Theorem qusinv 13442

Description: Value of the group inverse operation in a quotient group. (Contributed by Mario Carneiro, 18-Sep-2015.)

Hypotheses

Ref	Expression
qusgrp.h	|- H = ( G /.s ( G ~QG S ) )
qusinv.v	|- V = ( Base ` G )
qusinv.i	|- I = ( invg ` G )
qusinv.n	|- N = ( invg ` H )

Assertion

Ref	Expression
qusinv	|- ( ( S e. (NrmSGrp ` G ) /\ X e. V ) -> ( N ` [ X ] ( G ~QG S ) ) = [ ( I ` X ) ] ( G ~QG S ) )

Proof of Theorem qusinv

Step	Hyp	Ref	Expression
1		nsgsubg 13411	. . . . . 6 |- ( S e. (NrmSGrp ` G ) -> S e. (SubGrp ` G ) )
2		subgrcl 13385	. . . . . 6 |- ( S e. (SubGrp ` G ) -> G e. Grp )
3	1, 2	syl 14	. . . . 5 |- ( S e. (NrmSGrp ` G ) -> G e. Grp )
4		qusinv.v	. . . . . 6 |- V = ( Base ` G )
5		qusinv.i	. . . . . 6 |- I = ( invg ` G )
6	4, 5	grpinvcl 13250	. . . . 5 |- ( ( G e. Grp /\ X e. V ) -> ( I ` X ) e. V )
7	3, 6	sylan 283	. . . 4 |- ( ( S e. (NrmSGrp ` G ) /\ X e. V ) -> ( I ` X ) e. V )
8		qusgrp.h	. . . . 5 |- H = ( G /.s ( G ~QG S ) )
9		eqid 2196	. . . . 5 |- ( +g ` G ) = ( +g ` G )
10		eqid 2196	. . . . 5 |- ( +g ` H ) = ( +g ` H )
11	8, 4, 9, 10	qusadd 13440	. . . 4 |- ( ( S e. (NrmSGrp ` G ) /\ X e. V /\ ( I ` X ) e. V ) -> ( [ X ] ( G ~QG S ) ( +g ` H ) [ ( I ` X ) ] ( G ~QG S ) ) = [ ( X ( +g ` G ) ( I ` X ) ) ] ( G ~QG S ) )
12	7, 11	mpd3an3 1349	. . 3 |- ( ( S e. (NrmSGrp ` G ) /\ X e. V ) -> ( [ X ] ( G ~QG S ) ( +g ` H ) [ ( I ` X ) ] ( G ~QG S ) ) = [ ( X ( +g ` G ) ( I ` X ) ) ] ( G ~QG S ) )
13		eqid 2196	. . . . . 6 |- ( 0g ` G ) = ( 0g ` G )
14	4, 9, 13, 5	grprinv 13253	. . . . 5 |- ( ( G e. Grp /\ X e. V ) -> ( X ( +g ` G ) ( I ` X ) ) = ( 0g ` G ) )
15	3, 14	sylan 283	. . . 4 |- ( ( S e. (NrmSGrp ` G ) /\ X e. V ) -> ( X ( +g ` G ) ( I ` X ) ) = ( 0g ` G ) )
16	15	eceq1d 6637	. . 3 |- ( ( S e. (NrmSGrp ` G ) /\ X e. V ) -> [ ( X ( +g ` G ) ( I ` X ) ) ] ( G ~QG S ) = [ ( 0g ` G ) ] ( G ~QG S ) )
17	8, 13	qus0 13441	. . . 4 |- ( S e. (NrmSGrp ` G ) -> [ ( 0g ` G ) ] ( G ~QG S ) = ( 0g ` H ) )
18	17	adantr 276	. . 3 |- ( ( S e. (NrmSGrp ` G ) /\ X e. V ) -> [ ( 0g ` G ) ] ( G ~QG S ) = ( 0g ` H ) )
19	12, 16, 18	3eqtrd 2233	. 2 |- ( ( S e. (NrmSGrp ` G ) /\ X e. V ) -> ( [ X ] ( G ~QG S ) ( +g ` H ) [ ( I ` X ) ] ( G ~QG S ) ) = ( 0g ` H ) )
20	8	qusgrp 13438	. . . 4 |- ( S e. (NrmSGrp ` G ) -> H e. Grp )
21	20	adantr 276	. . 3 |- ( ( S e. (NrmSGrp ` G ) /\ X e. V ) -> H e. Grp )
22		eqid 2196	. . . 4 |- ( Base ` H ) = ( Base ` H )
23	8, 4, 22	quseccl 13439	. . 3 |- ( ( S e. (NrmSGrp ` G ) /\ X e. V ) -> [ X ] ( G ~QG S ) e. ( Base ` H ) )
24	8, 4, 22	quseccl 13439	. . . 4 |- ( ( S e. (NrmSGrp ` G ) /\ ( I ` X ) e. V ) -> [ ( I ` X ) ] ( G ~QG S ) e. ( Base ` H ) )
25	7, 24	syldan 282	. . 3 |- ( ( S e. (NrmSGrp ` G ) /\ X e. V ) -> [ ( I ` X ) ] ( G ~QG S ) e. ( Base ` H ) )
26		eqid 2196	. . . 4 |- ( 0g ` H ) = ( 0g ` H )
27		qusinv.n	. . . 4 |- N = ( invg ` H )
28	22, 10, 26, 27	grpinvid1 13254	. . 3 |- ( ( H e. Grp /\ [ X ] ( G ~QG S ) e. ( Base ` H ) /\ [ ( I ` X ) ] ( G ~QG S ) e. ( Base ` H ) ) -> ( ( N ` [ X ] ( G ~QG S ) ) = [ ( I ` X ) ] ( G ~QG S ) <-> ( [ X ] ( G ~QG S ) ( +g ` H ) [ ( I ` X ) ] ( G ~QG S ) ) = ( 0g ` H ) ) )
29	21, 23, 25, 28	syl3anc 1249	. 2 |- ( ( S e. (NrmSGrp ` G ) /\ X e. V ) -> ( ( N ` [ X ] ( G ~QG S ) ) = [ ( I ` X ) ] ( G ~QG S ) <-> ( [ X ] ( G ~QG S ) ( +g ` H ) [ ( I ` X ) ] ( G ~QG S ) ) = ( 0g ` H ) ) )
30	19, 29	mpbird 167	1 |- ( ( S e. (NrmSGrp ` G ) /\ X e. V ) -> ( N ` [ X ] ( G ~QG S ) ) = [ ( I ` X ) ] ( G ~QG S ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2167   ` cfv 5259  (class class class)co 5925   [cec 6599   Basecbs 12703   +g cplusg 12780   0gc0g 12958   /._s cqus 13002   Grpcgrp 13202   invgcminusg 13203  SubGrpcsubg 13373  NrmSGrpcnsg 13374   ~_QG cqg 13375

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 4149  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-addcom 7996  ax-addass 7998  ax-i2m1 8001  ax-0lt1 8002  ax-0id 8004  ax-rnegex 8005  ax-pre-ltirr 8008  ax-pre-lttrn 8010  ax-pre-ltadd 8012

This theorem depends on definitions:  df-bi 117  df-3or 981  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 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-tp 3631  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-er 6601  df-ec 6603  df-qs 6607  df-pnf 8080  df-mnf 8081  df-ltxr 8083  df-inn 9008  df-2 9066  df-3 9067  df-ndx 12706  df-slot 12707  df-base 12709  df-sets 12710  df-iress 12711  df-plusg 12793  df-mulr 12794  df-0g 12960  df-iimas 13004  df-qus 13005  df-mgm 13058  df-sgrp 13104  df-mnd 13119  df-grp 13205  df-minusg 13206  df-subg 13376  df-nsg 13377  df-eqg 13378

This theorem is referenced by:  qussub  13443