Theorem subginv 13041

Description: The inverse of an element in a subgroup is the same as the inverse in the larger group. (Contributed by Mario Carneiro, 2-Dec-2014.)

Hypotheses

Ref	Expression
subg0.h	|- H = ( G ↾s S )
subginv.i	|- I = ( invg ` G )
subginv.j	|- J = ( invg ` H )

Assertion

Ref	Expression
subginv	|- ( ( S e. (SubGrp ` G ) /\ X e. S ) -> ( I ` X ) = ( J ` X ) )

Proof of Theorem subginv

Step	Hyp	Ref	Expression
1		subg0.h	. . . . 5 |- H = ( G ↾s S )
2	1	subggrp 13037	. . . 4 |- ( S e. (SubGrp ` G ) -> H e. Grp )
3	1	subgbas 13038	. . . . . 6 |- ( S e. (SubGrp ` G ) -> S = ( Base ` H ) )
4	3	eleq2d 2247	. . . . 5 |- ( S e. (SubGrp ` G ) -> ( X e. S <-> X e. ( Base ` H ) ) )
5	4	biimpa 296	. . . 4 |- ( ( S e. (SubGrp ` G ) /\ X e. S ) -> X e. ( Base ` H ) )
6		eqid 2177	. . . . 5 |- ( Base ` H ) = ( Base ` H )
7		eqid 2177	. . . . 5 |- ( +g ` H ) = ( +g ` H )
8		eqid 2177	. . . . 5 |- ( 0g ` H ) = ( 0g ` H )
9		subginv.j	. . . . 5 |- J = ( invg ` H )
10	6, 7, 8, 9	grprinv 12923	. . . 4 |- ( ( H e. Grp /\ X e. ( Base ` H ) ) -> ( X ( +g ` H ) ( J ` X ) ) = ( 0g ` H ) )
11	2, 5, 10	syl2an2r 595	. . 3 |- ( ( S e. (SubGrp ` G ) /\ X e. S ) -> ( X ( +g ` H ) ( J ` X ) ) = ( 0g ` H ) )
12	1	a1i 9	. . . . . 6 |- ( S e. (SubGrp ` G ) -> H = ( G ↾s S ) )
13		eqidd 2178	. . . . . 6 |- ( S e. (SubGrp ` G ) -> ( +g ` G ) = ( +g ` G ) )
14		id 19	. . . . . 6 |- ( S e. (SubGrp ` G ) -> S e. (SubGrp ` G ) )
15		subgrcl 13039	. . . . . 6 |- ( S e. (SubGrp ` G ) -> G e. Grp )
16	12, 13, 14, 15	ressplusgd 12587	. . . . 5 |- ( S e. (SubGrp ` G ) -> ( +g ` G ) = ( +g ` H ) )
17	16	adantr 276	. . . 4 |- ( ( S e. (SubGrp ` G ) /\ X e. S ) -> ( +g ` G ) = ( +g ` H ) )
18	17	oveqd 5892	. . 3 |- ( ( S e. (SubGrp ` G ) /\ X e. S ) -> ( X ( +g ` G ) ( J ` X ) ) = ( X ( +g ` H ) ( J ` X ) ) )
19		eqid 2177	. . . . 5 |- ( 0g ` G ) = ( 0g ` G )
20	1, 19	subg0 13040	. . . 4 |- ( S e. (SubGrp ` G ) -> ( 0g ` G ) = ( 0g ` H ) )
21	20	adantr 276	. . 3 |- ( ( S e. (SubGrp ` G ) /\ X e. S ) -> ( 0g ` G ) = ( 0g ` H ) )
22	11, 18, 21	3eqtr4d 2220	. 2 |- ( ( S e. (SubGrp ` G ) /\ X e. S ) -> ( X ( +g ` G ) ( J ` X ) ) = ( 0g ` G ) )
23	15	adantr 276	. . 3 |- ( ( S e. (SubGrp ` G ) /\ X e. S ) -> G e. Grp )
24		eqid 2177	. . . . 5 |- ( Base ` G ) = ( Base ` G )
25	24	subgss 13034	. . . 4 |- ( S e. (SubGrp ` G ) -> S C_ ( Base ` G ) )
26	25	sselda 3156	. . 3 |- ( ( S e. (SubGrp ` G ) /\ X e. S ) -> X e. ( Base ` G ) )
27	6, 9	grpinvcl 12921	. . . . . . . 8 |- ( ( H e. Grp /\ X e. ( Base ` H ) ) -> ( J ` X ) e. ( Base ` H ) )
28	27	ex 115	. . . . . . 7 |- ( H e. Grp -> ( X e. ( Base ` H ) -> ( J ` X ) e. ( Base ` H ) ) )
29	2, 28	syl 14	. . . . . 6 |- ( S e. (SubGrp ` G ) -> ( X e. ( Base ` H ) -> ( J ` X ) e. ( Base ` H ) ) )
30	3	eleq2d 2247	. . . . . 6 |- ( S e. (SubGrp ` G ) -> ( ( J ` X ) e. S <-> ( J ` X ) e. ( Base ` H ) ) )
31	29, 4, 30	3imtr4d 203	. . . . 5 |- ( S e. (SubGrp ` G ) -> ( X e. S -> ( J ` X ) e. S ) )
32	31	imp 124	. . . 4 |- ( ( S e. (SubGrp ` G ) /\ X e. S ) -> ( J ` X ) e. S )
33	25	sselda 3156	. . . 4 |- ( ( S e. (SubGrp ` G ) /\ ( J ` X ) e. S ) -> ( J ` X ) e. ( Base ` G ) )
34	32, 33	syldan 282	. . 3 |- ( ( S e. (SubGrp ` G ) /\ X e. S ) -> ( J ` X ) e. ( Base ` G ) )
35		eqid 2177	. . . 4 |- ( +g ` G ) = ( +g ` G )
36		subginv.i	. . . 4 |- I = ( invg ` G )
37	24, 35, 19, 36	grpinvid1 12924	. . 3 |- ( ( G e. Grp /\ X e. ( Base ` G ) /\ ( J ` X ) e. ( Base ` G ) ) -> ( ( I ` X ) = ( J ` X ) <-> ( X ( +g ` G ) ( J ` X ) ) = ( 0g ` G ) ) )
38	23, 26, 34, 37	syl3anc 1238	. 2 |- ( ( S e. (SubGrp ` G ) /\ X e. S ) -> ( ( I ` X ) = ( J ` X ) <-> ( X ( +g ` G ) ( J ` X ) ) = ( 0g ` G ) ) )
39	22, 38	mpbird 167	1 |- ( ( S e. (SubGrp ` G ) /\ X e. S ) -> ( I ` X ) = ( J ` X ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   e. wcel 2148   ` cfv 5217  (class class class)co 5875   Basecbs 12462   ↾_s cress 12463   +g cplusg 12536   0gc0g 12705   Grpcgrp 12877   invgcminusg 12878  SubGrpcsubg 13027

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-cnex 7902  ax-resscn 7903  ax-1cn 7904  ax-1re 7905  ax-icn 7906  ax-addcl 7907  ax-addrcl 7908  ax-mulcl 7909  ax-addcom 7911  ax-addass 7913  ax-i2m1 7916  ax-0lt1 7917  ax-0id 7919  ax-rnegex 7920  ax-pre-ltirr 7923  ax-pre-ltadd 7927

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-riota 5831  df-ov 5878  df-oprab 5879  df-mpo 5880  df-pnf 7994  df-mnf 7995  df-ltxr 7997  df-inn 8920  df-2 8978  df-ndx 12465  df-slot 12466  df-base 12468  df-sets 12469  df-iress 12470  df-plusg 12549  df-0g 12707  df-mgm 12775  df-sgrp 12808  df-mnd 12818  df-grp 12880  df-minusg 12881  df-subg 13030

This theorem is referenced by:  subginvcl  13043  subgsub  13046  subgmulg  13048