Theorem qusinv 13773

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 13742	. . . . . 6 |- ( S e. (NrmSGrp ` G ) -> S e. (SubGrp ` G ) )
2		subgrcl 13716	. . . . . 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 13581	. . . . 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 2229	. . . . 5 |- ( +g ` G ) = ( +g ` G )
10		eqid 2229	. . . . 5 |- ( +g ` H ) = ( +g ` H )
11	8, 4, 9, 10	qusadd 13771	. . . 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 1372	. . 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 2229	. . . . . 6 |- ( 0g ` G ) = ( 0g ` G )
14	4, 9, 13, 5	grprinv 13584	. . . . 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 6716	. . 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 13772	. . . 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 2266	. 2 |- ( ( S e. (NrmSGrp ` G ) /\ X e. V ) -> ( [ X ] ( G ~QG S ) ( +g ` H ) [ ( I ` X ) ] ( G ~QG S ) ) = ( 0g ` H ) )
20	8	qusgrp 13769	. . . 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 2229	. . . 4 |- ( Base ` H ) = ( Base ` H )
23	8, 4, 22	quseccl 13770	. . 3 |- ( ( S e. (NrmSGrp ` G ) /\ X e. V ) -> [ X ] ( G ~QG S ) e. ( Base ` H ) )
24	8, 4, 22	quseccl 13770	. . . 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 2229	. . . 4 |- ( 0g ` H ) = ( 0g ` H )
27		qusinv.n	. . . 4 |- N = ( invg ` H )
28	22, 10, 26, 27	grpinvid1 13585	. . 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 1271	. 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 1395   e. wcel 2200   ` cfv 5318  (class class class)co 6001   [cec 6678   Basecbs 13032   +g cplusg 13110   0gc0g 13289   /._s cqus 13333   Grpcgrp 13533   invgcminusg 13534  SubGrpcsubg 13704  NrmSGrpcnsg 13705   ~_QG cqg 13706

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-addcom 8099  ax-addass 8101  ax-i2m1 8104  ax-0lt1 8105  ax-0id 8107  ax-rnegex 8108  ax-pre-ltirr 8111  ax-pre-lttrn 8113  ax-pre-ltadd 8115

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-tp 3674  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-er 6680  df-ec 6682  df-qs 6686  df-pnf 8183  df-mnf 8184  df-ltxr 8186  df-inn 9111  df-2 9169  df-3 9170  df-ndx 13035  df-slot 13036  df-base 13038  df-sets 13039  df-iress 13040  df-plusg 13123  df-mulr 13124  df-0g 13291  df-iimas 13335  df-qus 13336  df-mgm 13389  df-sgrp 13435  df-mnd 13450  df-grp 13536  df-minusg 13537  df-subg 13707  df-nsg 13708  df-eqg 13709

This theorem is referenced by:  qussub  13774