R e. V -> ( ( R ↾s x ) e. Grp <-> ( O ↾s x ) e. Grp ) )
25	8, 24	3anbi13d 1351	. . 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 13879	. . . 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 2232	. . . . 5 |- ( Base ` O ) = ( Base ` O )
29	28	issubg 13879	. . . 4 |- ( x e. (SubGrp ` O ) <-> ( O e. Grp /\ x C_ ( Base ` O ) /\ ( O ↾s x ) e. Grp ) )
30	4	sseq2d 3267	. . . . 5 |- ( R e. V -> ( x C_ ( Base ` R ) <-> x C_ ( Base ` O ) ) )
31	30	3anbi2d 1354	. . . 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 2230	1 |- ( R e. V -> (SubGrp ` R ) = (SubGrp ` O ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2203   _Vcvv 2812   i^i cin 3209   C_ wss 3210   ` cfv 5351  (class class class)co 6049   Basecbs 13201   ↾_s cress 13202   +g cplusg 13279   Grpcgrp 13702  SubGrpcsubg 13873  opp_rcoppr 14200

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 2205  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-cnex 8214  ax-resscn 8215  ax-1cn 8216  ax-1re 8217  ax-icn 8218  ax-addcl 8219  ax-addrcl 8220  ax-mulcl 8221  ax-addcom 8223  ax-addass 8225  ax-i2m1 8228  ax-0lt1 8229  ax-0id 8231  ax-rnegex 8232  ax-pre-ltirr 8235  ax-pre-lttrn 8237  ax-pre-ltadd 8239

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 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-tpos 6475  df-pnf 8306  df-mnf 8307  df-ltxr 8309  df-inn 9234  df-2 9292  df-3 9293  df-ndx 13204  df-slot 13205  df-base 13207  df-sets 13208  df-iress 13209  df-plusg 13292  df-mulr 13293  df-0g 13460  df-mgm 13558  df-sgrp 13604  df-mnd 13619  df-grp 13705  df-subg 13876  df-oppr 14201

This theorem is referenced by:  opprsubrngg  14345  isridlrng  14617  isridl  14639