Theorem subginv 13251

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 13247	. . . 4 |- ( S e. (SubGrp ` G ) -> H e. Grp )
3	1	subgbas 13248	. . . . . 6 |- ( S e. (SubGrp ` G ) -> S = ( Base ` H ) )
4	3	eleq2d 2263	. . . . 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 2193	. . . . 5 |- ( Base ` H ) = ( Base ` H )
7		eqid 2193	. . . . 5 |- ( +g ` H ) = ( +g ` H )
8		eqid 2193	. . . . 5 |- ( 0g ` H ) = ( 0g ` H )
9		subginv.j	. . . . 5 |- J = ( invg ` H )
10	6, 7, 8, 9	grprinv 13123	. . . 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 2194	. . . . . 6 |- ( S e. (SubGrp ` G ) -> ( +g ` G ) = ( +g ` G ) )
14		id 19	. . . . . 6 |- ( S e. (SubGrp ` G ) -> S e. (SubGrp ` G ) )
15		subgrcl 13249	. . . . . 6 |- ( S e. (SubGrp ` G ) -> G e. Grp )
16	12, 13, 14, 15	ressplusgd 12746	. . . . 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 5935	. . 3 |- ( ( S e. (SubGrp ` G ) /\ X e. S ) -> ( X ( +g ` G ) ( J ` X ) ) = ( X ( +g ` H ) ( J ` X ) ) )
19		eqid 2193	. . . . 5 |- ( 0g ` G ) = ( 0g ` G )
20	1, 19	subg0 13250	. . . 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 2236	. 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 2193	. . . . 5 |- ( Base ` G ) = ( Base ` G )
25	24	subgss 13244	. . . 4 |- ( S e. (SubGrp ` G ) -> S C_ ( Base ` G ) )
26	25	sselda 3179	. . 3 |- ( ( S e. (SubGrp ` G ) /\ X e. S ) -> X e. ( Base ` G ) )
27	6, 9	grpinvcl 13120	. . . . . . . 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 2263	. . . . . 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 3179	. . . 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 2193	. . . 4 |- ( +g ` G ) = ( +g ` G )
36		subginv.i	. . . 4 |- I = ( invg ` G )
37	24, 35, 19, 36	grpinvid1 13124	. . 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 1249	. 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 1364   e. wcel 2164   ` cfv 5254  (class class class)co 5918   Basecbs 12618   ↾_s cress 12619   +g cplusg 12695   0gc0g 12867   Grpcgrp 13072   invgcminusg 13073  SubGrpcsubg 13237

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-i2m1 7977  ax-0lt1 7978  ax-0id 7980  ax-rnegex 7981  ax-pre-ltirr 7984  ax-pre-ltadd 7988

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-pnf 8056  df-mnf 8057  df-ltxr 8059  df-inn 8983  df-2 9041  df-ndx 12621  df-slot 12622  df-base 12624  df-sets 12625  df-iress 12626  df-plusg 12708  df-0g 12869  df-mgm 12939  df-sgrp 12985  df-mnd 12998  df-grp 13075  df-minusg 13076  df-subg 13240

This theorem is referenced by:  subginvcl  13253  subgsub  13256  subgmulg  13258