Theorem opprsubgg 14161

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 14152	. . . . 5 |- ( R e. V -> ( Base ` R ) = ( Base ` O ) )
5		eqid 2231	. . . . . . 7 |- ( +g ` R ) = ( +g ` R )
6	2, 5	oppraddg 14153	. . . . . 6 |- ( R e. V -> ( +g ` R ) = ( +g ` O ) )
7	6	oveqdr 6056	. . . . 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 13663	. . . 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 2806	. . . . . . . 8 |- x e. _V
13	12	a1i 9	. . . . . . 7 |- ( R e. V -> x e. _V )
14	10, 1, 11, 13	ressbasd 13213	. . . . . 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 14150	. . . . . . 7 |- ( R e. V -> O e. _V )
17	15, 4, 16, 13	ressbasd 13213	. . . . . 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 13275	. . . . . . 7 |- ( R e. V -> ( +g ` R ) = ( +g ` ( R ↾s x ) ) )
21	15, 6, 13, 16	ressplusgd 13275	. . . . . . 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 6056	. . . . 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 13663	. . . 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 13823	. . . 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 13823	. . . 4 |- ( x e. (SubGrp ` O ) <-> ( O e. Grp /\ x C_ ( Base ` O ) /\ ( O ↾s x ) e. Grp ) )
30	4	sseq2d 3258	. . . . 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 2229	1 |- ( R e. V -> (SubGrp ` R ) = (SubGrp ` O ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2202   _Vcvv 2803   i^i cin 3200   C_ wss 3201   ` cfv 5333  (class class class)co 6028   Basecbs 13145   ↾_s cress 13146   +g cplusg 13223   Grpcgrp 13646  SubGrpcsubg 13817  opp_rcoppr 14144

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 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-addcom 8175  ax-addass 8177  ax-i2m1 8180  ax-0lt1 8181  ax-0id 8183  ax-rnegex 8184  ax-pre-ltirr 8187  ax-pre-lttrn 8189  ax-pre-ltadd 8191

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-tpos 6454  df-pnf 8258  df-mnf 8259  df-ltxr 8261  df-inn 9186  df-2 9244  df-3 9245  df-ndx 13148  df-slot 13149  df-base 13151  df-sets 13152  df-iress 13153  df-plusg 13236  df-mulr 13237  df-0g 13404  df-mgm 13502  df-sgrp 13548  df-mnd 13563  df-grp 13649  df-subg 13820  df-oppr 14145

This theorem is referenced by:  opprsubrngg  14289  isridlrng  14561  isridl  14583