Theorem opprsubgg 14119

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 2232	. . . . 5 |- ( R e. V -> ( Base ` R ) = ( Base ` R ) )
2		opprbas.1	. . . . . 6 |- O = (oppr ` R )
3		eqid 2231	. . . . . 6 |- ( Base ` R ) = ( Base ` R )
4	2, 3	opprbasg 14110	. . . . 5 |- ( R e. V -> ( Base ` R ) = ( Base ` O ) )
5		eqid 2231	. . . . . . 7 |- ( +g ` R ) = ( +g ` R )
6	2, 5	oppraddg 14111	. . . . . 6 |- ( R e. V -> ( +g ` R ) = ( +g ` O ) )
7	6	oveqdr 6049	. . . . 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 13621	. . . 4 |- ( R e. V -> ( R e. Grp <-> O e. Grp ) )
9		eqidd 2232	. . . . 5 |- ( R e. V -> ( Base ` ( R ↾s x ) ) = ( Base ` ( R ↾s x ) ) )
10		eqidd 2232	. . . . . . 7 |- ( R e. V -> ( R ↾s x ) = ( R ↾s x ) )
11		id 19	. . . . . . 7 |- ( R e. V -> R e. V )
12		vex 2805	. . . . . . . 8 |- x e. _V
13	12	a1i 9	. . . . . . 7 |- ( R e. V -> x e. _V )
14	10, 1, 11, 13	ressbasd 13171	. . . . . 6 |- ( R e. V -> ( x i^i ( Base ` R ) ) = ( Base ` ( R ↾s x ) ) )
15		eqidd 2232	. . . . . . 7 |- ( R e. V -> ( O ↾s x ) = ( O ↾s x ) )
16	2	opprex 14108	. . . . . . 7 |- ( R e. V -> O e. _V )
17	15, 4, 16, 13	ressbasd 13171	. . . . . 6 |- ( R e. V -> ( x i^i ( Base ` R ) ) = ( Base ` ( O ↾s x ) ) )
18	14, 17	eqtr3d 2266	. . . . 5 |- ( R e. V -> ( Base ` ( R ↾s x ) ) = ( Base ` ( O ↾s x ) ) )
19		eqidd 2232	. . . . . . . 8 |- ( R e. V -> ( +g ` R ) = ( +g ` R ) )
20	10, 19, 13, 11	ressplusgd 13233	. . . . . . 7 |- ( R e. V -> ( +g ` R ) = ( +g ` ( R ↾s x ) ) )
21	15, 6, 13, 16	ressplusgd 13233	. . . . . . 7 |- ( R e. V -> ( +g ` R ) = ( +g ` ( O ↾s x ) ) )
22	20, 21	eqtr3d 2266	. . . . . 6 |- ( R e. V -> ( +g ` ( R ↾s x ) ) = ( +g ` ( O ↾s x ) ) )
23	22	oveqdr 6049	. . . . 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 13621	. . . 4 |- ( R e. V -> ( ( R ↾s x ) e. Grp <-> ( O ↾s x ) e. Grp ) )
25	8, 24	3anbi13d 1350	. . 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 13781	. . . 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 2231	. . . . 5 |- ( Base ` O ) = ( Base ` O )
29	28	issubg 13781	. . . 4 |- ( x e. (SubGrp ` O ) <-> ( O e. Grp /\ x C_ ( Base ` O ) /\ ( O ↾s x ) e. Grp ) )
30	4	sseq2d 3257	. . . . 5 |- ( R e. V -> ( x C_ ( Base ` R ) <-> x C_ ( Base ` O ) ) )
31	30	3anbi2d 1353	. . . 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 2229	1 |- ( R e. V -> (SubGrp ` R ) = (SubGrp ` O ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1004   = wceq 1397   e. wcel 2202   _Vcvv 2802   i^i cin 3199   C_ wss 3200   ` cfv 5326  (class class class)co 6021   Basecbs 13103   ↾_s cress 13104   +g cplusg 13181   Grpcgrp 13604  SubGrpcsubg 13775  opp_rcoppr 14102

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8126  ax-resscn 8127  ax-1cn 8128  ax-1re 8129  ax-icn 8130  ax-addcl 8131  ax-addrcl 8132  ax-mulcl 8133  ax-addcom 8135  ax-addass 8137  ax-i2m1 8140  ax-0lt1 8141  ax-0id 8143  ax-rnegex 8144  ax-pre-ltirr 8147  ax-pre-lttrn 8149  ax-pre-ltadd 8151

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-fv 5334  df-riota 5974  df-ov 6024  df-oprab 6025  df-mpo 6026  df-tpos 6414  df-pnf 8219  df-mnf 8220  df-ltxr 8222  df-inn 9147  df-2 9205  df-3 9206  df-ndx 13106  df-slot 13107  df-base 13109  df-sets 13110  df-iress 13111  df-plusg 13194  df-mulr 13195  df-0g 13362  df-mgm 13460  df-sgrp 13506  df-mnd 13521  df-grp 13607  df-subg 13778  df-oppr 14103

This theorem is referenced by:  opprsubrngg  14247  isridlrng  14518  isridl  14540