Theorem subginv 17601

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	. . . . . 6 |- H = ( G ↾s S )
2	1	subggrp 17597	. . . . 5 |- ( S e. (SubGrp ` G ) -> H e. Grp )
3	2	adantr 481	. . . 4 |- ( ( S e. (SubGrp ` G ) /\ X e. S ) -> H e. Grp )
4	1	subgbas 17598	. . . . . 6 |- ( S e. (SubGrp ` G ) -> S = ( Base ` H ) )
5	4	eleq2d 2687	. . . . 5 |- ( S e. (SubGrp ` G ) -> ( X e. S <-> X e. ( Base ` H ) ) )
6	5	biimpa 501	. . . 4 |- ( ( S e. (SubGrp ` G ) /\ X e. S ) -> X e. ( Base ` H ) )
7		eqid 2622	. . . . 5 |- ( Base ` H ) = ( Base ` H )
8		eqid 2622	. . . . 5 |- ( +g ` H ) = ( +g ` H )
9		eqid 2622	. . . . 5 |- ( 0g ` H ) = ( 0g ` H )
10		subginv.j	. . . . 5 |- J = ( invg ` H )
11	7, 8, 9, 10	grprinv 17469	. . . 4 |- ( ( H e. Grp /\ X e. ( Base ` H ) ) -> ( X ( +g ` H ) ( J ` X ) ) = ( 0g ` H ) )
12	3, 6, 11	syl2anc 693	. . 3 |- ( ( S e. (SubGrp ` G ) /\ X e. S ) -> ( X ( +g ` H ) ( J ` X ) ) = ( 0g ` H ) )
13		eqid 2622	. . . . . 6 |- ( +g ` G ) = ( +g ` G )
14	1, 13	ressplusg 15993	. . . . 5 |- ( S e. (SubGrp ` G ) -> ( +g ` G ) = ( +g ` H ) )
15	14	adantr 481	. . . 4 |- ( ( S e. (SubGrp ` G ) /\ X e. S ) -> ( +g ` G ) = ( +g ` H ) )
16	15	oveqd 6667	. . 3 |- ( ( S e. (SubGrp ` G ) /\ X e. S ) -> ( X ( +g ` G ) ( J ` X ) ) = ( X ( +g ` H ) ( J ` X ) ) )
17		eqid 2622	. . . . 5 |- ( 0g ` G ) = ( 0g ` G )
18	1, 17	subg0 17600	. . . 4 |- ( S e. (SubGrp ` G ) -> ( 0g ` G ) = ( 0g ` H ) )
19	18	adantr 481	. . 3 |- ( ( S e. (SubGrp ` G ) /\ X e. S ) -> ( 0g ` G ) = ( 0g ` H ) )
20	12, 16, 19	3eqtr4d 2666	. 2 |- ( ( S e. (SubGrp ` G ) /\ X e. S ) -> ( X ( +g ` G ) ( J ` X ) ) = ( 0g ` G ) )
21		subgrcl 17599	. . . 4 |- ( S e. (SubGrp ` G ) -> G e. Grp )
22	21	adantr 481	. . 3 |- ( ( S e. (SubGrp ` G ) /\ X e. S ) -> G e. Grp )
23		eqid 2622	. . . . 5 |- ( Base ` G ) = ( Base ` G )
24	23	subgss 17595	. . . 4 |- ( S e. (SubGrp ` G ) -> S C_ ( Base ` G ) )
25	24	sselda 3603	. . 3 |- ( ( S e. (SubGrp ` G ) /\ X e. S ) -> X e. ( Base ` G ) )
26	7, 10	grpinvcl 17467	. . . . . . . 8 |- ( ( H e. Grp /\ X e. ( Base ` H ) ) -> ( J ` X ) e. ( Base ` H ) )
27	26	ex 450	. . . . . . 7 |- ( H e. Grp -> ( X e. ( Base ` H ) -> ( J ` X ) e. ( Base ` H ) ) )
28	2, 27	syl 17	. . . . . 6 |- ( S e. (SubGrp ` G ) -> ( X e. ( Base ` H ) -> ( J ` X ) e. ( Base ` H ) ) )
29	4	eleq2d 2687	. . . . . 6 |- ( S e. (SubGrp ` G ) -> ( ( J ` X ) e. S <-> ( J ` X ) e. ( Base ` H ) ) )
30	28, 5, 29	3imtr4d 283	. . . . 5 |- ( S e. (SubGrp ` G ) -> ( X e. S -> ( J ` X ) e. S ) )
31	30	imp 445	. . . 4 |- ( ( S e. (SubGrp ` G ) /\ X e. S ) -> ( J ` X ) e. S )
32	24	sselda 3603	. . . 4 |- ( ( S e. (SubGrp ` G ) /\ ( J ` X ) e. S ) -> ( J ` X ) e. ( Base ` G ) )
33	31, 32	syldan 487	. . 3 |- ( ( S e. (SubGrp ` G ) /\ X e. S ) -> ( J ` X ) e. ( Base ` G ) )
34		subginv.i	. . . 4 |- I = ( invg ` G )
35	23, 13, 17, 34	grpinvid1 17470	. . 3 |- ( ( G e. Grp /\ X e. ( Base ` G ) /\ ( J ` X ) e. ( Base ` G ) ) -> ( ( I ` X ) = ( J ` X ) <-> ( X ( +g ` G ) ( J ` X ) ) = ( 0g ` G ) ) )
36	22, 25, 33, 35	syl3anc 1326	. 2 |- ( ( S e. (SubGrp ` G ) /\ X e. S ) -> ( ( I ` X ) = ( J ` X ) <-> ( X ( +g ` G ) ( J ` X ) ) = ( 0g ` G ) ) )
37	20, 36	mpbird 247	1 |- ( ( S e. (SubGrp ` G ) /\ X e. S ) -> ( I ` X ) = ( J ` X ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   ` cfv 5888  (class class class)co 6650   Basecbs 15857   ↾_s cress 15858   +g cplusg 15941   0gc0g 16100   Grpcgrp 17422   invgcminusg 17423  SubGrpcsubg 17588

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-rep 4771  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-rmo 2920  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-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-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-subg 17591

This theorem is referenced by:  subginvcl  17603  subgsub  17606  subgmulg  17608  zringlpirlem1  19832  prmirred  19843  psgninv  19928  subgtgp  21909  clmneg  22881  qrngneg  25312