Theorem subginv 13632

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 13628	. . . 4 |- ( S e. (SubGrp ` G ) -> H e. Grp )
3	1	subgbas 13629	. . . . . 6 |- ( S e. (SubGrp ` G ) -> S = ( Base ` H ) )
4	3	eleq2d 2277	. . . . 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 2207	. . . . 5 |- ( Base ` H ) = ( Base ` H )
7		eqid 2207	. . . . 5 |- ( +g ` H ) = ( +g ` H )
8		eqid 2207	. . . . 5 |- ( 0g ` H ) = ( 0g ` H )
9		subginv.j	. . . . 5 |- J = ( invg ` H )
10	6, 7, 8, 9	grprinv 13498	. . . 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 2208	. . . . . 6 |- ( S e. (SubGrp ` G ) -> ( +g ` G ) = ( +g ` G ) )
14		id 19	. . . . . 6 |- ( S e. (SubGrp ` G ) -> S e. (SubGrp ` G ) )
15		subgrcl 13630	. . . . . 6 |- ( S e. (SubGrp ` G ) -> G e. Grp )
16	12, 13, 14, 15	ressplusgd 13076	. . . . 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 5984	. . 3 |- ( ( S e. (SubGrp ` G ) /\ X e. S ) -> ( X ( +g ` G ) ( J ` X ) ) = ( X ( +g ` H ) ( J ` X ) ) )
19		eqid 2207	. . . . 5 |- ( 0g ` G ) = ( 0g ` G )
20	1, 19	subg0 13631	. . . 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 2250	. 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 2207	. . . . 5 |- ( Base ` G ) = ( Base ` G )
25	24	subgss 13625	. . . 4 |- ( S e. (SubGrp ` G ) -> S C_ ( Base ` G ) )
26	25	sselda 3201	. . 3 |- ( ( S e. (SubGrp ` G ) /\ X e. S ) -> X e. ( Base ` G ) )
27	6, 9	grpinvcl 13495	. . . . . . . 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 2277	. . . . . 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 3201	. . . 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 2207	. . . 4 |- ( +g ` G ) = ( +g ` G )
36		subginv.i	. . . 4 |- I = ( invg ` G )
37	24, 35, 19, 36	grpinvid1 13499	. . 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 1250	. 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 1373   e. wcel 2178   ` cfv 5290  (class class class)co 5967   Basecbs 12947   ↾_s cress 12948   +g cplusg 13024   0gc0g 13203   Grpcgrp 13447   invgcminusg 13448  SubGrpcsubg 13618

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-addcom 8060  ax-addass 8062  ax-i2m1 8065  ax-0lt1 8066  ax-0id 8068  ax-rnegex 8069  ax-pre-ltirr 8072  ax-pre-ltadd 8076

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-pnf 8144  df-mnf 8145  df-ltxr 8147  df-inn 9072  df-2 9130  df-ndx 12950  df-slot 12951  df-base 12953  df-sets 12954  df-iress 12955  df-plusg 13037  df-0g 13205  df-mgm 13303  df-sgrp 13349  df-mnd 13364  df-grp 13450  df-minusg 13451  df-subg 13621

This theorem is referenced by:  subginvcl  13634  subgsub  13637  subgmulg  13639  mplnegfi  14582