Theorem subginv 13311

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 13307	. . . 4 |- ( S e. (SubGrp ` G ) -> H e. Grp )
3	1	subgbas 13308	. . . . . 6 |- ( S e. (SubGrp ` G ) -> S = ( Base ` H ) )
4	3	eleq2d 2266	. . . . 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 2196	. . . . 5 |- ( Base ` H ) = ( Base ` H )
7		eqid 2196	. . . . 5 |- ( +g ` H ) = ( +g ` H )
8		eqid 2196	. . . . 5 |- ( 0g ` H ) = ( 0g ` H )
9		subginv.j	. . . . 5 |- J = ( invg ` H )
10	6, 7, 8, 9	grprinv 13183	. . . 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 2197	. . . . . 6 |- ( S e. (SubGrp ` G ) -> ( +g ` G ) = ( +g ` G ) )
14		id 19	. . . . . 6 |- ( S e. (SubGrp ` G ) -> S e. (SubGrp ` G ) )
15		subgrcl 13309	. . . . . 6 |- ( S e. (SubGrp ` G ) -> G e. Grp )
16	12, 13, 14, 15	ressplusgd 12806	. . . . 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 5939	. . 3 |- ( ( S e. (SubGrp ` G ) /\ X e. S ) -> ( X ( +g ` G ) ( J ` X ) ) = ( X ( +g ` H ) ( J ` X ) ) )
19		eqid 2196	. . . . 5 |- ( 0g ` G ) = ( 0g ` G )
20	1, 19	subg0 13310	. . . 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 2239	. 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 2196	. . . . 5 |- ( Base ` G ) = ( Base ` G )
25	24	subgss 13304	. . . 4 |- ( S e. (SubGrp ` G ) -> S C_ ( Base ` G ) )
26	25	sselda 3183	. . 3 |- ( ( S e. (SubGrp ` G ) /\ X e. S ) -> X e. ( Base ` G ) )
27	6, 9	grpinvcl 13180	. . . . . . . 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 2266	. . . . . 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 3183	. . . 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 2196	. . . 4 |- ( +g ` G ) = ( +g ` G )
36		subginv.i	. . . 4 |- I = ( invg ` G )
37	24, 35, 19, 36	grpinvid1 13184	. . 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 2167   ` cfv 5258  (class class class)co 5922   Basecbs 12678   ↾_s cress 12679   +g cplusg 12755   0gc0g 12927   Grpcgrp 13132   invgcminusg 13133  SubGrpcsubg 13297

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-pre-ltirr 7991  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-pnf 8063  df-mnf 8064  df-ltxr 8066  df-inn 8991  df-2 9049  df-ndx 12681  df-slot 12682  df-base 12684  df-sets 12685  df-iress 12686  df-plusg 12768  df-0g 12929  df-mgm 12999  df-sgrp 13045  df-mnd 13058  df-grp 13135  df-minusg 13136  df-subg 13300

This theorem is referenced by:  subginvcl  13313  subgsub  13316  subgmulg  13318