Theorem qussub 13774

Description: Value of the group subtraction 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 )
qussub.p	|- .- = ( -g ` G )
qussub.a	|- N = ( -g ` H )

Assertion

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

Proof of Theorem qussub

Step	Hyp	Ref	Expression
1		qusgrp.h	. . . . 5 |- H = ( G /.s ( G ~QG S ) )
2		qusinv.v	. . . . 5 |- V = ( Base ` G )
3		eqid 2229	. . . . 5 |- ( Base ` H ) = ( Base ` H )
4	1, 2, 3	quseccl 13770	. . . 4 |- ( ( S e. (NrmSGrp ` G ) /\ X e. V ) -> [ X ] ( G ~QG S ) e. ( Base ` H ) )
5	4	3adant3 1041	. . 3 |- ( ( S e. (NrmSGrp ` G ) /\ X e. V /\ Y e. V ) -> [ X ] ( G ~QG S ) e. ( Base ` H ) )
6	1, 2, 3	quseccl 13770	. . 3 |- ( ( S e. (NrmSGrp ` G ) /\ Y e. V ) -> [ Y ] ( G ~QG S ) e. ( Base ` H ) )
7		eqid 2229	. . . 4 |- ( +g ` H ) = ( +g ` H )
8		eqid 2229	. . . 4 |- ( invg ` H ) = ( invg ` H )
9		qussub.a	. . . 4 |- N = ( -g ` H )
10	3, 7, 8, 9	grpsubval 13579	. . 3 |- ( ( [ X ] ( G ~QG S ) e. ( Base ` H ) /\ [ Y ] ( G ~QG S ) e. ( Base ` H ) ) -> ( [ X ] ( G ~QG S ) N [ Y ] ( G ~QG S ) ) = ( [ X ] ( G ~QG S ) ( +g ` H ) ( ( invg ` H ) ` [ Y ] ( G ~QG S ) ) ) )
11	5, 6, 10	3imp3i2an 1207	. 2 |- ( ( S e. (NrmSGrp ` G ) /\ X e. V /\ Y e. V ) -> ( [ X ] ( G ~QG S ) N [ Y ] ( G ~QG S ) ) = ( [ X ] ( G ~QG S ) ( +g ` H ) ( ( invg ` H ) ` [ Y ] ( G ~QG S ) ) ) )
12		eqid 2229	. . . . 5 |- ( invg ` G ) = ( invg ` G )
13	1, 2, 12, 8	qusinv 13773	. . . 4 |- ( ( S e. (NrmSGrp ` G ) /\ Y e. V ) -> ( ( invg ` H ) ` [ Y ] ( G ~QG S ) ) = [ ( ( invg ` G ) ` Y ) ] ( G ~QG S ) )
14	13	3adant2 1040	. . 3 |- ( ( S e. (NrmSGrp ` G ) /\ X e. V /\ Y e. V ) -> ( ( invg ` H ) ` [ Y ] ( G ~QG S ) ) = [ ( ( invg ` G ) ` Y ) ] ( G ~QG S ) )
15	14	oveq2d 6017	. 2 |- ( ( S e. (NrmSGrp ` G ) /\ X e. V /\ Y e. V ) -> ( [ X ] ( G ~QG S ) ( +g ` H ) ( ( invg ` H ) ` [ Y ] ( G ~QG S ) ) ) = ( [ X ] ( G ~QG S ) ( +g ` H ) [ ( ( invg ` G ) ` Y ) ] ( G ~QG S ) ) )
16		nsgsubg 13742	. . . . . . 7 |- ( S e. (NrmSGrp ` G ) -> S e. (SubGrp ` G ) )
17		subgrcl 13716	. . . . . . 7 |- ( S e. (SubGrp ` G ) -> G e. Grp )
18	16, 17	syl 14	. . . . . 6 |- ( S e. (NrmSGrp ` G ) -> G e. Grp )
19	2, 12	grpinvcl 13581	. . . . . 6 |- ( ( G e. Grp /\ Y e. V ) -> ( ( invg ` G ) ` Y ) e. V )
20	18, 19	sylan 283	. . . . 5 |- ( ( S e. (NrmSGrp ` G ) /\ Y e. V ) -> ( ( invg ` G ) ` Y ) e. V )
21	20	3adant2 1040	. . . 4 |- ( ( S e. (NrmSGrp ` G ) /\ X e. V /\ Y e. V ) -> ( ( invg ` G ) ` Y ) e. V )
22		eqid 2229	. . . . 5 |- ( +g ` G ) = ( +g ` G )
23	1, 2, 22, 7	qusadd 13771	. . . 4 |- ( ( S e. (NrmSGrp ` G ) /\ X e. V /\ ( ( invg ` G ) ` Y ) e. V ) -> ( [ X ] ( G ~QG S ) ( +g ` H ) [ ( ( invg ` G ) ` Y ) ] ( G ~QG S ) ) = [ ( X ( +g ` G ) ( ( invg ` G ) ` Y ) ) ] ( G ~QG S ) )
24	21, 23	syld3an3 1316	. . 3 |- ( ( S e. (NrmSGrp ` G ) /\ X e. V /\ Y e. V ) -> ( [ X ] ( G ~QG S ) ( +g ` H ) [ ( ( invg ` G ) ` Y ) ] ( G ~QG S ) ) = [ ( X ( +g ` G ) ( ( invg ` G ) ` Y ) ) ] ( G ~QG S ) )
25		qussub.p	. . . . . 6 |- .- = ( -g ` G )
26	2, 22, 12, 25	grpsubval 13579	. . . . 5 |- ( ( X e. V /\ Y e. V ) -> ( X .- Y ) = ( X ( +g ` G ) ( ( invg ` G ) ` Y ) ) )
27	26	3adant1 1039	. . . 4 |- ( ( S e. (NrmSGrp ` G ) /\ X e. V /\ Y e. V ) -> ( X .- Y ) = ( X ( +g ` G ) ( ( invg ` G ) ` Y ) ) )
28	27	eceq1d 6716	. . 3 |- ( ( S e. (NrmSGrp ` G ) /\ X e. V /\ Y e. V ) -> [ ( X .- Y ) ] ( G ~QG S ) = [ ( X ( +g ` G ) ( ( invg ` G ) ` Y ) ) ] ( G ~QG S ) )
29	24, 28	eqtr4d 2265	. 2 |- ( ( S e. (NrmSGrp ` G ) /\ X e. V /\ Y e. V ) -> ( [ X ] ( G ~QG S ) ( +g ` H ) [ ( ( invg ` G ) ` Y ) ] ( G ~QG S ) ) = [ ( X .- Y ) ] ( G ~QG S ) )
30	11, 15, 29	3eqtrd 2266	1 |- ( ( S e. (NrmSGrp ` G ) /\ X e. V /\ Y e. V ) -> ( [ X ] ( G ~QG S ) N [ Y ] ( G ~QG S ) ) = [ ( X .- Y ) ] ( G ~QG S ) )

Syntax hints:   -> wi 4   /\ w3a 1002   = wceq 1395   e. wcel 2200   ` cfv 5318  (class class class)co 6001   [cec 6678   Basecbs 13032   +g cplusg 13110   /._s cqus 13333   Grpcgrp 13533   invgcminusg 13534   -gcsg 13535  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-1st 6286  df-2nd 6287  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-sbg 13538  df-subg 13707  df-nsg 13708  df-eqg 13709

This theorem is referenced by: (None)