Theorem opprsubg 18636

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
opprsubg	|- (SubGrp ` R ) = (SubGrp ` O )

Proof of Theorem opprsubg

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		opprbas.1	. . . . . 6 |- O = (oppr ` R )
2		eqid 2622	. . . . . 6 |- ( Base ` R ) = ( Base ` R )
3	1, 2	opprbas 18629	. . . . 5 |- ( Base ` R ) = ( Base ` O )
4		eqid 2622	. . . . . 6 |- ( +g ` R ) = ( +g ` R )
5	1, 4	oppradd 18630	. . . . 5 |- ( +g ` R ) = ( +g ` O )
6	3, 5	grpprop 17438	. . . 4 |- ( R e. Grp <-> O e. Grp )
7		biid 251	. . . 4 |- ( x C_ ( Base ` R ) <-> x C_ ( Base ` R ) )
8		vex 3203	. . . . . . 7 |- x e. _V
9		eqid 2622	. . . . . . . 8 |- ( R ↾s x ) = ( R ↾s x )
10	9, 2	ressbas 15930	. . . . . . 7 |- ( x e. _V -> ( x i^i ( Base ` R ) ) = ( Base ` ( R ↾s x ) ) )
11	8, 10	ax-mp 5	. . . . . 6 |- ( x i^i ( Base ` R ) ) = ( Base ` ( R ↾s x ) )
12		eqid 2622	. . . . . . . 8 |- ( O ↾s x ) = ( O ↾s x )
13	12, 3	ressbas 15930	. . . . . . 7 |- ( x e. _V -> ( x i^i ( Base ` R ) ) = ( Base ` ( O ↾s x ) ) )
14	8, 13	ax-mp 5	. . . . . 6 |- ( x i^i ( Base ` R ) ) = ( Base ` ( O ↾s x ) )
15	11, 14	eqtr3i 2646	. . . . 5 |- ( Base ` ( R ↾s x ) ) = ( Base ` ( O ↾s x ) )
16	9, 4	ressplusg 15993	. . . . . . 7 |- ( x e. _V -> ( +g ` R ) = ( +g ` ( R ↾s x ) ) )
17	12, 5	ressplusg 15993	. . . . . . 7 |- ( x e. _V -> ( +g ` R ) = ( +g ` ( O ↾s x ) ) )
18	16, 17	eqtr3d 2658	. . . . . 6 |- ( x e. _V -> ( +g ` ( R ↾s x ) ) = ( +g ` ( O ↾s x ) ) )
19	8, 18	ax-mp 5	. . . . 5 |- ( +g ` ( R ↾s x ) ) = ( +g ` ( O ↾s x ) )
20	15, 19	grpprop 17438	. . . 4 |- ( ( R ↾s x ) e. Grp <-> ( O ↾s x ) e. Grp )
21	6, 7, 20	3anbi123i 1251	. . 3 |- ( ( R e. Grp /\ x C_ ( Base ` R ) /\ ( R ↾s x ) e. Grp ) <-> ( O e. Grp /\ x C_ ( Base ` R ) /\ ( O ↾s x ) e. Grp ) )
22	2	issubg 17594	. . 3 |- ( x e. (SubGrp ` R ) <-> ( R e. Grp /\ x C_ ( Base ` R ) /\ ( R ↾s x ) e. Grp ) )
23	3	issubg 17594	. . 3 |- ( x e. (SubGrp ` O ) <-> ( O e. Grp /\ x C_ ( Base ` R ) /\ ( O ↾s x ) e. Grp ) )
24	21, 22, 23	3bitr4i 292	. 2 |- ( x e. (SubGrp ` R ) <-> x e. (SubGrp ` O ) )
25	24	eqriv 2619	1 |- (SubGrp ` R ) = (SubGrp ` O )

Syntax hints:   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   i^i cin 3573   C_ wss 3574   ` cfv 5888  (class class class)co 6650   Basecbs 15857   ↾_s cress 15858   +g cplusg 15941   Grpcgrp 17422  SubGrpcsubg 17588  opp_rcoppr 18622

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-subg 17591  df-oppr 18623

This theorem is referenced by:  opprsubrg  18801